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+{"text":"\n\\section{Introduction}\n\nQuantum interference plays a key role in mesoscopic transport phenomena  where impurities or dots are employed as 'shunts' for transferring particles, energy and information without degrading   phase coherence in the process \\cite{beenakker1991quantum,datta1997electronic,nazarov2009quantum}. In recent years a novel route to investigate this field of \\emph{quantum transport} emerged by employing ultracold atoms confined by optical or magnetic potentials  \\cite{chien2015quantum}. The ability to control and manipulate the effective dimensionality and geometry of the systems, the possibility to tune the inter-particle interaction strength, to add or eliminate disorder and to choose between fermionic or bosonic quantum particles made ultracold atoms an ideal testing ground for quantum transport phenomena \\cite{PhysRevA.75.023615,PhysRevLett.103.140405}. In these systems effects are accessible which were out of reach or very challenging to investigate in solid state. E.g. the periodic velocity change of a quantum particle moving in a lattice under the action of a constant driving force, known as Bloch oscillation, is difficult to observe in condensed matter systems due to impurity scattering but has beautifully been demonstrated with ultra-cold atoms in optical lattices \\cite{dahan1996bloch,morsch2001bloch}. Transport experiments of ultra-cold Fermi atoms through point contacts \\cite{brantut2012conduction,krinner2015observation} verified the quantization of conductance predicted by the Landauer theory of transport, which has previously been observed only in electronic systems. Both bosonic and fermionic superfluids can be created using ultra-cold atoms and frictionless flow has been observed \\cite{onofrio2000observation,desbuquois2012superfluid}. Persistent currents in ring geometries have been  realized in atomic superfluids \\cite{ryu2007observation,ramanathan2011superflow} and cold-atom analogues of Josephson junctions have been constructed \\cite{levy2007ac,ryu2013experimental} with the potential for an atomtronic analogue of a SQUID. Finally the coupling between particle and heat transport has been observed in fermionic cold atoms providing a cold-atom analogue of the thermoelectric effect \\cite{brantut2013thermoelectric}. However, despite of all experimental advances in the field, the creation and precise control of superfluid currents remains a challenge in atomtronics. Besides moving potential barriers or time-dependent artificial gauge fields, currents are typically generated by a difference of chemical potentials between the ends of a channel, i.e by fixing \"voltage\" rather than \"current\".\n\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{Fig1.pdf}\n\\caption{Scheme for controlling superfluid flow in a one dimensional interacting Bose gas using one a) or two b) noisy impurities, with or without an external current at velocity $v$. c) Induced superfluid current at a noisy point contact of noise strength $\\sigma$ in a moving condensate at velocity $v$. For weak noise the current grows monotonically with \n$\\sigma$, but for stronger noise the system enters a Zeno regime, where the current decreases. For the two largest velocities the system enters a regime of dynamical instabilities beyond a certain value of $\\sigma$, where \nthe current is no longer stationary  and thus not shown.\n}\n\\label{fig:illustration}\n\\end{figure}\n\nIn the present work we suggest and analyze a different method to create and manipulate the superfluid flow in a one-dimensional quasi-condensate of Bose atoms, see \\cref{fig:illustration}. Importantly   here we control the superfluid current directly rather than fixing chemical potentials. In particular we make use of the interaction of the condensate with quantum impurities that are coupled to the Boson's density with a fluctuating, i.e. noisy strength. Analyzing the system we identify different dynamical regimes, including a linear response regime, a Zeno-regime \\cite{Misra77} with negative differential current to noise-strength characteristics, and a regime\nwith dynamical instabilities characterized by continuous soliton emission. \n\nImpurities in interacting systems have been instrumental to develop our understanding of the extended pattern of correlations in  quantum many particle systems,\nby employing them as probes, tunable perturbations,  or even seeds for entanglement.\nIn unitary quantum dynamics, examples range from the 'catastrophic' effect of a scattering potential intruding in a Fermi sea~\\cite{anderson1967infrared,Schmidt_2018}, to strongly entangled magnetic impurities coupled to fermionic or bosonic reservoirs, or the dressing of static and moving particles in Fermi gases or Bose-Einstein condensates as it occurs in polaron formation. \nThe last decade has also witnessed a growth of attention towards the dissipative counterpart of the problem of quantum impurities embedded in interacting extended quantum systems~\\cite{Mueller2021,Wasak2021,Tonielli2020,Krapivsky2020,Kunimi2019,Zezyulin2021,Barmettler2011}. \n Pioneering results of  one decade ago illustrates e.g. the action of a localized dissipative potential on a macroscopic matter wave  by shining an electron beam on an atomic BEC \\cite{Brazhnyi2009,Barontini13}.\nAtomic losses induced by local dissipation were monitored as a function of noise strength, providing a proxy for a many-body version of the Zeno effect. \nThe stabilisation of dark solitons by engineered losses has been studied in \\cite{Baals2021}. \nFluctuations in the condensate can build up strong correlations with localized dissipation, resulting in a suppression of transport at large noise strength which can be regarded as non-equilibrium phase transition \\cite{Labouvie2016,Mink2022}.\n\n\nIn this work, we illustrate how density rearrangements provoked by local dephasing can be utilized to control coherent superflows in a one-dimensional Bose condensate.\nSpecifically, we consider a static or uniformly moving condensate coupled to a noisy local impurity.\nThe noisy point contact acts as a source of incoherent, i.e. non-condensed atoms, which due to total particle-number conservation creates a superfluid flow towards the impurity. The superfluid flow  increases monotonically  with growing noise up to some critical value at which the system becomes dominated by the\nquantum Zeno effect which leads to a reduction of transport corresponding to a negative differential current - noise characteristics.\nWe furthermore demonstrate that the archetypal effect of transport suppression due to Zeno effect is drastically altered in a moving rather than a static condensate.\nIn particular, we observe a lowering of the critical threshold of noise strength for entering the Zeno regime when the background speed of the condensate is increased. This is shown in~\\cref{fig:illustration}c where the onset of the Zeno regime \ndrifts towards smaller values of dissipation strength. \nAs outreach, we demonstrate complete tunability of a supercurrent in a static condensate by a pair of\nnoise point contacts.\n \n \n\n\\section{Model}\n\\label{sec:model}\n\nWe consider a homogeneous one-dimensional Bose gas with weak repulsive interactions ($g>0$) and boson mass $m$. \nWe study the effect of a noisy point contact in a Bose gas moving relative to the impurity \nwith fixed velocity $v$.\nThe impurity-BEC coupling is modeled by a Gaussian white noise process $\\eta(t)$, with mean $\\overline{\\eta(t)} = 0$ and variance $\\overline{\\eta(t) \\eta(t^\\prime)} = \\delta(t-t^\\prime)$,  multiplied by a local potential $V(x+vt)$, whose profile will be specified subsequently. \nSince we consider a weakly interacting condensate \\cite{pethick2002}, quantified by a small Lieb-Liniger parameter $\\gamma = g m \/ n  \\ll 1$, where $n$ is the average boson density in the 1D gas, we apply a phase-space description of the quantum Bose field using the Glauber-P distribution \\cite{gardiner1985}.\nDue to the action of the noisy point contact, we cannot ignore fluctuations even in the limit of a highly occupied condensate mode at very low temperatures.  \nWithin the phase space approach normal-ordered correlations of the Bose field operator $\\hat \\psi(x,t)$ are \ngiven by stochastic averages of a c-number field $\\psi(x,t)$. The time evolution \nof $\\psi(x,t)$ in the rest frame of the moving Bose gas is then determined by a Gross-Pitaevskii-type equation with an additional stochastic term\n(SGPE) \\cite{Stoof2001,Cockburn2009}\n\\begin{equation}\n\\begin{aligned}\n    i\\; \\text{d} \\psi(x,t) =&  \\Big[ - \\frac{\\partial_x^2}{2 m}  + g |\\psi(x,t)|^2 \\Big]\\psi(x,t) \\,\\text{d} t  \\\\ \n    & + \\;V(x+vt) \\, \\psi(x,t) \\, \\circ \\text{d} W. \\label{eq:SGPE0}\n\\end{aligned}\n\\end{equation}\nHere   $dW = \\eta(t) \\text{d} t $ is a infinitesimal Wiener process \\cite{gardiner1985}. \nSince the delta-correlated white noise $\\eta(t)$ results from colored noise in the limit of small correlation times, \\cref{eq:SGPE0} has to be interpreted as a Stratonovich stochastic differential equation \\cite{gardiner1985}, denoted by $\\circ$.\n\n\n\nIn order to gauge away the explicit time dependence of the potential $V(x+vt)$, we apply a Galilean transformation to the reference frame where the point contact is at rest.\nThis results into a SGPE with static potential and with a spatial gradient term proportional to $v$:\n\\begin{equation}\n\\begin{aligned}\n    i\\; \\text{d} \\phi(x,t) =&  \\Big[ - \\frac{\\partial_x^2}{2 m} -i v\\; \\partial_x + g |\\phi(x,t)|^2 \\Big]\\phi(x,t) \\,\\text{d} t  \\\\ \n    & + \\;V(x) \\, \\phi(x,t) \\,  \\circ \\text{d} W. \\label{eq:SGPE1}\n\\end{aligned}\n\\end{equation}\n$\\overline{\\phi}(x,t)$ describes the average Bose field in the rest-frame of the impurity, which includes both a quantum mechanical average and one over classical fluctuations induced by the noisy point contact. We refer to $\\overline{\\phi}$ as the coherent  amplitude of the Bose field. \n\n\n\n\\section{Noisy point contact in a static BEC}\n\\label{sec:static}\n\nWe start our analysis by reviewing the physics of a single  point contact  placed at $x=0$ in a static BEC ($v=0$). \nThe effect of the noisy impurity on the Bose gas shares at a first sight some similarities with the physics of local losses in Bose wires~\\cite{sels2020,Zezyulin2012,Brazhnyi2009,Barontini13}:  they both scatter particles out of the macroscopically populated ground state $\\overline{\\phi}(x,t)$.\nHowever, the dissipative impurity considered here conserves the total number of particles, which is crucial for potential applications in atomtronic devices. \n\nIn order to compare with the dynamics resulting from local losses, we first analyse the coherent amplitude \n$\\overline{\\phi}$.\nTherefore we consider the noise average of the SGPE \\cref{eq:SGPE1} \n\\begin{align}\n    i \\frac{\\text{d}}{\\text{d} t} \\overline{\\phi} = -\\frac{\\partial_x^2}{2m} \\overline{\\phi} \\,+\\, g \\,\\overline{|\\phi|^2 \\phi} \\,-\\, \\frac{i}{2} V(x)^2 \\,\\overline{\\phi} \\label{eq:average}.\n\\end{align}\nWhile the fluctuating potential vanishes on average it does have an effect on the average field $\\overline{\\phi}$. This is because it is a \\textit{multiplicative} noise and the field $\\phi(t)$ at a given time depends on the noise such that $\\overline{\\phi(x,t)   dW} \\neq 0$ (Stratonovich calculus \\cite{gardiner1985}). As a result of this, the average field experiences an effective loss, which physically describes nothing else than the scattering of particles out of the condensate into excited modes of the Bose gas, for more details see Appendix \\ref{sec:ito}.\n\n\\cref{eq:average} matches the evolution of the noise-averaged amplitude subject to local \\emph{particle loss} (cf.~\\cite{sels2020,Zezyulin2012,Brazhnyi2009,Barontini13}  with the identification $V(x)^2 = 2 \\sigma \\delta(x)$. \nWe consider\n this potential  as the limit of a Gaussian potential $V_l(x)^2 = 2 \\sigma \/ \\sqrt{\\pi l^2} \\; \\exp(-x^2\/l^2)$, with the length $l$ acting as a regulator, such that $V(x)$ itself is well defined. If $l$ is  chosen smaller than the healing length of the Bose gas $\\xi = 1 \/ \\sqrt{2 g n m}\\gg l$ the internal structure of the impurity potential becomes irrelevant.\n\nAs shown in \\cite{sels2020,Zezyulin2012,Brazhnyi2009,Barontini13} the effective local loss in \\cref{eq:average}  will induce currents.\nThis can be seen most easily from the\n continuity equation of the modulus of the average field $|\\overline{\\phi}|^2 $, which \n contains the coherent current \n\\begin{equation}\nj_\\text{coh} = \\frac{1}{m}\\, \\text{Im} (\\bar{\\phi}^* \\partial_x \\bar{\\phi} ).\n\\end{equation}\nNote that here the noise is averaged over the individual fields first and then bilinear combinations are formed. \n$j_\\text{coh}$ is in general not equal to the  average total particle current, which is defined by deriving the continuity equation for $\\phi^* \\phi$ from\nthe original SGPE, \\cref{eq:SGPE1}, and performing the noise average afterwards.\nThe total current reads\n\\begin{equation}\n    j_\\text{tot} = \\frac{1}{m}\\, \\overline{\\bigl.\\text{Im} (\\phi^* \\partial_x \\phi)}.\n\\end{equation}\nWe analyze both  currents as well as their difference, which we refer to as the incoherent current. It describes the flow of particles in excited modes of the Bose field\ncreated by the local noise.\nTo evaluate analytically the dynamics of \\cref{eq:average} we assume that the nonlinear term factorizes under average $\\overline{|\\phi(x,t)|^2 \\phi(x,t)} \\simeq |\\overline{\\phi(x,t)}|^2 \\overline{\\phi(x,t)}$; this approximation turns out to be in excellent agreement with numerics provided the coherent state $\\overline{\\phi(x,t)}$ describing the mean-field dynamics of the Bose gas is macroscopically populated. \nWe show the adequacy of this approximation by solving the full SGPE \\cref{eq:SGPE1} and evaluating the coherent $|\\overline{\\phi(x,t)}|^2$ and total density $\\overline{|\\phi(x,t)|^2}$ (cf. with  \\cref{fig:density_v=0}). \n\nFor weak dissipation the system is in a linear-response phase and the analytic solution of   \\cref{eq:average} reads \n\\begin{equation}\n    \\overline{ \\phi(x,t)} = \\sqrt{n_0} \\exp( - i m \\sigma |x| - i \\mu t ),\n\\end{equation}\n (cf. also \\cite{sels2020}). \nAfter switching on the local noise the system will assume this quasi-stationary state within a spatial region which grows in time with the local speed of sound $c_0 =\\sqrt{gn_0\/m} $. The density of the condensate in this area is reduced to $n_0<n$ and the constant phase gradient describes a coherent current\n\\begin{equation}\n    j_\\text{coh} = - n_0 \\sigma \\;\\text{sgn}(x)\\qquad\\textrm{for}\\quad \\sigma < \\sigma_c\n\\end{equation}\n flowing towards the point contact. As $j_\\text{coh}$ is proportional to the noise strength $\\sigma$, the regime is called \n  \"linear-response regime\". Here the chemical potential is $\\mu = g n_0 + m \\sigma^2 \/2$. \n\nAbove a  critical noise strength \\cite{sels2020}\n\\begin{equation}\n\\sigma_c = \\frac{2}{3} c = \\frac{2}{3} \\sqrt{2} g n \\xi \n\\end{equation}\n the system crosses over into a Zeno phase \\cite{Misra77}, where the current ceases to further increase  with the strength of the dissipation.  The critical point is reached when the velocity of the coherent flow $u_0 = j_\\text{coh}\/n_0$ attains  the local speed of sound. \nFor $\\sigma>\\sigma_c$ a grey soliton (a local density depletion of the size of the healing length \\cite{pethick2002}) forms at the position of the point contact, cf. with \\cref{fig:density_v=0}b.  \nThe density reduction associated with the formation of the grey soliton decreases the scattering rate at the point contact, which results in a reduction of the coherent current, which in turn determines self-consistently the depth of the grey soliton. As a consequence the functional dependence of the coherent current from the noise strenth changes from a linear increase $\\sigma$ to an inverse scaling:\n\\begin{equation}\n   j_\\text{coh} =  - n_0 \\frac{c_0^2}{\\sigma} \\; \\text{sgn}(x)\\qquad\\textrm{for}\\quad \\sigma>\\sigma_c.\n\\end{equation}\nThis is characteristic of the Zeno phase in extended systems~\\cite{Misra77}:\nat strong enough dissipation transport across the dissipative impurity is impeded as a result of the frequent  measurement of the observable to which the noise couples  at $x=0$. Outside  the depleted area (which travels at the sound speed) the density $n_0<n$ remains constant. \n\nIn contrast to the case of local loss, the noisy potential conserves the total particle number, which at first glance seems at odds with a current of particles flowing towards the point contact, while the particle  density remains constant over time. \nSimulating the dynamics of the total SGPE \\cref{eq:SGPE1} one finds, however,  that the total current vanishes in the area of constant density, see insets of \\cref{fig:density_v=0}. \nThis shows that the noisy point contact scatters particles out of the \ncondensate state, causing a coherent inward flow. At the same time it is a source of particles in excited modes of the Bose field\nleading to an incoherent outward flow of particles. \nSince these particles are removed from the coherent state, the noise affects the coherent amplitude of the Bose gas similarly to a local loss.\nThis means that a local non-unitary rearrangement of the system generates a coherent superfluid flow. In the next Sections, we harness this mechanism to engineer the coherent transport properties of the Bose gas by using purely incoherent point contacts. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{Fig2.pdf}\n\\caption{ \\textbf{Snapshots of density close to a static noisy point contact} in a) the linear-response phase  $\\sigma = 0.2 gn \\xi$ and b) in the Zeno phase $\\sigma = 8 gn \\xi$. Solid lines are  total densities $\\overline{|\\phi(x,t)|^2}$ while dashed lines are the coherent ones $|\\overline{\\phi(x,t)}|^2$. The density of incoherent particles is low. The insets show the coherent (dashed) and total (solid) particle current. The point contact scatters particles out of the coherent state, leading to a coherent current towards the noise contact and an incoherent counterflow, which exactly balances the coherent flow. The dotted vertical lines mark the size of the 'sound' cone, moving at the average speed of sound $c = \\sqrt{gn\/m}$. Colors match the ones of the related density profiles evaluated at the same time. }\n\\label{fig:density_v=0}\n\\end{figure}\n\n\n\n\\section{noisy point contact in a moving BEC}\n\\label{sec:moving}\n\n\n\\begin{figure*}[t!]\n    \\centering\n    \\includegraphics[width=.95\\textwidth]{Fig3.pdf}\n    \\caption{a) \\textbf{Phase diagram of a noisy point contact} of strength $\\sigma$ in a BEC moving  with velocity $v$. We plot the total coherent current flowing towards the impurity on the left and on the  right of the point contact, which   equals  the scattering rate of the dissipative impurity. \n    %\n   We average in space over the interval $x \\in \\pm [2,5 ] \\xi$ and in time over $t \\in [10 , 20] \/ gn$, where the intervals are chosen to be within the 'sound' cone, but large enough to average over multiple oscillations in the dynamically unstable phase III. The red line marks the estimated transition between linear response and Zeno regime which agrees well with the local maximum for fixed $v$, see Appendix \\ref{sec:crit} for details. The white dashed line marks the transition form the Zeno to the soliton regime. The analytical result $\\sigma_c = 2 c \/3 $ of the transition form normal to Zeno phase at $v=0$ \\cite{sels2020} is shown by the black circle.\n   b)-d) Density close to the noisy point contact at different times. Shown are in b) the normal phase I for $\\sigma = 0.2 gn\\xi$ and $v = 0.2 c$, in c) the Zeno phase II for $\\sigma = 0.7 gn\\xi$ and $v = 0.2 c$; and in d) the soliton phase III for $\\sigma = 2 gn\\xi$ and $v = 0.8 c$. The times marked in color are the same in all three plots. Solid, dashed and dotted lines are chosen as in \\cref{fig:diss_v}. The red dotted line in d) is the profile of a grey soliton fitted to the simulated density.   }\n    \\label{fig:diss_v}\n    \\end{figure*}\n\nIn this section we generalize our results to a noisy point contact in an externally imposed coherent current\nor equivalently to a noisy impurity moving at a constant velocity $v$ relative to the condensate.\nAn important difference with respect to a static point contact is the emergence of a third dynamical phase (which we label 'phase III' in the following), that is characterized by the absence of stationary particle flows at the point contact. This phase occurs in addition to the linear-response I and Zeno phases II. \nWe characterize the phases by evaluating the coherent current on the left and on the right of the noisy point contact; their difference is equal to the change in the number of particles of the coherent fraction of the field and therefore proportional to the scattering rate of particles off the dissipative impurity, cf.  Appendix \\ref{sec:coh_particle}\nSince we work in the frame in which the point contact is stationary there appears an additional term in the expression of $j_\\text{coh}$\n\\begin{align}\n    j_\\text{coh} = \\frac{1}{m}\\, \\text{Im}\\big( \\bar{\\phi^*} \\partial_x \\bar{\\phi} \\big)  + v \\; \\bar{\\phi^*}\\,\\bar{\\phi}.\\label{eq:augmented_current}\n\\end{align}\nThe scattering rate of particles out of the condensate and thus the total coherent current flowing towards the impurity depends both on the velocity $v$ of the point contact and on the noise strength $\\sigma$ as shown in \\cref{fig:diss_v}a. We distinguish the three phases, depending on whether this current increases or decreases as the noise strength changes.\n\n\\subsubsection{Linear-response regime}\n\nPhase I, cf. with \\cref{fig:diss_v}b, is akin to the linear-response phase of a static noisy  point contact, since the scattering rate increases with increasing dissipation strength, inducing an increasing coherent current flowing towards the noise source. \n\nDue to the motion of the condensate relative to the contact, the coherent currents on the left and on the right side are unequal in magnitude.\nThis results also in a different density on the left $n_l$ and on the right $n_r$ of the point contact. \nOur numerical simulations show that, like in the static case,   a quasi-stationary state is established in an area growing over time, but with  different velocities $(c_l - v)$ for $x<0$ and $(c_r+v)$ for $x>0$. \nThe two halves of the system are characterized by different velocities for two distinct reasons:    the speed of sound $c_{l,r}= \\sqrt{g n_{l,r}\/m}$ is different as a result of density differences on the two sides of the dissipative impurity, and   the velocity $v$ breaks the directional symmetry in the 1D gas.   \n\n\\subsubsection{Zeno regime}\n\n\nUpon increasing the dissipation strength, the system undergoes a transition into the Zeno phase II (\\cref{fig:diss_v}c), however this occurs at a smaller critical value as in the static case. \n    An estimate for the crossover point can be obtained as follows: The transition to the Zeno regime occurs when the local speed of sound $c(x)$ and the velocity of the coherent current $u$ become equal\n    \\begin{align}\n        c(x) \\equiv \\sqrt{\\frac{g n(x)}{m}} = u(x)\\equiv \\frac{j_\\textrm{coh}(x)}{n(x)}, \\label{eq:crit_cond}\n    \\end{align}\n    %\n     at any point in the system.\n    %\n    The reduction of the critical noise strength in a moving condensate can then be traced back to two effects.\n    %\n    First the coherent current is modified by to the background flow at velocity $v$.\n    %\n    Second the local speed of sound is smaller on one side of the contact when compared to the stationary case, because of the reduced density. \n    %\n    The overall coherent current in the system can therefore become supersonic already at a smaller critical dissipation strength.\n    %\n    As explained in detail in Appendix \\ref{sec:crit} one can derive an approximate expression for the transition point by utilizing \\cref{eq:crit_cond}. The result is marked by the red line in \\cref{fig:diss_v}a and agrees very well with the observed local maximum of the current. \n    %\n\nAs in the static case a grey soliton forms in the Zeno phase II at the position of the point contact and the smaller density leads to a decrease of the scattering rate with increasing dissipation strength. However, due to the motion of the condensate relative to the point contact the coherent current cannot go to zero but must always stays finite, allowing for the onset of a new phase III. \n\n\n\\subsubsection{Soliton-emission regime}\n\nThe minimum density of the grey soliton close to the  point contact  would drop to zero for strong dissipation $\\sigma \\gg gn\\xi$\nobstructing any particle current at $x=0$.\nHowever, the external flow forces particles to pass the noise contact, which can no longer be facilitated by a grey solition solution if $\\sigma$ increases. This then leads to instabilities and a continuous train of solitons is formed moving in the direction of the external current, see \\cref{fig:diss_v}d.\nThe system becomes dynamically unstable, when the external current becomes so large that the condition for the\nself-consistent formation of a grey soliton \ncan not be fulfilled any more. \nThe minimum density in a stable grey soliton is related to the velocity $u$ of the total coherent current passing it by $n_\\text{min}\/n_0 = u^2 \/ c_0^2$ \\cite{pethick2002}. \nA similar effect of a  continuous creation of solitons also occurs in the case of a constant repulsive potential in a moving condensate \\cite{Hakim97}. It happens when the Bose gas density is locally reduced to an extent that a constant coherent current (superfluid flow) cannot be sustained anymore. \n\nTo verify that the moving density oscillations are indeed soliton trains, we fit the analytic expression for a grey soliton wave function \\cite{pethick2002} to it, which agrees well with the observed density, see red dotted line in \\cref{fig:diss_v}d. \n\n\n\nIn summary, a moving Bose gas responds to a local noisy impurity like a stationary Bose gas, resulting in a linear response I and a Zeno phase II with renormalized transition points between the phases. The key difference is the formation of a soliton phase III, which only exists in the presence of an external current, preventing the formation of a quasi stationary state close the impurity and a constant current flow.   Different from the Zeno regime the \"shooting\" of solitons leads again to an increase in the time-averaged number of scattered particles with growing dissipation strength. \n\n\n\n \n\\section{controlling superfluid flow with two noisy contacts}\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=.95\\textwidth]{Fig4.png}\n    \\caption{\\textbf{Phase diagram of a configuration of two noisy point contacts.} a) Scattering rate out of the condensate at the right point contact for  impurity separation $r = 10 \\xi$, plotted for different noise strengths $\\sigma_l$ and  $\\sigma_r$. The current is averaged over time $t \\in [25-35] \/ gn$ and space $x \\in [-4.5,4.5 ] \\xi$; intervals are chosen  as in \\cref{fig:diss_v}a. The dashed lines mark the border between different phases as in Fig.~\\ref{fig:diss_v}, calculated by assuming a single contact in motion, see main text. b)-d) Density in the vicinity of two dissipative point contacts at distance $r = 10$ at equal dissipation strength $\\sigma_r =  \\sigma_l = \\sigma$. Their positions are marked with the red dotted lines. Parameters are chosen for both contacts to be in   b) the normal phase $\\sigma = 0.2 gn \\xi$, c) the Zeno phase $\\sigma = .5gn \\xi$ and d) the soliton phase $\\sigma = 2.2 gn \\xi$. Solid and dashed lines are chosen as in \\cref{fig:diss_v}}\n    \\label{fig:bi1}\n\\end{figure*}\n\nIn this section we show how superfluid flow can be controlled  using a pair of noisy point contacts. Each contact creates a coherent current of particles flowing towards it, which is balanced out by an incoherent one. \nAfter a time $t = r\/c $, where $r$ is the distance between the  contacts, the coherent current created by one  reaches  the other contact.\nEach of the two dissipative impurities thus experiences an effective coherent flow generated by the other impurity, and thus can sustain one of the three previously discussed phases. In the following we determine the phase diagram of the wire depending on the noise strength of the left ($\\sigma_l$) and right ($\\sigma_r$) noisy contacts. \nEvaluating the resulting currents in between the  contacts we will demonstrate that a segment with two noisy defects at its edges, can act as a current shunt.\\\\\n\nWe assume the noises $\\text{d} W_r$ and $\\text{d} W_l$ acting on the left and right impurity to be uncorrelated   $\\overline{\\text{d} W_r\\, \\text{d} W_l} = 0$, such that the time evolution is determined by the SGPE \n\\begin{equation}\n\\begin{aligned}\n    \\text{d} \\phi(x,t) =& -i \\Big[ - \\frac{\\partial_x^2}{2 m}  + g |\\phi(x,t)|^2 \\Big]\\phi(x,t) \\,\\text{d} t  \\\\\n    & - i \\;\\sqrt{2 \\sigma_l \\; \\delta(x+r\/2)} \\; \\phi(x,t) \\, \\circ \\text{d} W_l \\\\ \n    & - i \\;\\sqrt{2 \\sigma_r \\; \\delta(x-r\/2)} \\; \\phi(x,t) \\, \\circ \\text{d} W_r. \\label{eq:MF-bi}\n\\end{aligned}\n\\end{equation}\nWe consider, in the following, a separation of the contacts larger than the healing length $r \\gg \\xi$;  the latter is, in fact, the minimum  length over which a coherent current can be established \\cite{pethick2002}, and therefore a necessary requirement to apply the tools developed in the previous Sections.\nThe scattering out of the condensate at the right noisy contact is plotted in \\cref{fig:bi1}a for different noise strengths. For a fixed noise strength of the left contact ($\\sigma_l$), the number of scattered particles at the right impurity grows upon increasing the noise strength $\\sigma_r$ in the linear-response phase \\cref{fig:bi1}b, and then it shows Zeno physics above a critical value of $\\sigma_r$, see \\cref{fig:bi1}c. \nUpon further increasing the noise strength on the right point contact, the effect of solitons 'shooting' discussed in the previous Section sets in, leading again to an increase of scattered particles when averaged over time; solitons move downstream towards the other point contact resulting in an oscillatory density patter in space and time between them (cf. \\cref{fig:bi1}d).\\\\\n\nWe now show that the  critical thresholds for the dissipation strength of two contacts can be approximated using the results for a single moving defect. \nLet us assume that the left contact is placed into a initially static gas. This then leads to an onset of a coherent current, as discussed in Sec.~\\ref{sec:static}. \nWe determine the velocity of this current by interpolating the results in \\cref{fig:diss_v}a at zero velocity ($v=0$). \nThe right contact is then placed into this background current; we further assume that its presence will not affect the scattering rate at the left  impurity and that the system near the right impurity is determined by its own dissipation strength $\\sigma_r$ and the velocity of the coherent background current.\nUnder these assumption we can determine the system response following the lines of Sec~\\ref{sec:moving}, and estimate the crossovers in the setup of a pair of noise contacts. \nThese crossover points are marked with the white dashed lines in \\cref{fig:bi1}a and we recognize that they agree well with the observed extrema especially for small values of $\\sigma_l\/gn\\xi$. \nThis shows that only the coherent current is relevant for characterizing the steady state of the system under the noisy drive of the two impurities.   \nFor larger values of $\\sigma_l$ the assumption of a constant coherent background current created by the left impurity no longer holds and the scattering rate out of the condensate at the left contact depends also on the noise strength of the right one. This  explains the poorer agreement of the numerical results with the above physical picture for larger values of $\\sigma_l$.\n\n \n\n\n \nA possible application of the system of two noise contacts is the creation of a coherent current in the space between them. \nWe note that in the proposed scheme  the current is controlled directly and not via differences in chemical potentials.\nIn \\cref{fig:bi2} the coherent current between the contacts, averaged over space and a finite time interval, is plotted as function of the two noise strengths. Note that it is anti-symmetric since the exchange of the two interaction strengths leads to a reversal of the current.\nFor a small sum $\\sigma_+ = \\sigma_r + \\sigma_l \\ll gn \\xi$ both contacts are in the normal phase and the scattering rate of each contact is independent from the noise strength of the other. The coherent current in between the contacts is therefore the sum of two independent contacts. This is shown in the inset of \\cref{fig:bi2}, were the coherent current is normalized to $g n_0^2 \\xi_0$, with $n_0$ being the average density between the contacts, depending weakly on $\\sigma_{l,r}$, and $\\xi_0 = 1\/\\sqrt{2 g m n_0 }$ is the corresponding healing length.  For small  $\\sigma_+\\equiv\\sigma_l+\\sigma_r$ the normalized current depends only on the difference $\\sigma_- = \\sigma_r -\\sigma_l$ and it is equal to the current created by a single contact at dissipation strength $\\sigma_-$.\nFor larger $\\sigma_+$ at least one of the two contacts is not in the linear response phase, which results in a different slope and a non monotonous dependency on $\\sigma_-$. \n\\\\\n\n\n\n\n\\begin{figure}[h]\n   \n    \\includegraphics[width=.45\\textwidth]{Fig5.pdf}\n    \\caption{\\textbf{Coherent current between two noisy impurities} as a function of the individual noise strengths $\\sigma_r$ and $\\sigma_l$. The current is averaged over time $t \\in [25-35] \/ gn$ and space $x \\in [-4.5,4.5 ] \\xi$. \n    %\n    The inset shows the coherent current as a function of the difference $\\sigma_-$ and sum $\\sigma_+\/gn\\xi$ (different colors) of the two noise strengths. \n    The  current is rescaled to the density $n_0$ and the healing $\\xi_0= 1\/\\sqrt{2gn_0 m}$ of the Bose gas inbetween the two contacts.\n    The plot shows that the current does not depend on  $\\sigma_+$, for small $\\sigma_+ \\leq gn\\xi$. The black dashed line is the coherent current created by a single stationary point contact at noise strength $\\sigma_-$, which agrees well with the two-contacts result  at small $\\sigma_+$.}\n    \\label{fig:bi2}\n\\end{figure}\n    \n    \n    \n\\section{Experimental implementation and perspectives}\n\n\nIn this work we have revisited the Zeno crossover for particle currents traversing a moving noisy defect. We have shown that the speed of the impurity can be used as a knob to boost transport suppression. \nAs a possible experimental implementation  we envisage the use of noisy in-situ potentials, to control superfluid flows. Such potentials can be realized with two-color time-dependent optical potentials and tailored conservative potentials. We first note that it is crucial to have a vanishing mean of $V(x)$ for all positions $x$  (see section \\,\\ref{sec:model}. This is important in order to avoid residual repulsive or attractive potentials, which interfere with the effect of the dephasing. This condition can be fulfilled by using two laser beams, which are red- and blue-detuned with respect to an atomic transition \\cite{Jiang2021}. Both beams have to share the same spatial mode, which can be ensured by guiding them through the same optical fiber. For the defect considered in this work, it is sufficient to use Gaussian beams, which are focues onto the atoms with a high numerical aperture objective. To achieve a defect size, which is smaller than the healing length (as assumed in this work), one has to find a proper combination of numerical aperture (NA=0.4 or higher is necessary for most parameter settings), a short wavelength (higher energy atomic transitions are the better choice as not much optical power is needed to create the necessary potential height) and atomic density and interaction in order to enlarge the healing length. In 1D (as considered here) or 2D configurations, the  Rayleigh length should be larger than the thickness of the sample in order to treat the impurity as independent of the perpendicular direction.\n\nRegarding the time dependence of the optical potential, a large bandwidth of the modulation is another necessity. Modulating the intensity with acousto-optical modulators typically results in a bandwidth of more than 1 MHz. This is much faster than any intrinsic timescale (interaction energy, kinetic energy, potential energy, transverse confinement) of a typical experimental setting. The corresponding correlation time of less than 1\\textmu s is therefore short enough to provide an effective $\\delta$-correlated noise potential. In order to provide white or colored noise in the defect, both laser beams have to be driven with an arbitrary waveform generator, whose temporal signals are either inherently provided by the function generator or are computer generated, providing the required correlation functions. We note that experimentally, it is straightforward to generate much more complex correlation functions for the defect potential, thus bridging noisy defects and Floquet driven defects. \n\nMeasurements of the superfluid density in a quantum gas experiment are always challenging since in most schemes it is the total atomic density which is imaged. In the case of 1D systems, heterodyning with a twin system is the method of choice in order to access the motion of the superfluid as well as its amplitude \\cite{Hofferberth2007}. To this end, one has to prepare a twin 1D system aside with the system under investigation. Upon measuring, one lets both systems  interfere with each other and the fringe distance encodes the local velocity of the atoms, while the fringe contrast encodes the amplitude of the superfluid density. \n\nFrom the theory side, it would be interesting to extend the control of transport properties through the segment in systems without a macroscopic condensate occupation. For instance, studying the effect of two Markovian time-dependent noisy fields coupled to local  densities in an interacting fermionic wire. The non-interacting case could be solved exactly  as for the single impurity~\\cite{Dolgirev2020}, while the RG-scattering theory of Refs.~\\cite{Froeml2020}  could be used to assess the role of  strong quantum fluctuations in enhancing or eradicating the semi-classical effect discussed in this paper. We expect that studying real time dynamics of the problem with bosonization could serve equally well for this purpose.\nFor what concerns the results disussed in our work, we expect that  adding quantum fluctuations on top of the macroscopic occupation of the Bose gas, would not significantly alter the dynamics discussed in the paper. On one hand, quantum effects   would become sizeable only on times that are parametrically large in the condensate occupation. \nOn the other hand, the region traversed  by the density waves produced by the impurity can be regarded effectively as a driven-open systems and therefore subject to decoherence: the energy is pumped into the system via the noisy contact (which is held at infinite temperature) and dissipated by the 'bath'  given by the rest of the system  which stays at zero temperature, till the heat front will reach it.\nThe dynamics  within the 'sound' cone will therefore wash out quantum fluctuations through decoherence as any other open quantum system would.\nIn a semi-classical quantum trajectory description it is in fact impossible to distinguish the noise averaging used to derive the dynamics in our work, from sampling over a probability distribution function given  by the quantum fluctuations inherent in the initial state:\nthe trajectories sampled from the classical noise imprinted by the impurity would quickly dephase those arising from quantum fluctuations.\n\nAnother interesting direction would consist in generalizing the setup of our work to interacting quantum spin chains in view of applications to  spintronics.\n\n\n\\section{acknowledgements}\nWe   thank R. Barnett, J. Jager, S. Kelly and D. Sels for fruitful discussions. \nM.W., H.O. and M.F. acknowledge financial support by the DFG through SFB\/TR 185, Project No.277625399. \nJ. M. acknowledges financial support by the DFG through  the grant HADEQUAM-MA7003\/3-1.\nJ.M. and M.F acknowledge support from the Dynamics and Topology Centre funded by the State of Rhineland Palatinate.\nM.W. is supported by the Max Planck Graduate Center with the Johannes Gutenberg-Universit\u00e4t Mainz.\n \n\\section*{Appendix}\n\n    \\subsection{Derivation of the noise averaged SGPE} \n    \\label{sec:ito}\n    In the following we derive the noise average of the SGPE \\cref{eq:MF-bi}, which can be written as\n    \\begin{align}\n        d \\phi(x,t) = A[\\phi,\\phi^*] \\text{d} t + B[\\phi] \\circ \\text{d} W,\n    \\end{align}\n    where \n    \\begin{equation}\n    \\begin{aligned}\n        A[\\phi,\\phi^*] &= -i \\Big[ - \\frac{\\partial_x^2}{2 m} -i v\\; \\partial_x + g |\\phi(x,t)|^2 \\Big]\\phi(x,t) \\\\\n        B[\\phi] &= -i V(x) \\phi(x,t).\n    \\end{aligned}\n    \\end{equation}\n    This equation is a Stratonovich stochastic differential equation, where the noise is correlated with $\\phi(x,t)$, so that $\\overline{B[\\phi] \\circ \\text{d} W} \\neq 0$. To evaluate the noise average we transform the Stratonovich into an Ito equation, where the noise and field are not correlated $\\overline{B[\\phi] \\text{d} W} = 0$, see \\cite{gardiner1985}. The Ito equation is then given by\n    \\begin{align}\n        d \\phi(x,t) = \\Big\\{ A[\\phi,\\phi^*] +\\frac{1}{2}   B[\\phi] \\frac{\\delta}{\\delta \\phi(x,t)}  B[\\phi] \\Big\\}\\text{d} t + B[\\phi]  \\text{d} W.\n    \\end{align}\n    The noise average results in the Gross-Pitaevskii equation for the coherent state $ \\overline{ \\phi(x,t)}$\n    \n    \\begin{align}\n         \\frac{\\text{d}}{\\text{d} t} \\overline{\\phi} =-i \\Big[ -\\frac{\\partial_x^2}{2m} -i v\\; \\partial_x -\\, \\frac{i}{2} V(x)^2\\Big] \\overline{\\phi} \\,-i \\, g \\,\\overline{|\\phi|^2 \\phi}  \\label{eq:average_appendix}.\n    \\end{align}\n    The complex potential shows that the local noise scatters particles out of the coherent state $\\overline{\\phi}$, resulting in the incoherent current flowing away form the noise source.  \\\\\n    \n   \n    \\subsection{Coherent particle number} \n    \\label{sec:coh_particle}\n    In this section we show that the jump in the coherent current between the left and right sides of a noise contact is equal to the change in the number of particles of the coherent fraction of the field.\n    We start by deriving a continuity equation for the modulus of the average field $n_\\text{coh}(x,t) = \\overline{\\phi}^*\\, \\overline{\\phi}$ from the noise averaged mean-field equation \\cref{eq:average_appendix}\n    \\begin{align}\n        \\partial_t n_\\text{coh}(x,t) + \\partial_x j_\\text{coh}(x,t) =  - 2 \\sigma \\delta(x) n_\\text{coh}(x,t) \\label{eq:cont_coh}.\n    \\end{align}\n    Note that since the left hand side of this equation is nonzero $N_\\text{coh} = \\int_{-L\/2}^{L\/2} dx\\, n_\\text{coh}(x,t)$ is not conserved. The local noise scatters particles out of the coherent state. This is follows from integrating \\cref{eq:cont_coh} over the whole system\n    \\begin{align}\n        \\dot{N}_\\text{coh} &= \\int_{-L\/2}^{L\/2} \\text{d} x \\, \\partial_t\\, n_\\text{coh}(x,t)\n       \n        =       -\\, \\sigma \\, n_\\text{coh}(0,t),\n    \\end{align}\n    where the current term vanishes because we use periodic boundary conditions.\n    Integrating \\cref{eq:cont_coh} again, but over a small interval around the impurity shows\n    \\begin{align}\n        \\Delta j_\\text{coh} = \\int_{-\\epsilon}^{\\epsilon} \\text{d} x \\, \\partial_x j_\\text{coh}  = - 2 \\sigma \\; n_\\text{coh}(0,t), \\label{eq:Dj_sigma_n0}\n    \\end{align}\n    where we used that $n_\\text{coh}(x,t)$ is constant close to the impurity in the long time limit, which we showed by simulating the dynamics of the total SGPE \\cref{eq:SGPE1}. This yields\n    \\begin{align}\n        \\dot{N}_\\text{coh} = \\Delta j_\\text{coh}.\n        \\label{eq:Ndot_Dj}\n    \\end{align}\\\\\n    \n    \n\n   \n    \\subsection{Estimate of the transition point between linear response and Zeno regimes}\n    \\label{sec:crit}\n    \n    \n    In the following we estimate the linear response to Zeno transition of a noisy point contact in an external driven current of velocity $v$. We do so by deriving four equations containing the local speeds of sound $c_i = \\sqrt{g n_i\/m}$ and current velocities $u_i = j_{\\text{coh},i} \/ n_i$ at the left $(i=l)$ and right side $(i=r) $ of the noise contact, which determine the crossover point. \\\\\n    The system undergoes a transition, once the current velocity is equal to the speed of sound $c_i = |u_i|$ on either of the two sides of the contact. For $v >0$ the simulation show $c_l < c_r$ and $|u_l| > |u_r|$, see \\cref{fig:crit}, causing the critical condition to be fulfilled first on the left side  \n    \\begin{align}\n        c_l =  u_l, \\label{eq:crit1}\n    \\end{align}\n    which is the first equation we use. We determine the other three by analyzing the system in the linear response regime and assuming that the conditions are still valid at the critical point.\n    \n      Since the state in the depleted area is quasi stationary, the chemical potential  $\\mu = gn_i + \\frac{1}{2} m u_i^2 - \\frac{1}{2} m v^2 $ on both sides of the contact must be equal, from which the second equation is derived\n    \\begin{align}\n        c_l^2 + \\frac{1}{2} u_l^2 = c_r^2 + \\frac{1}{2} u_r^2. \\label{eq:crit2}\n    \\end{align}\n    \n    \n    \\begin{figure}[tb]\n       \n        \\includegraphics[width=.45\\textwidth]{Fig6.pdf}\n        \\caption{\n        a) Qualitative sketch of the density profile (i.e. local speed of sound) at a dissipative point contact in a constant background current $vn$, within the linear response regime. The size of the blue arrows indicates the strength of the current at their position. b) Simulation of the coherent current in the linear response regime ($\\sigma = 0.2 gn\\xi$, $v = 0.2 c$ and $t = 20\/gn$). The two lines agree close to the contact, since it induces equally strong currents on both sides.}\n        \\label{fig:crit}\n    \\end{figure}\n    %\n    \n    \n  \n    For the third equation we utilize the numerical evidence, that the contact induces equally strong currents on both sides, which is either added to or subtracted form the background current $v n$, see \\cref{fig:crit}b. This symmetry can be written as\n    \\begin{align}\n       j_{\\text{coh},r} + j_{\\text{coh},l} &= 2 v n \n       \\,\\, \\Rightarrow\\,\\,  c_l^2 u_l + c_r^2 u_r = 2 v c^2. \\label{eq:crit3}\n    \\end{align}\n    %\n    At last we derive an equation for the difference of the currents, which is equal to the change in the number of particles of the coherent fraction of the field $\\dot{N}_\\text{coh}$, see \\cref{eq:Ndot_Dj}. We assume that these particles are only removed from the ''transition area'' in between the quasi stationary state at density $n_i$ and the unperturbed area at density $n$. To estimate it we approximate the density profile as being linear, as illustrated in \\cref{fig:crit}a). This results in\n    \\begin{equation}\n    \\begin{aligned}\n        \\dot{N}_\\text{coh} =&- (n-n_l) (c_l+c-2v)\/2\\\\ &-(n-n_r)  (c_r+c+2v)\/2.\n    \\end{aligned}\n    \\end{equation}\n    The fourth equation is then given by\n    \\begin{equation}\n    \\begin{aligned}\n        &2c_l^2 u_l - 2c_r^2 u_r\\\\\n        =&\\;(c^2-c_l^2) (c_l+c-2v) + (c^2-c_r^2)  (c_r+c+2v). \\label{eq:crit4}\\\\\n    \\end{aligned}\n    \\end{equation}\n   \n    \\\\%\n    To determine the critical values we solve \\cref{eq:crit1,eq:crit2,eq:crit3,eq:crit4} numerically and in order to calculate the corresponding critical noise strength $\\sigma_c$  we use \\cref{eq:Dj_sigma_n0}, with the approximation $n_\\text{coh}(0,t) = (n_r+n_l)\/2$. This eventually yields\n    \\begin{align}\n        \\sigma_c  =  \\frac{c_l^2 u_l -c_r^2 u_r }{c_l^2 + c_r^2}.\n    \\end{align}\n    %\n    The critical dissipation strength derived in this way agrees very well with the local maximum in the coherent current, which we calculated numerically, see \\cref{fig:diss_v}a. In the stationary case ($v = 0$) we get $\\sigma_c = 0.74 c$, which is only slightly larger as the exact value $\\sigma_c = 2 c \/3$ \\cite{sels2020}.\n    \n\n\n\\bibliographystyle{apsrev4-2}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Supplementary Material}\n\n\\subsection{Determining the Effective Unitary Interaction}\n\nCentral to our discussion in the main text is the measurement operator $\\ensuremath{\\Upsilon_{\\! h}^{\\phantom{\\dagger}}}$, which is used to describe the operation to the mechanical resonator via the optomechanical interaction and then single photon detection. $\\ensuremath{\\Upsilon_{\\! h}^{\\phantom{\\dagger}}} = \\bra{1,0}\\ensuremath{U_{\\textrm{eff}}}\\ket{0,0}$, where the ket is the initial state of light at the cavity resonance for the two orthogonal polarizations used, the bra describes a $h$-polarization photon detection with no $v$ photon detection, and $\\ensuremath{U_{\\textrm{eff}}}$ is the effective optomechanical interaction including the manipulations to the optical field made after interaction with the mechanical resonator. In this supplementary we provide a discussion how $\\ensuremath{U_{\\textrm{eff}}}$ is obtained.\n\nThe time evolutions described in Eq.~(2) of the main text are generated by the beam-splitter and two-mode-squeezing effective interaction Hamiltonians. In the former case $a$ accumulates correlation with $b$ and in the latter case $a$ accumulates correlation with $b^\\dagger$. For vacuum on the input of mode $a$, the expectation of the number operator in the output of mode $a$ for the beam-splitter and two-mode-squeezing interactions are\n\\begin{displaymath}\n\\sin^2\\! \\ensuremath{{\\textstyle \\frac{\\theta}{2}}} \\, \\mean{b^\\dagger b}, \\quad \\textrm{and} \\quad\n\\sinh^2\\! r\\, \\mean{b b^\\dagger},\n\\end{displaymath}\nrespectively, where $\\sin^2(\\ensuremath{{\\textstyle \\frac{\\theta}{2}}})$ is the (intensity) reflectivity of the beam-splitter and $r$ is the squeezing parameter. In the optomechanical scheme we have considered, the mean photon number scattered by the optomechanical interaction for the beam-splitter and two-mode-squeezing interactions are\n\\begin{displaymath}\n\\mean{n_h} = (1 - e^{-2G_h\\tau})\\langle\\ensuremath{b^{\\dagger}_{0}} \\ensuremath{b^{\\phantom{\\dagger}}_{0}}\\rangle, \\,\\, \\textrm{and} \\,\\,\n\\mean{n_v} = (e^{2G_v\\tau} - 1)\\langle\\ensuremath{b^{\\phantom{\\dagger}}_{0}} \\ensuremath{b^{\\dagger}_{0}}\\rangle,\n\\end{displaymath}\nrespectively. For small $\\ensuremath{{\\textstyle \\frac{\\theta}{2}}}$, $r$, and $G\\tau$ we then have\n\\begin{displaymath}\n\\ensuremath{{\\textstyle \\frac{\\theta}{2}}} = \\sqrt{2G_h\\tau}, \\quad \\textrm{and} \\quad\nr = \\sqrt{2G_v\\tau},\n\\end{displaymath}\nfor the effective optomechanical beam-splitter and two-mode-squeezing parameters, respectively. It is noted here that computing the mean number output in mode $b$ can also be performed to yield these parameters. As both the beam-splitter and two-mode-squeezing processes are driven simultaneously, we expect that the effective optomechanical unitary take the form $\\ensuremath{U_{\\textrm{eff}}} = \\exp \\left[-\\frac{i}{\\hbar}(H_{BS} + H_{SQ})\\tau\\right]$, where $H_{BS} \\propto a^\\dagger b + a b^\\dagger$ and $H_{SQ} \\propto ab + a^\\dagger b^\\dagger$ are the beam-splitter and two-mode-squeezing Hamiltonians respectively. To first order in the beam-splitter and squeezing parameters the effective unitary describing the cavity optomechanical interaction is then\n\\begin{displaymath}\n\\ensuremath{U_{\\textrm{eff}}} = 1 + (\\ensuremath{{\\textstyle \\frac{\\theta}{2}}} a_h^\\dagger b e^{-i\\phi} - r a_v^\\dagger b^\\dagger e^{i\\varphi} - \\textrm{H.c.}).\n\\end{displaymath}\nFinally, to obtain the effective unitary used for the measurement operator, the polarization manipulations to the optical fields, as discussed in the main text,  must be performed.\n\n\n\\subsection{Arbitrary Quantum State Transformation}\nIn the main text we introduced a scheme for \\emph{arbitrary quantum state transformation} that generates a target state from a known input state. Here we further discuss our protocol and provide a specific quantum state transformation example.\n\nThe protocol works as follows. For a known initial state\n\\begin{displaymath}\n\\ket{\\psi} = \\sum_{n=0}^N \\psi_n \\ket{n},\n\\end{displaymath}\nwhich has no excitation beyond $N$ quanta (or has been approximated by truncation at this level), any target state of the form\n\\begin{displaymath}\n\\ket{\\phi} = \\sum_{n=0}^N \\phi_n \\ket{n},\n\\end{displaymath}\ncan be generated by applying a controllably weighted superposition of identity and subtraction $N$ times, i.e.\n\\begin{equation}\n\\Phi = \\prod_{j=1}^N (\\mu_j + \\nu_j b)\/\\sqrt{2} = \\sum_{i=0}^N C_i b^i.\n\\label{eq:munuC}\n\\end{equation}\nApplying this operation to the initial state we have\n\\begin{displaymath}\n\\Phi \\ket{\\psi} = \\sum_{i=0}^N \\sum_{k=0}^N C_i \\psi_k \\sqrt{\\frac{k!}{(k-i)!}}\\ket{k-i},\n\\end{displaymath}\nwhere we have used $b\\ket{n} = \\sqrt{n}\\ket{n-1}$. \n\nThe operation $\\Phi$ is a non-unitary process and the un-normalized matrix elements of the state after application of $\\Phi$ are\n\\begin{equation}\n\\label{eq:Suppphin}\n\\bra{n}\\Phi\\ket{\\psi} = \\sum_{i=0}^{N-n} C_i \\, \\psi_{i+n} \\sqrt{\\frac{(i+n)!}{n!}}.\n\\end{equation}\nThe target state $\\ket{\\phi}$ is reached when $\\bra{n}\\Phi\\ket{\\psi} = \\phi_n$. Provided that $\\psi_N \\neq 0$ a set of coefficients $C_i$ fulfilling $\\bra{n}\\Phi\\ket{\\psi} = \\phi_n$ can be determined via matrix inversion. Once a set of coefficients $C_i$ is determined, a set of complex coefficients $\\mu_j$ and $\\nu_j$ that satisfy (\\ref{eq:munuC}) can also readily be determined via matrix inversion.\n\nWe now provide a specific example of a quantum state transformation. Starting with an initial state $\\ket{\\psi} = \\ket{4}$ we wish to reach the target state $\\ket{\\phi} = (\\ket{1} + \\ket{4})\/\\sqrt{2}$. This target state can be reached with three applications of identity and subtraction. Solving (\\ref{eq:Suppphin}) we find that $C_0~=~\\sqrt{24} C_3$ and $C_1 = C_2 = 0$. As identity has been used with each application we set $\\mu = 1$ and obtain\n\\begin{align}\n\\nu_1 + \\nu_2 + \\nu_3 &= 0, \\nonumber\\\\\n\\nu_1 \\nu_2 + \\nu_1 \\nu_3 + \\nu_2 \\nu_3  &= 0, \\nonumber\\\\\n\\nu_1 \\nu_2 \\nu_3 \\sqrt{24} &= 1. \\nonumber\\\\\n \\nonumber\n\\end{align}\nThese equations can be readily solved exactly to provide the relative amplitudes between identity and subtraction to produce the target state. Numerical approximations to the solutions and the intermediate states during the quantum state transformation process are shown in Fig.~\\ref{Fig:TransformExample}.\n\n\\begin{widetext}\n\n\\begin{figure*}[h!]\n\\includegraphics[width=1.0\\hsize]{3_StateTransform.pdf\n\\caption{\nAn example quantum state transformation. Shown are Wigner functions (blue-cyan: positive, red-yellow: negative, larger ticks mark the origin and they increment by unity) of an initial Fock state (left) to a target state (right). The target state is reached by a sequence of three operations of a controllably weighted superposition of identity and subtraction. The relative amplitude between identity and subtraction for each step is shown.\n}\n\\label{Fig:TransformExample}\n\\end{figure*}\n\n\\end{widetext}\n\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nCausal mediation analysis seeks to determine the role that an intermediate variable plays in transmitting the effect from an exposure to an outcome. An indirect effect refers to the effect that goes through the intermediate variable in mediation analysis; a direct effect is a measure of the effect that does not. The study of causal mediation has in recent years enjoyed an explosion in popularity \\citep{robins1992identifiability,robins1999testing,robins2003semantics,pearl2001direct,\navin2005identifiability,taylor2005counterfactual,petersen2006estimation,ten2007causal,\nalbert2008mediation,goetgeluk2008estimation,van2008direct,vanderweele2009marginal,\nvanderweele2009conceptual,vanderweele2010odds,imai2010general,\nimai2010identification,albert2011generalized,tchetgen2011causal,vanderweele2011causal,albert2012mediation,tchetgen2012semiparametric,\nwang2012estimation,shpitser2013counterfactual,tchetgen2013inverse,tchetgen2014estimation,wang2013estimation,albert2015sensitivity,hsu2015surrogate}, not only in terms of theoretical developments, but also in practice, most notably in the fields of epidemiology and social sciences. This strand of work is based on ideas originating from \\cite{robins1992identifiability} and \\cite{pearl2001direct} grounded in the language of potential outcomes \\citep{splawa1990application, rubin1974estimating, rubin1978bayesian} to give nonparametric definitions of effects involved in mediation analysis, allowing for settings where interactions and nonlinearities may be present.\n\nConsider an intervention which sets the exposure of interest for all persons in the population to one of two possible values, a reference value or an active value. The total effect of such an intervention corresponds to the change of the counterfactual outcome mean if the exposure were set to the active value compared with if it were set to the reference value. \\cite{robins1992identifiability} formalized the concept of effect decomposition of the total effect into direct and indirect effects by defining pure direct and indirect effects. \\cite{pearl2001direct} relabeled these effects as natural direct and indirect effects. The pure direct effect ($\\textsc{pde}$) corresponds to the change in the counterfactual outcome mean under an intervention which changes a person's exposure status from the reference value to the active value, while maintaining the person's mediator to the value it would have had under the exposure reference value. In contrast, the natural indirect effect ($\\textsc{nie}$) corresponds to the change in the average counterfactual outcome under an intervention that sets a person's exposure value to the active value, while changing the value of the mediator from the value it would have had under the reference exposure value, to its value under the active exposure value. The $\\textsc{pde}$ and $\\textsc{nie}$ sum to give the total effect.\n\nIdentification of these natural effects has been somewhat controversial as it requires assumptions that may be overly restrictive for many applications in the health sciences. First, identification invokes a so-called cross-world-counterfactuals-independence assumption, which by virtue of involving counterfactuals under conflicting interventions on the exposure, can neither be enforced experimentally nor tested empirically \\citep{pearl2001direct, robins2010alternative}. Secondly, a necessary assumption for identification rules out the presence of exposure-induced confounding of the mediator's effect on the outcome, even if all confounders are observed. While this assumption is in principle testable provided no unmeasured confounding, more often than not, post-exposure covariates are altogether ignored in routine application, in which case mediation analyses may be invalid. These issues have recently been considered, and some work has been done on partial or point identification under a weaker assumption. Specifically, on the one hand \\cite{robins2010alternative} and \\cite{tchetgen2014identification} provide conditions for point identification of the pure direct effect when a confounder is directly affected by the exposure. On the other hand, \\cite{robins2010alternative} give bounds for the pure direct effect for binary mediator without making the cross-world-counterfactual-independence assumption, but assuming no exposure-induced confounding of the mediator-outcome relation, and \\cite{tchetgen2014bounds} extend these bounds to account for exposure-induced confounding. Bounds are commonly employed in causal inference when structural assumptions are not sufficiently strong to give point identification of a causal parameter of interest \\citep{robins1989analysis,balke1997probabilistic,zhang2003estimation,kaufman2005improved,cheng2006bounds,cai2008bounds,sjolander2009bounds,taguri2015principal}. We build on this previous work to provide a number of new nonparametric bounds for the pure direct effect allowing for a polytomous mediator when either (i) exposure-induced confounding is present, or (ii) one does not assume that cross-world counterfactuals of the mediating and outcome variables are independent, or (iii) both (i) and (ii) hold.\n\nWe apply these bounds to data from the Harvard PEPFAR program in Nigeria, where we evaluate the extent to which the effects of antiretroviral therapy on virological failure are mediated by a patient's adherence. We show that PEPFAR results are sensitive to the choice of assumptions made, consequently, we counsel investigators employing these effects to exercise caution in considering the basis for point identification and to explicitly state the assumptions required for them to be valid. Where assumptions are empirically untestable, they should be argued for on the basis of scientific understanding, and ideally the alternative should be explored by employing partial identification bounds given both here and elsewhere. While some work has been done to develop sensitivity analyses for unmeasured confounding of the mediator \\citep{tchetgen2011causal,tchetgen2012semiparametric,vansteelandt2012natural}, sensitivity analyses for ranges of plausible associations between cross-world counterfactuals remain undeveloped. Further development of sensitivity analyses of both forms would be highly beneficial for practical use, and is fertile ground for future work. We hope that the work presented here will inspire deeper consideration and transparency regarding underlying identifying assumptions in the practice of mediation analysis.\n\n\\section{Preliminaries}\nBy way of introduction, the directed acyclic graph ($\\textsc{dag}$) displayed in Fig. \\ref{fig:1}.(a) illustrates the simplest possible mediation setting, where $A$ is defined to be the exposure taking either baseline value $a^*$ or comparison value $a$, $M$ is defined to be the (potential) mediator, and $Y$ is defined to be the outcome.\n\\begin{figure}\n\\centering\n\\begin{tabular}{ccc}\n\\\\\n\\\\\n\\begin{tikzpicture}[->,>=stealth',baseline={(A)},scale=1, line width=1pt]\n\\tikzstyle{every state}=[draw=none]\n\\node[shape=circle, draw, inner sep=1mm] (A) at (0,0) {$A$};\n\\node[shape=circle, draw, inner sep=1mm] (M) at (2,0) {$M$};\n\\node[shape=circle, draw, inner sep=1mm] (Y) at (4,0) {$Y$};\n\n  \\path \n\t(A)  edge               (M)  \n\t(A)  edge  [bend right]  (Y)\n\t(M)  edge              (Y)\n\t;\n\\end{tikzpicture}\n& \n&\n\\begin{tikzpicture}[->,>=stealth',baseline={(A)},scale=1, line width=1pt]\n\\tikzstyle{every state}=[draw=none]\n\\node[shape=semicircle, draw, inner sep=1mm, shape border rotate=90, inner sep=1.5mm] (A) at (0,0) {$A$};\n\\node[shape=semicircle, draw, shape border rotate=270, color=red, inner sep=1.85mm] (a) at (.75,0) {$\\tilde{a}$};\n\\node[shape=semicircle, draw, inner sep=1mm, shape border rotate=90, inner sep=.5mm] (M) at (3,0) {$M(\\tilde{a})$};\n\\node[shape=semicircle, draw, shape border rotate=270, color=red, inner sep=2mm] (m) at (4,0) {$\\tilde{m}$};\n\\node[shape=ellipse, draw, inner sep=1mm] (Y) at (6.25,0) {$Y(\\tilde{a},\\tilde{m})$};\n\n  \\path \n\t(a)  edge               (M)  \n\t(m)  edge               (Y)  \n\t(a)  edge  [bend right=60]   (Y)  \n\t;\n\\end{tikzpicture}\n\\end{tabular}\n\\caption{(a) The three-node mediation directed acyclic graph in a setting with no confounding. The nodes represent random variables, and the arrows represent possible causal effects of one random variable on another. (b) The single-world intervention graph in the setting of (a) under the intervention setting $A$ to $\\tilde{a}$ and $M$ to $\\tilde{m}$. The black nodes represent random variables under this intervention, the red nodes represent the level an intervened random variable takes under this intervention, and the arrows represent possible causal effects of one variable under this intervention on another.}\n\\label{fig:1}\n\\end{figure}\nThis $\\textsc{dag}$ assumes randomization of the exposure, which for expositional simplicity we maintain throughout. The graph also encodes no unobserved confounding of the effect of $M$ on $Y$ given $A$. The effect along the path $A\\rightarrow Y$ on the diagram is generally referred to as direct with respect to $M$, and the effect along the path $A\\rightarrow M\\rightarrow Y$ on the diagram is generally referred to as indirect with respect to $M$.\n\nFurther elaboration of the specific type of direct and indirect effect under consideration necessitates counterfactual definitions. Let $Y(a)$ denote a subject's outcome if treatment $A$ were set, possibly contrary to fact, to $a$. In the context of mediation, there will also be potential outcomes for the intermediate variable. Counterfactuals $M(a)$ and $Y(m,a)$ are defined similarly. In order to link these with the observed data, we adopt the standard set of consistency assumptions that\n\\begin{align*}\n&\\textrm{if } A=a\\textrm{, then } M(a)=M\\textrm{ with probability one,}\\\\\n&\\textrm{if } A=a\\textrm{ and } M=m\\textrm{, then } Y(m,a)=Y\\textrm{ with probability one, and}\\\\\n&\\textrm{if } A=a\\textrm{, then } Y(a)=Y\\textrm{ with probability one.}\\\\\n\\end{align*}\nIn terms of counterfactuals, the randomization assumption encoded by the $\\textsc{dag}$ in Fig. \\ref{fig:1}.(a) is $\\{Y(a,m),M(a)\\}\\mbox{\\ensuremath{\\perp\\!\\!\\!\\perp}} A$ for all $a$ and $m$; the assumption of no unobserved confounding of $M$ given $A$ is $Y(a,m)\\mbox{\\ensuremath{\\perp\\!\\!\\!\\perp}} M(a)\\mid A=a$ for all $a$ and $m$. Finally, we will consider as well defined the nested counterfactual $Y\\{a,M(a^*)\\}$, i.e., the counterfactual outcome under an intervention which sets the exposure to the comparison value $a$, and the mediator to the value it would have taken under the conflicting baseline exposure value $a^*$.\n\nWe may now define the pure\/natural direct effect and natural indirect effect \\citep{robins1992identifiability, pearl2001direct}, which form the following decomposition of the average causal effect:\n\\begin{align*}\n&E\\left\\{ Y(a)\\right\\} -E\\left\\{ Y(a^*)\\right\\}  \\\\\n&=\\overset{\\mathrm{total\\text{ }effect}}{\\overbrace{E\\left[\nY\\{a,M(a)\\}\\right] -E\\left[ Y\\{a^*,M(a^*)\\}\\right] }} \\\\\n&=\\overset{\\mathrm{natural\\text{ }indirect\\text{ }effect}}{\\overbrace{E%\n\\left[ Y\\{a,M(a)\\}\\right] -E\\left[ Y\\{a,M(a^*)\\}\\right] }}+\\overset{%\n\\mathrm{pure\\text{ }direct\\text{ }effect}}{\\overbrace{E\\left[\nY\\{a,M(a^*)\\}\\right] -E\\left[ Y\\{a^*,M(a^*)\\}\\right) }}.\\newline\n\\end{align*}\nThe terms $E\\{Y(a)\\}=E\\left[Y\\left\\{a,M(a)\\right\\}\\right]$, for all $a$, are identified under randomization of $A$. The parameter $\\gamma_0\\equiv E[Y\\{a,M(a^*)\\}]$ would be identified if one were to interpret the $\\textsc{dag}$ in Fig. \\ref{fig:1}.(a) as a nonparametric structural equation model with independent errors ($\\textsc{npsem-ie}$). Structural equations provide a nonparametric algebraic interpretation of this $\\textsc{dag}$ corresponding to three equations, one for each variable in the graph. Each random variable on the graph is associated with a distinct, arbitrary function, denoted $g$, and a distinct random disturbance, denoted $\\varepsilon$, each with a subscript corresponding to its respective random variable. Each variable is generated by its corresponding function, which depends only on all variables that affect it directly (i.e., its parents on the graph), and its corresponding random disturbance, as follows:\n\\begin{align*}\nA&=g_A(\\varepsilon_A)\\\\\nM&=g_M(A,\\varepsilon_M)\\\\\nY&=g_Y(A,M,\\varepsilon_Y).\n\\end{align*}\nUnder particular interventions, these structural equations naturally encode dependencies of counterfactuals. Consider, for example, two interventions, one setting $A=a^*$, and another setting $A=a$ and $M=m$. The structural equations then become\n\\begin{equation*}\n\\begin{aligned}[c]\nA&=a^*\\\\\nM(a^*)&=g_M(a^*,\\varepsilon_M)\\\\\nY(a^*)&=g_Y(a^*,M(a^*),\\varepsilon_Y)\n\\end{aligned}\n\\qquad\\qquad\\qquad\\qquad\\qquad\n\\begin{aligned}[c]\nA&=a\\\\\nM(a)&=m\\\\\nY(a,m)&=g_Y(a,m,\\varepsilon_Y).\n\\end{aligned}\n\\end{equation*}\n\nThis formulation places no a priori restriction on the distribution of counterfactuals. The key assumption of the $\\textsc{npsem-ie}$ is that the random disturbances are mutually independent. This allows us to make independence statements regarding counterfactuals under various, possibly-conflicting interventions. In particular, this model implies that for all $m$, (i) $\\{M(a),Y(a,m)\\}\\mbox{\\ensuremath{\\perp\\!\\!\\!\\perp}} A$, (ii) $Y(a,m)\\mbox{\\ensuremath{\\perp\\!\\!\\!\\perp}} M(a)\\mid A=a$, and (iii) $Y(a,m)\\mbox{\\ensuremath{\\perp\\!\\!\\!\\perp}} M(a^*)\\mid A=a$, which in turn suffice for identification of $\\gamma_0$ \\citep{pearl2001direct}. Independence statements such as (iii) are known as cross-world counterfactual statements if $a$ is not equal to $a^*$, due to their comparison of interventions that could never occur in the same world simultaneously. Independence condition (iii) can be seen to hold under the model by considering the $\\textsc{npsem-ie}$ under a specific intervention and noting that the only source of randomness in $Y(a,m)=g_Y(a,m,\\varepsilon_Y)$ is $\\varepsilon_Y$ and the only source of randomness in $M(a^*)=g_M(a^*,\\varepsilon_M)$ is $\\varepsilon_M$. Thus, the cross-world-counterfactual-independence statement follows directly from independence of exogenous disturbances. However, such an independence is neither experimentally verifiable nor enforceable \\citep{robins2010alternative}.\n\nThis issue has been discussed extensively \\citep{robins2010alternative,richardson2013single}, and in large part motivated the development of the single-world intervention graphs ($\\textsc{swig}$s) of \\cite{richardson2013single}. These causal graphs manage to elucidate this issue by graphically representing the counterfactuals themselves, allowing independence statements of counterfactuals to be read directly from the graph. Consider the $\\textsc{swig}$ in Fig. \\ref{fig:1}.(b). By $d$-separation, it is clear that (i) $Y(a,m)\\mbox{\\ensuremath{\\perp\\!\\!\\!\\perp}} M(a)$ for all $a$ and $m$, however no such statement can be made from the graph about $Y(a,m)$ and $M(a^*)$ when $a\\neq a^*$. Under this $\\textsc{swig}$, independence between $Y(a,m)$ and $M(a^*)$ is not assumed, and hence $\\gamma_0$ is not point identified. \\cite{robins2010alternative} provide the following bounds for its partial identification in the setting where $M$ is binary and $\\textsc{swig}$ independence assumptions $M(a)\\mbox{\\ensuremath{\\perp\\!\\!\\!\\perp}} A$ and $Y(a,m)\\mbox{\\ensuremath{\\perp\\!\\!\\!\\perp}} \\{M(a),A\\}$ hold for all $a$ and $m$:\n\\begin{align*}\n\\max &\\{0,\\mathrm{pr}(M=0\\mid A=a^*)+E(Y\\mid M=0,A=a)-1\\}\\\\\n+&\\max \\{0,\\mathrm{pr}(M=1\\mid A=a^*)+E(Y\\mid M=1,A=a)-1\\}\\\\\n&\\qquad\\qquad\\qquad\\qquad\\leq \\gamma_0 \\leq \\\\\n\\min &\\{\\mathrm{pr}(M=0\\mid A=a^*),E(Y\\mid M=0,A=a)\\}\\\\\n+&\\min \\{\\mathrm{pr}(M=1\\mid A=a^*),E(Y\\mid M=1,A=a)\\}.\n\\end{align*}\nIn Section 2, we extend this result to the setting of a polytomous $M$.\n\nAs previously mentioned, another often-overlooked condition required for identification of $\\gamma_0$ is that there is no confounder of the mediator's effect on the outcome that is affected by the exposure. Such a confounder is present in the setting illustrated in the $\\textsc{dag}$ in Fig. \\ref{fig:2}.(a).\n\\begin{figure}\n\\centering\n\\begin{tabular}{ccc}\n\\\\\n\\\\\n\\begin{tikzpicture}[->,>=stealth',baseline={(A)},scale=1, line width=1pt]\n\\tikzstyle{every state}=[draw=none]\n\\node[shape=circle, draw, inner sep=1mm] (A) at (0,0) {$A$};\n\\node[shape=circle, draw, inner sep=1mm] (R) at (1.25,0) {$R$};\n\\node[shape=circle, draw, inner sep=1mm] (M) at (2.5,0) {$M$};\n\\node[shape=circle, draw, inner sep=1mm] (Y) at (3.75,0) {$Y$};\n\n  \\path \n\t(A)  edge               (R)  \n\t(R)  edge               (M)\n\t(M)  edge               (Y)\n\t(A)  edge  [bend left]  (M)\n\t(A)  edge  [bend left=45]  (Y)\n\t(R)  edge  [bend right]  (Y)\n\n\t;\n\\end{tikzpicture}\n& \n&\n\\begin{tikzpicture}[->,>=stealth',baseline={(A)},scale=1, line width=1pt]\n\\tikzstyle{every state}=[draw=none]\n\\node[shape=semicircle, draw, inner sep=1mm, shape border rotate=90, inner sep=1.5mm] (A) at (0,0) {$A$};\n\\node[shape=semicircle, draw, shape border rotate=270, color=red, inner sep=1.85mm] (a) at (.75,0) {$\\tilde{a}$};\n\\node[shape=ellipse, draw, inner sep=1mm] (R) at (2.75,0) {$R(\\tilde{a})$};\n\\node[shape=semicircle, draw, inner sep=1mm, shape border rotate=90, inner sep=.5mm] (M) at (4.75,0) {$M(\\tilde{a})$};\n\\node[shape=semicircle, draw, shape border rotate=270, color=red, inner sep=2mm] (m) at (5.75,0) {$\\tilde{m}$};\n\\node[shape=ellipse, draw, inner sep=1mm] (Y) at (7.75,0) {$Y(\\tilde{a},\\tilde{m})$};\n\n  \\path \n\t(a)  edge               (R)  \n\t(R)  edge               (M)\n\t(m)  edge               (Y)  \n\t(a)  edge  [bend left=60]   (Y)  \n\t(R)  edge  [bend right=60]   (Y)  \n\t(a)  edge  [bend left]   (M)  \n\t;\n\\end{tikzpicture}\n\n\\end{tabular}\n\\caption{(a) A mediation directed acyclic graph in which $R$ is an exposure-induced confounder. The nodes represent random variables, and the arrows represent possible causal effects of one random variable on another. (b) The single-world intervention graph in the setting of (a) that has been intervened on to set $A$ to $\\tilde{a}\\in\\{a,a^*\\}$ and $M$ to $\\tilde{m}$. The black nodes represent random variables under this intervention, the red nodes represent the level an intervened random variable takes under this intervention, and the arrows represent possible causal effects of one variable under this intervention on another.}\n\\label{fig:2}\n\\end{figure}\nGenerally, even under an $\\textsc{npsem-ie}$ interpretation of this $\\textsc{dag}$, $\\gamma_0$ will not be identified in this setting. This is readily seen by considering the following representation under this model given by \\cite{robins2010alternative}:\n\\begin{align}\n\\gamma_0=\\sum\\limits_{r,r^*}&E\\left\\{ E(Y\\mid M,R=r,A=a)\\mid R=r^*,A=a^*\\right\\}\\mathrm{pr}\\left\\{R(a)=r,R(a^*)=r^*\\right\\}.\n\\end{align}\nClearly the joint probability term can never be identified from observed data, since we will never be able to observe $R(a)$ and $R(a^*)$ for the same individual.\n\nA few conditions for identification have been proposed. \\cite{robins2010alternative} give two. The first is that $R(a)\\mbox{\\ensuremath{\\perp\\!\\!\\!\\perp}} R(a^*)$, in which case the troublesome term in (1) will factor, giving\n\\begin{align*}\n\\gamma_0 = \\sum\\limits_{r^*,r}&E\\left\\{ E(Y\\mid M,R=r,A=a)\\mid R=r^*,A=a^*\\right\\}\\mathrm{pr}(R=r^*\\mid A=a^*)\\\\\n&\\times\\mathrm{pr}(R=r\\mid A=a).\n\\end{align*}\nIt seems biologically unlikely, however, that in a scenario in which $A$ affects $R$, the counterfactual $R$ under $A=a$ would not be predictive of the counterfactual $R$ under $A=a^*$. The other condition is that the counterfactual outcome under one exposure value is a deterministic function of the counterfactual for the other treatment, i.e., $R(a)=g\\{R(a^*)\\}$. In this case,\n\\begin{align*}\n\\gamma_0 = \\sum\\limits_{r^*,r}&E\\left\\{ E(Y\\mid M,R=r,A=a)\\mid R=r^*,A=a^*\\right\\}\\mathrm{pr}(R=r^*\\mid A=a^*)I\\{r=g(r^*)\\}.\n\\end{align*}\nThe above assumption is implied by rank preservation \\citep{robins2010alternative}, which is unlikely to hold in social and health sciences as it rules out individual-level effect heterogeneity \\citep{tchetgen2014identification}. As none of these conditions are experimentally verifiable, the authors themselves ``do not advocate blithely adopting such assumptions in order to preserve identification of the $\\textsc{pde}$ in [this setting]\" \\citep{robins2010alternative}.\n\n\\cite{tchetgen2014identification} give two testable conditions for identification of $\\gamma_0$ when $R$ is present. The first is of $A$--$R$ monotonicity, i.e., for Bernoulli $R$, $R(a)\\geq R(a^*)$. If $R$ is a vector of Bernoulli random variables whose structural equations have independent errors, and if monotonicity holds for each element,\n\\[\\gamma_0=\\sum\\limits_{r,r^*}E\\left\\{ E(Y\\mid M,R=r,A=a)\\mid R=r^*,A=a^*\\right\\}\\prod\\limits_{j=1}^k f_j(r_j,r^*_j,a,a^*)\\]\nwhere\n\\begin{align*}\nf_j(r_j,r_j^*,a,a^*)=\\left\\{\n\\begin{array}{rl}\n\\mathrm{pr}(R_j=1\\mid A=a^*) & \\textrm{ if } r_j^*=r_j=1,\\\\\n\\mathrm{pr}(R_j=1\\mid A=a)-\\mathrm{pr}(R_j=1\\mid A=a^*) & \\textrm{ if } r_j^*=0 \\textrm{ and } r_j=1,\\\\\n0 & \\textrm{ if } r_j^*=1 \\textrm{ and } r_j=0,\\\\\n\\mathrm{pr}(R_j=0\\mid A=a) & \\textrm{ if } r_j^*=r_j=0.\n\\end{array}\\right.\n\\end{align*}\nTheir second condition is no $M$--$R$ additive mean interaction, i.e.,\n\\[E(Y\\mid m,r,a)-E(Y\\mid m^*,r,a)-E(Y\\mid m,r^*,a)+E(Y\\mid m^*,r^*,a)=0,\\]\nfor all levels $m$ and $m^*$ of $M$ and $r$ and $r^*$ of $R$. For discrete $M$ and $R$, this yields\n\\begin{align*}\n\\gamma_0 = &\\sum_m \\left\\{E(Y\\mid m,r^*,a)-E(Y\\mid m^*,r^*,a)\\right\\}\\mathrm{pr}(M=m\\mid A=a^*)\\\\\n&+\\sum_r \\left\\{E(Y\\mid m^*,r,a)-E(Y\\mid m^*,r^*,a)\\right\\}\\mathrm{pr}(R=r\\mid A=a)\\\\\n&+E(Y\\mid m^*,r^*,a).\n\\end{align*}\n\nEschewing the cross-world-counterfactual assumptions of the $\\textsc{npsem-ie}$ , \\cite{tchetgen2014bounds} extend the bounds of \\cite{robins2010alternative} to allow for the presence of an exposure-induced confounder when the mediator is binary:\n\\begin{align*}\n\\max &\\left\\{0,\\mathrm{pr}(M=0\\mid A=a^*)+\\sum_r E(Y\\mid M=0,R=r,A=a)\\mathrm{pr}(R=r\\mid A=a)-1\\right\\}\\\\\n+\\max& \\left\\{0,\\mathrm{pr}(M=1\\mid A=a^*)+\\sum_r E(Y\\mid M=1,R=r,A=a)\\mathrm{pr}(R=r\\mid A=a)-1\\right\\}\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\leq \\gamma_0 \\leq \\\\\n\\min &\\left\\{\\mathrm{pr}(M=0\\mid A=a^*),\\sum_r E(Y\\mid M=0,R=r,A=a)\\mathrm{pr}(R=r\\mid A=a)\\right\\}\\\\\n+\\min& \\left\\{\\mathrm{pr}(M=1\\mid A=a^*),\\sum_r E(Y\\mid M=1,R=r,A=a)\\mathrm{pr}(R=r\\mid A=a)\\right\\}.\n\\end{align*}\nWe extend these bounds as well to allow for polytomous $M$ in Section 3. Additionally, we construct bounds for $\\gamma_0$ under an $\\textsc{npsem-ie}$ that account for a discrete exposure-induced confounder, but require no further assumption.\n\n\\section{New partial identification results}\nWe begin by extending the bounds of \\cite{robins2010alternative} and \\cite{tchetgen2014bounds} to settings with discrete mediator and outcome. Proofs can be found in the Appendix.\n\n\\begin{theorem}\n\\label{theorem1}\nUnder the $\\textsc{swig}$ in either Fig. \\ref{fig:1}.(b) or Fig. \\ref{fig:2}.(b) with discrete $M$ and $Y$ and arbitrary $R$,\n\\begin{align*}\n\\sum\\limits_{m,y}&y\\left(\\max\\left[0,\\mathrm{pr}\\{M(a^*)=m\\}+\\mathrm{pr}\\{Y(a,m)=y\\}-1\\right]I(y>0)\\right.\\\\\n&\\left.+\\min\\left[\\mathrm{pr}\\{M(a^*)=m\\},\\mathrm{pr}\\{Y(a,m)=y\\}\\right]I(y<0)\\right)\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\leq \\gamma_0 \\leq\\\\\n\\sum\\limits_{m,y}&y\\left(\\max\\left[0,\\mathrm{pr} \\{M(a^*)=m\\}+\\mathrm{pr}\\{Y(a,m)=y\\}-1\\right]I(y<0)\\right.\\\\\n&\\left.+\\min\\left[\\mathrm{pr}\\{M(a^*)=m\\},\\mathrm{pr}\\{Y(a,m)=y\\}\\right]I(y>0)\\right).\n\\end{align*}\n\\end{theorem}\n\nThe upper and lower bounds coincide when $Y(a,m)$ or $M(a^*)$ is degenerate, which follows from the properties of joint probability mass functions. The upper bound is achieved only if $Y(a,m)$ and $M(a^*)$ are comonotone for each $m$, i.e., if $F_{Y(a,m),M(a^*)}(y,m)=\\min\\left[F_{Y(a,m)}(y),F_{M(a^*)}(m)\\right]$ for each $m$, where $F_X(\\cdot)$ denotes the joint (or marginal) cumulative distribution function of the random vector (or scalar) $X$. The lower bound is achieved only if they are countermonotone for each $m$, i.e., if $F_{Y(a,m),M(a^*)}(y,m)=\\max\\left\\{0,F_{Y(a,m)}(y)+F_{M(a^*)}(m)-1\\right\\}$ for each $m$. A straightforward application of the $g$-formula under the $\\textsc{dag}$s in Fig. \\ref{fig:1} and \\ref{fig:2} yields the following corollaries:\n\\begin{corollary}\nFor polytomous $M$ and $Y$, $\\gamma_0$ is partially identified under the $\\textsc{swig}$ in Fig. \\ref{fig:1}.(b) by the bounds in Theorem 1 with $\\mathrm{pr}\\{M(a^*)=m\\}=\\mathrm{pr}(M=m\\mid a^*)$ and $\\mathrm{pr}\\{Y(a,m)=y\\}=\\mathrm{pr}(Y=y\\mid m,a)$. It is partially identified under the $\\textsc{swig}$ in Fig. \\ref{fig:2}.(b) by the same bounds, but with $\\mathrm{pr}\\{M(a^*)=m\\}=\\mathrm{pr}(M=m\\mid a^*)$ and $\\mathrm{pr}\\{Y(a,m)=y\\}=\\sum_r\\mathrm{pr}(Y=y\\mid m,r,a)\\mathrm{pr}(R=r\\mid a)$.\n\\end{corollary}\nThe second part of the corollary continues to hold even if there were a hidden common cause of $R$ and $Y$ as in Fig. \\ref{fig:3}, since the same $g$-formula applies in this setting.\n\\begin{figure}\n\\centering\n\\begin{tabular}{ccc}\n\\\\\n\\\\\n\\begin{tikzpicture}[->,>=stealth',baseline={(A)},scale=1, line width=1pt]\n\\tikzstyle{every state}=[draw=none]\n\\node[shape=circle, draw, inner sep=1mm] (A) at (0,0) {$A$};\n\\node[shape=circle, draw, inner sep=1mm] (R) at (1.25,0) {$R$};\n\\node[shape=circle, draw, inner sep=1mm] (M) at (2.5,0) {$M$};\n\\node[shape=circle, draw, inner sep=1mm] (Y) at (3.75,0) {$Y$};\n\\node[shape=circle, draw, inner sep=1mm, color=gray] (H) at (2.5,1.25) {$H$};\n\n  \\path \n\t(A)  edge               (R)  \n\t(R)  edge               (M)\n\t(M)  edge               (Y)\n\t(A)  edge  [bend right]  (M)\n\t(A)  edge  [bend right=45]  (Y)\n\t(R)  edge  [bend left=35]  (Y)\n\t(H)  edge\t[bend right]               (R)\n\t(H)  edge\t[bend left]               (Y)\n\t;\n\\end{tikzpicture}\n& \n&\n\\begin{tikzpicture}[->,>=stealth',baseline={(A)},scale=1, line width=1pt]\n\\tikzstyle{every state}=[draw=none]\n\\node[shape=semicircle, draw, inner sep=1mm, shape border rotate=90, inner sep=1.5mm] (A) at (0,0) {$A$};\n\\node[shape=semicircle, draw, shape border rotate=270, color=red, inner sep=1.85mm] (a) at (.75,0) {$\\tilde{a}$};\n\\node[shape=ellipse, draw, inner sep=1mm] (R) at (2.75,0) {$R(\\tilde{a})$};\n\\node[shape=semicircle, draw, inner sep=1mm, shape border rotate=90, inner sep=.5mm] (M) at (4.75,0) {$M(\\tilde{a})$};\n\\node[shape=semicircle, draw, shape border rotate=270, color=red, inner sep=2mm] (m) at (5.75,0) {$\\tilde{m}$};\n\\node[shape=ellipse, draw, inner sep=1mm] (Y) at (7.75,0) {$Y(\\tilde{a},\\tilde{m})$};\n\\node[shape=circle, draw, inner sep=1mm, color=gray] (H) at (5.25,2) {$H$};\n\n  \\path \n\t(a)  edge               (R)  \n\t(R)  edge               (M)\n\t(m)  edge               (Y)  \n\t(H)  edge\t[bend right]               (R)  \n\t(H)  edge\t[bend left]               (Y)  \n\t(a)  edge  [bend right=60]   (Y)  \n\t(R)  edge  [bend left=45]   (Y)  \n\t(a)  edge  [bend right]   (M)  \n\t;\n\\end{tikzpicture}\n\\end{tabular}\n\\caption{(a) A mediation directed acyclic graph in which an unobserved variable $H$ affects $R$, an exposure-induced confounder, and $Y$. The black nodes represent observed random variables, and the arrows represent possible causal effects of one random variable on another. (b) The single-world intervention graph in the setting of (a) that has been intervened on to set $A$ to $\\tilde{a}\\in\\{a,a^*\\}$ and $M$ to $\\tilde{m}$. The black nodes represent random variables under this intervention, the red nodes represent the level an intervened random variable takes under this intervention, and the arrows represent possible causal effects of one variable under this intervention on another. In each panel, the gray node represents a hidden random variable}\n\\label{fig:3}\n\\end{figure}\nWhereas the previous results invoked no cross-world-counterfactual independences under the $\\textsc{swig}$ interpretation of the $\\textsc{dag}$ in Fig. \\ref{fig:2}.(a), sharper bounds are available under Pearl's $\\textsc{npsem-ie}$ interpretation of these $\\textsc{dag}$s, as derived in the following result.\n\n\\begin{theorem}\n\\label{theorem2}\nFor discrete $R$ taking values in $\\{1,\\hdots,p\\}$, let $B$ be the $p^2\\times (p-1)^2$ matrix\n\\[\\left[\n\\begin{array}{ccccc}\nI_{p-1} & 0_{(p-1)\\times (p-1)} & \\cdots & 0_{(p-1)\\times (p-1)} & 0_{(p-1)\\times (p-1)} \\\\\n-1_{p-1}^T & 0_{p-1}^T & \\cdots & 0_{p-1}^T & 0_{p-1}^T\\\\\n0_{(p-1)\\times (p-1)} & I_{p-1} & \\cdots & 0_{(p-1)\\times (p-1)}  & 0_{(p-1)\\times (p-1)}\\\\\n0_{p-1}^T & -1_{p-1}^T & \\cdots & 0_{p-1}^T & 0_{p-1}^T\\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\\n0_{(p-1)\\times (p-1)} & 0_{(p-1)\\times (p-1)} & \\cdots & I_{p-1} & 0_{(p-1)\\times (p-1)}\\\\\n0_{p-1}^T & 0_{p-1}^T & \\cdots & -1_{p-1}^T & 0_{p-1}^T \\\\\n0_{(p-1)\\times (p-1)} & 0_{(p-1)\\times (p-1)} & \\cdots & 0_{(p-1)\\times (p-1)} & I_{p-1}\\\\\n0_{p-1}^T & 0_{p-1}^T & \\cdots & 0_{p-1}^T & -1_{p-1}^T \\\\\n-I_{p-1} & -I_{p-1} & \\cdots & -I_{p-1} & -I_{p-1}\\\\\n&& 1_{(p-1)^2}^T &&\n\\end{array}\\right],\\]\n$d$ be the $p^2$-dimensional vector\n\\[\n\\left[\n\\begin{array}{c}\n0_{p-1}\\\\\n\\mathrm{pr}\\left(R=1\\mid A=a\\right)\\\\\n0_{p-1}\\\\\n\\mathrm{pr}\\left(R=2\\mid A=a\\right)\\\\\n\\vdots\\\\\n0_{p-1}\\\\\n\\mathrm{pr}\\left(R=p-1\\mid A=a\\right)\\\\\n\\mathrm{pr}\\left(R=1\\mid A=a^*\\right)\\\\\n\\mathrm{pr}\\left(R=2\\mid A=a^*\\right)\\\\\n\\vdots\\\\\n\\mathrm{pr}\\left(R=p-1\\mid A=a^*\\right)\\\\\n\\mathrm{pr}\\left(R=p\\mid A=a\\right)+\\mathrm{pr}\\left(R=p\\mid A=a^*\\right)-1\\\\\n\\end{array}\\right],\n\\]\nand $x$ be the vectorization of the matrix $\\left[E\\left\\{ E(Y\\mid M,R=r,A=a)\\mid R=r^*,A=a^*\\right\\}\\right]_{r,r^*}$. Under a $\\textsc{npsem-ie}$ corresponding to the $\\textsc{dag}$ in Fig. \\ref{fig:2}.(a) where $M$ and $Y$ can be either continuous or discrete, $\\gamma_0$ is partially identified by $\\left[x^T(B\\delta_L+d),x^T(B\\delta_U+d)\\right]$, where $\\delta_L$ and $\\delta_U$ are the minimizing and maximizing solutions respectively to the linear programming problem with objective function $x^TB\\delta$ subject to the constraints\n\\[\\left[\n\\begin{array}{c}\nI_{(p-1)^2} \\\\\n-I_{(p-1)^2}\n\\end{array}\\right]\n\\delta\\leq\n\\left[\n\\begin{array}{c}\n\\min \\{\\mathrm{pr}(R=1\\mid A=a),\\mathrm{pr}(R=1\\mid A=a^*)\\}\\\\\n\\min \\{\\mathrm{pr}(R=1\\mid A=a),\\mathrm{pr}(R=2\\mid A=a^*)\\}\\\\\n\\vdots\\\\\n\\min \\{\\mathrm{pr}(R=p\\mid A=a),\\mathrm{pr}(R=p-1\\mid A=a^*)\\}\\\\\n\\min \\{\\mathrm{pr}(R=p\\mid A=a),\\mathrm{pr}(R=p\\mid A=a^*)\\}\\\\\n\\min \\{0,1-\\mathrm{pr}(R=1\\mid A=a)-\\mathrm{pr}(R=1\\mid A=a^*)\\}\\\\\n\\min \\{0,1-\\mathrm{pr}(R=1\\mid A=a)-\\mathrm{pr}(R=2\\mid A=a^*)\\}\\\\\n\\vdots\\\\\n\\min \\{0,1-\\mathrm{pr}(R=p\\mid A=a)-\\mathrm{pr}(R=p-1\\mid A=a^*)\\}\\\\\n\\min \\{0,1-\\mathrm{pr}(R=p\\mid A=a)-\\mathrm{pr}(R=p\\mid A=a^*)\\}\n\\end{array}\\right]\\]\nand $\\delta\\geq 0$. \n\\end{theorem}\n\nSimilar to the previous result, these bounds coincide if either $R(a)$ or $R(a^*)$ is degenerate. The upper bound is achieved when $R(a)$ and $R(a^*)$ are comonotone; the lower bound is achieved when they are countermonotone. While these bounds are not available in closed form, they can be readily solved using standard software, such as with the lp\\_solve function, which uses the revised simplex method and is accessible from a number of languages, including R, MATLAB, Python, and C. While the method used by this software is not guaranteed to converge at a polynomial rate \\citep{klee1970good}, it is quite efficient in most cases \\citep{schrijver1998theory}. The following corollary shows that these bounds reduce to a closed form when $R$ is binary.\n\n\\begin{corollary}\nUnder a $\\textsc{npsem-ie}$ corresponding to the $\\textsc{dag}$ in Fig. \\ref{fig:2}.(a) with binary $R$,\n\\begin{align*}\n\\min\\limits_{\\pi_{11}\\in\\Pi}\\sum\\limits_{r,r^*}&E\\left\\{ E(Y\\mid M,R=r,A=a)\\mid R=r^*,A=a^*\\right\\}h(r,r^*,\\pi_{11})\\\\\n&\\qquad\\qquad\\qquad\\qquad\\leq\\gamma_0\\leq\\\\\n\\max\\limits_{\\pi_{11}\\in\\Pi}\\sum\\limits_{r,r^*}&E\\left\\{ E(Y\\mid M,R=r,A=a)\\mid R=r^*,A=a^*\\right\\}h(r,r^*,\\pi_{11})\\\\\n\\end{align*}\nwhere $\\Pi$ is the set\n\\begin{align*}\n\\{&\\max\\left\\{0,\\mathrm{pr}(R=1\\mid A=a)+\\mathrm{pr}(R=1\\mid A=a^*)-1\\right\\},\\\\\n&\\min\\left\\{\\mathrm{pr}(R=1\\mid A=a),\\mathrm{pr}(R=1\\mid A=a^*)\\right\\}\\}\n\\end{align*}\nand\n\\begin{align*}\nh(r,r^*,\\pi_{11})=\\left\\{\n\\begin{array}{rl}\n\\pi_{11} & \\textrm{ if } r^*=r=1,\\\\\n\\mathrm{pr}(R=1\\mid A=a)-\\pi_{11} & \\textrm{ if } r^*=0 \\textrm{ and } r=1,\\\\\n\\mathrm{pr}(R=1\\mid A=a^*)-\\pi_{11} & \\textrm{ if } r^*=1 \\textrm{ and } r=0,\\\\\n1-\\mathrm{pr}(R=1\\mid A=a)-\\mathrm{pr}(R=1\\mid A=a^*)+\\pi_{11} & \\textrm{ if } r^*=r=0.\n\\end{array}\\right.\n\\end{align*}\n\\end{corollary}\n\nUnder $A-R$ monotonicity with binary $R$, the identifying functional given by \\cite{tchetgen2014identification} is recovered at the upper bound in Corollary 2. All results given here can be extended to settings with observed pre-exposure confounders, which we denote $C$. In Corollary 1, one must first perform conditional inference given C, then subsequently average over the conditional bounds. This is in fact valid due to Jensen's inequality, because the constraints on the marginal joint probabilities are already implied by the constraints enforced on the conditional joint distributions, so no further constraints need be considered. However, Jensen's inequality does not apply in the case of Theorem 2, so controlling for $C$ requires estimating two pairs of candidate bounds and selecting the larger of the lower bounds and the smaller of the upper bounds. When $p$ is of moderate size, $\\delta$ can be solved for each covariate pattern of $C$, i.e., without modeling the dependence of the cross-world-counterfactual joint distribution on $C$. Averaging the resulting conditional bounds gives the first pair of bounds. The second pair results from replacing each probability in the theorem with an average over the probabilities conditional on $C$ and doing the same with $x$.\n\n\n\\section{Application to Harvard PEPFAR data set}\nWe now consider an application to a data set collected by the Harvard President's Emergency Plan for AIDS Relief (PEPFAR) program in Nigeria. The data set consists of previously antiretroviral therapy (ART)-na{\\\"i}ve, HIV-1 infected adult patients who began ART in the program and were followed at least one year following initiation. Patients without reliable viral load data at two of the hospitals were excluded. Only complete cases initially prescribed to either TDF+3TC\/FTC+NVP or AZT+3TC+NVP\\footnote{3TC=lamivudine, AZT=zidovudine, FTC=emtricitabine, NVP=nevirapine, TDF=tenofovir} were considered for this analysis. Thus, the data set we consider consists of 6627 patients, 1919 of whom were prescribed to TDF+3TC\/FTC+NVP, and the remaining 4708 prescribed to AZT+3TC+NVP.\n\nThere has accumulated evidence of a differential effect on virologic failure between these two first-line antiretroviral treatment regimens \\citep{tang2012review}. Virologic failure is defined by the World Health Organization as repeat viral load above 1000 copies\/mL. We base this on measurements at 12 and 18 months of ART duration in our analysis.\n\nA natural question of scientific interest is what role adherence plays in mediating this differential effect. We are primarily interested in learning about the scientific mechanism of this effect on the individual level. The natural indirect effect best captures this mechanism in that it captures an isolated effect difference mediated by adherence by, in a sense, deactivating effect differences along all other possible causal pathways. We specifically examine the effect through adherence over the second six months since treatment assignment, i.e., the six months prior to the first viral load measurement. Identification is complicated by the presence of treatment toxicity, which clearly affects adherence directly, and has the potential to modify the effect of the treatment assignment on virologic failure. Thus, toxicity measured at six months after treatment assignment is an exposure-induced confounder of the effect of the mediator on the outcome. Further, toxicity and virologic failure are likely to be rendered dependent by unobserved underlying biological common causes as in Fig. \\ref{fig:3}, where $H$ represents these hidden biological mechanisms. Because we define the mediator to be adherence over the second six months, adherence over the first six months is also an exposure-induced confounder along with toxicity, and must be accounted for. Had we defined the mediator to be adherence over the full year, measurement of the mediator and toxicity would have overlapped, violating the principle of temporal ordering.\n\nLet $C$ denote the vector consisting of baseline covariates sex, age, marital status, WHO stage, hepatitis C virus, hepatitis B virus, CD4+ cell count, and viral load. Let $A$ be an indicator of ART assignment taking levels $a^*$ for TDF+3TC\/FTC+NVP and $a$ for AZT+3TC+NVP; $R$ be a vector of two indicator variables, one of the presence of any lab toxicity, and one of adherence exceeding 95\\%, both over the first six months following initiation of therapy; $M$ be an indicator of adherence exceeding 95\\% over the subsequent six months; and $Y$ be an indicator of virologic failure at one year, i.e., repeat viral load above 1000 copies\/mL at one year and at 18 months.\n\nHere we estimate the natural indirect effect of $A$ on $Y$ through $M$, as defined above, on the risk difference scale using the various sets of identifying assumptions given above. Throughout, inference is performed using maximum likelihood for point estimation and a weighted bootstrap \\citep{rao1992approximation,van1996weak} for confidence intervals, which appropriately accounts for the rare outcome. The results are summarized in Fig. \\ref{fig:4}.\n\\begin{figure\n\\centering\n\\includegraphics[scale=.55]{plot4}\n\\caption{A plot showing the estimated natural indirect effect of ART assignment on virologic failure with respect to adherence under the various assumptions. The assumptions vary across the horizontal axis, with the first part of the label indicating the assumption regarding the exposure-induced confounder, $R$, and the second part indicating the assumption regarding cross-world counterfactuals. For the assumptions regarding $R$, ``Ignore\" means that the presence of $R$ is ignored altogether, ``Monoton.\" means the $A$--$R$ monotonicity assumption in Section 1, ``No M*R\" means the no $M$--$R$ interaction assumption in Section 1, and ``None\" means that $R$ was accounted for without additional assumptions. For the assumptions regarding cross-world counterfactuals, ``IE\" means a $\\textsc{npsem-ie}$ was assumed, and ``None\" means no cross-world-counterfactuals independences were assumed. When the assumptions give partial identification, the two dots represent the point estimates of the upper and lower bound for the natural indirect effect, and the vertical bar represents the bootstrap 95\\% confidence interval for the interval. When the assumptions give full identification, the single dot represents the point estimate of the natural indirect effect, and the vertical bar represents its bootstrap 95\\% confidence interval.}\n\\label{fig:4}\n\\end{figure}\nIt is immediately apparent that inference is sensitive to which identifying assumptions are made. Consider an investigator who might be willing to rely on cross-world-counterfactual independences. If she decides to ignore the presence of toxicity, she might likely conclude that there is a very small, yet significant negative indirect effect. Conversely, were she to make the no $M$--$R$ interaction assumption, she would find a significant positive indirect effect with considerable uncertainty. In fact, an empirical test of this assumption reveals that it is unlikely to apply. Likewise, the data suggest that the required assumption of independent errors of the components of $R$ is also unlikely to hold. Nonetheless, we present both results for the sake of comparison. Results are fairly imprecise under monotonicity, and do not show a significant effect.\n\nAnother investigator unwilling to impose cross-world-counterfactual-independence assumptions is left with little to say as the bounds are wide, and include the null hypothesis of no $\\textsc{nie}$, regardless of how toxicity is handled. Interestingly, the bounds that result from making no assumptions about the joint distribution of the cross-world $R$ counterfactuals are narrower than the bounds that result from ignoring $R$. That is, the bounds themselves appear narrower; the variances of the interval estimates appear to be comparable. This is because even though we do not impose any restrictions on the distribution of $R$ or its counterfactuals a priori, observing $R$ is clearly informative. The bounds accounting for $R$ have the added advantage of being the only identifying formula that remains valid when toxicity and virologic suppression are affected by an unobserved common cause, as in Fig. \\ref{fig:3}.\n\nFinally, incorporating $R$ results in narrower interval estimates than not imposing the $\\textsc{npsem-ie}$, even if $R$ were ignored. Thus, cross-world-counterfactual-independences appear to have stronger empirical implications in the current analysis than assumptions regarding exposure-induced confounders. The general trend in these results is that little is gained in terms of precision by assumptions regarding $R$. In fact, the confidence interval for the bounds resulting from the independent errors assumption and no assumption regarding $R$ is narrower than the confidence interval for the estimate that results from assuming monotonicity, despite the fact that the $\\textsc{nie}$ is point-identified in the latter case. The na\\\"{\\i}ve assumption that $R$ is not a confounder is the only assumption about $R$ under which precision is gained.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nOne of the most prominent features of the atomic nucleus is that it \norganizes itself into various types of geometrical shapes \\cite{BM} that often \nevolve gradually as  functions of the nucleon number within an isotopic \nor isotonic chain. In some instances, however, such  shape evolution \ntakes place abruptly at particular nucleon numbers. This phenomenon is \nknown as a (quantum) shape phase transition \\cite{cejnar2010}. Over the \npast decades, numerous experiments have been carried out to measure  \nobservables signaling such  phase transitions \\cite{cejnar2010}. \nTheoretical calculations have also been carried out within several  \nframeworks \\cite{niksic2007,robledo2009,li2010,cejnar2010,shimizu2017}. \nTypical examples of shape phase transitions are, the \nspherical-to-axially-deformed \\cite{iachello2001} and the \nspherical-to-$\\gamma$-soft \\cite{iachello2000} ones. Other types of  \ntransitions include the one that occurs between prolate and oblate \nconfigurations going through a transitional $\\gamma$-soft shape \n\\cite{jolie2001}. \n\n\nNuclear shape transitions have been  well studied for even-even nuclei. \nThere is, however, a wealth of experimental information for odd-mass \nsystems that remains to be analyzed from a theoretical perspective. \nWithin this context, it is particularly interesting to \nconsider the nature of  phase transitions in those odd-mass nuclei and \nhow they correlate with the ones in the neighbouring even-even systems \n\\cite{iachello2011}. However, the theoretical description of  odd-mass \nnuclei tends to be more cumbersome than for even-even systems, as one \nneeds to take into account both collective and single-particle degrees \nof freedom on an equal footing \\cite{bohr1953}. \n\n\nThe aim of this paper is to study the effect of the odd particle on the \nprolate-to-oblate shape transition in  neutron-rich nuclei with mass \nnumber $A\\approx 190$. To this end, we have selected the  odd-mass  \nsystems $^{185-199}$Pt $^{185-193}$Os and $^{185-195}$Ir. Their \neven-even neighbors $^{186-200}$Pt and $^{186-194}$Os, are considered \nto be good examples of the prolate-to-oblate shape transition. In many cases \n$\\gamma$-soft shapes  are also found. Therefore, they represent  a \nstringent test for nuclear structure models. In order to describe  \nspectroscopic properties, we have  resorted to the recently developed \nmethod of Ref.~\\cite{nomura2016odd}, based on the nuclear energy \ndensity functional  (EDF) theory and the particle-core coupling scheme \n\\cite{iachello1979,IBFM}. The method has already been applied to study the \nspherical-to-axially-deformed \\cite{nomura2016qpt,nomura2017odd-2} and \nspherical-to-$\\gamma$-soft \\cite{nomura2017odd-1,nomura2017odd-3} shape \nphase transitions as well as octupole correlations in neutron-rich Ba \nnuclei \\cite{nomura2018oct}. The robustness of the method has been \nstudied using both non-relativistic \\cite{nomura2017odd-2} and \nrelativistic \\cite{nomura2017odd-1} EDFs.\n\n\nIn this work, the even-even Pt and Os nuclei are described within the \nneutron-proton interacting boson model (IBM-2) \\cite{OAI,IBM} built on \nthe neutron (proton) $s_{\\nu}$ and $d_{\\nu}$ ($s_{\\pi}$ and $d_{\\pi}$) \nbosons, which represent correlated $J^{\\pi}=0^+$ and $2^+$ pairs of \nvalence neutrons (protons) \\cite{OAI}. On the other hand, the \nparticle-core coupling is considered within the neutron-proton \ninteracting boson-fermion model (IBFM-2) \\cite{alonso1984,IBFM}. Similar to our previous \nstudies \\cite{nomura2017odd-2,nomura2017odd-3}, which were based on the \nsimpler IBM-1 model \\cite{nomura2017odd-2}, the strength parameters for \nthe IBM-2 Hamiltonian, the single-particle energies and  the occupation \nprobabilities of the odd particle, are determined by  constrained \nself-consistent mean-field calculations based on the Gogny-D1M  EDF \n\\cite{D1M,Gogny}. The coupling constants of the boson-fermion \ninteraction are the only free parameters of the model. They are specifically fitted \nto  reproduce the low-lying excitation spectrum for each odd-mass \nnucleus. The IBFM-2 phenomenology has already been considered  in this \nregion of the nuclear chart \\cite{arias1986}. However, in this work we \nresort to a microscopic input obtained from the Gogny-D1M EDF \nframework, i.e., the IBM-2 Hamiltonian parameters, single-particle \nenergies and occupation probabilities are  determined within the HFB \nframework. In a previous study \\cite{nomura2011sys}, we have already \nconsidered even-even nuclei in this mass region, including the Pt and \nOs ones studied here, with the IBM-2 Hamiltonian parameters derived \nfrom HFB calculations based on the  Gogny-D1M EDF. In this work, we \nwill take the IBM-2 Hamiltonian parameters for  the even-even nuclei \nfrom Ref~\\cite{nomura2011sys} and focus on the remaining ones to study the \nodd-mass nuclei. Let us also mention that other theoretical frameworks, \nlike the symmetry-projected generator coordinate method (GCM) for odd \nmass systems \\cite{bally2014,borrajo2016} and the large-scale shell \nmodel \\cite{caurier2005,shimizu2017}, could be employed. However, they \nare computationally much more demanding for heavier and\/or open-shell \nnuclei. Hence, computationally feasible schemes, such as the \nparticle-vibration coupling \\cite{bohr1953}, represent a more feasible \nalternative and have often been considered in the literature, e.g., \\cite{colo2017}.\n\n\nThe paper is organized as follows. In Sec.~\\ref{sec:model}, we briefly \noutline the theoretical framework used in this study. There, we will \nalso discuss the Gogny-HFB deformation energy surfaces as well as  the  \nparameters of the Hamiltonian. Then, in Sec.~\\ref{sec:sys}, we discuss  \nthe spectroscopic properties of the considered nuclei. We briefly \nreview the results obtained for even-even nuclei in \nSec.~\\ref{sec:even}. The systematics of the low-lying yrast levels in \nthe odd-mass nuclei is presented in Sec.~\\ref{sec:odd}. More detailed \nlevel schemes and electromagnetic properties for  some selected \nodd-mass nuclei are discussed in Sec.~\\ref{sec:detail}. As yet another \nsignature of the prolate-to-oblate shape transition, in \nSec.~\\ref{sec:def}, we consider effective $\\beta$ and $\\gamma$ \ndeformations. Finally, Sec.~\\ref{sec:summary} is devoted to the \nconcluding remarks.\n\n\n\n\n\\section{Building the interacting boson-fermion Hamiltonian\\label{sec:model}}\n\n\n\nThe IBFM-2 Hamiltonian is comprised of the IBM-2 Hamiltonian \n$\\hat H_{\\rm B}$ \\cite{nomura2011sys}, the Hamiltonian for the odd nucleon $\\hat H_{\\rm F}$,\nand the boson-fermion interaction $\\hat H_{\\rm BF}$: \n\\begin{eqnarray}\n\\label{eq:ham}\n \\hat H_{\\rm IBFM} = \\hat H_{\\rm B} + \\hat H_{\\rm F} + \\hat H_{\\rm BF}.\n\\end{eqnarray}\n\nIn this expression, the doubly-magic nucleus $^{208}$Pb is taken as the \ninert core. In the IBM-2, the number of neutron (proton) bosons \n$N_{\\nu}$ ($N_\\pi$) is equal to half the number of valence neutrons \n(protons) and is counted as the number of holes in the latter half of a \ngiven major shell. In the present case, all the bosons are  hole-like \nand therefore $2\\leq N_\\nu\\leq 9$ and $N_\\pi= 2$ for $^{186-200}$Pt and \n$4\\leq N_\\nu\\leq 8$ and $N_\\pi= 3$ for $^{186-194}$Os. The strength \nparameters for the IBM-2 Hamiltonian for the even-even nuclei \n$^{186-200}$Pt and $^{186-194}$Os have been previously determined \n\\cite{nomura2008} by mapping  the $(\\beta,\\gamma)$-deformation energy \nsurface, computed within the constrained Gogny-D1M HFB approach, onto \nthe expectation value of the IBM-2 Hamiltonian in the boson condensate \nstate \\cite{ginocchio1980}. For a more detailed account of the whole \nprocedure, the reader is referred to \nRefs.~\\cite{nomura2008,nomura2011pt,nomura2011sys}. The parameters \nobtained via the mapping procedure can be found in Table I of \nRef.~\\cite{nomura2011sys}.\n \n\n\n\n\\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[width=\\linewidth]{pes-d1m.pdf}\n\\caption{(Color online) The Gogny-D1M HFB deformation energy surfaces in the\n $(\\beta,\\gamma)$-deformation space for the \n $^{186-194}$Pt and  $^{186-194}$Os nuclei are plotted  up to 3 MeV from the\n global minimum. The energy difference between the neighbouring\n contours is 100 keV.  \n } \n\\label{fig:pes}\n\\end{center}\n\\end{figure}\n\nIn Fig.~\\ref{fig:pes} we have depicted the Gogny-D1M energy surfaces \nfor those even-even Pt and Os nuclei corresponding to the \nprolate-oblate transitional regions. Results for other nuclei can be \nfound in Ref.~\\cite{robledo2009}. Note, that in Ref.~\\cite{robledo2009} \nwe have resorted to the Gogny-D1S \\cite{D1S} EDF. However, the \nmean-field surfaces obtained with the parametrization D1S are \nessentially the same as the ones provided by the  parameter set D1M. \nThe minimum of each of the energy surfaces in the figures changes \ngradually, as a function of neutron number, from near prolate to \nshallow triaxial  (around $^{188}$Pt and $^{190}$Os) and then near \noblate  (around $^{192}$Pt and $^{194}$Os). The IBM-2 energy surfaces, \nobtained via the mapping procedure, can be found in \nRef.~\\cite{nomura2011sys}. \n \n\n\nThe second term in Eq.~(\\ref{eq:ham}) reads, $\\hat H_{\\rm \nF}=\\sum_{j}\\epsilon_{j\\tau} (a_{\\tau,j}^{\\dagger}\\times \\tilde \na_{\\tau,j})^{(0)}$, with $\\epsilon_j$ being the single-particle \nenergies of the orbitals for the odd neutron ($\\tau=\\nu$) or proton \n($\\tau=\\pi$). As fermionic valence space for the odd-$N$ Pt and Os \nnuclei, we have taken the whole neutron $N=82-126$ major shell: \n$3p_{1\/2}$, $3p_{3\/2}$, $2f_{5\/2}$, $2f_{7\/2}$, $1h_{9\/2}$ for \nnegative-parity states and $1i_{13\/2}$ for positive-parity states. For \nthe odd-$Z$ Ir isotopes we have taken the whole proton $Z=50-82$ major \nshell: $3s_{1\/2}$, $2d_{3\/2}$, $2d_{5\/2}$, $1g_{7\/2}$ for positive \nparity and $1h_{11\/2}$ for negative-parity states.  Note that all the \nvalence particles are treated here as holes. Therefore, for an odd-mass \nnucleus with mass $A$, its even-even neighbor with mass $A+1$ is taken \nas the even-even boson core. \n\n\n\n\n\\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[width=\\linewidth]{{para.ptos}.pdf}\n\\caption{(Color online) The strength parameters\n $\\Gamma_\\nu$, $\\Lambda_\\nu$ and $A_\\nu$, employed\n for the odd-$N$ Pt and Os nuclei are depicted  as functions of the neutron number.} \n\\label{fig:para-ptos}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[width=\\linewidth]{{para.ir}.pdf}\n\\caption{(Color online) The strength parameters $\\Gamma_\\pi$,\n $\\Lambda_\\pi$ and $A_\\pi$ for the\n odd-$Z$ Ir isotopes are depicted  as functions of the neutron number. } \n\\label{fig:para-ir}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[width=\\linewidth]{{spe.ptos}.pdf}\n\\caption{(Color online) Single-particle energies (plotted with respect to\n that of the $3p_{1\/2}$ orbital) and occupation\n probabilities employed for the odd-$N$ Pt and Os nuclei as functions of the neutron number. } \n\\label{fig:spe-ptos}\n\\end{center}\n\\end{figure}\n\n\n\n\\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[width=\\linewidth]{{spe.ir}.pdf}\n\\caption{(Color online) The same as in  Fig.~\\ref{fig:spe-ptos} but for the\n odd-$Z$ Ir isotopes. The single-particle energies are plotted with respect to\n $3s_{1\/2}$ orbital.} \n\\label{fig:spe-ir}\n\\end{center}\n\\end{figure}\n\n\nFor the boson-fermion interaction term $\\hat H_{\\rm BF}$ in\nEq.~(\\ref{eq:ham}), we use an expression similar to the one used in previous studies\n\\cite{scholten1985,arias1986}: \n\\begin{eqnarray}\n\\label{eq:ham-bf}\n \\hat H_{\\rm BF} = &&\\Gamma_\\nu\\hat Q_{\\pi}^{(2)}\\cdot\\hat q_{\\nu}^{(2)} +\n \\Gamma_\\pi\\hat Q_{\\nu}^{(2)}\\cdot\\hat q_{\\pi}^{(2)}  \\nonumber \\\\ \n&&+\n  \\Lambda_\\nu\\hat V_{\\pi\\nu} + \\Lambda_\\pi\\hat V_{\\nu\\pi} + A_\\nu\\hat n_{d\\nu}\\hat\n  n_{\\nu} + A_\\pi\\hat n_{d\\pi}\\hat\n  n_{\\pi}, \n\\end{eqnarray}\nwhere  the first and second terms are the quadrupole dynamical terms,\nwith the bosonic \nquadrupole operator for proton $\\hat Q^{(2)}_{\\pi}$  and \nneutron $\\hat Q^{(2)}_{\\nu}$, respectively. The fermionic quadrupole \noperator for the odd neutron or proton reads: \n\\begin{eqnarray}\n\\hat\nq^{(2)}_\\tau=\\sum_{jj'}\\gamma_{jj'}(a^\\+_{j\\tau}\\times\\tilde\na_{j'\\tau})^{(2)},\n\\end{eqnarray} \nwhere $\\gamma_{jj'}=(u_ju_{j'}-v_jv_{j'})Q_{jj'}$ and  $Q_{jj'}=\\langle\nj||Y^{(2)}||j'\\rangle$ represents\nthe matrix element of the fermionic \nquadrupole operator in the considered single-particle basis.\nThe third and fourth terms in Eq.~(\\ref{eq:ham-bf}) are the exchange\ninteractions. They are  introduced to \naccount for the fact that bosons are built from nucleon pairs and are\ngiven by \\cite{alonso1984,arias1986}\n\\begin{eqnarray}\n\\label{Rayner-new-label}\n \\hat V_{\\pi\\nu} =&& -(s_{\\pi}^\\+\\tilde d_{\\pi})^{(2)}\n\\cdot\n\\Bigg\\{\n\\sum_{jj'j''}\n\\sqrt{\\frac{10}{N_\\nu(2j+1)}}\\beta_{jj'}\\beta_{j''j} \\nonumber \\\\\n&&:((d_{\\nu}^\\+\\times\\tilde a_{j''\\nu})^{(j)}\\times\n(a_{j\\nu}^\\+\\times\\tilde s_\\nu)^{(j')})^{(2)}:\n\\Bigg\\} + (H.c.), \\nonumber \\\\\n\\end{eqnarray}\nwith a similar expression for  $\\hat V_{\\nu\\pi}$.\nIn Eq.~(\\ref{Rayner-new-label}), $\\beta_{jj'}=(u_jv_{j'}+v_ju_{j'})Q_{jj'}$.\nThe fifth and sixth terms in Eq.~(\\ref{eq:ham-bf}) are the monopole\ninteractions with $\\hat n_{d\\nu}$ and $\\hat n_{d\\pi}$ the number\noperators for neutron and proton $d$ bosons, respectively, while the\nnumber operator for the odd fermion $\\hat\nn_{\\tau}=\\sum_j(-\\sqrt{2j+1})(a^\\+_{j\\tau}\\times\\tilde \na_{j\\tau})^{(0)}$. \n\nThe boson-fermion Hamiltonian $\\hat H_{\\rm BF}$ in Eq.~(\\ref{eq:ham-bf})\nhas been justified from microscopic considerations based on the\ngeneralized seniority scheme \\cite{IBFM,scholten1985}. \nBoth the quadrupole dynamic and exchange terms act predominantly\nbetween protons and neutrons (i.e., between odd neutron and proton\nbosons and between odd proton and neutron bosons) \\cite{alonso1984},\nwhile the monopole interaction acts between like-particles \n(i.e., between odd neutron and neutron bosons and between odd proton and\nproton bosons) \\cite{IBFM}. \n\nThe single-particle energies $\\epsilon_{j\\tau}$ and occupation \nprobabilities $v^2_j$ of the odd nucleon at the  $j$ orbital are \nobtained from Gogny-D1M HFB calculations constrained to quadrupole \nmoment zero. Note that this is a standard HFB calculation without \nblocking. However, the particle number is constrained to odd $N$ or \n$Z$. For more details, see Ref.~\\cite{nomura2017odd-2}. \n\n\nThe coupling constants $\\Gamma_\\nu$, $\\Gamma_\\pi$, $\\Lambda_\\nu$,\n$\\Lambda_\\pi$, $A_\\nu$, and $A_{\\pi}$ in Eq.~(\\ref{eq:ham-bf}) are treated as free\nparameters. They have been  \nfitted to reproduce the energies of\nthe lowest-lying states in each of the odd-mass nuclei, separately for\nnormal-parity and unique-parity configurations. Those \nparameters have been  plotted in  Figs.~\\ref{fig:para-ptos} (odd-$N$ Pt\nand Os nuclei) and \\ref{fig:para-ir} (odd-$Z$ Ir nuclei). \nAs can be seen in\nFigs.~\\ref{fig:para-ptos}(a) and \\ref{fig:para-ptos}(b), the\n$\\Gamma_\\nu$ values for both the normal-parity (denoted by pfh) and\nunique-parity (i13) configurations stay  rather  constant as a function of\nneutron number.\nThe $\\Lambda_\\nu$ values, in Figs.~\\ref{fig:para-ptos}(c) and\n\\ref{fig:para-ptos}(d) also look rather insensitive to variations in neutron\nnumber except for the abrupt change in the $\\Lambda_\\nu$ value\nfor the $i_{13\/2}$ configuration (see, Fig.~\\ref{fig:para-ptos}(d)) in going from\n$^{185}$Pt to $^{187}$Pt.  As shown later in Fig.~\\ref{fig:level-pt-i13}, \nthe abrupt change observed is very likely a consequence of the\nevolution of the low-lying positive-parity level structure when going from\none nucleus to the other. \nFurthermore, as can be seen from Figs.~\\ref{fig:para-ptos}(e) and\n\\ref{fig:para-ptos}(f),  the monopole strength\n$A_\\nu$ is chosen to be zero in many of the \nstudied nuclei. Nevertheless,  a relatively large\nvalue is needed for the transitional regions, i.e., $^{189,191}$Pt and\n$^{189,191}$Os. \nThe $\\Gamma_\\pi$, $\\Lambda_\\pi$ and $A_\\pi$ values for the odd-$Z$ Ir isotopes are\nshown in Fig.~\\ref{fig:para-ir}. They are \nalso nearly constant or change only moderately as functions of the neutron number $N$. Since the monopole \ninteraction turns out to play a\nmajor role for most of the considered Ir nuclei, its strength parameter is\nmuch larger in magnitude than in the case of the odd-mass Pt and Os nuclei. \n\n\nIn addition, we plot in Figs.~\\ref{fig:spe-ptos} and \\ref{fig:spe-ir} the single-particle\nenergies and occupation probabilities used in the present study for the\nconsidered odd-mass nuclei. \n\n\nThe  IBFM Hamiltonian of Eq.~(\\ref{eq:ham}), with  parameters \ndetermined via the mapping procedure, has been diagonalized to obtain \nexcitation spectra and electromagnetic transition rates. For the later,  \nE2 and M1 operators similar to those in Refs \n\\cite{nomura2017odd-2,nomura2017odd-3} have been used. The effective \nbosonic charge  has been taken to be the same for both protons and \nneutrons, with a value $e_{\\rm B}^\\nu=e_{\\rm B}^\\pi=0.15\\,e$b fitted to \nreproduce the experimental $B(E2;2^+_1\\rightarrow 0^+_1)$ transition \nrate in $^{196}$Pt. For the fermion effective charges we have used the \nvalues $e_{\\rm F}^\\nu=0.3\\,e$b and $e_{\\rm F}^{\\pi}=0.5\\,e$b. Moreover, \nfor the bosonic $g$-factor we have also taken the same value for \nprotons and neutrons $g_{\\rm B}^\\nu=g_{\\rm B}^\\pi=0.3\\,\\mu_N$ that is \nfitted to reproduce the magnetic moment of the $2^+_1$ state in \n$^{196}$Pt. For the fermionic $g$-factors, we have adopted \n$g_l=1.0\\,\\mu_N$ for the odd proton and $g_l=0\\,\\mu_N$ for the odd \nneutron. The free values of $g_s$ have been quenched by 30 \\%. \n\n\n\n\n\n\\section{Results for spectroscopic properties\\label{sec:sys}}\n\nIn this section, we discuss the results of this study. First, in \nSec.~\\ref{sec:even}, we briefly discuss the results obtained for \neven-even nuclei. The systematics of the \nlow-lying yrast levels in the odd-mass nuclei is presented in\nSec.~\\ref{sec:odd}. A more detailed analysis of the level\nschemes and electromagnetic properties for some selected odd-mass nuclei \nis presented in Sec.~\\ref{sec:detail}.\nFinally, in Sec.~\\ref{sec:def}, we consider \neffective $\\beta$ and $\\gamma$ deformation parameters as another \nsignature of the\nprolate-to-oblate shape transition.\n\n \n\\subsection{Even-even nuclei\\label{sec:even}}\n\n\n\\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[width=\\linewidth]{{energies.ptos}.pdf}\n\\caption{(Color online) Spectroscopic properties of the even-even\n $^{186-200}$Pt and $^{186-194}$Os nuclei plotted as functions of the\n neutron number: the energy ratios\n $R_{42}$ and $R_{4\\gamma}$, spectroscopic quadrupole\n moment  $Q_{2^+_1}$ and the $B(E2)$ value\n $B_{2\\gamma}$. For more details, see the main\n text. Open circles, connected by lines, represent the theoretical values\n while solid diamonds represent the experimental data taken from\n Refs.~\\cite{data,stone2005}. The symmetry limits $R_{42}=2.0$ (U(5)), \n2.5 (O(6)) and 3.3 (SU(3)), and $R_{4\\gamma}=1.0$ (O(6)) of the IBM \n\\cite{IBM} are also indicated.} \n\\label{fig:even}\n\\end{center}\n\\end{figure}\n\n \n\nLet us first consider how the IBM-2 Hamiltonian, extracted from the \nGogny-D1M HFB calculations via the mapping procedure, describes the \nspectroscopic properties of the even-even nuclei. To this end, in \nFig.~\\ref{fig:even}, we have plotted spectroscopic properties of  \n$^{186-200}$Pt and $^{186-194}$Os as functions of the neutron number \n$N$. The symmetry limits $R_{42}=2.0$ (U(5)), 2.5 (O(6)) and 3.3 \n(SU(3)) and $R_{4\\gamma}=1.0$ (O(6)) of the IBM \\cite{IBM}, \nare also indicated in the figure. As can be seen in \nFig.~\\ref{fig:even}(a),\nboth the theoretical and experimental $R_{42}=E(4^+_1)\/E(2^+_1)$ ratios \nfor Pt isotopes do not change too much and are located around the O(6) limit \n$R_{42}\\approx 2.5$. On the other hand, from Fig.~\\ref{fig:even}(a), \none observes that the theoretical ratio $R_{42}$ systematically \noverestimates the experimental one for the heavier isotopes. This might be \na consequence of the pronounced ground state deformation  in the \ncorresponding Gogny-D1M energy surfaces when approaching the neutron \nclosed shell $N=126$. As a result, the IBM-2 model provides a more \nrotational-like spectrum. The $R_{42}$, displayed in \nFig.~\\ref{fig:even} for Os isotopes, changes rather fast  with $N$ \ncompared to the Pt isotopes. The energy ratio \n$R_{4\\gamma}=E(4^+_1)\/E(2^+_2)$, depicted in Figs.~\\ref{fig:even}(b) \nand \\ref{fig:even}(f), can be regarded as a  signature of \n$\\gamma$-softness. The predicted $R_{4\\gamma}$ values for both Pt and \nOs nuclei exhibit a peak at around $^{188}$Pt ($N=110$) and $^{192}$Os \n($N=116$), being close to the O(6) limit of $R_{4\\gamma}=1.0$. This \nindicates that those nuclei represent the most $\\gamma$-soft among the \nconsidered systems. Indeed, the  Gogny-D1M energy surface for \n$^{188}$Pt exhibits the most pronounced triaxial minimum at \n$\\gamma\\approx 30^{\\circ}$ (see, Fig.~\\ref{fig:pes}). As can be seen \nfrom Fig.~\\ref{fig:even}(b), both theoretically and experimentally, the \nratio $R_{4\\gamma}$ remains rather constant in the case of Pt isotopes. \nHowever, the experimental $R_{4\\gamma}$ values are systematically \nunderestimated for $N\\geq 112$. This may be due to a similar reason as \nwith the discrepancy in $R_{42}$  already mentioned.\n\n\nThe spectroscopic quadrupole moment $Q_{2^+_1}$ for the $2^+_1$ state,\ndisplayed in Figs.~\\ref{fig:even}(c) and \\ref{fig:even}(g), represents\na useful measure of whether\nthe nucleus is prolate or oblate. As can be seen in Fig.~\\ref{fig:even}(c),\nthe theoretical $Q_{2^+_1}$ value is negative \nfor $^{186,188}$Pt while it is positive for $^{190-200}$Pt. This is \nconsistent with the prolate-to-oblate shape transition observed at the \nmean-field level in Fig.~\\ref{fig:pes}. \nA similar observation can be made for the Os isotopes in Fig.~\\ref{fig:even}(g).\nFurthermore, the $B_{2\\gamma}=B(E2; 2^+_2\\rightarrow 2^+_1)$ transition\nprobability provides a stringent test for\n$\\gamma$-softness. For both Pt and Os isotopes, it shows a\npeak at $N=110$ (Fig.~\\ref{fig:even}(d)) and 116 (Fig.~\\ref{fig:even}(h)). \n\n\n\n\\subsection{Systematics of low-energy excitation spectra in odd-mass nuclei\\label{sec:odd}}\n\n\n\\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[width=\\linewidth]{{oddpt.level.pfh}.pdf}\n\\caption{(Color online) The theoretical and experimental low-lying\nnegative-parity ($\\pi=-1$) yrast states in the\nodd-$N$ isotopes $^{185-199}$Pt are plotted as \nfunctions of the neutron number.} \n\\label{fig:level-pt-pfh}\n\\end{center}\n\\end{figure}\n\n\n\n\\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[width=\\linewidth]{{oddpt.level.i13}.pdf}\n\\caption{(Color online) The same as in Fig.~\\ref{fig:level-pt-pfh} but for the\npositive-parity states.} \n\\label{fig:level-pt-i13}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[width=\\linewidth]{{oddos.level.pfh}.pdf}\n\\caption{(Color online) The  same as in Fig.~\\ref{fig:level-pt-pfh}, \nbut for the isotopes $^{185-193}$Os.} \n\\label{fig:level-os-pfh}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[width=\\linewidth]{{oddos.level.i13}.pdf}\n\\caption{(Color online) The same as in Fig.~\\ref{fig:level-pt-i13} \nbut for the isotopes $^{185-193}$Os.} \n\\label{fig:level-os-i13}\n\\end{center}\n\\end{figure}\n\n\n\n\\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[width=\\linewidth]{{oddir.level.h11}.pdf}\n\\caption{(Color online) The same as in Fig.~\\ref{fig:level-pt-pfh} \nbut for the isotopes $^{185-195}$Ir.} \n\\label{fig:level-ir-h11}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[width=\\linewidth]{{oddir.level.sdg}.pdf}\n\\caption{(Color online) The same as in Fig.~\\ref{fig:level-pt-i13} \nbut for the isotopes $^{185-195}$Ir.} \n\\label{fig:level-ir-sdg}\n\\end{center}\n\\end{figure}\n\n\nIn Figs.~\\ref{fig:level-pt-pfh} to \\ref{fig:level-ir-sdg}, we have \nplotted the energy systematics of the low-lying positive- and \nnegative-parity yrast states in the odd-$N$ nuclei $^{185-199}$Pt, \n$^{185-193}$Os and $^{185-195}$Ir as functions of $N$. For the three \nisotopic chains, our calculations  describe reasonably well the \nexperimental trend. For the odd-$N$ Pt isotopes, in \nFigs.~\\ref{fig:level-pt-pfh} and \\ref{fig:level-pt-i13}, the observed \nchange in the ground state's spin in going from $N=107$ to 109 for both \nparities, can be regarded as a signature of  structural evolution and \ncorrelates well with the shape transition that occurs in the \ncorresponding even-even systems. Indeed, the Gogny-D1M energy surfaces \n(see, Fig.~\\ref{fig:pes}) suggest the transition from prolate \n($^{186}$Pt) to triaxial shapes ($^{188}$Pt). For $N=109-113$, both \ntheoretically and experimentally as well as for both parities, a \nsimilar low-lying level structure is observed. However, as seen from \nFig.~\\ref{fig:level-pt-pfh}(a), another signature of the shape \ntransition appears in the case of the negative-parity states for \nodd-$N$ Pt isotopes, i.e., at the neutron number $N=113$ many states \nare found below 0.3 MeV excitation energy while those levels higher \nthan the $J={5\/2}^-$ one go up rapidly for larger $N$. This also \ncorrelates well with the  Gogny-D1M energy surfaces obtained for \neven-even nuclei (see, Fig.~\\ref{fig:pes}) which exhibit a gradual \nchange of the global minimum from shallow triaxial ($^{192}$Pt) to \noblate  ($^{194}$Pt). \n\n\nThe results obtained for odd-$N$ Os isotopes are shown in \nFigs.~\\ref{fig:level-os-pfh} and \\ref{fig:level-os-i13}. For the \nnegative-parity states, in Fig.~\\ref{fig:level-os-pfh}, both \nexperimentally and theoretically the low-lying level structure below \n0.3 MeV excitation energy changes significantly  around $N=115$, \nincluding a change in the ground state's spin. Once more, this agrees \nwell with the Gogny-D1M energy surfaces (see, Fig.~\\ref{fig:pes}) \nsuggesting a transition from a triaxial shape at $^{192}$Os ($N=116$) \nto an oblate-soft one at $^{194}$Os ($N=118$). At $N=109$ and 111, the predicted \nground-state spins for positive parity states \n(Fig.~\\ref{fig:level-os-i13}(a)) do not coincide with the experiment \n(Fig.~\\ref{fig:level-os-i13}(b)). This results from the fact, that the \nboson-fermion parameters for those nuclei have been chosen so as to \nreproduce the overall level structure up to the spin $J\\sim{19\/2}^+$. \nHowever, we have also verified that if one attempts to reproduce the \nexperimental ground-state's spin for $^{185,187}$Os, the whole spectrum \nbecomes too compressed.\n\n\nSimilar observations apply to the results for the odd-$Z$ Ir nuclei, \ndepicted in Figs.~\\ref{fig:level-ir-h11} and \\ref{fig:level-ir-sdg}. \nFor example,  both the theoretical and experimental negative-parity \nspectra in Figs.~\\ref{fig:level-ir-h11}(a) and \n\\ref{fig:level-ir-h11}(b), respectively, suggest a rapid structural \nchange in going from $N=110$ to 112. At those neutron numbers, the \ncorresponding even-even Pt core nuclei undergo a structural change in \ntheir energy surfaces and spectroscopic properties (see, Figs.~\\ref{fig:pes} and \\ref{fig:even}). \n\n\n\n\n\\subsection{Detailed level schemes for selected odd-mass nuclei\\label{sec:detail}}\n\n\n\n\n\n\n\\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[width=\\linewidth]{{195pt.basic-crop}.pdf}\n\\caption{(Color online) Comparison between the theoretical and\nexperimental \\cite{data} low-lying positive- and negative-parity \nspectra for  $^{195}$Pt.} \n\\label{fig:195pt}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[width=\\linewidth]{{189os.basic-crop}.pdf}\n\\caption{(Color online) The same as in Fig.~\\ref{fig:195pt} \nbut for $^{189}$Os.} \n\\label{fig:189os}\n\\end{center}\n\\end{figure}\n\n\n\n\\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[width=\\linewidth]{{191ir.basic-crop}.pdf}\n\\caption{(Color online) The same as in Fig.~\\ref{fig:195pt}\nbut for $^{191}$Ir.} \n\\label{fig:191ir}\n\\end{center}\n\\end{figure}\n\n\nIn this section, we present a more detailed analysis of the nuclei \n$^{195}$Pt, $^{189}$Os and $^{191}$Ir, taken as illustrative examples. \nFor them, abundant experimental information, especially for \nelectromagnetic properties, is available for a detailed comparison with \nthe theory predictions. \n\nAs can be seen from Fig.~\\ref{fig:195pt}, our calculations reproduce \nreasonably well the experimental negative-parity yrast states for \n$^{195}$Pt. However, the predicted  non-yrast levels tend to be \noverestimated like, for example, those experimental levels around \n$\\approx$0.2 MeV excitation energy. The discrepancies occur mainly \nbecause the single-particle energies and $v^2_j$ values used in the \ncalculations may not be realistic enough to reproduce those levels. On \nthe other hand, the agreement between the theoretical and experimental \npositive-parity levels is reasonable. \n\nIn Table~\\ref{tab:195pt}, we compare the predicted and experimental \ntransition rates $B(E2)$ and $B(M1)$ as well as the spectroscopic \nquadrupole $Q_{J}$ and magnetic $\\mu_J$ moments. The overall agreement \nis reasonably good. However, there are also some noticeable \ndiscrepancies. For instance, both the  $B(E2; \n{5\/2}^-_2\\rightarrow{3\/2}^-_1)$ and $B(M1; \n{5\/2}^-_2\\rightarrow{3\/2}^-_1)$ values are significantly smaller than \nthe experimental ones. The dominant components of the  IBFM-2 wave \nfunction for the ${5\/2}^-_2$ state are $3p_{3\/2}$ (39 \\%) and \n$2f_{5\/2}$ (45 \\%), while those for the ${3\/2}^-_1$ state are \n$3p_{1\/2}$ (45 \\%) and $3p_{3\/2}$ (37 \\%). Such a  difference in the \ncomposition of the wave functions result in a small overlap between the\ntwo states that is the main responsible of the too small \nE2 and M1 transition rates predicted. \n\n\n\n\\begin{table}[htb]\n\\caption{\\label{tab:195pt}%\nThe theoretical $B(E2)$ and $B(M1)$ transition probabilities  (in\nWeisskopf units) and the $Q_J$ (in $e$b units) and $\\mu_J$ (in $\\mu_N$ \nunits) values for $^{195}$Pt are compared with the available\nexperimental data \\cite{data,stone2005}. }\n\\begin{center}\n\\begin{tabular}{p{2.5cm}cccc}\n\\hline\\hline\n\\multirow{2}{*}{} & \\multicolumn{2}{c}{$B(E2)$ (W.u.)} &\n \\multicolumn{2}{c}{$B(M1)$ (W.u.)} \\\\\n\\cline{2-3} \n\\cline{4-5}\n          & Th.         & Exp.    & Th.         & Exp.     \\\\\n\\hline\n${3\/2}^-_1\\rightarrow {1\/2}^-_1$ & 24 & 11.5(15) & 0.034 & 0.0168(19) \\\\\n${3\/2}^-_2\\rightarrow {1\/2}^-_1$ & 1.8 & 4.5(13) & 0.0066 & 0.00033(11) \\\\\n${3\/2}^-_3\\rightarrow {1\/2}^-_1$ & 7.8 & 30(7) & 0.029 & 0.024(4) \\\\\n${3\/2}^-_4\\rightarrow {1\/2}^-_1$ & 2.2 & 0.22(7) & 0.0060 & 0.0036(7) \\\\\n${5\/2}^-_1\\rightarrow {1\/2}^-_1$ & 21 & 8.9(7) & - & - \\\\\n${5\/2}^-_2\\rightarrow {1\/2}^-_1$ & 14 & 49(7) & - & - \\\\\n${5\/2}^-_3\\rightarrow {1\/2}^-_1$ & 0.00043 & 1.3(9) & - & - \\\\\n${3\/2}^-_4\\rightarrow {1\/2}^-_2$ & 2.2 & $<$37 & 0.0083 & $>$0.00054 \\\\\n${3\/2}^-_2\\rightarrow {3\/2}^-_1$ & 15 & 0.05$^{+106}_{-5}$ & 0.014 & 0.0030(8) \\\\\n${3\/2}^-_4\\rightarrow {3\/2}^-_1$ & 4.1 & 0.07(6) & 0.027  & 0.0013(3) \\\\\n${5\/2}^-_1\\rightarrow {3\/2}^-_1$ & 10 & 4.8(19) & 0.012 & 0.0269(21) \\\\\n${5\/2}^-_2\\rightarrow {3\/2}^-_1$ & 0.082 & 11(6) & 0.0015 & 0.019(3) \\\\\n${5\/2}^-_3\\rightarrow {3\/2}^-_1$ & 11 & 38(20) & 0.041 & 0.038(17) \\\\\n${5\/2}^-_4\\rightarrow {3\/2}^-_1$ & - & - & 0.00050 & $<0.013$ \\\\\n${7\/2}^-_2\\rightarrow {3\/2}^-_1$ & 4.1 & 29(10) & - & - \\\\\n${5\/2}^-_4\\rightarrow {3\/2}^-_3$ & - & - & 0.0017 & $<0.017$\\\\\n${7\/2}^-_2\\rightarrow {3\/2}^-_3$ & 2.2 & 7(3) & - & - \\\\\n${7\/2}^-_3\\rightarrow {3\/2}^-_3$ & 11 & 26(17) & - & - \\\\\n${5\/2}^-_3\\rightarrow {5\/2}^-_1$ & 0.57 & 0.0097 & 0.0044 &\n 0.026(12)\\\\\n${7\/2}^-_2\\rightarrow {5\/2}^-_1$ & - & - & 0.031 & 0.014(5) \\\\\n${9\/2}^-_1\\rightarrow {5\/2}^-_1$ & 45 & 35(8) & - & - \\\\\n${5\/2}^-_3\\rightarrow {5\/2}^-_2$ & - & - & 0.057 & 0.030(15) \\\\\n${5\/2}^-_4\\rightarrow {5\/2}^-_2$ & 0.33 & $<60$ & - & - \\\\\n${7\/2}^-_3\\rightarrow {5\/2}^-_2$ & 0.12 & $<210$ & 0.00077 & $<0.077$ \\\\\n${9\/2}^-_2\\rightarrow {5\/2}^-_2$ & 34 & 30(8) & - & - \\\\\n${7\/2}^-_2\\rightarrow {5\/2}^-_3$ & 0.11 & $<3.9\\times 10^{3}$ & 0.00042 & $<0.14$ \\\\\n\\hline\n\\multirow{2}{*}{} & \\multicolumn{2}{c}{$Q_J$ ($e$b)} &\n \\multicolumn{2}{c}{$\\mu_J$ ($\\mu_N$)} \\\\\n\\cline{2-3} \n\\cline{4-5}\n          & Theo.         & Exp.    & Theo.         & Exp.     \\\\\n\\hline\n${1\/2}^-_1$ & - & - & +0.46  & +0.60952(6) \\\\\n${3\/2}^-_1$ & +0.46 & & -0.37 & -0.62(6) \\\\\n${3\/2}^-_3$ & +0.083 & & -0.59 & +0.16(3) \\\\\n${5\/2}^-_1$ & +0.75 & & +0.90 & +0.90(6) \\\\\n${5\/2}^-_2$ & +0.20 & & +1.06 & +0.52(5) \\\\\n${5\/2}^-_3$ & +0.41 & & -0.062 & +0.39(10) \\\\\n${5\/2}^-_4$ & -0.48 & & +0.87 & +1.6(6) \\\\\n${7\/2}^-_2$ & +0.50 & & +0.84 & +0.55(8) \\\\\n${7\/2}^-_3$ & +0.26 & & +0.78 & +1.4(4) \\\\\n${7\/2}^-_5$ & +0.11 & & +0.76 & +1.2(3) \\\\\n${9\/2}^-_2$ & +0.70 & & +1.53 & +1.55(12) \\\\\n${9\/2}^-_3$ & +0.76 & & +0.59 & +1.52(16) \\\\\n${13\/2}^+_1$ & +0.79 & +1.4(6) & -1.31 & -0.606(105) \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\nFor  $^{189}$Os, as seen in Fig.~\\ref{fig:189os}, the obtained  level \nstructure is similar to the one  for $^{195}$Pt. However, the \nnegative-parity spectrum differs from the experimental one, for \nexample, with respect to the ground-state spin. In addition, the very \nlow-lying ${9\/2}^-_1$ level near the ground state could not be \nreproduced. Empirically, the ${9\/2}^-_1$ state arises from the \n${1h_{9\/2}}$ orbital coming closer to the Fermi surface \\cite{data}. However, in \nthe calculations  the single-particle energy for the $1h_{9\/2}$ orbital \nlies much higher than all the other orbitals (see, \nFig.~\\ref{fig:spe-ptos}(c)). Let us remark, that such a feature cannot \nbe controlled via three boson-fermion interaction strengths alone.\n\n\n\nIn Table~\\ref{tab:189os} we compare the electromagnetic properties \nobtained for $^{189}$Os with the available experimental data \n\\cite{data,stone2005}. Most of the discrepancy  is found for those \ntransitions that involve non-yrast states. Note that, indeed, the \nenergy levels corresponding to those states are neither properly \nreproduced. \n\n\n\n\n\\begin{table}[htb]\n\\caption{\\label{tab:189os}%\nThe same as in Table~\\ref{tab:195pt} but for $^{189}$Os. }\n\\begin{center}\n\\begin{tabular}{p{2.5cm}cccc}\n\\hline\\hline\n\\multirow{2}{*}{} & \\multicolumn{2}{c}{$B(E2)$ (W.u.)} &\n \\multicolumn{2}{c}{$B(M1)$ (W.u.)} \\\\\n\\cline{2-3} \n\\cline{4-5}\n          & Theo.         & Exp.    & Theo.         & Exp.     \\\\\n\\hline\n${3\/2}^-_2\\rightarrow {1\/2}^-_1$ & 1.9 & 27(16) & 0.00044 & 0.048(8) \\\\\n${5\/2}^-_1\\rightarrow {1\/2}^-_1$ & 61 & 24(3) & -  & - \\\\\n${5\/2}^-_2\\rightarrow {1\/2}^-_1$ & 4.8 & 25$^{+5}_{-8}$ & - & - \\\\\n${5\/2}^-_3\\rightarrow {1\/2}^-_1$ & 0.0029 & 0.6(4) & - & - \\\\\n${1\/2}^-_1\\rightarrow {3\/2}^-_1$ & 134  & 27(7) & 0.0070 & 0.042(3) \\\\\n${3\/2}^-_2\\rightarrow {3\/2}^-_1$ & 45  & 14(3) & 0.014 & 0.0032(6) \\\\\n${5\/2}^-_1\\rightarrow {3\/2}^-_1$ & 7.7 & 100(10) & 0.019 & 0.0026(2) \\\\\n${5\/2}^-_2\\rightarrow {3\/2}^-_1$ & 10 & 10$^{+3}_{-4}$ & 0.00058 &  0.0005$^{+3}_{-4}$ \\\\\n${5\/2}^-_3\\rightarrow {3\/2}^-_1$ & 3.9  & 1.5(3) & 0.0021 & $8.9\\times 10^{-5}$(23) \\\\\n${7\/2}^-_1\\rightarrow {3\/2}^-_1$ & 82 & 18.2(11) & - & - \\\\\n${7\/2}^-_2\\rightarrow {3\/2}^-_1$ & 0.94 & 38(2) & - & - \\\\\n${5\/2}^-_2\\rightarrow {3\/2}^-_2$ & 9.0  & 17$^{+6}_{-7}$ & 0.018 & 0.0012$^{+4}_{-5}$ \\\\\n${5\/2}^-_3\\rightarrow {3\/2}^-_2$ & 50 & 0.53(50) & 0.0013 & $<7.0\\times\n 10^{-5}$ \\\\\n${7\/2}^-_1\\rightarrow {3\/2}^-_2$ & 0.00077  & 1.75(22) & - & - \\\\\n${7\/2}^-_2\\rightarrow {3\/2}^-_2$ & 7.7 & 5(3) & - & - \\\\\n${3\/2}^-_2\\rightarrow {5\/2}^-_1$ & 13 & 80$^{+90}_{-40}$ & 0.00057  &\n 0.011(7) \\\\\n${5\/2}^-_2\\rightarrow {5\/2}^-_1$ & 32 & $<16$ & 0.0018 & 0.00087(63) \\\\\n${5\/2}^-_3\\rightarrow {5\/2}^-_1$ & 0.49 & 1.05(33) & 0.0097 & $<3.3\\times\n 10^{-5}$ \\\\\n${7\/2}^-_1\\rightarrow {5\/2}^-_1$ & 10 & 14(6) & 0.00059 & 0.0008(4) \\\\\n${7\/2}^-_2\\rightarrow {5\/2}^-_1$ & 7.0 & 43(2) & - & - \\\\\n${5\/2}^-_3\\rightarrow {7\/2}^-_2$ & - & - & 0.029  & 0.0099(21) \\\\\n${5\/2}^-_3\\rightarrow {9\/2}^-_1$ & 1.9 & 41(8) & - & - \\\\\n${7\/2}^-_1\\rightarrow {9\/2}^-_1$ & 1.5 & $<$2.2 & 0.026 & 0.00107(17) \\\\\n${7\/2}^-_2\\rightarrow {9\/2}^-_1$ & 18 & 6$^{+2}_{-1}$ & 0.000 &\n 0.00025$^{+11}_{-14}$ \\\\\n\\hline\n\\multirow{2}{*}{} & \\multicolumn{2}{c}{$Q_J$ ($e$b)} &\n \\multicolumn{2}{c}{$\\mu_J$ ($\\mu_N$)} \\\\\n\\cline{2-3} \n\\cline{4-5}\n          & Theo.         & Exp.    & Theo.         & Exp.     \\\\\n\\hline\n${1\/2}^-_1$ & - & - & +0.45 & +0.23(3) \\\\\n${3\/2}^-_1$ & -0.53 & +0.98(6) & +0.17 & +0.6599 \\\\\n${5\/2}^-_1$ & -1.03 & -0.63(2) & +0.96 & +0.988(6) \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\nIn Fig.~\\ref{fig:191ir} we compare the excitation spectra for \n$^{191}$Ir.  Both the positive- \nand negative-parity states are rather well described. The\nelectromagnetic properties computed for \nthis nucleus are given in Table~\\ref{tab:191ir}. Although the \ncorresponding energy levels are reasonably well described, some \ntransition strengths, like the $B(E2; {1\/2}^+_1\\rightarrow {3\/2}^+_1)$ \none, are significantly underestimated. \nThe reason is \nthat the IBFM-2 wave functions of the ${3\/2}^+_1$ and ${1\/2}^+_1$ \nstates are mainly built from the $2d_{3\/2}$ (54 \\%) and \n$3s_{1\/2}$ (58 \\%) single-particle configurations, respectively. As a \nconsequence, the E2 matrix element between the two states becomes too \nsmall. Keeping in mind that the employed model contains only three free \nparameters for each nucleus, the predicted electromagnetic properties \nin Table~\\ref{tab:191ir} for $^{191}$Ir, together with those for \n$^{195}$Pt (Table~\\ref{tab:195pt}) and $^{189}$Os \n(Table~\\ref{tab:189os}), appear to be rather reasonable. \n\n\n\n\\begin{table}[htb]\n\\caption{\\label{tab:191ir}%\nThe same as in Table~\\ref{tab:195pt}, but for $^{191}$Ir.}\n\\begin{center}\n\\begin{tabular}{p{2.2cm}cccc}\n\\hline\\hline\n\\multirow{2}{*}{} & \\multicolumn{2}{c}{$B(E2)$ (W.u.)} &\n \\multicolumn{2}{c}{$B(M1)$ (W.u.)} \\\\\n\\cline{2-3} \n\\cline{4-5}\n          & Theo.         & Exp.    & Theo.         & Exp.     \\\\\n\\hline\n${1\/2}^+_2\\rightarrow {1\/2}^+_1$ & - & - & 0.00096 & $<0.0038$ \\\\\n${3\/2}^+_2\\rightarrow {1\/2}^+_1$ & 2.4 & 58(9) & 0.00064 & 0.066(8) \\\\\n${3\/2}^+_3\\rightarrow {1\/2}^+_1$ & 15 & 0.39(10) & 0.00016 & 0.00204(21) \\\\\n${5\/2}^+_1\\rightarrow {1\/2}^+_1$ & 0.37 & 10.4(13) & - & - \\\\\n${5\/2}^+_2\\rightarrow {1\/2}^+_1$ & 5.5 & 39(7) & - & - \\\\\n${1\/2}^+_1\\rightarrow {3\/2}^+_1$ & 0.32 & 20.9(7) & 3.3$\\times 10^{-6}$ & 0.000474(14) \\\\\n${1\/2}^+_2\\rightarrow {3\/2}^+_1$ & 13 & $<2.7$ & 0.0070 & $<0.000277$(23) \\\\\n${3\/2}^+_2\\rightarrow {3\/2}^+_1$ & 24 & 15.7(20) & 0.029 & 0.00205(24) \\\\\n${3\/2}^+_4\\rightarrow {3\/2}^+_1$ & 0.27 & 2.52(25) & 4.9$\\times 10^{-5}$  & 0.0040(4) \\\\\n${3\/2}^+_5\\rightarrow {3\/2}^+_1$ & 0.23 & - & 8.2$\\times 10^{-5}$ & $\\approx 0.17$ \\\\\n${5\/2}^+_1\\rightarrow {3\/2}^+_1$ & 49 & 96.2(24) & 0.11 & 0.0259(6) \\\\\n${5\/2}^+_2\\rightarrow {3\/2}^+_1$ & 9.6 & 1.7(4) & 0.044 & 0.0060(9) \\\\\n${5\/2}^+_3\\rightarrow {3\/2}^+_1$ & 0.011 & - & 0.0087 & $\\approx 0.17$ \\\\\n${7\/2}^+_1\\rightarrow {3\/2}^+_1$ & 21 & 41.9(20) & - & - \\\\\n${7\/2}^+_2\\rightarrow {3\/2}^+_1$ & 8.1 & 8.9(11) & - & - \\\\\n${3\/2}^+_3\\rightarrow {3\/2}^+_2$ & 7.2 & 0.013(10) & 0.067 & 0.0080(7) \\\\\n${5\/2}^+_2\\rightarrow {3\/2}^+_2$ & 1.9 & 3.9(9) & 0.0015 & 0.056(8) \\\\\n${1\/2}^+_2\\rightarrow {5\/2}^+_1$ & 3.4 & $<1.3$ & - & - \\\\\n${3\/2}^+_2\\rightarrow {5\/2}^+_1$ & 24 & 27(9) & 0.00055  & 0.0053(11) \\\\\n${3\/2}^+_3\\rightarrow {5\/2}^+_1$ & 4.7 & 0.60(9) & 0.00034  & 0.0071(8) \\\\\n${7\/2}^+_1\\rightarrow {5\/2}^+_1$ & 6.8 & 29.8(15) & 0.0071 & 0.0296(15) \\\\\n${7\/2}^+_2\\rightarrow {5\/2}^+_1$ & 13 & 18(3) & 0.022 & 0.0117(18) \\\\\n${9\/2}^+_1\\rightarrow {5\/2}^+_1$ & 33 & 72(11) & - & - \\\\\n${3\/2}^+_3\\rightarrow {5\/2}^+_2$ & 0.022 & 10(4) & 0.0052 & 0.0031(5) \\\\\n${11\/2}^+_1\\rightarrow {7\/2}^+_1$ & 23 & 70(4) & - & - \\\\\n${3\/2}^-_1\\rightarrow {7\/2}^-_1$ & 80 & $>38$ & - & - \\\\\n${7\/2}^-_1\\rightarrow {11\/2}^-_1$ & 78 & 56(5) & - & - \\\\\n\\hline\n\\multirow{2}{*}{} & \\multicolumn{2}{c}{$Q_J$ ($e$b)} &\n \\multicolumn{2}{c}{$\\mu_J$ ($\\mu_N$)} \\\\\n\\cline{2-3} \n\\cline{4-5}\n          & Theo.         & Exp.    & Theo.         & Exp.     \\\\\n\\hline\n${1\/2}^+_1$ & - & - & +1.20 & +0.600(6) \\\\\n${3\/2}^+_1$  & +0.32 & +0.816(9) & +0.29 & +0.1507(6) \\\\\n${5\/2}^+_1$ &  & - & +1.37 & +0.81(6) \\\\\n${7\/2}^+_1$ &  & - & +0.99 & +1.40(6) \\\\\n${9\/2}^+_1$ &  & - & +2.11 & +2.4(2) \\\\\n${11\/2}^-_1$ &  & - & +6.66 & +6.03(4) \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\subsection{Signatures of shape phase transitions\\label{sec:def}}\n\n\n\n\n\\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[width=0.7\\linewidth]{{even.qinvar}.pdf}\n\\caption{(Color online) Effective $\\beta$ and $\\gamma$ deformation\nparameters for the even-even nuclei $^{186-200}$Pt and $^{186-194}$Os \nobtained from the E2 transition matrix elements.} \n\\label{fig:def-even}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[width=\\linewidth]{{odd.qinvar.all}.pdf}\n\\caption{(Color online) Effective $\\beta$ and $\\gamma$ deformation\n parameters for the odd-$N$ nuclei $^{185-199}$Pt and $^{185-193}$Os and\n the odd-$Z$ nuclei \n $^{185-191}$Ir for the $J^{\\pi}={1\/2}^\\pi_1$, ${3\/2}^\\pi_1$ and\n ${5\/2}^\\pi_1$ states for normal-parity configurations ($\\pi=-1$ for Pt\n and Os, and $\\pi=+1$ for Ir) and\n $J^\\pi={11\/2}^\\pi_1$ state for unique-parity configurations ($\\pi=+1$ for Pt\n and Os, and $\\pi=-1$ for Ir). } \n\\label{fig:def-odd}\n\\end{center}\n\\end{figure}\n\n\n\nAs yet another signature of the  prolate-to-oblate shape phase \ntransitions, we consider the quadrupole shape invariants \n\\cite{cline1986} (denoted as q-invariants) $q_m$ ($m=2,3,\\ldots$) \nobtained from the E2 matrix elements. These quantities have already \nbeen shown \\cite{nomura2017odd-1} to be  good signatures of shape phase \ntransitions involving $\\gamma$-softness.  For our purpose in this work, \nthe relevant $q_m$'s read\n \\begin{eqnarray}\n\\label{eq:q2}\nq_2=\\sum_{J'}q_2(J')\n\\end{eqnarray}\nwith \n\\begin{eqnarray}\n\\label{eq:q2a}\n q_2(J')=\\sum_{i}^n\\langle J||\\hat Q||J^{\\prime}_i\\rangle\\langle\n  J^{\\prime}_i||\\hat Q||J\\rangle,\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\label{eq:q3}\nq_3=-\\sqrt{\\frac{7}{10}}\n\\sum_{J'J''}\n\\sum_{ij}^{n}\n\\langle J||\\hat Q||J'_i\\rangle\n\\langle J'_i||\\hat Q||J''_j\\rangle\n\\langle J''_j||\\hat Q||J\\rangle.\n\\end{eqnarray}\nIn Eqs.~(\\ref{eq:q2a}) and (\\ref{eq:q3}), all possible E2 transition matrix elements\namong the states  $J$, $J'$ and $J''$, that satisfy the E2 selection\nrule, have been considered. \nThe indices $i$ ($j$) in the sums are ordered according to increasing excitation\nenergies of the $J'$ ($J''$) levels and run up to $n=\\infty$. However, we have\nconfirmed that only a few of the lowest transitions contribute \nto the q-invariants significantly \n\\cite{nomura2017odd-1}. In the case of even-even systems, the q-invariants for the\n$0^+_1$ ground state (i.e, $J=0^+_1$ and $J^{\\prime}=2^+$)  have been \ncomputed. The effective deformation parameters\n$\\beta_{\\rm eff}$ and $\\gamma_{\\rm  eff}$ are obtained\nfrom the $q_2$ and $q_3$ values by the formulas\n\\begin{eqnarray}\n\\label{eq:eff_beta}\n &&\\beta_{\\rm eff} = \\frac{4\\pi}{3ZR^2_0}\\sqrt{\\sum_{J'}\\frac{1}{2J^{\\prime}+1}(J^{\\prime}2J0|JJ)^{-2}q_2(J')} \\\\\n\\label{eq:eff_gamma}\n &&\\gamma_{\\rm eff} = \\frac{1}{3}\\arccos{\\frac{q_3}{q_2^{3\/2}}}\n\\end{eqnarray}\nwhere $R_0=1.2 A^{1\/3}$ fm and $(J^{\\prime}2J0|JJ)$  represents a  Clebsch-Gordan\ncoefficient. \n\n\nIn Fig.~\\ref{fig:def-even}, we have depicted the $\\beta_{\\rm eff}$ and \n$\\gamma_{\\rm eff}$ values for the even-even Pt and Os nuclei. A \nmonotonic decrease of $\\beta_{\\rm eff}$ is observed in \nFig.~\\ref{fig:def-even}(a) as one approaches the  $N=126$ shell \nclosure. This agrees well with  the gradual shift, from  $\\beta\\approx \n0.20$ to $\\beta\\approx 0$, in the global minima of the Gogny-D1M energy \nsurfaces (see, Fig.~\\ref{fig:pes}). On the other hand, the $\\gamma_{\\rm \neff}$ value, plotted in Fig.~\\ref{fig:def-even}(b), exhibits a faster \nchange with $N$, jumping from  $\\gamma_{\\rm eff}\\approx 25^{\\circ}$ (at \n$N=110$) to  $\\gamma_{\\rm eff}\\approx 40^{\\circ}$ (from $N=110$ onward) \nin Pt isotopes. Furthermore, the rate change is slower in the Os \nisotopes from $N=114$ up to 118. This behavior of $\\gamma_{\\rm eff}$ \nconfirms that the prolate-to-oblate shape transition takes place. It is \nalso consistent with the systematics of the Gogny-D1M energy surfaces. \n\n\nSimilar plots of $\\beta_{\\rm eff}$ and\n$\\gamma_{\\rm eff}$, are shown in Fig.~\\ref{fig:def-odd}  for \nseveral configurations  close to the ground states of odd-mass\nnuclei. As can be seen in\nFigs.~\\ref{fig:def-odd}(a) (odd-$N$ Pt), \\ref{fig:def-odd}(c) (odd-$N$ Os) and\n\\ref{fig:def-odd}(e) (odd-$Z$ Ir),  the deformation $\\beta_{\\rm eff}$ \nfor each state typically shows a smooth behavior\nas a function of $N$. This correlates well with \nthe results obtained for even-even systems (see, Fig.~\\ref{fig:def-even}).\nOn the other hand, as in the case of even-even systems, a rapid\nchange of the $\\gamma_{\\rm eff}$ value from below to above $\\gamma_{\\rm\neff}\\approx 30^{\\circ}$ occurs in some of the states and in each of the \nisotopic chains  shown in panels (b), (d) and (f) of\nFig.~\\ref{fig:def-odd}, i.e., in going from $N=109$ to 111 in odd-$N$ Pt\n(for the ${3\/2}^+_1$ and ${5\/2}^+_1$ states), in going from \n$N=113$ to 117 in odd-$N$ Os (for the ${3\/2}^+_1$ and ${5\/2}^+_1$\nstates) and in going from $N=110$ to 112 in odd-$Z$ Ir (for the ${11\/2}^-_1$ state). \nFor those nuclei where $\\gamma_{\\rm eff}$ changes abruptly, the associated\neven-even isotopes also show signs of a \nprolate-to-oblate shape transition (see, Fig.~\\ref{fig:def-even}(b)). \n\n\n\n\\section{Summary and concluding remarks\\label{sec:summary}}\n\n\n\nIn this paper, we have studied the prolate-to-oblate shape phase \ntransition in neutron-rich odd-mass nuclei with mass $A\\approx 190$. \nSpectroscopic properties have been computed  within a recently \ndeveloped method where most of the parameters of the effective \nHamiltonian of the IBFM are obtained from an EDF. To this end, the \n$(\\beta,\\gamma)$-deformation energy surfaces for the even-even core \nnuclei $^{186-200}$Pt and $^{186-194}$Os, spherical single-particle \nenergies and occupation probabilities for the corresponding odd-mass \nsystems, have been computed within a microscopic EDF framework based on  \nconstrained mean field HFB configurations obtained with the Gogny-D1M \nparametrization. These quantities have then been used to determine the \nIBFM-2 Hamiltonian. The diagonalization of the IBFM-2 Hamiltonian \nallows to study the properties of $^{185-195}$Pt, $^{185-193}$Os and \n$^{185-195}$Ir. A few coupling constants, for the boson-fermion \ninteraction, have been specifically fitted to the low-energy excitation \nspectra for each odd-mass nucleus. However,  those parameters turned \nout to be almost constant or exhibit a gradual variation with nucleon \nnumber. \n\n\nOur calculations account reasonably well for the spectroscopic \nproperties of the studied odd-mass nuclei. In particular, we have \nidentified a clear signature of a shape phase transitions by analyzing \nthe systematic trend of the several calculated observables for the \nodd-mass nuclei. For instance, the evolution of the low-lying yrast \nstates as well as the effective $\\gamma$ deformation parameter exhibits \nsignificant structural changes at some specific neutron numbers. Our \nresults point to the robustness of the prolate-to-oblate shape \ntransitions in both even-even and odd-mass nuclei in this particular \nmass region. The present study could be extended further to another \ninteresting case, such as those odd-mass nuclei in neutron-deficient Pb \nand Hg regions, which are characterized by a spectacular case of shape \ncoexistence phenomena. This would require a major extension of the \npresent method, and work along this line is in progress. \n\n\n\n\\acknowledgments\nThis work was supported in part by the QuantiXLie Centre of Excellence, a project\nco-financed by the Croatian Government and European Union through the\nEuropean Regional Development Fund - the Competitiveness and Cohesion\nOperational Programme (Grant KK.01.1.1.01.0004).\nThe  work of LMR was \nsupported by Spanish Ministry of Economy and Competitiveness (MINECO)\nGrants No. FPA2015-65929-P and FIS2015-63770-P.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction \\label{INTRO}}\n\nThe Askey scheme \\cite{KLS} is a hierarchy of hypergeometric and $q$-hypergeometric  orthogonal poynomials, which includes many families of classical polynomials, including the Hermite, Laguerre, Gegenbauer, and Jacobi polynomials. Each family depends on a number of auxiliary parameters, and lower families in the hierarchy can be obtained from higher families by a suitable specialization of parameters. \n\n\nIf $\\cP=\\{P_0,P_1,\\ldots\\}$ and $\\cP'=\\{P_0',P_1',\\ldots\\}$ are two sequences of polynomials satisfying $\\deg(P_n)=\\deg(P'_n)=n$, then one has an expansion of the form\n\\begin{equation}\n\\label{cnk}\nP_n=\\sum\\nolimits_{m\\le n} c_{m,n}P_m'.\n\\end{equation}\n(The coefficients $c_{m,n}$ depend on the ordered bases $\\cP$ and $\\cP'$ and so might be more fully denoted by $ c_{m,n}^{\\cP,\\cP'}.$\nHowever to keep our notation more spare, we usually make clear the bases by context and omit the superscripts.)\n\nNow suppose $\\cP$ and $\\cP'$ are from the \\emph{same} family in the Askey-Wilson scheme and differ in only \\emph{one} auxiliary parameter. In this case one can sometimes obtain an explicit formula for the ``connection' coefficients'' \n$c_{m,n}$ \nin \\eqref{cnk}  in terms of Pochammer symbols and their $q$-analogs, which are defined as follows:\n\\begin{align}\n(a)_n = \\prod\\nolimits_{k=0}^{n-1} (a+k),\\quad \n& (a_1,\\ldots,a_r)_n= \\prod\\nolimits_{i=1}^{r} (a_i)_n, \\\\\n(a|q)_n = \\prod\\nolimits_{k=0}^{n-1} (1-aq^k),\\quad \n& (a_1,\\ldots,a_r|q)_n= \\prod\\nolimits_{i=1}^{r} (a_i|q)_n.\n\\end{align} \nFor example, one has the following classical result:\n\\ifJOLT \\begin{Theorem} \\else \\begin{thms} \\fi\\label{JACOBICONTHM}( 3.40  in \\cite{INTREP})\n\\noindent If $\\cP=\\{P_n^{(\\gamma,\\beta)}(x)\\}$ and $\\cP'=\\{P_n^{(\\alpha,\\beta)}(x)\\}$ are two sequences of Jacobi polynomials differing in one parameter, then one has\n\\begin{multline}\n\\label{JACOBICON}\nP_n^{(\\gamma,\\beta)}(x)=\\\\\n\\sum_{k=0}^n \\frac{ (\\beta+k+1 )_{ n - k} ( \\gamma-\\alpha)_{ n - k}  (\\beta+\\gamma+n+1 )_{ k } ( 2k+\\alpha+\\beta+1) } {(\\alpha+\\beta+k+1 )_{n+1 } (n-k)! } P_k^{(\\alpha,\\beta)}(x).\n\\end{multline}\n\\ifJOLT \\end{Theorem} \\else \\end{thms} \\fi\nSimilarly equation (3.42) in  \\cite{INTREP} tells us for Gegenbauer polynomials\n\\begin{equation}\n\\label{GEGENCON}\nC_n^{\\lambda}(x) =\n \\sum_{k=0}^{\\left[ \\frac{n}{2} \\right]} \\frac{ (\\lambda - \\nu )_{ k} ( \\lambda)_{ n - k } ( n - 2k + \\nu ) } { ( \\nu )_{ n - k + 1} k!} C_{n-2k}^{\\nu}(x). \n\\end{equation}\n(Equations (3.40) and (3.42) in \\cite{INTREP} and elsewhere are often expressed in terms of Gamma functions, but the identity\n$$\n\\Gamma(k+\\alpha) = (\\alpha)_k \\Gamma(\\alpha) \\hspace{5mm} \\text{ for } k \\in \\mathbb{N}\n$$\nreadily leads to formulas \\eqref{JACOBICON} and \\eqref{GEGENCON}.)\n\nA similar result holds for the Askey-Wilson polynomials $P_n(z;a,b,c,d|q)$, which are a 4 parameter $q$-hypergeometric family at the top of the Askey hierarchy.\n\n\\ifJOLT \\begin{Theorem} \\else \\begin{thms} \\fi\\label{PTCFRM}(\\cite{MEMOIRS},Askey-Wilson)\nIf $\\cP= \\{P_n(z;a,b,c,d|q)\\}=\\{P_n\\}\n\\text{ and }$ \\\\\n$\\cP'= \\{P_n(z;e,b,c,d|q)\\}=\\{P_n'\\}$ are two sequences of Askey-Wilson polynomials differing in one parameter, then one has\n\\begin{eqnarray}\n\\label{AWCONSUM} P_n &=&\\sum_{m\\le n}  c_{m,n} (a, e ; b, c, d) P_m'  \\hspace{10mm} \\text{ where }\\\\\n\\label{AWCON} c_{m,n}(a, e ; b, c, d)&=&  \\frac{ (q^{n-m+1} | q )_m(bc q^m, bd q^m, cd q^m, ae^{-1}  | q )_{n-m}   }  { (q  | q )_m (abcdq^{n+m-1}, bcdeq^{2m}  | q )_{n-m} } e^{n-m}. \n\\end{eqnarray}\n\\ifJOLT \\end{Theorem} \\else \\end{thms} \\fi\nHere we follow the notation of \\cite{ASKVOL} and regard the Askey-Wilson polynomials as Laurent polynomials in $z$, symmetric under the inversion $z\\mapsto z^{-1}$. This is related to the ordinary polynomial variable $x$ of Askey-Wilson \\cite{MEMOIRS} by $2x= z +z^{-1}$. Our normalization of $P_n$ as a monic Laurent polynomial in $z$ is also different from \\cite{MEMOIRS}. \n\nAs explained in \\cite{ASKVOL}, the Askey-Wilson polynomials admit nonsymmetric analogs $E_r=E_r(z; a,b,c,d|q)$, defined for any integer $r$, which are eigenfunctions of certain $q$-difference operators. While $P_n=P_n(z; a,b,c,d|q)$ can be obtained from $E_{\\pm n}$ by a suitable symmetrization operator, the $E_r$ themselves are not symmetric under $z\\mapsto z^{-1}$.  Also, while the $P_n$ depend symmetrically on $a,b,c \\text{ and }d$, the $E_r$ only have the symmetries $a \\leftrightarrow b$ and $c \\leftrightarrow d$.  \n\n\nWe define the ``zig-zag'' order $\\prec$ on integers as follows: \n\\begin{equation}\n0 \\prec -1 \\prec 1 \\prec -2 \\prec 2 \\prec \\cdots.\\label{zigzag}\n\\end{equation}\nWe say a Laurent polynomial $F(z)$ has degree \nan integer \n$r$ if there is a constant $c\\ne0$ such that $F(z)-cz^r$ is in the span of $\\{z^s, s\\prec r\\}$.\nSuch a Laurent polynomial is called monic if that leading coefficient $c$ is $1.$\n If $\\cF=\\{F_0,F_{-1},F_{1},\\ldots \\}$ and $\\cF'=\\{F'_0,F'_{-1},F'_{1},\\ldots \\}$  are two families of Laurent polynomials satisfying $\\deg(F_r)=\\deg(F_r')=r$, then one can again consider connection coefficients (e.g. denoted by $b_{r,s}$) such that\n\\begin{equation}\n\\label{crsE}\nF_s=\\sum\\nolimits_{r\\preceq s} b_{r,s}F_r'.\n\\end{equation}\n\nThe nonsymmetric Askey-Wilson polynomial satisfy the degree condition, and \nour main results are \nformulas \nof this form\ngiving the connection coefficients \nfor two sequences of such polynomials differing in one parameter. \nIn view of the $a \\leftrightarrow b$ and $c \\leftrightarrow d$ symmetries, there are only two distinct cases to consider: (1) the parameter $a$ is replaced by $e$, say, and (2) the parameter $c$ is replaced by $g$, say.\n\n\n\nThe first case\nis a change of basis relationship  from basis $\\{E_r(z; a,b,c,d|q)\\}$ \nto basis $\\{E_r(z; e,b,c,d|q)\\}.$ \nWe often refer to the \ncorresponding matrix transforming components relative to these bases \nas a {\\em transition matrix.} \n\n\\ifJOLT \\begin{Theorem} \\else \\begin{thms} \\fi\n\\label{ETC}\nLet $\\cF=\\{E_r(z;a,b,c,d|q)\\}=\\{E_r\\}$ and $\\cF'= \\{E_r(z;e,b,c,d|q)\\}=\\{E_r'\\}$ be sequences of nonsymmetric Askey-Wilson polynomials. Then\n$$\n\\displaystyle E_s =\\sum_{r \\preceq s} d_{r,s}c_{|r|,|s|} E_r'\n$$\nwhere $c_{m,n}$ is the symmetric connection coefficient $c_{m,n}(a, e; b,c,d)$ as in \\eqref{AWCON}, and\n\\[ \\quad  d_{r,s}= \\begin{cases}  \n\\displaystyle  \\frac{q^{s-r} (abcdq^{s+r-1}| q)_1}{(abcdq^{2s-1}| q)_1} & \\text{ if }   r \\ge 0, s\\ge 0 \\\\ \n\\displaystyle   \\frac{ (q^{-(r+s) }|q)_{ 1 } } {( q^{-s}, cdq^{-(s+1) }|q)_{ 1 }} & \\text{ if }   r \\ge 0, s< 0 \\\\ \n\\displaystyle  \\frac{bcdeq^{s-(r+1)} (q^{-r}, cdq^{-(r+1)}, ae^{-1}q^{s+r }|q)_{1 } } { (abcdq^{2s-1}, bcdeq^{-(2r+1) }|q)_{ 1}}  & \\text{ if }   r < 0, s\\ge 0 \\\\ \n\\displaystyle  \\frac{ (q^{-r}, cdq^{-(r+1)}, bcdeq^{-(r+s+1) }|q)_{1 } } { (q^{-s}, cdq^{-(s+1)}, bcdeq^{-(2r+1) }|q)_{ 1} }  & \\text{ if }   r < 0, s< 0  \\\\\n\\end{cases} \\]\n\\ifJOLT \\end{Theorem} \\else \\end{thms} \\fi\n\n\\noindent {\\bf Remark:} It \nis\nworth noting that if one were to replace  $E_r(z; a,b,c,d|q)$ above by $\\psi(r) E_r(z; a,b,c,d|q)$   then\nthere is some simplification in the resultant $d_{r,s}$ formulas when $\\psi(r)$ is defined to be \n\\begin{equation} \\label{psi}\n\\psi(r)= \\begin{cases}  \n\\hfil 1 & \\text{if }   r \\ge 0\\\\\n(1-q^{-r})(1- cdq^{-(r+1)}) & \\text{if }  r < 0. \n\\end{cases}\n\\end{equation}\n(This corresponds to  a change of normalization of the $E_r$ for $r<0$.)\n\nWe now consider the case where the parameter $c$ is replaced by $g$. \n\\ifJOLT \\begin{Theorem} \\else \\begin{thms} \\fi\n\\label{ETC2}\nLet $\\cF=\\{E_r(z;a,b,c,d|q)\\}=\\{E_r\\}$ and $\\cF''= \\{E_r(z;a,b,g,d|q)\\}=\\{E_r''\\}$ be sequences of nonsymmetric Askey-Wilson polynomials.\nThen\n$$\nE_s =\\sum_{r \\preceq s}  d_{r,s}'' c_{|r|,|s|}'' E_r''  \n$$\nwhere\n \\begin{enumerate}\n\\item $c_{m,n}''$ is the symmetric connection coefficient $c_{m,n}(c, g; a, b, d)$\n\\item $d_{r,s}''$ is $d^c_{r,s}(c, g; a,b,d )$ and\n\\end{enumerate}\n\\[ \\quad d^c_{r,s}(c, g; a, b, d )= \\begin{cases}  \n\\displaystyle  \\frac{ ( abq^s, abcdq^{r+s-1} | q)_1  }  { ( abq^r, abcdq^{2s-1} | q)_1 }  & \\text{ if }   r \\ge 0, s\\ge 0 \\\\ \n\\displaystyle  -\\frac{  abq^r ( q^{-r -s}  | q)_1 }  { ( q^{-s}, abq^r | q)_1  }   & \\text{ if }   r \\ge 0, s< 0 \\\\ \n\\displaystyle  -  \\frac{ dq^{-r-1}g (q^{-r},  abq^s, cg^{-1} q^{s+r}  | q)_1 }  { ( abcd q^{2s-1}, abdgq^{-2r-1} | q)_1 }  & \\text{ if }   r < 0, s\\ge 0 \\\\  \n\\displaystyle   \\frac{ ( q^{-r} , abdg q^{ -r-s-1} | q)_1  }  { ( q^{-s} , abdg q^{-2r-1}  | q)_1 }  & \\text{ if }   r < 0, s< 0.  \\\\\n\\end{cases} \\]\n\\ifJOLT \\end{Theorem} \\else \\end{thms} \\fi\n\n\n\\noindent {\\bf Acknowledgement:} The research of SS was partially supported by NSF grants DMS-$1939600$ and $2001537$, as well as Simons Foundation grant $509766.$ The research of BS was partially supported by Simons Foundation grant $89086.$\n\n\n\n\\begin{center} \\bf Motivation and Applications \\end{center} \nFor certain values of the parameters, formulas such as (\\ref{JACOBICON}) in Theorem \\ref{JACOBICONTHM} and (\\ref{GEGENCON})  can be interpreted as describing \naspects  \nof\n branching theorems for spherical\nrepresentations of  compact classical Lie groups. \nSpherical representations of compact groups are isomorphic to irreducible subrepresentations of \n \\[L^2(G\/K) = \\oplus_ \\mu W_\\mu\\] for a compact symmetric space $G\/K.$ \nHere $G$ acts on the left. (See  \\cite{HELG2} and chapters 1 and 2 of \\cite{MODSPHFCNS}.) Each $W_\\mu$ contains a \n$K$-bi-invariant function\n $f_\\mu \\in C^\\infty(K,\\ G \/K)$. In  representation theory  $ f_\\mu$ is referred  to as a spherical function for $G\/K,$ but sometimes  in the literature the term zonal spherical function is used. Denoting by $T$ the one dimensional maximal  torus of a rank $1$ symmetric space, we have $G\/K=KT$ by the maximal torus theorem for compact symmetric spaces and thus we can consider $f_\\mu$ as a function on $T.$ \nAs explained in detail within chapter $3$ of \\cite{MODSPHFCNS},\nGegenbauer and Jacobi polynomials can be viewed (up to scaling) as spherical functions on the rank $1$ symmetric spaces \n$S^n$ and $CP^n.$\n\nTable $1$ in \\cite{JACCROSS} extends these observations to the other compact simply connected  rank $1$ symmetric spaces, the quaternionic projective space,\nand the Cayley plane. This table also writes down the classical polynomial parameter values associated with these geometric examples.\nFor the sphere $S^n,$ the parameter $\\nu$ in the Gegenbauer polynomial $C_k^{\\nu}(x)$ is $\\displaystyle \\frac{n-1}{2}.$ For the complex projective space\n$CP^n,$ the parameters in the Jacobi polynomial  $P_k^{(\\alpha,\\beta)}$ are given by $\\alpha=n-1$ and $\\beta=0.$ \n\n\\medskip\nRecall how  branching of spherical representations from $G$ to $G'$ can give rise to connection coefficient formulas (\\ref{GEGENCON}) and\n(\\ref{JACOBICON}). The $G' \\subset G$ situations will be $SO(2n) \\subset SO(2n+1)$ and $SU(n-1) \\subset SU(n).$\nWe start with a spherical representation $\\tau $ of $G,$ realized as a subrepresentation of $L^2(G\/K)$.\nWe then want to consider  $G' \\subset G$ with an associated rank $1$ symmetric space $G'\/K'.$ Since they are both rank $1$ symmetric spaces,\ntheir maximal tori in the sense of  symmetric spaces may be identified, so we may view \n\\[G'\/K'=K'T.\\] \nLet $f_{\\tau}$ denote the spherical function associated to $\\tau.$\nWhen we restrict $\\tau$ to $G'$ we obtain $\\oplus \\tau_{\\alpha}'.$\nFor each summand $\\tau_\\alpha'$ we have the corresponding spherical function $f_\\alpha'$ and \na relationship\n\\[ f_\\tau(x) = \\sum c_{\\alpha} f_{\\alpha}'(x) \\]\nwhere the $c_{\\alpha}$ are non-negative constants. \n(Since spherical functions with respect to $G\/K$ restricted to $T$ are multiples of matrix elements $<v,\\tau(g)v>$ for $v$ a $K$-invariant vector of $\\tau,$\nthe restricted functions  $f_{\\alpha}'(x)$ are spherical functions with respect to $G'\/K'.)$\n\nFor $SO(2n) \\subset SO(2n+1)$ and $SU(n-1) \\subset SU(n),$ this translates to the formulas (\\ref{GEGENCON}) and (\\ref{JACOBICON})  for \n\nGegenbauer and Jacobi functions respectively.  So  $c_{\\alpha} $ becomes\na  connection coefficient in a formula  like\n(\\ref{GEGENCON}) and\n(\\ref{JACOBICON}).\n\n\n\nThe function connection coefficient relations (\\ref{GEGENCON}) and (\\ref{JACOBICON}) can also be viewed as being related to calculating integrals\nof a product of two such special polynomials with different parameters; as such one might for example use\nRodriguez' formula and integration by parts to determine them. \n\nIn this paper we shall generalize these classical relations by purely \ncombinatorial methods. Although there could also be relations to deformations of algebraic structures\nsuch as double affine Hecke algebras via their generators and eigenfunctions, and also possibly to the\ntheory of spherical functions for certain quantum groups, we shall not go into these. \n\n\n\n\n\\begin{center} \\bf  Structure of the proof of Theorem \\ref{ETC}\\end{center} \nOne natural approach to proving this theorem would be to use the vector-valued reformulation of the $E_r$ in \\cite{VECVAL}, which extends earlier results in \\cite{ZHED}. \n\nOur proof\nhere uses an interesting alternative. We start with proving the special case of $a$ and $e$ differing by a factor of $q.$ This is a $q$-shift analogous to a shift by $1$ in one of the classical integer parameters. Since transition functions describing change of basis matrices satisfy a natural co-cycle condition,\nwe can  establish the case of $a$ and $e$ differing by an integral power of $q$ by showing that our asserted expressions for the transition\nfunctions also satisfy the co-cycle condition. And then, by observing that everything involved is given by rational functions agreeing at infinitely many values,\nwe obtain the theorem for arbitrary $a$ and $e.$\n\n\\begin{center} \\bf The Co-cycle Condition in Detail and Proof Plan A \\end{center}\nThe partition of powers of $z$ into non-negative vs. negative (corresponding to the same notions on the root system $C_1$ sitting inside the\naffine root system $\\tilde{C}_1$) gives a direct sum decomposition of Laurent polynomials\n$$\\mathcal{R} = \\mathcal{R}^0 \\oplus \\mathcal{R}^1$$\nwhere we can choose ordered bases\n$$\\begin{aligned}\n E_0,E_1,E_2,\\ldots  \\text{ for } &  \\mathcal{R}^0 \\\\\nE_{-1},E_{-2},\\ldots \\text{ for } & \\mathcal{R}^1.\\\\\n\\end{aligned}$$\n\n\nWe view connection coefficient relations like those given in Theorem \\ref{ETC} as describing a change of basis in the space of Laurent polynomials, perhaps truncated\nin degree, so as to reference a finite dimensional subspace. \n\nUnless otherwise specified, we will henceforth treat the common parameters $b,c,d,\\text{ and } q$ as unchanged and drop them from the\nargument lists of the $E_r$ (and associated connection coefficients.)\n For example $E_n(a)$ is a shorthand for $E_n(z; a,b,c,d|q).$\n\nWe use $\\boldsymbol{\\mathcal{T}}(a,e)$ for the `true' transition matrix from components relative to\nthe $\\{E_r(a)\\}$ basis to components relative the $\\{E_r(e)\\}$ basis. The notation $T(a,e)$ will refer to the transition matrix specified by the formulas\nin Theorem \\ref{ETC}.\nThus proving Theorem \\ref{ETC} amounts to showing $\\boldsymbol{\\mathcal{T}}(a,e)=T(a,e).$\n\nAs described fully in Appendix A, this block decomposition and choice of ordered bases corresponds to $T(a,e)$ having the block decomposition\n$$\nT=\\begin{bmatrix}\nT ^{00} & T^{01} \\\\\nT^{10} & T^{11} \\\\\n\\end{bmatrix}\n$$\nwith\n\n$$\\begin{aligned}\nT^{00}=&\\begin{bmatrix}\n\\tau_{0,0}  & \\tau_{0,1} & \\tau_{0,2}  &  \\tau_{0,3}  & \\ldots\\\\\n0  & \\tau_{1,1} & \\tau_{1,2}  &  \\tau_{1,3}  & \\ldots \\\\\n0  & 0 & \\tau_{2,2}  & \\tau_{2,3} & \\ldots \\\\\n0  & 0 & 0  & \\ddots & \\vdots \\\\\n\\end{bmatrix}\n&\nT^{01}=&\\begin{bmatrix}\n\\sigma_{0, -1}  & \\sigma_{0,-2} & \\sigma_{0,-3}  &  \\sigma_{0,-4}  & \\ldots\\\\\n0  & \\sigma_{1,-2} & \\sigma_{1,-3}  &  \\sigma_{1,-4}  & \\ldots \\\\\n0  & 0 & \\sigma_{2,-3}  & \\sigma_{2,-4} & \\ldots \\\\\n0  & 0 & 0  & \\ddots & \\vdots \\\\\n\\end{bmatrix}\n\\\\\nT^{10}=&\n\\begin{bmatrix}\n0  & \\sigma_{-1, 1} & \\sigma_{-1, 2}  &  \\sigma_{-1, 3}  & \\ldots\\\\\n0  & 0 & \\sigma_{-2,2}  &  \\sigma_{-2,3}  & \\ldots \\\\\n0  & 0 & 0  & \\sigma_{-3,3} & \\ldots \\\\\n0  & 0 & 0  & \\ddots & \\vdots \\\\\n\\end{bmatrix}\n&\nT^{11}=&\\begin{bmatrix}\n\\tau_{-1, -1}  & \\tau_{-1,-2} & \\tau_{-1,-3}  &  \\tau_{-1,-4}  & \\ldots \\\\\n0  & \\tau_{-2,-2} & \\tau_{-2,-3}  &  \\tau_{-2,-4}  & \\ldots \\\\\n0  & 0 & \\tau_{-3,-3}  & \\tau_{-3,-4} & \\ldots \\\\\n0  & 0 & 0  & \\ddots & \\vdots \\\\\n\\end{bmatrix}.\n\\end{aligned}$$\nHere $\\tau_{r,s}$ and $\\sigma_{r,s}$ (zero unless $r \\preceq s$) are the products of the $c$'s and $d$'s defined by\n\\begin{align}\n\\label{TAUDEF} \\tau_{r,s}&=&d_{r,s}c_{|r|,|s|} & \\hspace{10mm} & \\text{if } (r \\ge 0 \\text{ and } s \\ge 0)  \\text{ or } (r <  0 \\text{ and } s < 0)\\\\\n\\label{SIGMADEF} \\sigma_{r,s}&=& d_{r,s}c_{|r|,|s|} & \\hspace{10mm} & \\text{if } (r \\ge 0 \\text{ and } s < 0)  \\text{ or } (r <  0 \\text{ and } s > 0). \n\\end{align}\nWe think of this transition matrix as acting on the left on column vectors of components relative to one basis and producing\na column vector of components relative to the other basis.\n\n\n\n\n\n\nThe `true' transition function $\\boldsymbol{\\mathcal{T}}(a,e)$ satisfies the co-cycle condition\n\\begin{equation}\n\\label{COCYC}\n\\boldsymbol{\\mathcal{T}}(a,e) = \\boldsymbol{\\mathcal{T}}(f,e)\\boldsymbol{\\mathcal{T}}(a,f)\n\\end{equation}\nsince each side describes a valid way to go from  $a$-coordinates to $e$-coordinates. In particular this means a `discrete' co-cycle condition; namely for any non-negative integer $p:$\n\\begin{equation}\n\\label{COCYCAQ}\n\\boldsymbol{\\mathcal{T}}(a,aq^{p+1}) = \\boldsymbol{\\mathcal{T}}(aq^p,aq^{p+1})\\boldsymbol{\\mathcal{T}}(a,aq^p).\n\\end{equation}\n\nOur  proof of Theorem \\ref{ETC}  has three steps which we refer to as\n\\begin{equation} \n\\label{PLANA}\n\\text{ \\em (PROOF PLAN A).  }\n\\end{equation}\n\n\\begin{enumerate}\n\\item Show that the entries of both $T(a,e)$ and  $\\boldsymbol{\\mathcal{T}}(a,e) $ are rational functions of $e$ with coefficients in the filed $\\mathbb{Q}(a,b,c,d,q).$\n\\item Show $\\boldsymbol{\\mathcal{T}}(a,aq)=T(a,aq).$\n\\item Show $T$ also satisfies the discrete co-cycle condition $$T(a,aq^{p+1}) = T(aq^p,aq^{p+1})T(a,aq^p)$$ for any $p \\in \\mathbb{N}.$\n\\end{enumerate}\nSince both $T(a,a)$ and $\\boldsymbol{\\mathcal{T}}(a,a)$ are the identity, and  equation (\\ref{COCYCAQ}) says $\\boldsymbol{\\mathcal{T}}$ satisfies the discrete\nco-cycle condition, it is immediate from \nparts $2$ and $3$ above\nthat  $\\boldsymbol{\\mathcal{T}}(a,e)$ and $T(a,e)$ agree whenever $e=aq^p$ for non-negative $p.$  Now using \npart $1$ of Proof Plan A,\nwe see that each entry of the two matrices is a rational function of $e$ agreeing with the other at infinitely many points. So they must agree\n$\\big($as rational functions with coefficients in $\\mathbb{Q}(a,b,c,d,q) \\big)$ for all $e.$ \n\n\n\n\n\\begin{center} \\bf Structure of the Paper \\end{center} \n\n\nWe have already stated our main results above, explained how these kind of results can relate to branching theorems in\nrepresentation theory, and described the basic approach of the proof. \n\nThe heart of the proof of \nTheorem \\ref{ETC} are the last two steps of \\eqref{PLANA}, Proof Plan A. These  are carried out in sections \n\\ref{TAQ} and \\ref{COCYCPRF}. We have written out the verification of these two steps in a detailed step-by-step way,\nand so these two sections constitute about one third of the main body of our paper.\n\nSection \\ref{PREL} briefly recalls the basic double affine Hecke algebra (DAHA) and Noumi representation points of view about\nthe nonsymmetric Askey-Wilson polynomials. Much of what we use is based on the approach of \\cite{ANNALS}, which was\nspecialized to the one variable case in \\cite{ASKVOL}, and further enhanced (including some notational adjustments) in\n\\cite{SIGMA}. At certain points, we need a little more detail than was recorded in the statements of the theorems proved there,\nand so explain how those come from this earlier work. \n\nThe zig-zag order leads to a filtration of the Laurent polynomials and Section \\ref{ALMSYM} exploits some aspects of this.\nSome of the results are conveniently expressed in terms of what we call `almost symmetric' basis elements. These are based on Laurent \npolynomials which are either symmetric or skew-symmetric under one of the involutions $z \\mapsto z^{-1}$  or  $z \\mapsto qz^{-1}.$ \n\nThe recursive description of the $\\{E_r\\},$ equations \\eqref{SRCNX}, \nplays a key role. \nIt immediately gives us the first step of \\eqref{PLANA}, Proof Plan A.\nThe filtration properties of this `zig-zag recursion'\nallow us to obtain explicit formulas for the three highest zig-zag degree terms of each $E_r.$ The coefficients of these terms with\nrespect to the appropriate almost symmetric basis elements are determined in the later parts of Section \\ref{ALMSYM}. Our determination of the last of these, Theorem \\ref{CDG2COEFFS}, is a little involved, and so we have also included a detailed step-by-step presentation\nof the argument.\n\nProperties of the filtration allow us to quickly express $\\boldsymbol{\\mathcal{T}}(a,aq)$ in terms of the  three highest zig-zag degree terms of the $E_r.$ Thus the later parts of Section \\ref{ALMSYM} are exactly what we need to\ncarry out the second step of \\eqref{PLANA}, Proof Plan A, in Section \\ref{TAQ}.\n\nIt is natural to compare our nonsymmetric Askey-Wilson connection coefficient results (Theorems \\ref{ETC} and \\ref{ETC2})  to\nthe corresponding long known symmetric case, Theorem \\ref{PTCFRM}. That is why those former statements are in terms of products \nlike $d_{r,s}c_{|r|,|s|}.$ However, for carrying out the proof here, it is cumbersome to be constantly writing out these products\nand so an alternate notation for these products is introduced in Section \\ref{TAUSIGMA}.\n\nBecause of the natural $2 \\times 2$ block structure of our transition matrices, verifying the co-cycle condition in Section \\ref{COCYCPRF}\nhas four somewhat similar pieces. It also turns out that those verifications can be carried out  more simply by first simplifying\nsome ratios, and that is done at the beginning of Section \\ref{COCYCPRF}.\n\nUsing our result, we also give a re-proof of Askey and Wilson's Theorem \\ref{PTCFRM} in \nSection \\ref{REPROOF}\nof our paper. Here the $d_{r,s}c_{|r|,|s|}$ representation of the connection coefficients in Theorem \\ref{ETC} greatly simplifies the exposition.\n\nWhile the considerations are very elementary, making explicit how our conventions lead to the precise matrices we use is\nimportant for being able to verify correctness of our arguments. This is done in Section \\ref{APPA}, Appendix A.\n\nSection \\ref{HAT}, Appendix B, a summary table, includes a few DAHA related formulas that we do not use in this paper, but which \nare of the same nature as ones we fully justify in the main body.\n\n\\ifLONG\n\nThe technique for proving Theorem \\ref{ETC2} is extremely close to the one for Theorem \\ref{ETC}, so we omit including the details\nwithin the main body of this paper. \n\nHowever, in this longer version of our paper, the details (about a third of this longer form)  are available in Appendices C1 (Section \\ref{SHIFTC1}), C2 (Section \\ref{SHIFTC2}), and C3 (Section \\ref{COCYCPRFC})\nlocated just before the bibliography here.\n\\else\nSince the technique is so close to the one we fully describe for Theorem \\ref{ETC}, we omit the details of the proof of Theorem \\ref{ETC2}. \n\n\\fi\n\nWe mention\none striking feature, however. In the proof of the discrete co-cycle condition for the shift-c case (Propositions \\ref{T00PROP}, \\ref{T01PROP}, \\ref{T10PROP}, \\text{ and }\\ref{T11PROP} below in the shift-a case), the polynomials $p_2,$ whose vanishing in the last quarter of the proofs we are demonstrating, can in a natural\nway be chosen to be {\\em identical } to the $p_2$ of the shift-a case.\n\n\n\n\n\n\\section{Preliminaries \\label{PREL}}\n\n\nFirst we recall the double affine Hecke algebra (DAHA) point of view and the Noumi representation on Laurent polynomials.\n\nLet $\\mathcal{R}$ denote the Laurent polynomials in one variable $z$ with coefficients in a field such as $\\mathbb{Q}(a,b,c,d,q).$ \n\nLet $\\mathbb{F}$ be the field $\\mathbb{Q}(q^{\\frac{1}{2}}, t_0^{\\frac{1}{2}}, t_1^{\\frac{1}{2}},u_0^{\\frac{1}{2}},u_1^{\\frac{1}{2}})$. As described in  \\cite{ANNALS} and \\cite{ASKVOL}, the Noumi representation \\cite{NOUMI} is a faithful representation\nof a double affine Hecke algebra (DAHA)  $\\mathcal{H}$ with coefficients in $\\mathbb{F}.$  Here $\\mathcal{H}$ is the $\\mathbb{F}$-algebra with generators $T_0,T_1,U_0,U_1$\nand relations\n$$T_0 \\sim t_0,\\ T_1 \\sim t_1,\\ U_0 \\sim u_0,\\ U_1 \\sim u_1, \\text{ and } T_1T_0U_0U_1=q^{-\\frac{1}{2}}$$\nwhere the meaning of $F \\sim f$ is $F-F^{-1}=f^{\\frac{1}{2}} - f^{-\\frac{1}{2}}.$ The scalars $q,a,b,c,d,$ $t_0,t_1,u_0,u_1$ are related by\n\n\n\\begin{equation}\n\\label{SCALARS}\n\\begin{array}{rrrr}\n \\displaystyle t_0= - \\frac{cd}{q} & t_1=-ab & \\displaystyle u_0= - \\frac{c}{d} & \\displaystyle u_1=-\\frac{a}{b} \\\\\n\\displaystyle a=t_1^{\\frac{1}{2}} u_1^{\\frac{1}{2}}  &\\displaystyle b = - t_1^{\\frac{1}{2}} u_1^{-\\frac{1}{2}}  &\n\\displaystyle c=q^{\\frac{1}{2}} t_0^{\\frac{1}{2}} u_0^{\\frac{1}{2}}  &\\displaystyle d = -q^{\\frac{1}{2}} t_0 ^{\\frac{1}{2}} u_0 ^{-\\frac{1}{2}} \\\\\n\\end{array}\n\\end{equation}\nFor our purposes, we mostly want to reason about Laurent polynomials with {\\em rational} coefficients $\\mathbb{Q}(a,b,c,d,q).$ So we \nwork with elements $\\widetilde{T}_0,\\widetilde{T}_1, \\text{ and } \\widetilde{U}_0$ (an idea described in \\cite{SIGMA}) that are multiples\nof the usual $T_0,T_1, \\text{ and } U_0:$\n$$\n\\widetilde{T}_0= t_0^{\\frac{1}{2}}T_0 \\hspace{10mm} \n\\widetilde{T}_1= t_1^{\\frac{1}{2}}T_1 \\hspace{10mm} \n\\widetilde{U}_0= (qt_0)^{\\frac{1}{2}}U_0.\n$$\nThe usual nicely symmetric DAHA inversion formulas, e.g.   \n$$T_1^{-1}= T_1 -t_1^{\\frac{1}{2}} +t_1^{-\\frac{1}{2}},$$\nnow translate (as in \\cite{SIGMA}) to ones like\n$$\n\\left(\\widetilde{T_1}+1\\right)   \\left( \\widetilde{T_1}-t_1 \\right)=0; \\hspace{10mm} \\text{i.e. }  \\widetilde{T_1}^{-1} = \\frac{\\widetilde{T}_1}{t_1} -1+ \\frac{1}{t_1}.\n$$\n\nWith these adjustments, the Noumi representation of the DAHA $\\mathcal{H}$ on a Laurent polynomial $f \\in \\mathcal{R}$ looks like:\n\\begin{eqnarray} \n\\label{T0DEF} \\left[ \\widetilde{T}_0 f\\right](z)&=& t_0 f(z)+ \\displaystyle \\frac{(z-c)(z-d)}{(z^2-q)}\\left[ f\\big(\\frac{q}{z}\\big)-f(z) \\right] \\\\\n\\label{T1DEF}  \\left[\\widetilde{T}_1 f\\right](z)&=& t_1 f(z)+\\displaystyle \\frac{(1-az)(1-bz)}{(1-z^2)} \\left[ f \\big( \\frac{1}{z}\\big)-f(z) \\right] \\\\\n\\label{U0DEF} \\widetilde{U}_0 f &=&t_0    \\widetilde{T}_0^{-1}X f   \\ =  \\left( \\widetilde{T}_0X+(1-t_0)X\\right) f\n\\end{eqnarray}\nsince $t_0 \\widetilde{T}_0^{-1}=\\widetilde{T}_0 + (1-t_0) $\nand we use the multiplication operator\n$$ \\left[ Xf \\right](z)= z\\big( f(z) \\big). $$\nWe call the operator $X$ because these DAHA preliminaries are often expressed in terms of a {\\em Laurent series} variable $x$ while\nwe consistently use $z$ for the Laurent variable and its related $x=\\frac{1}{2}(z+z^{-1})$ for the ordinary polynomial variable.\n\nThe nonsymmetric Askey-Wilson polynomials are eigenvectors of \n$$ \\widetilde{Y}= \\widetilde{T}_1 \\widetilde{T}_0 \\text{ (as well as of } Y=T_1T_0)$$\nwith $\\widetilde{Y} E_r=\\widetilde{\\mu}_r E_r$ and\n$$\n\\widetilde{\\mu}_r = \\begin{cases} (t_0t_1)q^r &\\ \\text{if }r\\ge 0 \\\\  q^r &\\ \\text{if }r<0.\\end{cases}\\\\\n$$\n$\\big(\\widetilde{\\mu}_r =(t_0t_1)^{\\frac{1}{2}}q^{\\overline{r}}\\big),$ where \n\\begin{equation}\n\\label{QBARDEF} q^{\\overline{r}}= \\begin{cases} (t_0t_1)^{\\frac{1}{2}} q^r &\\ \\text{if }r\\ge 0 \\\\  (t_0t_1)^{-\\frac{1}{2}} q^r &\\ \\text{if }r<0\\end{cases}\n\\end{equation}\nis the notation used in \\cite{ASKVOL} for the eigenvalues of $Y=T_1T_0.$ \n\n\nThe definition of the $E_r$ on page 278 of \\cite{ANNALS} is via creation operators $\\mathcal{S}_0 \\text{ and } \\mathcal{S}_1.$ Up to normalization,\nfor $n \\ge 0,$ $\\mathcal{S}_0$ takes $E_n$ to $E_{-(n+1)}$ and $\\mathcal{S}_1$ takes $E_{-(n+1)}$ to $E_{n+1}.$ \nThis is the original recursive description of the nonsymmetric Askey-Wilson polynomials.\n\nTheorem 4.1 on page 402 of\n\\cite{ASKVOL} makes this (in arbitrary rank)  more explicit. The proof of that theorem introduces a variant  $\\mathcal{S'}_0$ of $\\mathcal{S}_0$ which also takes\n$E_{-(n+1)}$ to a multiple of $E_{n+1}.$ It is not explicitly noted in the theorem, but the proof makes clear that the formulas stated are simply the result of\napplying the operators $\\mathcal{S'}_0$ and $\\mathcal{S}_1.$\n\nUsing our usual $\\widetilde{U_0} \\text{ and } \\widetilde{T_1},$ the corresponding creation operators are\n\\begin{eqnarray*}\n\\widetilde{\\mathcal{S'}}_0=& [\\widetilde{Y},\\widetilde{U}_0]  \\hspace{10mm}  \\widetilde{\\mathcal{S}}_1=& [\\widetilde{T}_1,\\widetilde{Y}]    \n\\end{eqnarray*}\nwith $\\widetilde{\\mathcal{S'}}_0=t_0(qt_1)^{\\frac{1}{2}}\\mathcal{S'}_0$ and $\\widetilde{\\mathcal{S}}_1=(t_0)^{\\frac{1}{2}} t_1\\mathcal{S}_1.$\n\nThe normalization condition on $E_r$ is that the highest zig-zag degree term $z^r$ has coefficient $1;$ we  refer to this as $E_r$ being zig-zag monic.  We introduce the following $\\widetilde{\\zeta}_{i,r}$ notation for the explicit rescaling factors ($r$ of any sign):\n$$\n\\widetilde{\\mathcal{S'}}_0E_r= \\widetilde{\\zeta'}_{0,-(r+1)}   E_{-(r+1)} \\hspace{10mm} \\widetilde{\\mathcal{S}}_1E_{-(r+1)}= \\widetilde{\\zeta}_{1,r+1}  E_{r+1} .\n$$\nFor $n \\ge 0$ we will determine $\\widetilde{\\zeta'}_{0,-(n+1)}$ and $\\widetilde{\\zeta}_{1,n}$ below in Proposition \\ref{SCALE}. \n\nIn those terms, Theorem 4.1 of \\cite{ASKVOL} for the rank 1 case translates to:\n\n\\begin{eqnarray}\n\\label{SRCN0}\n\n E_{-(n+1)}=\\left[  \\widetilde{\\zeta'}_{0,-(n+1)}  \\right]^{-1} \\widetilde{\\mathcal{S'}}_0E_n& =t_0\\left[ \\widetilde{\\zeta'}_{0,-(n+1)}  \\right]^{-1} \\left[\\left(\\displaystyle  \\frac{ \\widetilde{a}_{-(n+1)} } {t_0} \\right) \\widetilde{U}_0+\\widetilde{b}_{-(n+1)} \\right]E_{n} \\quad&\\\\   \n\\label{SRCN1}\nE_n= \\left[ \\widetilde{\\zeta}_{1,n}  \\right]^{-1} \\widetilde{\\mathcal{S}}_1E_{-n}&=t_1\\left[ \\widetilde{\\zeta}_{1,n}  \\right]^{-1} \\left[\\left(\\displaystyle  \\frac{ \\widetilde{c}_{n} } {t_1} \\right) \\widetilde{T}_1+\\widetilde{d}_{n} \\right]E_{-n} \\quad (n \\neq 0) &\n\\end{eqnarray} \nwhere\n\\begin{equation*}\n\\begin{aligned}\n\\displaystyle\\frac{\\widetilde{a}_n}{t_0}= & \\begin{cases}\\displaystyle \\frac{(abcdq^{2n}| q)_1}{cdq^n} & \\text{\\ if\\ } n\\geq 0 \\\\\n                                          \\displaystyle  - \\frac{1-abcdq^{-2(n+1)}}{cdq^{-(n+1)}}&\\text{\\ if\\ } n <0 \\\\ \\end{cases} \\\\\n\\displaystyle \\widetilde{b}_n=& \\begin{cases}  abcdq^{n}\\left(\\displaystyle \\frac{1}{c}+\\frac{1}{d}\\right)-(a+b)& \\text{\\ if\\ } n\\geq 0 \\\\\n                                          q^{- (n  + 1)}\\left(\\displaystyle \\frac{1}{c}+\\frac{1}{d}\\right)-(a+b)&\\text{\\ if\\ } n <0 \\\\ \\end{cases} \\\\\n\\frac{\\widetilde{c}_n}{t_1}=&  \\begin{cases}\\displaystyle  0 & \\text{\\ if\\ } n=0\\\\\n                                         \\displaystyle  -  \\frac{(abcdq^{2n-1}| q)_1}{abq^{n}} & \\text{\\ if\\ } n>0 \\\\                                         \n                                         \\displaystyle  \\frac{1-abcdq^{-2n-1}}{abq^{-n}} &\\text{\\ if\\ } n <0 \\\\ \\end{cases} \\\\\n\\displaystyle \\widetilde{d}_n= &  \\begin{cases} \\displaystyle \\frac{q-abcd}{q} & \\text{\\ if\\ } n=0\\\\\n                                           \\displaystyle - \\frac{ (abq^n| q)_1 + \\left(abcdq^{n-1}| q\\right)_1}{abq^{n}} & \\text{\\ if\\ } n>0 \\\\                                           \n                                           \\displaystyle  \\frac{-cdq^{-n}(ab+1)+(cd+q)}{q}&\\text{\\ if\\ } n <0 \\\\ \\end{cases}  \\\\                                 \n\\end{aligned}\n\\end{equation*}\nTheorem 1.2 in \\cite{ASKVOL} stated this, but as mentioned in \\cite{SIGMA}, has some typos.\nThe formulas in equations (\\ref{SRCN0}) and (\\ref{SRCN1}) hold for any sign of $n$ above, but we emphasize, in this paper, the $n \\ge 0$ cases because they,\ntogether with $E_0=1,$ give a straightforward way to determine the $E_r$ inductively, for $r$ increasing in the zig-zag order sense. And it is easier to\nprove, as we do later in Proposition \\ref{SCALE}, the (also simpler) formulas for $\\widetilde{\\zeta'}_{0,-(n+1)}$ and $\\widetilde{\\zeta}_{1,-n}$ in those $n \\ge 0$ cases.\n\n\\begin{description} \n\\item[Comment on Some Awkward Looking Factors:] $\\displaystyle \\frac{\\widetilde{a}_n}{t_0}$ and $\\displaystyle \\frac{\\widetilde{c}_n}{t_1}$ as well as the $t_i$ factors in relating $\\mathcal{S'}_0$ (respectively $\\mathcal{S}_1$)  to $\\displaystyle \\frac{\\widetilde{a}_n}{t_0} + \\widetilde{b}_n$ (respectively $\\displaystyle \\frac{\\widetilde{c}_n}{t_1} + \\widetilde{d}_n$) arise   to make $\\widetilde{a}_n$ and $\\widetilde{c}_n$\ndifferences of eigenvalues of $\\widetilde{Y}$ just as $a_n$ and $c_n$ are differences of eigenvalues of $Y;$ e.g. $\\widetilde{c}_n = \\widetilde{\\mu}_{-n} - \\widetilde{\\mu}_{n}$ in analogy to $c_n= q^{\\overline{-n}} - q^{\\overline{n}}$ (correcting a typo on page 397 of \\cite{ASKVOL}.)\n\\item[More Detail on the Translation from \\cite{ASKVOL}:] \nThe starting point is Theorem 4.2 of \\cite{ASKVOL} using the value $n=1$ ($1$ variable case) there. We now review the elements in the proof and\napplication of this theorem.\n\nBesides DAHA identity manipulation, the proof is based on three things:\n\\begin{enumerate}\n\\item The relations among eigenvalues of $Y=T_1T_0$ that correspond to the intertwining identities\n\\begin{equation}\n\\label{INTERTWINE} Y\\mathcal{S}_1=\\mathcal{S}_1Y^{-1} \\hspace{15mm} Y\\mathcal{S}_0= q^{-1}\\mathcal{S}_0 Y^{-1}  \\text{ where } \\mathcal{S}_0= q^{\\frac{1}{2}} Y \\mathcal{S'}_0.\n\\end{equation}\n\\item The creation operator definition, up to normalization, in \\cite{ANNALS} of the $E_r.$\n\\item The fact that the creation operators $\\mathcal{S}_0$ and $\\mathcal{S}_1$ have squares which, when restricted to their natural invariant $2$-dimensional subspaces,  are multiples of the identity.\n\\end{enumerate}\n\nThe eigenvalue relations may either be viewed:\n\\begin{enumerate}\n\\item  As consequences of the formula (\\ref{QBARDEF}) (above) for $q^{\\overline{n}},$ originally given in \\cite{ASKVOL}.\n\\item Or as applications of Theorem 5.1 in \\cite{ANNALS}. In particular, in Theorem 5.1, one can choose \n$\\tilde{\\nu}= 1+0\\delta \\in \\mathbb{Z} \\times  \\mathbb{Z} \\delta$\nsatisfying\n\\begin{eqnarray*}\ns_0(1) = -1 -\\delta & & s_1(1)=-1.\n\\end{eqnarray*}\n\\end{enumerate}\nThe statements about $\\mathcal{S}_0^2$ and $\\mathcal{S}_1^2$ are Corollary 5.2 for $n=1$ in \\cite{ANNALS}. Here one has to keep in\nmind that the $E_r$ are eigenvectors of $Y=T_1T_0.$\nThe following table summarizes many relationships between the original objects in \\cite{ANNALS}, \\cite{ASKVOL} and our notation here.\n\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|} \\hline\n$\\widetilde{T}_0=t_0^{\\frac{1}{2}} T_0$ & $\\widetilde{T}_1=t_1^{\\frac{1}{2}} T_1$ & $\\widetilde{Y}=(t_0t_1)^{\\frac{1}{2}} Y$\\\\ \\hline\n$U_0=q^{-\\frac{1}{2}} T_0^{-1}X$ & $\\widetilde{U}_0=t_0 \\widetilde{T}_0^{-1} X$ & $\\widetilde{U}_0 = (qt_0)^{\\frac{1}{2}} U_0$\\\\ \\hline\n$U_1=(T_1X)^{-1}$ & $\\widetilde{U}_1=t_1(\\widetilde{T}_1X)^{-1}$ & $\\widetilde{U}_1 = t_1^{\\frac{1}{2}} U_1$\\\\ \\hline\n$T_1T_0U_0U_1=q^{-\\frac{1}{2}}$ & $\\widetilde{T}_1\\widetilde{T}_0\\widetilde{U}_0\\widetilde{U}_1=t_0t_1$ & \\\\ \\hline\n$\\mathcal{S}_1=[T_1,Y] $ & $\\widetilde{\\mathcal{S}}_1= [\\widetilde{T}_1,\\widetilde{Y}]$ &$\\widetilde{\\mathcal{S}}_1= t_0^{\\frac{1}{2}}t_1\\mathcal{S}_1$ \\\\ \\hline\n$\\mathcal{S'}_0=[Y,U_0] $ & $\\widetilde{\\mathcal{S}}_0'= [\\widetilde{Y},\\widetilde{U}_0]$ &$\\widetilde{\\mathcal{S}}_0'= t_0(qt_1)^{\\frac{1}{2}}\\mathcal{S'}_0$ \\\\ \\hline\n& & $\\mathcal{S'}_0 E_{-(n+1)}= \\left( a_{n}U_0 + b_{n} \\right) E_{-(n+1)}$ \\\\ \\hline\n& & $\\mathcal{S}_1 E_{-n}= \\left( c_{n}T_1 + d_{n} \\right) E_{-n}$  \\\\ \\hline\n$\\mathcal{S}_0=[Y,U_1^{-1}] $ & $\\widetilde{\\mathcal{S}}_0= [\\widetilde{Y},\\widetilde{U}_1^{-1}]$ &$\\widetilde{\\mathcal{S}}_0= (t_0)^{\\frac{1}{2}}\\mathcal{S}_0$ \\\\ \\hline\n$\\displaystyle \\frac{\\widetilde{a}_n}{t_0}=\\left(\\frac{t_1}{t_0}\\right)^{\\frac{1}{2}}a_n$& $\\widetilde{b}_n=(qt_1)^{\\frac{1}{2}}b_n$ & $\\displaystyle \\frac{\\widetilde{a}_n}{t_0}\\widetilde{U}_0+\\widetilde{b}_n =(qt_1)^{\\frac{1}{2}}\\left( a_nU_0+b_n \\right)$ \\\\ \\hline\n$\\displaystyle \\frac{\\widetilde{c}_n}{t_1}=\\left(\\frac{t_0}{t_1}\\right)^{\\frac{1}{2}}c_n$ &$\\widetilde{d}_n=t_0^{\\frac{1}{2}}d_n$ & $\\displaystyle \\frac{\\widetilde{c}_n}{t_1}\\widetilde{T}_1+\\widetilde{d}_n =t_0^{\\frac{1}{2}}\\left( c_nT_1+d_n \\right)$ \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\n\nTo get the $a_n$ and $b_n$ (as in the above table)  from Theorem 4.2 in \\cite{ASKVOL} (and then the asserted $\\widetilde{a}_n \\text{ and } \\widetilde{b}_n),$  use $\\lambda=\\lambda_1=n, \\mu=\\mu_1=-(n+1).$ Here the $b_n$ comes from the statement about $c_0$ in Theorem 4.2.\n\nTo get the $c_n$ and $d_n$ (as in the above table)  from Theorem 4.2 in \\cite{ASKVOL} (and then the asserted $\\widetilde{c}_n \\text{ and } \\widetilde{d}_n),$  use $\\lambda=\\lambda_1=n,\\mu = \\mu_1=- n$  Here the $d_n$ comes from the statement about (a different)  $c_n$  for $n=1$ in Theorem 4.2.\n\nExpressing everything in terms of $a,b,c,d,q,\\text{ and } n$ (of any sign) via equations (\\ref{SCALARS}) gives the asserted formulas for \n$\\widetilde{a}_n,\\widetilde{b}_n,\\widetilde{c}_n, \\text{ and } \\widetilde{d}_n.$ \n\n\\end{description}\n\\section{Almost Symmetric Bases and their Applications\\label{ALMSYM}}\n\nFor any integer $n,$ let $\\mathcal{R}_n$ denote the elements of the Laurent polynomials $\\mathcal{R}$ of zig-zag degree at most $n$ in the zig-zag order.\nThis gives rise to a filtration\n$$\n\\mathcal{R}_0 \\subset  \\mathcal{R}_{-1} \\subset \\mathcal{R}_1 \\subset \\mathcal{R}_{-2} \\subset \\mathcal{R}_2 \\subset \\ldots \n$$\nwith both $E_n$ and $z^n$ projecting to the same nonzero generator of the $1$-dimensionsal $\\mathcal{R}_n\/\\mathcal{R}_{n-}.$ Here $n-$ denotes\nthe  predecessor of $n$ in the zig-zag order.\n\nIn thinking about the operators $\\widetilde{T}_i,$ it is natural to relate them to skew-symmetrization. If we denote the involutions by $s_i:\\mathcal{R} \\to \\mathcal{R},$\nnamely \n\\begin{eqnarray*}\n\\left[ s_1(f) \\right](z)=f\\left(z^{-1} \\right) &\\hspace{15mm} &\\left[ s_0(f) \\right](z)=f\\left(qz^{-1} \\right)\n\\end{eqnarray*}\nand the corresponding skew (respectively q-skew) symmetrizations by\n$$\n\\Lambda_1(f)= \\frac{1}{2} (1- s_1)\\left(f\\right)  \\hspace{15mm} \\text{(respectively } \\Lambda_0(f)= \\frac{1}{2} (1- s_0)\\left(f\\right) )\n$$\nthen $\\Lambda_1$ (respectively $\\Lambda_0$) has eigenfunctions which we will denote for $n \\ge 0$ as follows:\n$$\\begin{aligned}\n\\text{Eigenvalue } 1: \\ \\ \\ f_n =& (z+z^{-1})^n \\\\\n\\hspace{5mm} & \\text{\\big(respectively } f_{n\\_\\text{q}}= (z+qz^{-1})^n.\\big) \\\\\n\\text{Eigenvalue } -1: \\ \\ \\ g_n =&(z-z^{-1}) (z+z^{-1})^{n-1} \\\\\n\\hspace{5mm} & \\text{\\big(respectively } g_{n\\_\\text{q}}= (z-qz^{-1})(z+qz^{-1})^{n-1}.\\big)\n\\end{aligned}$$\nUnfortunately neither  $f_{n+1}$  nor $g_{n+1}$ belong to $\\mathcal{R}_{-(n+1)},$ but their difference does. So for $n \\ge 0$ we define\n$$\nh_{n+1}= z^{-1} (z+z^{-1})^n \\hspace{15mm}  \\text{(respectively } h_{n+1,\\_\\text{q}}= qz^{-1} (z+qz^{-1})^n.)\n$$\nNow \n$ \\{ f_0,h_1,f_1,h_2,f_2,\\ldots\\}$ and $ \\{ f_{0\\_\\text{q}},h_{1\\_\\text{q}},f_{1\\_\\text{q}},h_{2\\_\\text{q}},f_{2\\_\\text{q}},\\ldots\\}$ both\nform bases compatible with the filtration\n$$\n\\mathcal{R}_0 \\subset  \\mathcal{R}_{-1} \\subset \\mathcal{R}_1 \\subset\\ldots \\ .\n$$\nWe refer to these as the {\\em almost symmetric} and {\\em almost q-symmetric} bases respectively.\n\nThe operator $\\widetilde{T}_0$ involves division by $z^2-q.$ Easy argument shows that for any Laurent polynomial $f$, its q-skew\nsymmetrization $\\Lambda_0(f)$ is divisible by $z^2-q.$ To understand the operators $\\widetilde{T}_0,\\widetilde{U}_0\\text{ and } \\widetilde{T}_1$  more fully,\nwe define Laurent polynomials $f_{\\text{skew}},f_{\\text{q\\_skew}},f_{\\text{skew\\_rdcd}}, f_{\\text{q\\_skew\\_rdcd}}, f_{\\text{sym}},\\text{ and } f_{\\text{q\\_sym}}, $ for any Laurent polynomial $f$ by:\n\\begin{eqnarray*}\nf_{\\text{skew}} =  \\Lambda_1(f) &\\hspace{15mm}& f_{\\text{q\\_skew}} =  \\Lambda_0(f) \\\\\n f_{\\text{skew}} (z) = (z^2-1) \\left[ f_{\\text{skew\\_rdcd}}(z) \\right] &\\hspace{15mm} &f_{\\text{q\\_skew}} (z) = (z^2-q) \\left[ f_{\\text{q\\_skew\\_rdcd}}(z) \\right] \\\\\n f_{\\text{sym}} = \\frac{1}{2} (1+ s_1)\\left(f\\right) &  & f_{\\text{q\\_sym}} = \\frac{1}{2} (1+ s_0)\\left(f\\right).\n\\end{eqnarray*} \nThus\n$$\nf= f_{\\text{sym}} +f_{\\text{skew}} = f_{\\text{q\\_sym}} +f_{\\text{\\_skew}}.\n$$\nIn terms of these we have:\n\\ifJOLT \\begin{Proposition} \\else \\begin{props} \\fi\n\\label{U_0_T_1}\n\\ \n\n\n\\begin{equation}\n\\label{U0A}\n\\begin{split}\n\\widetilde{U}_0f =& \\left(\\frac{(c+d) z - cd } {z } \\right)f -2q\\left(\\frac{(z-c)(z-d)} {z}\\right) f_{\\text{q\\_skew\\_rdcd}}\\\\\n=& - 2qz  f_{\\text{q\\_skew\\_rdcd}} +\\left[ c+d+  \\left(\\frac{cd } {2q } \\right)\\left(z - \\frac{q} {z }\\right)\\right. \\\\\n &  \\hspace{15mm}  \\left.  - \\left(\\frac{cd } {2q }\\right) \\left(z + \\frac{q} {z }\\right)\\right]\\left(f + 2q f_{\\text{q\\_skew\\_rdcd}}\\right) \\\\\n\\end{split}\n\\end{equation}\n\\begin{equation}\n\\label{T1A}\n\\begin{split}\n \\widetilde{T}_1 f =& -ab f + 2\\left[(1-az)(1-bz)\\right]   f_{\\text{skew\\_rdcd}}\\\\\n\n = &-ab f + \\left[ -2(a+b) +\\left(ab+1 \\right) \\left(z+\\frac{1}{z} \\right) + \\left(ab - 1 \\right) \\left(z -\\frac{1}{z} \\right) \\right] \\left( zf_{\\text{skew\\_rdcd}}\\right)\n\\end{split}\n\\end{equation}\n\\ifJOLT \\end{Proposition} \\else \\end{props} \\fi\n\\begin{proof}\nTo establish (\\ref{U0A}), using the definitions (\\ref{T0DEF})  and (\\ref{U0DEF}), we start with\n\\begin{multline*}\n\\Big( \\widetilde{U}_0 f \\Big)(z) = \\Big( \\widetilde{T}_0 + (1-t_0) \\Big) \\Big(  z \\left[ f(z) \\right] \\Big) \\\\\n=(1-t_0) z \\left[ f(z) \\right] + t_0 \\Big( z \\left[ f(z) \\right] \\Big) +\\Big(\\frac{(z-c)(z-d) } {z^2-q } \\Big) \\Big( \\left( \\frac{q}{z} \\right) \\left[ f \\left(\\frac{q}{z} \\right) \\right] -z \\left[ f(z) \\right] \\Big) \\\\\n\\end{multline*}\nSince \n$$\n\\frac{1}{z} - \\frac{z}{z^2-q} = -\\frac{q}{z(z^2-q)},\n$$\nthis gives\n\\begin{multline*}\n(\\widetilde{U}_0f)(z)  =zf(z) - (z-c)(z-d) \\left[ \\frac{1}{z} f\\left( \\frac{q}{z}\\right)\\right. \\ -  \\frac{z}{z^2-q} \\left(\\left. f\\left( \\frac{q}{z}\\right)-f(z)\\right)\\right] \\\\\n= z\\left[ f_{\\text{q\\_sym}}(z) + f_{\\text{q\\_skew}}(z)\\right]- (z-c)(z-d)\\left[ \\frac{1}{z}\\left( f_{\\text{q\\_sym}}(z) - f_{\\text{q\\_skew}}(z)\\right)\\right.\\\\\n\\left. -\\frac{z}{z^2-q}\\left( - 2 f_{\\text{q\\_skew}}(z)\\right) \\right]\n\\end{multline*}\n\\begin{multline*}\n\\hphantom{(\\widetilde{U}_0f)(z)}= z\\left[ f_{\\text{q\\_sym}}(z) + f_{\\text{q\\_skew}}(z)\\right]- \\frac{(z-c)(z-d)} {z}\\Big[ [ \\left( f_{\\text{q\\_sym}}(z) - f_{\\text{q\\_skew}}(z)\\right)\\\\\n\\left. + \\frac{2z^2}{z^2-q}\\left( f_{\\text{q\\_skew}}(z)\\right) \\right]\\\\\n= z\\left[ f_{\\text{q\\_sym}}(z) + f_{\\text{q\\_skew}}(z)\\right]- \\frac{(z-c)(z-d)} {z} \\Big[  f_{\\text{q\\_sym}}(z)  +\\frac{z^2+q}{z^2 - q} \\left(f_{\\text{q\\_skew}}(z)\\right)\\Big]\\\\\n\\end{multline*}\n\\begin{multline*}\n\\hphantom{(\\widetilde{U}_0f)(z) } = z\\left[ f_{\\text{q\\_sym}}(z) + f_{\\text{q\\_skew}}(z)\\right]- \\frac{(z-c)(z-d)} {z}\\Big[  f_{\\text{q\\_sym}}(z) + f_{\\text{q\\_skew}}(z) \\\\\n\\left. + \\frac{2q}{z^2-q}\\left( f_{\\text{q\\_skew}}(z)\\right) \\right]\n\\end{multline*}\n\\begin{multline*}\n\\hphantom{(\\widetilde{U}_0f)(z) }=\\left(z - \\frac{(z-c)(z-d)} {z} \\right)f  - \\left(2q \\frac{(z-c)(z-d)} {z}\\right) f_{\\text{q\\_skew\\_rdcd}}(z)\\\\\n=\\left( \\frac{(c+d)z -cd} {z} \\right)f  - \\left(2q \\frac{(z-c)(z-d)} {z}\\right) f_{\\text{q\\_skew\\_rdcd}}(z)\n\\end{multline*}\nThe proof of the second form of (\\ref{U0A}) comes from further observing\n$$\n \\frac{(c+d) z - cd } {z }  = c+d +  \\left(\\frac{cd } {2q } \\right)\\left(z - \\frac{q} {z }\\right) - \\left(\\frac{cd } {2q }\\right) \\left(z + \\frac{q} {z }\\right)\n$$\nand\n$$\n\\frac{(z-c)(z-d)} {z}= z - \\left(\\frac{(c+d) z - cd } {z } \\right).\n$$\n\nTo establish the first form of  (\\ref{T1A}), using the definition (\\ref{T1DEF}), we start with\n\\begin{multline*}\n \\left( \\widetilde{T}_1 f \\right)(z)= t_1\\left[  f(z) \\right] +\\left( \\displaystyle \\frac{(1-az)(1-bz)}{(1-z^2)} \\right) \\left( f \\big( \\frac{1}{z}\\big)-f(z) \\right) \\\\\n =-ab \\left[  f(z) \\right] +\\left( \\displaystyle \\frac{(1-az)(1-bz)}{(1-z^2)} \\right) \\left[ -2 f_{\\text{\\_skew}}(z) \\right] \\\\\n  =-ab \\left[  f(z) \\right] +\\left( \\displaystyle \\frac{(1-az)(1-bz)}{(1-z^2)} \\right) \\left( -2(z^2-1) \\right) \\left[ f_{\\text{\\_skew\\_rdcd}}(z) \\right] \\\\\n    =-ab \\left[  f(z) \\right] +2(1-az)(1-bz)  \\left[ f_{\\text{\\_skew\\_rdcd}}(z) \\right] \n \\end{multline*}\n which is the first form of (\\ref{T1A}).\n \nTo get the second form, we use\n$$\n2\\frac{(1-az)(1-bz)}{z}=(ab+1)\\Big(z + \\frac{1}{z} \\Big) + (ab-1) \\Big(z - \\frac{1}{z} \\Big) -2(a+b).\n$$\n\\end{proof}\n\nAn immediate corollary is that $\\widetilde{U}_0$ and $\\widetilde{T}_1$ behave very nicely on our filtered bases:\n\\ifJOLT \\begin{Corollary} \\else \\begin{cors} \\fi\n\\label{T_U_BS}\n$$\\begin{aligned}\n\\widetilde{U}_0(f_{n\\_\\text{q}})&= &  -\\left(\\frac{cd}{q}\\right) h_{n+1,\\_\\text{q}}& + (c+d) f_{n\\_\\text{q}}\\\\\n\\widetilde{U}_0(h_{n\\_\\text{q}})& = &q f_{n-1,\\_\\text{q}} \\\\\n\\widetilde{T}_1(f_n) &=& -ab f_n  \\\\\n\\widetilde{T}_1(h_n) &=& -ab f_n - h_n &+(a+b) f_{n -1} \\\\\n\\end{aligned}$$\n\\ifJOLT \\end{Corollary} \\else \\end{cors} \\fi\n\\begin{proof}\n\\begin{enumerate}\n\\item For $f=f_{n\\text{\\_q}}=\\displaystyle \\left(z + \\frac{q} {z }\\right)^n,$ we have $f_{\\text{q\\_skew\\_rdcd}}=0.$ So\n\\begin{multline*}\n\\widetilde{U}_0(f)=\\Big((c+d) +  \\left(\\frac{cd } {2q } \\right)\\left(z - \\frac{q} {z }\\right) - \\left(\\frac{cd } {2q }\\right) \\left(z + \\frac{q} {z }\\right) \\Big)f_{n\\text{\\_q}} \\\\\n=\\left( (c+d) - \\left( \\frac{2q}{z} \\right) \\left( \\frac{cd } {2q }\\right) \\right) f_{n\\text{\\_q}}\\\\\n= (c+d) f_{n\\text{\\_q}} -  \\left( \\frac{cd } {q }\\right) h_{n+1,\\text{\\_q}}.\\\\\n\\end{multline*}\n\\item For $f=h_{n\\text{\\_q}}= \\displaystyle \\left( \\frac{q}{z} \\right) \\left(z + \\frac{q} {z }\\right)^{n-1},$\n$$\nf_{\\text{q\\_skew}} = \\displaystyle \\left(  \\frac{1}{2} \\right)  \\left( \\frac{q}{z}  - z\\right) \\left(z + \\frac{q} {z }\\right)^{n-1} = \\displaystyle- \\left( \\frac{1}{2z} \\right) (z^2-q) \\left(z + \\frac{q} {z }\\right)^{n-1}\n$$\nSo\n$$\nf_{\\text{q\\_skew\\_rdcd}} = \\displaystyle  - \\left( \\frac{1}{2z} \\right) \\left(z + \\frac{q} {z }\\right)^{n-1}=  \\displaystyle  - \\left( \\frac{1}{2q} \\right) h_{n\\text{\\_q}} = \\displaystyle - \\frac{f}{2q}.\n$$\nThus \n$$\nf+2q f_{\\text{q\\_skew\\_rdcd}} =0 \\text{ and } z f_{\\text{q\\_skew\\_rdcd}}  = \\displaystyle - \\left(  \\frac{1}{2} \\right)  \\left(z + \\frac{q} {z }\\right)^{n-1}=  \\displaystyle - \\left(  \\frac{1}{2} \\right) f_{n-1,\\text{\\_q}}.\n$$\nThis gives us\n$$\n\\widetilde{U}_0(h_{n\\text{\\_q}})= -2q \\left(  -\\frac{1}{2} f_{n-1,\\text{\\_q}}\\right) +0=qf_{n-1,\\text{\\_q}}.\n$$\n\\item For $f=f_n= \\displaystyle \\left( z + \\frac{1}{z} \\right)^n,$ we have $f_{\\text{skew\\_rdcd}} =0.$ So by (\\ref{T1A}),\n$$\n\\widetilde{T}_1 f_n=-ab f_n.\n$$\n\\item For $f=h_n= \\displaystyle\\left( \\frac{1}{z} \\right) \\left( z + \\frac{1}{z} \\right)^n,$\n$$\nf_{\\text{skew}} =\\displaystyle \\left( \\frac{1}{2} \\right) \\left( \\frac{1}{z}-z \\right) \\left( z + \\frac{1}{z} \\right)^{n-1}= \\displaystyle -\\left( \\frac{1}{2z} \\right) \\big(z^2-1\\big) \\left( z + \\frac{1}{z} \\right)^{n-1}.\n$$\nSo \n$$\nf_{\\text{skew\\_rdcd}} = \\displaystyle -\\left( \\frac{1}{2z} \\right) \\left( z + \\frac{1}{z} \\right)^{n-1}= \\displaystyle - \\frac{h_n}{2}  \\text{ and } zf_{\\text{skew\\_rdcd}}=\\displaystyle - \\frac{f_{n-1}}{2} .\n$$\nThus by  (\\ref{T1A}),\n\\begin{multline*}\n\\widetilde{T}_1 h_n = -(ab ) h_n  + \\Big\\{ -2(a+b) +2ab \\left( z + \\frac{1}{z} \\right) -2(ab-1) \\left(\\frac{1}{z}  \\right) \\Big\\} \\left(- \\frac{f_{n-1}}{2} \\right) \\\\\n= -(ab ) h_n  + (a+b) f_{n-1} -(ab) f_n + (ab-1) h_n \\\\\n= -(ab) f_n -h_n +  (a+b) f_{n-1} \n\\end{multline*}\n\n\\end{enumerate}\n\\end{proof}\n\n\\noindent {\\bf Remark:} Corollary \\ref{T_U_BS} shows, for $n \\ge 0:$\n\\begin{enumerate}\n\\item $\\widetilde{U}_0$ maps $\\mathcal{R}_n$ to $\\mathcal{R}_{-(n+1)}$ and $\\mathcal{R}_{-(n+1)}$ to $\\mathcal{R}_{-(n+1)}.$ \n\\item $\\widetilde{T}_1$ maps $\\mathcal{R}_{-(n+1)}$ to $\\mathcal{R}_{n+1}$ and $\\mathcal{R}_{n}$ to $\\mathcal{R}_{n}.$\n\\end{enumerate}\n\nBecause of Corollary \\ref{T_U_BS}  and the recursion (\\ref{SRCN0}) and (\\ref{SRCN1}), we will need to convert between the almost symmetric bases in low zig-zag\nco-degree. \n\\ifJOLT \\begin{Proposition} \\else \\begin{props} \\fi\n\\label{COB}\n\\begin{eqnarray*}\nf_n = f_{n\\text{\\_q}} + (q^{-n}-1)h_{n\\text{\\_q}} \\text{ mod } \\mathcal{R}_{-(n-1)} & &\nh_n = q^{-n}h_{n\\text{\\_q}} \\text{ mod } \\mathcal{R}_{-(n-1)} \\\\\nh_{n+1\\text{\\_q}} = q^{n+1}h_{n+1} \\text{ mod } \\mathcal{R}_{n-1} & &\nf_{n\\text{\\_q}} = f_{n} + (q^{n}-1)h_{n}  \\text{ mod } \\mathcal{R}_{n-1}\n\\end{eqnarray*}\n\\ifJOLT \\end{Proposition} \\else \\end{props} \\fi\n\\begin{proof}\n\\begin{equation*}\nh_n = z^{-1} (z+z^{-1})^{n-1} = z^{-n}= q^{-n}h_{n\\text{\\_q}}  \\text{ mod } \\mathcal{R}_{-(n-1)}  \n\\end{equation*}\n\\begin{multline*}\nf_n= (z+z^{-1})^n = z^n + z^{-n} = z^n + q^nz^{-n} + (1-q^n) z^{-n}  \\\\ = z^n + q^nz^{-n} +  (q^{-n}-1) q^nz^{-n}\\\\ = f_{n\\text{\\_q}} + (q^{-n}-1)h_{n\\text{\\_q}}  \\text{ mod } \\mathcal{R}_{-(n-1)}  \\\\\nh_{n+1\\text{\\_q}}= qz^{-1} (z+qz^{-1})^n=q^{n+1}z^{-(n+1)}= q^{n+1}h_{n+1}  \\text{ mod } \\mathcal{R}_{n-1} \\\\\nf_{n+1\\text{\\_q}}=  (z+qz^{-1})^{n+1}=z^{n+1} +q^{n+1} z^{-(n+1)} \\\\=z^{n+1} + z^{-(n+1)} +(q^{n+1}-1)z^{-(n+1)}  \\\\ = f_{n+1} + (q^{n+1}-1)h_{n+1}   \\text{ mod } \\mathcal{R}_{n-1}\n\\end{multline*}\n\n\n\\end{proof}\n\nThe almost symmetric bases allow us to make the rescalings involved in the recursive computation of the $E_n$ explicit. This also makes the rationality of\nthe formulas transparent.\n\n\nSince the recursion in equations (\\ref{SRCN0}) and (\\ref{SRCN1}) is in terms of $\\widetilde{U}_0 \\text{ and } \\widetilde{T}_1,$ Corollary \\ref{T_U_BS} makes an expansion\nof the $E_r$ in terms of the  almost symmetric bases natural. For $n \\ge 0,$ we will use the notation (remembering e.g. $h_{m+1} \\in \\mathcal{R}_{-(m+1)}$):\n\\begin{eqnarray*}\nE_n & =& \\sum_{m=0}^n \\lambda_{m,n} f_m +  \\sum_{m=0}^{n-1} \\mu_{-(m+1), n} h_{m+1} \\\\\nE_n &=& \\sum_{m=0}^n \\lambda_{m,n\\text{\\_q}} f_{m\\text{\\_q}} +  \\sum_{m=0}^{n-1} \\mu_{-(m+1), n} h_{m+1,\\text{\\_q}} \\\\\nE_{-(n+1)} &= & \\sum_{m=0}^n \\lambda_{-(m+1),-(n+1)} h_{m+1} +  \\sum_{m=0}^{n} \\mu_{m, -(n+1)} f_{m} \\\\\nE_{-(n+1)}& = &\\sum_{m=0}^n \\lambda_{-(m+1),-(n+1)\\text{\\_q}} h_{m+1\\text{\\_q}} +  \\sum_{m=0}^{n} \\mu_{m, -(n+1)\\text{\\_q}} f_{m\\text{\\_q}}.\n\\end{eqnarray*}\n\n\nUsing Corollary \\ref{COB}, we quickly see\n\\ifJOLT \\begin{Proposition} \\else \\begin{props} \\fi\n\\label{TOFROMQ}\nFor $n \\ge 0:$\n\\begin{eqnarray*}\n\\lambda_{-(n+1),-(n+1)}&=&\\lambda_{-(n+1),-(n+1)\\text{\\_q}} q^{n+1} \\\\\n \\mu_{n,-(n+1)} &=& \\mu_{n,-(n+1)\\text{\\_q}}\\\\\n\\lambda_{-n,-(n+1)} &=&  \\mu_{n,-(n+1)\\text{\\_q}} (q^{n}-1)  + \\lambda_{-n,-(n+1)\\text{\\_q}} q^{n} \\ ( n \\ge1) \n\\end{eqnarray*}\n\\begin{eqnarray*}\n\\lambda_{n,n\\text{\\_q}}& = &\\lambda_{n,n}\\\\\n\\mu_{-(n+1),n+1\\text{\\_q}} &=& \\lambda_{n+1,n+1}(q^{-(n+1)}-1)+\\mu_{-(n+1),n+1}q^{-(n+1)} \\\\\n{} \\lambda_{n,n+1\\text{\\_q}} &=& \\lambda_{n,n+1}.\n\\end{eqnarray*}\n\\ifJOLT \\end{Proposition} \\else \\end{props} \\fi\n\\begin{proof}\nModulo $\\mathcal{R}_{n-1}:$\n\\begin{multline*}\nE_{-(n+1)}= \\lambda_{-(n+1),-(n+1)\\text{\\_q}} h_{n+1,\\text{\\_q}}+ \\mu_{n,-(n+1)\\text{\\_q}} f_{n,\\text{\\_q}}+ \\lambda_{-n,-(n+1)\\text{\\_q}} h_{n,\\text{\\_q}} \\\\\n               = \\lambda_{-(n+1),-(n+1)\\text{\\_q}} \\left[ q^{n+1}h_{n+1}  \\right] + \\mu_{n,-(n+1)\\text{\\_q}}\\left[ f_{n} + (q^{n}-1)h_{n} \\right] \\\\\n               + \\lambda_{-n,-(n+1)\\text{\\_q}} \\left[ q^{n}h_{n} \\right] \\\\\n               =\\left[\\lambda_{-(n+1),-(n+1)\\text{\\_q}}  q^{n+1} \\right] h_{n+1} + \\left[ \\mu_{n,-(n+1)\\text{\\_q}}  \\right] f_n \\\\\n               + \\left[   \\mu_{n,-(n+1)\\text{\\_q}} (q^{n}-1)  + \\lambda_{-n,-(n+1)\\text{\\_q}} q^{n} \\right] h_n \n\\end{multline*}\nfrom which we read off the first three statements.\n\nModulo $\\mathcal{R}_{-(n-1)}:$\n\\begin{multline*}\nE_n = \\lambda_{n,n} f_n +   \\mu_{-n,n} h_n +  \\lambda_{n-1,n} f_{n-1} \\\\\n       = \\lambda_{n,n} \\left[  f_{n\\text{\\_q}} + (q^{-n}-1)h_{n\\text{\\_q}}\\right]  +   \\mu_{-n,n} \\left[ q^{-n}h_{n\\text{\\_q}} \\right] +  \\lambda_{n-1,n} \\left[ f_{n-1,\\text{\\_q}} \\right] \\\\\n       = \\left[  \\lambda_{n,n}  \\right] f_{n\\text{\\_q}} +  \\left[ \\lambda_{n,n}  (q^{-n}-1)+  \\mu_{-n,n}  q^{-n} \\right] h_{n\\text{\\_q}} + \\left[   \\lambda_{n-1,n} \\right] f_{n-1,\\text{\\_q}} \n\\end{multline*}\nfrom which we read off the last three statements.\n\n\\end{proof}\n\nThe zig-zag monic condition on the $E_r$  immediately implies\n\\ifJOLT \\begin{Proposition} \\else \\begin{props} \\fi\n\\label{LNN}\nFor any $n \\ge 0:$\n\\begin{eqnarray*}\n\\lambda_{n,n} =1& \\hspace{15mm} & \\lambda_{n,n\\text{\\_q}} =  1 \\hspace{10mm}\\\\\n\\lambda_{-(n+1),-(n+1)} = 1 &\\hspace{15mm}& \\lambda_{-(n+1),-(n+1)\\text{\\_q}}= q^{-(n+1)}.\n\\end{eqnarray*}\n\\ifJOLT \\end{Proposition} \\else \\end{props} \\fi\nFor our purpose, we need the explicit scaling factors arising in the zig-zag increasing cases of the creation operators acting on $E_r.$\nWe use the following notation for the exact coefficients:\n\\begin{equation}\n\\label{SRCNX}\n\\begin{aligned}\n {\\rm\\  (E_{negative} \\ case)\\ }\\ \\ \\ & E_{-(n+1)}&= \\ \\ \\ &\\left( { \\hat{a}_{-(n+1)} } \\widetilde{U}_0+\\hat{b}_{-(n+1)} \\right)E_{n}&\\ \\\\   \n{\\rm\\  (E_{positive} \\ case)\\ }\\ \\ \\ &E_n&=\\ \\ \\ & \\left(  { \\hat{c}_{n} }  \\widetilde{T}_1+\\hat{d}_{n} \\right)E_{-n}&  \\ \n\\end{aligned}\n\\end{equation}\n\n\\ifJOLT \\begin{Proposition} \\else \\begin{props} \\fi\n\\label{SCALE}\nFor $n \\ge 0:$\n\\begin{eqnarray*}\n\\hat{a}_{-(n+1)} = -\\frac{1}{cdq^n} &\n& \\hat{c}_n = - \\frac{1}{ab} \\hspace{15mm} \\\\\n\\widetilde{\\zeta'}_{0,-(n+1)} =  -cdq^{-1} (abcdq^{2n}| q)_1& \n& \\widetilde{\\zeta}_{1,n} = -abq^{-n}(abcdq^{2n-1}| q)_1  \\\\\n\\hat{b}_{-(n+1)} = \\frac{(c+d)- cdq^{n }(a+b)}{cdq^{n } (abcdq^{2n}| q)_1}&  \n& \\hat{d}_n = - \\frac{( abq^{n} | q )_1+ab(cdq^{n-1}| q)_1}{ab(1-abcdq^{2n-1})} \n\\end{eqnarray*}\n\\ifJOLT \\end{Proposition} \\else \\end{props} \\fi\n\n\\begin{proof}\nWe use the combination of the creation operator point of view with Corollary \\ref{T_U_BS}.\n\\begin{enumerate} \n\\item For the $\\hat{c}_n$ and $\\widetilde{\\zeta}_{1,n}$ determination, we start with\n$$\\begin{aligned}\nE_n= & \\left(   \\hat{c}_{n}  \\widetilde{T}_1+\\hat{d}_{n} \\right)E_{-n} \\\\\n     = &   \\hat{c}_{n}   \\widetilde{T}_1 h_n \\text{ mod } \\mathcal{R}_{-n}  \\\\\n     =& -ab \\hat{c}_n f_n \\text{ mod } \\mathcal{R}_{-n} \\\\\n     \\end{aligned}$$\n     Since $E_n = f_n \\text{ mod } \\mathcal{R}_{-n},$ this gives the asserted formula for $\\hat{c}_n.$ \n     This also means \n$$\n \\widetilde{T}_1 E_{-n} = \\left[ \\hat{c}_n \\right]^{-1} E_n \\text{ mod } E_{-n}  \n$$\nThen\n\\begin{multline*}\n\\widetilde{\\zeta}_{1, n} E_n = \\widetilde{\\mathcal{S}}_1 E_{-n} = [\\widetilde{T}_1,\\widetilde{Y}] E_{-n}= \\left(\\widetilde{\\mu}_{-n} -\\widetilde{\\mu}_n \\right)\\left(\\widetilde{T}_1 E_{-n}\\right) \\text{ mod } E_{-n} \\\\ =\\left(\\widetilde{\\mu}_{-n} -\\widetilde{\\mu}_n \\right) \\left[ \\hat{c}_n \\right]^{-1} E_n \n\\end{multline*}\ngives the $\\widetilde{\\zeta}_{1,n}$  determination.\n\\item For the $\\hat{a}_{-(n+1)} $ and $\\widetilde{\\zeta'}_{0,-(n+1)}$ determination, we start with\n$$\\begin{aligned}\nE_{-(n+1)}= & \\left(   \\hat{a}_{-(n+1)}  \\widetilde{U}_0+\\hat{b}_{-(n+1)} \\right)E_{n} \\\\\n     = &   \\hat{a}_{-(n+1)}   \\widetilde{U}_0 f_{n\\text{\\_q}} \\text{ mod } E_{n}  \\\\\n          = &   \\hat{a}_{-(n+1)} \\left(- \\frac{cd}{q}\\right)   h_{n+1\\text{\\_q}} \\text{ mod } \\mathcal{R}_{n}  \\\\\n     =& -cdq^n \\hat{a}_{-(n+1)} h_{n+1} \\text{ mod } \\mathcal{R}_{n}. \\\\\n\\end{aligned}$$\nSince $E_{-(n+1)} = h_{n+1} \\text{ mod } \\mathcal{R}_{n},$ this gives the asserted formula for $\\hat{a}_{-(n+1)}.$\nThis also means \n$$\n \\widetilde{U}_0 E_{n} = \\left[ \\hat{a}_{-(n+1)} \\right]^{-1} E_{-(n+1)} \\text{ mod } E_{n} . \n$$\nThen\n\\begin{multline*}\n\\widetilde{\\zeta'}_{0,-(n+1)} E_{-(n+1)} =\\widetilde{\\mathcal{S'}}_0 E_{n} = [\\widetilde{Y},\\widetilde{U}_0] E_{n}= \\left(\\widetilde{\\mu}_{-(n+1)} -\\widetilde{\\mu}_n \\right)\\left(\\widetilde{U}_0 E_{n}\\right) \\text{ mod } E_{n} \\\\=\\left(\\widetilde{\\mu}_{-(n+1)} -\\widetilde{\\mu}_n \\right) \\left[ \\hat{a}_{-(n+1)} \\right]^{-1} E_{-(n+1)} \n\\end{multline*}\ngives the $\\widetilde{\\zeta'}_{0,-(n+1)}$  determination.\n\\item The $\\hat{b}_{-(n+1)}$ and $\\hat{d}_n$ formulas come from equations (\\ref{SRCN0}) and (\\ref{SRCN1}) together with the previously translated formulas from \\cite{ASKVOL} for $\\widetilde{b}_{-(n+1)}$ and $\\widetilde{d}_n.$\n\\end{enumerate} \n\\end{proof}\n\nIn this paper, we only use the zig-zag increasing cases above and so have  included just the proofs of those.  However Appendix B  includes a table\nwith the zig--zag decreasing cases as well. (Formulas (4.11) and (4.12) of \\cite{ZHED} could be appealed to since they are equivalent to the determination of the \n$\\hat{c}_r,\\hat{d_r}$ for any sign of $r.$) We mention the simplified forms of those others as well because they may be of interest. They may \nbe established, e.g. in the $\\hat{a}_n,\\hat{b_n}$ case, either using Corollary 5.2 of \\cite{ANNALS} about $\\mathcal{S}_i^2$ or by noting that the relations\n\\begin{eqnarray*}\nE_{-(n+1)} = \\left( \\hat{a}_{-(n+1)} \\widetilde{U_0} + \\hat{b}_{-(n+1)} \\right) E_n  & \\hspace{15mm}  &E_n = \\left( \\hat{a}_n \\widetilde{U_0} + \\hat{b}_n \\right) E_{-(n+1)} \n\\end{eqnarray*}\nare inverse to each other. (Corollary \\ref{T_U_BS} also clarifies what happens in low zig-zag co-degree.)\n\nIf using Corollary 5.2 of \\cite{ANNALS}, one might first confirm, by thinking about the DAHA relation, that $t_i^{\\frac{1}{2}} \\text{ and } u_i^{\\frac{1}{2}}$\nare mapped by the involution $\\epsilon$ of that paper to what one might guess from its action on $T_i$ and $U_i.$ That then makes immediate\nwhat $\\epsilon(a)$ and $\\epsilon(c)$ are. Then the automorphism property, together with $t_0=-cdq^{-1} \\text{ and } t_1=-ab$ gives the\nneeded $\\epsilon(b)$ and $\\epsilon(d)$\n\nAn immediate corollary of proposition \\ref{SCALE} and recursion relations (\\ref{SRCNX})\nis the $\\boldsymbol{\\mathcal{T}}(a,e)$ part of the first step of (\\ref{PLANA}), Proof Plan A :\n\\ifJOLT \\begin{Corollary} \\else \\begin{cors} \\fi\n\\label{PLANA1}\nThe entries of both $T(a,e)$ and  $\\boldsymbol{\\mathcal{T}}(a,e) $ are rational functions of $e$ with coefficients in the filed $\\mathbb{Q}(a,b,c,d,q).$\n\\ifJOLT \\end{Corollary} \\else \\end{cors} \\fi\n(Rationality of entries of $T(a,e)$ is immediate from the formulas written down in theorem \\ref{ETC}, since $T(a,e)$ just refers to  the\nmatrix given by the formulas (\\ref{ETCFRM}), perse.)\n\n\nWe now work out the details of the low zig-zag co-degree expansion of the nonsymmetric polynomials in terms of the almost symmetric bases.\n\nThe recursive relations (\\ref{SRCNX})\nwith exact scaling factors implies the following for the low zig-zag co-degree almost symmetric basis coefficients:\n\\ifJOLT \\begin{Proposition} \\else \\begin{props} \\fi\n\\label{RCNL}\nFor $n \\ge 0:$\n\\begin{equation*}\n\\begin{aligned}\n\\lambda_{n+1,n+1} &\\ =&\\lambda_{-(n+1),-(n+1)} \\left[ -ab\\hat{c}_{n+1} \\right] \\\\% & & \n\\lambda_{-(n+1),-(n+1)\\text{\\_q}}&\\  = &\\lambda_{nn\\text{\\_q}} \\left[ \\left( -\\frac{cd}{q}\\right) \\hat{a}_{-(n+1)} \\right]  \\\\\n\\mu_{-(n+1),n+1} &\\ =&\\lambda_{-(n+1),-(n+1)} \\left[ -\\hat{c}_{n+1}+\\hat{d}_{n+1} \\right]  \\\\\n\\mu_{n,-(n+1)\\text{\\_q}} &\\ =&\\lambda_{nn\\text{\\_q}} \\left[ \\left( c+d\\right) \\hat{a}_{-(n+1)}+ \\hat{b}_{-(n+1)} \\right]  \\\\\n\\end{aligned}\n\\end{equation*} \n\\begin{multline*}\n\\lambda_{n,n+1} =\\lambda_{-(n+1),-(n+1)} \\left[ (a+b)\\hat{c}_{n+1} \\right] +\\mu_{n,-(n+1)} \\left[    (-ab)\\hat{c}_{n+1}+ \\hat{d}_{n+1}\\right]  \\\\\n+\\lambda_{-n,-(n+1)} \\left[ -ab \\hat{c}_{n+1}\\right] \\\\\n\\lambda_{-n,-(n+1)\\text{\\_q}} =  \\mu_{-n,n\\text{\\_q}}    \\hat{b}_{-(n+1)}+\\lambda_{n-1,n\\text{\\_q}} \\left[ \\left( -\\frac{cd}{q}\\right) \\hat{a}_{-(n+1)} \\right]\n\\end{multline*}  \n\\ifJOLT \\end{Proposition} \\else \\end{props} \\fi \n\\begin{proof}\nFor the first $2$ assertions on the left and the next to last one, recall that our standard notation is:\n\\begin{eqnarray*}\nE_{n+1}& =& \\lambda_{n+1,n+1} f_{n+1}+\\mu_{-(n+1),n+1} h_{n+1}+ \\lambda_{n,n+1} f_n \\\\\nE_{-(n+1)} &= &\\lambda_{-(n+1),-(n+1)} h_{n+1}+\\mu_{n,-(n+1)} f_{n}+ \\lambda_{-n,-(n+1)} h_n. \n\\end{eqnarray*}\nSo\n\\begin{multline*}\n\\left( \\hat{c}_{n+1} \\widetilde{T}_1 + \\hat{d}_{n+1} \\right) E_{-(n+1)}  \\text{ mod } \\mathcal{R}_{-n}  = \\\\\n\\lambda_{-(n+1),-(n+1)} \\left[  \\hat{c}_{n+1} \\left[ -abf_{n+1} -h_{n+1} +(a+b)f_n \\right]+  \\hat{d}_{n+1}h_{n+1}\\right] +\\\\ \n\\mu_{n,-(n+1)} \\left[  \\hat{c}_{n+1} \\left[ -abf_n \\right]+  \\hat{d}_{n+1}f_n\\right] +  \\lambda_{-n,-(n+1)} \\left[\\hat{c}_{n+1} \\left[ -abf_n \\right] \\right]\\\\\n=f_{n+1} \\left[ \\lambda_{-(n+1),-(n+1)} \\left[ -ab \\hat{c}_{n+1}  \\right] \\right] \\\\+ h_{n+1} \\left[ \\lambda_{-(n+1),-(n+1)} \\left[ -\\hat{c}_{n+1} +\\hat{d}_{n+1} \\right] \\right] +\\\\ \nf_{n} \\left[ \\lambda_{-(n+1),-(n+1)} \\left[ (a+b) \\hat{c}_{n+1}   \\right]  \\right. \\\\ +  \\left. \\mu_{n,-(n+1)} \\left[ -ab\\hat{c}_{n+1}+\\hat{d}_{n+1} \\right] + \\lambda_{-n,-(n+1)} \\left[ -ab \\hat{c}_{n+1} \\right] \\right]\n\\end{multline*}\nfrom which we can read off the three results.\n\nFor the other $3$ assertions, start with our standard notation of:\n\\begin{eqnarray*}\nE_{-(n+1)} &=& \\lambda_{-(n+1),-(n+1)\\text{\\_q}} h_{n+1\\text{\\_q}}+\\mu_{n,-(n+1)\\text{\\_q}} f_{n\\text{\\_q}}+ \\lambda_{-n,-(n+1)\\text{\\_q}} h_{n\\text{\\_q}} \\\\\nE_{n} &=& \\lambda_{n,n\\text{\\_q}} f_{n\\text{\\_q}}+\\mu_{-n,n\\text{\\_q}} h_{n\\text{\\_q}}+ \\lambda_{n-1,n\\text{\\_q}} f_{n-1,\\text{\\_q}}.\n\\end{eqnarray*}\nSo\n\\begin{multline*}\n\\left( \\hat{a}_{-(n+1)} \\widetilde{U}_0 + \\hat{b}_{-(n+1)} \\right) E_{n}  \\text{ mod } \\mathcal{R}_{n-1}  = \\\\\n \\lambda_{n,n\\text{\\_q}} \\left[  \\hat{a}_{-(n+1)}\\left[ \\left( -\\frac{cd}{q}\\right) h_{n+1\\text{\\_q}} +(c+d) f_{n\\text{\\_q}} \\right]  + \\hat{b}_{-(n+1)} f_{n\\text{\\_q}} \\right] +\\\\ \\mu_{-n,n\\text{\\_q}} \\left[ \\hat{a}_{-(n+1)}\\left[ qf_{n-1\\text{\\_q}} \\right]  + \\hat{b}_{-(n+1)}h_{n\\text{\\_q}} \\right] + \\lambda_{n-1,n\\text{\\_q}} \\left[ \\hat{a}_{-(n+1)}\\left[\\left( -\\frac{cd}{q}\\right) h_{n\\text{\\_q}}  \\right]  \\right]  \\\\\n =h_{n+1\\text{\\_q}} \\left[ \\lambda_{n,n\\text{\\_q}} \\left[  \\left( -\\frac{cd}{q}\\right) \\hat{a}_{-(n+1)} \\right] \\right] + f_{n\\text{\\_q}} \\left[ \\lambda_{n,n\\text{\\_q}} \\left[(c+d)\\hat{a}_{-(n+1)}+ \\hat{b}_{-(n+1)}  \\right]  \\right]\\\\ +h_{n\\text{\\_q}} \\left[ \\mu_{-n,n\\text{\\_q}}   \\left[ \\hat{b}_{-(n+1)} \\right] +\\lambda_{n-1,n\\text{\\_q}} \\left[  \\left( -\\frac{cd}{q}\\right) \\hat{a}_{-(n+1)}\\right] \\right]  \n\\end{multline*}\nfrom which we can read off the other three results.\n\\end{proof}\n\\ifJOLT \\begin{Proposition} \\else \\begin{props} \\fi\n\\label{ZZCD1}\nFor $n \\ge 0:$ \n\\begin{eqnarray}\n\\label{MUQN} \\mu_{n,-(n+1)} \\   = &\\displaystyle \\frac{ abq^{n}(c+d) - (a+b)  }{ (abcdq^{2n}| q)_1  } =  \\mu_{n,-(n+1)\\_\\text{q}} \\\\\n\\label{MUN} \\mu_{-(n+1),n+1}\\   = &\\displaystyle  - \\frac{ (q^{n+1}, cdq^{n}| q)_1 } {  (abcdq^{2n+1}| q)_1  } \\hspace{25mm}\\\\\n \\label{MUQP} \\mu_{-(n+1),n+1\\_\\text{q}} \\ = & \\displaystyle   \\frac{ cd(q^{n+1}, ab q^{n+1 }| q)_1  } {  q(abcdq^{2n+1}| q)_1 }.  \\hspace{25mm}\n\\end{eqnarray}\n\\ifJOLT \\end{Proposition} \\else \\end{props} \\fi\n\n\\begin{proof}\n\nBy proposition \\ref{RCNL}\n\\begin{multline}\n\\mu_{n,-(n+1)\\text{\\_q}} =\\lambda_{n,n\\text{\\_q}} \\left[ \\left( c+d\\right) \\hat{a}_{-(n+1)}+ \\hat{b}_{-(n+1)} \\right]= (c+d) \\left( -\\frac{1}{cdq^n}  \\right) +\\\\\n\\frac{(c+d)- cdq^{n }(a+b)}{cdq^{n } (abcdq^{2n}| q)_1} =\\frac{ (c+d) abcdq^{2n}- cdq^{n }(a+b)}{cdq^{n } (abcdq^{2n}| q)_1} = \\text{ as in }  (\\ref{MUQN}) \\text{ above.} \n\\end{multline}\nAnd by proposition \\ref{TOFROMQ} this is also the value of $\\mu_{n,-(n+1)} .$\n\nSimilarly \n\\begin{multline}\n\\mu_{-(n+1),n+1} =\\lambda_{-(n+1),-(n+1)} \\left[ -\\hat{c}_{n+1}+\\hat{d}_{n+1} \\right] \\\\ = - \\left( - \\frac{1}{ab}  \\right)+ \\left( - \\frac{( abq^{n+1} | q )_1+ab(cdq^{n}| q)_1}{ab(1-abcdq^{2n+1})} \\right) \\\\\n= \\frac{- abcdq^{2n+1} +abq^{n+1} -ab + abcdq^{n}  }{ab(1-abcdq^{2n+1}) } =-\\frac{(1-q^{n+1})(1-cdq^{n} )}{(1-abcdq^{2n+1})} \n\\end{multline}\nas in (\\ref{MUN}) above. \n\nAnd by proposition \\ref{TOFROMQ}\n\\begin{multline*}\n \\mu_{-(n+1),n+1\\text{\\_q}} = \\lambda_{n+1,n+1}(q^{-(n+1)}-1)+\\mu_{-(n+1),n+1}q^{-(n+1)} \\\\\n = (q^{-(n+1)}-1) + q^{-(n+1)} \\left(-\\frac{(1-q^{n+1})(1-cdq^{n} )}{(1-abcdq^{2n+1})}  \\right) \\\\\n  =\\left( \\frac{ 1-q^{n+1} }{q^{n+1} (1-abcdq^{2n+1})} \\right)\\left( -abcdq^{2n+1}+cdq^{n}\\right) = \\text{ as in }  (\\ref{MUQP}) \\text{ above.} \n\\end{multline*}\n\\end{proof}\n\n\\ifJOLT \\begin{Theorem} \\else \\begin{thms} \\fi\n\\label{CDG2COEFFS}\nThe zig-zag co-degree 2  coefficients  of the $E_r(a)$ are given, for $n \\geq 1$  by:\n\\begin{eqnarray*}\n\\lambda_{-n, -(n+1)\\_\\text{q}}= -\\frac{ (c+d)(q^n,abq^{n}| q)_1+ (a+b)(q^n,cdq^{n}| q)_1}{q^n(q,abcdq^{2n}| q)_{1} } \\\\\n\\lambda_{-n,-(n+1)} =  - \\frac{(c+d)(q^n,abq^{n+1}| q)_1 + q(a+b) (q^n,cdq^{n-1}| q)_1 }{(q,abcdq^{2n}| q)_1} \\\\\n \\lambda_{n-1,n} = \\lambda_{n-1,n\\_\\text{q}}\n\n =  - \\frac{ (c+d)(q^n,abq^{n}| q)_1 + q(a+b) (q^n,cdq^{n-1}| q)_1}{(q,abcdq^{2n-1}| q)_1}.\\\\\n\\end{eqnarray*}\n\\ifJOLT \\end{Theorem} \\else \\end{thms} \\fi \n\n\\begin{proof}\nWe prove these $\\lambda_{s,r} \\text{ and } \\lambda_{s,r\\text{\\_q}}$  formulas by induction on zig-zag degree $r.$\n\nNote in the case $r=-1,$  $\\lambda_{-0,-1}$ is not even defined (there is a constant term $\\mu_{0,-1}$ in $E_{-1}$), so\nwe can treat it as $0$ by convention; we are not assuming anything from this proposition in proving the $r=1$ case and \nso view this case as the start of the zig-zag induction.\n \nAssuming correctness for $r=-(n+1),$ we establish the $r=n+1 \\ge 1$ case: (The first case to prove is $s=0,r=1.)$ \n\n\n\\begin{multline}\n\\label{BOTLE2}\n\\lambda_{n,n+1} =\\lambda_{-(n+1),-(n+1)} \\left[ (a+b)\\hat{c}_{n+1} \\right] +\\mu_{n,-(n+1)} \\left[  (-ab)\\hat{c}_{n+1}+ \\hat{d}_{n+1}  \\right]\\\\+\\lambda_{-n,-(n+1)} \\left[ -ab \\hat{c}_{n+1}\\right] \\\\\n= \\left( 1 \\right) \\left[ (a+b) \\left[ - \\frac{1}{ab} \\right]  \\right] + \\left(\\frac{ abq^{n}(c+d) - (a+b)  }{ (abcdq^{2n}| q)_1  }  \\right) \\cdot \\\\\n\\Big\\{\\Big[  (-ab) \\left[ - \\frac{1}{ab}\\right] \\Big] - \\frac{( abq^{n+1} | q )_1+ab(cdq^{n}| q)_1}{ab(1-abcdq^{2n+1})} \\Big\\} \\\\\n+ \\left( - \\frac{(c+d)(q^n,abq^{n+1}| q)_1 + q(a+b) (q^n,cdq^{n-1}| q)_1 }{(q,abcdq^{2n}| q)_1} \\right) \\left[  (-ab) \\left[ - \\frac{1}{ab}\\right] \\right] \n\\end{multline}\nThe top line of the right hand side of the last equals sign combines to\n\\begin{multline}\n\\label{TOPL2}\n\\left( - \\frac{ 1} {ab (abcdq^{2n},abcdq^{2n+1}| q)_1} \\right) \\Big\\{ (a+b) (1-abcdq^{2n})(1-abcdq^{2n+1}) + \\\\\n\\left[ abq^{n}(c+d) - (a+b)\\right] \\big\\{ -ab(1-abcdq^{2n+1}) + (1-abq^{n+1}) +ab(1-cdq^n)\\big\\} \\Big\\}\\\\\n=\\left( - \\frac{ 1} {ab (abcdq^{2n},abcdq^{2n+1}| q)_1} \\right) \\Big\\{ (a+b) (1-abcdq^{2n})(1-abcdq^{2n+1}) + \\\\\n\\left[ abq^{n}(c+d) - (a+b)\\right] \\big\\{ (1-abq^{n+1}) (1-abcdq^{n})\\big\\} \\Big\\}\n\\end{multline}\n\\begin{multline}\n\\label{TOPL2B}\n=\\left( - \\frac{ 1} {ab (abcdq^{2n},abcdq^{2n+1}| q)_1} \\right) \\Big\\{ (a+b) \\big\\{  (1-abcdq^{2n})(1-abcdq^{2n+1}) - \\\\\n(1-abq^{n+1}) (1-abcdq^{n})\\big\\} +(c+d) \\big\\{abq^n (1-abq^{n+1}) (1-abcdq^{n}) \\big\\}.\n\\end{multline}\nFor $n=0$ (the start of the induction), this is all there is for $\\lambda_{0,1},$ and we note  this simplifies to the asserted formula.\nContinuing for $n \\ge1,$\ncombining equation (\\ref{TOPL2B}) with the bottom line of (\\ref{BOTLE2}) gives:\n\\begin{multline}\n\\label{CHUNK3}\n\\left( - \\frac{ 1} {ab (q,abcdq^{2n},abcdq^{2n+1}| q)_1} \\right) \\Big\\{ (a+b) \\big\\{ (1-q) (1-abcdq^{2n})(1-abcdq^{2n+1}) - \\\\\n(1-q)(1-abq^{n+1}) (1-abcdq^{n})+abq(1-abcdq^{2n+1})(q^n,cdq^{n-1}| q)_1) \\big\\} \\\\\n+(c+d)\\big \\{(1-q)abq^n (1-abq^{n+1}) (1-abcdq^{n}) +ab(1-abcdq^{2n+1})  (q^n,abq^{n+1}| q)_1)\\big\\}\\Big\\}.\n\\end{multline}\nWe view (\\ref{CHUNK3}) as \n$$\\left( - \\frac{ 1} {ab (q,abcdq^{2n},abcdq^{2n+1}| q)_1} \\right) \\Big\\{ (a+b)p_1 + (c+d)p_2 \\Big\\} $$\nwhere $p_1$ and $p_2$ are polynomials. We will finish this step in the inductive proof of the $ \\lambda_{n,n+1} $ formula in the theorem (with $n$ replaced by $n+1$) by showing\n$$\np_1=abq(1-abcdq^{2n})\\left[ (q^{n+1},cdq^{n}| q)_1\\right]   \\hspace{5mm} p_2=ab(1-abcdq^{2n})\\left[(q^{n+1},abq^{n+1}| q)_1 \\right].\n$$\nTo verify the $p_2$ claim:\n\\begin{multline*}\np_2=(1-q)abq^n (1-abq^{n+1}) (1-abcdq^{n}) +ab(1-abcdq^{2n+1})  (q^n,abq^{n+1}| q)_1) \\\\\n  = (abq^{n+1}| q)_1\\big\\{ (1-q)abq^n (1-abcdq^{n}) + ab(1-abcdq^{2n+1})  (1-q^n)\\big\\} \\\\\n  \\text{ (because the } abq^n \\text{ and } a^2b^2cdq^{2n+1} \\text{ terms cancel)} \\\\\n  = (abq^{n+1}| q)_1\\big\\{ -abq^{n+1}-a^2b^2cdq^{2n} +ab + a^2b^2cdq^{3n+1}  \\big\\} \\\\\n  =ab(abq^{n+1}| q)_1\\big\\{ (1-q^{n+1})(1-abcd q^{2n}) \\big\\} \n\\end{multline*}\nas asserted.\nThe $p_1$ simplification is a little more involved:\n\\begin{multline*}\np_1=(1-q) (1-abcdq^{2n})(1-abcdq^{2n+1}) -(1-q)(1-abq^{n+1}) (1-abcdq^{n})\\\\\n+abq(1-abcdq^{2n+1})(q^n,cdq^{n-1}| q)_1) \\\\\n=(1-q) (1-abcdq^{2n})(1-abcdq^{2n+1}) -(1-q)(1-abq^{n+1}) (1-abcdq^{2n}+abcdq^{2n}-abcdq^{n}) \\\\\n+abq(1-abcdq^{2n}+abcdq^{2n}-abcdq^{2n+1})(q^n,cdq^{n-1}| q)_1) \n\\end{multline*}\n\\begin{multline*}\n=(1-abcdq^{2n})\\big\\{(1-q) (1-abcdq^{2n+1}) -(1-q) (1-abq^{n+1})+abq(1-q^n)(1-cdq^{n-1}) \\big\\} \\\\\n-(1-q) (1-abq^{n+1}) \\left[ abcdq^n(q^n-1)\\right]+ \\left[ a^2b^2cdq^{2n+1}(1-q) \\right] (1-q^n) (1-cdq^{n-1})\n\\end{multline*}\n\\begin{multline*}\n=(1-abcdq^{2n})\\big\\{(1-q) abq^{n+1}(1-cdq^n)+abq(1-q^n)(1-cdq^{n-1}) \\big\\} \\\\\n+(1-q)  abcdq^n(1-q^n)\\big\\{(1-abq^{n+1}) +  abq^{n+1}(1-cdq^{n-1}) \\big\\}\\\\\n=(1-abcdq^{2n})\\big\\{ (1-q) abq^{n+1}(1-cdq^n) \\\\+abq(1-q^n)(1-cdq^{n-1})+abcdq^n(1-q)(1-q^n) \\big\\} \n\\end{multline*}\n\\begin{multline*}\n=(1-abcdq^{2n})\\big\\{ (1-q) abq^{n+1}(1-cdq^n)+abq(1-q^n)\\big[ (1-cdq^{n-1})+ cdq^{n-1}(1-q)\\big] \\big\\} \\\\\n=(1-abcdq^{2n})(1-cdq^n)\\big\\{(1-q) abq^{n+1}+abq(1-q^n)\\big\\} \\\\\n=abq(1-abcdq^{2n})(1-cdq^n)\\big\\{1-q^{n+1}\\big\\} \n\\end{multline*}\nas claimed. This finishes showing that the the zig-zag degree $-(n+1)$ case implies the asserted $\\lambda_{n,n+1}$ formula.\n\nWe have $\\lambda_{n,n+1} =\\lambda_{n,n+1\\text{\\_q}} $ by proposition \\ref{TOFROMQ} finishing induction starting as well as going from $r=-(n+1)$ to  $r=n+1.$ \n\nAssuming correctness for $r=n \\ge 1,$ we now establish the $r=-(n+1)$ case:\n\nBy proposition \\ref{RCNL}\n\\begin{multline*}\n\\lambda_{-n,-(n+1)\\text{\\_q}} = \\mu_{-n,n\\text{\\_q}}    \\hat{b}_{-(n+1)}+\\lambda_{n-1,n\\text{\\_q}} \\left[ \\left( -\\frac{cd}{q}\\right) \\hat{a}_{-(n+1)} \\right]  \\\\\n=   \\left(  \\frac{ cd(q^{n}, ab q^{n }| q)_1  } {  q(abcdq^{2n-1}| q)_1 } \\right) \\left[ \\frac{(c+d)- cdq^{n }(a+b)}{cdq^{n } (abcdq^{2n}| q)_1}  \\right] \\\\ + \\left(  -\\frac{ (c+d)(q^n,abq^{n}| q)_1 + q(a+b) (q^n,cdq^{n-1}| q)_1}{(q,abcdq^{2n-1}| q)_1}\\right) \\left[ \\left( -\\frac{cd}{q}\\right) \\left( -\\frac{1}{cdq^n} \\right)\\right] \\\\\n=  \\left(  \\frac{ (q^{n}| q)_1  } {  q^{n+1}(q,abcdq^{2n-1},abcdq^{2n}| q)_1 } \\right) \\Big\\{ (1-q)(abq^n| q)_1\\left[(c+d) -cdq^n(a+b) \\right]  \\\\\n-(1-abcdq^{2n}) \\left[ (c+d)(abq^{n}| q)_1 + q(a+b) (cdq^{n-1}| q)_1 \\right] \\Big\\}\n\\end{multline*}\n\\begin{multline*}\n=  \\left(  \\frac{ (q^{n}| q)_1  } {  q^{n+1}(q,abcdq^{2n-1},abcdq^{2n}| q)_1 } \\right) \\Big\\{(c+d) (abq^n| q)_1\\left[ abcdq^{2n}-q  \\right] \\\\\n+(a+b)\\left[ -(1-q)(1-abq^n)(cdq^n)-q(1-abcdq^{2n})(1-cdq^{n-1})\\right] \\Big\\} \\\\\n=  \\left(  \\frac{ (q^{n}| q)_1  } {  q^{n+1}(q,abcdq^{2n-1},abcdq^{2n}| q)_1 } \\right) \\Big\\{(c+d) (abq^n| q)_1\\left[-q(abcdq^{2n-1}| q)_1  \\right] \\\\\n+(a+b)\\left[ abcdq^{2n} +cdq^{n+1} -q -abc^2d^2q^{3n} \\right] \\Big\\} \\\\\n=  \\left(  \\frac{ (q^{n}| q)_1  } {  q^{n+1}(q,abcdq^{2n-1},abcdq^{2n}| q)_1 } \\right) \\Big\\{(c+d) (abq^n| q)_1\\left[-q(abcdq^{2n-1}| q)_1  \\right] \\\\\n+(a+b)\\left[ abcdq^{2n}(cdq^{n}| q)_1-q(cdq^{n}| q)_1 \\right] \\Big\\} \\\\\n=  - \\left(  \\frac{ (q^{n}| q)_1  } {  q^{n}(q,abcdq^{2n}| q)_1 } \\right) \\Big\\{ (c+d) (abq^n| q)_1 +(a+b)(cdq^{n}| q)_1 \\Big\\}  \n\\end{multline*}\nas asserted for the $r=-(n+1)$ case. \n\nNow that we have $\\lambda_{-n,-(n+1)\\text{\\_q}},$\n\\begin{multline*}\n\\lambda_{-n,-(n+1)}=\\mu_{n,-(n+1)\\text{\\_q}}(q^n-1)+\\lambda_{-n,-(n+1)\\text{\\_q}} q^{n} =(q^n-1)\\left( \\frac{ abq^{n}(c+d) - (a+b)  }{ (abcdq^{2n}| q)_1  }  \\right) \\\\\n+ q^n \\left( -\\frac{ (c+d)(q^n,abq^{n}| q)_1+ (a+b)(q^n,cdq^{n}| q)_1}{q^n(q,abcdq^{2n}| q)_{1} }  \\right) \\\\\n=-\\left( \\frac{1-q^n}{(q,abcdq^{2n}| q)_{1} } \\right)\\big\\{ abq^{n}(c+d)(1-q) - (a+b)(1-q) +(c+d)(1-abq^n)  \\\\  +(a+b)(1-cdq^n) \\big\\} \n=-\\left( \\frac{1-q^n}{(q,abcdq^{2n}| q)_{1} } \\right)\\big\\{  (c+d)(1-abq^{n+1})+(a+b)(q-cdq^n) \\big\\}. \n\\end{multline*}\nwhich agrees with the asserted $\\lambda_{-n,-(n+1)}  .$\n\\end{proof}\n\n\n\nThe formulas (\\ref{ETCFRM}) for $T(a,aq)$ involve factors $(ae^{-1}| q)_r.$ When $e=aq,$, these vanish for $r \\ge 2.$ Consequently the decomposition\n$$\nT(a,aq)=\\begin{bmatrix}\nT ^{00}(a,aq) & T^{01}(a,aq) \\\\\nT^{10}(a,aq) & T^{11}(a,aq)\\\\\n\\end{bmatrix}\n$$\nspecializes to\n$$\nT^{00}(a,aq)=\\begin{bmatrix}\n\\tau_{0,0}(a,aq)  & \\tau_{0,1}(a,aq)  &   0 &0\\\\\n0  &  \\tau_{1,1}(a,aq)  &\\tau_{1,2}(a,aq)     &\\ddots \\\\\n 0 &\\ddots   & \\ddots  &\\ddots \\\\\n\\end{bmatrix}\n$$\n$$\n\\ \\ T^{01}(a,aq)=\\begin{bmatrix}\n\\sigma_{0, -1} (a,aq)  & 0 & 0   \\\\\n0  & \\sigma_{1,-2}(a,aq)  & 0  &   \\\\\n0  &\\ddots & \\ddots  \\\\\n\\end{bmatrix}\n$$\n$$\nT^{10}(a,aq)=\n\\begin{bmatrix}\n0  & \\sigma_{-1,1} (a,aq) & 0  &  0 \\\\\n0  & 0  & \\sigma_{-2,2} (a,aq)  &   \\ddots \\\\\n0  & 0 &0  &  \\ddots \\\\\n\\end{bmatrix}\n$$\n$$\nT^{11}(a,aq)=\\begin{bmatrix}\n\\tau_{-1, -1} (a,aq)  & \\tau_{-1,-2} (a,aq) & 0  &  0  \\\\\n0  & \\tau_{-2,-2}(a,aq)  & \\tau_{-2,-3} (a,aq)  &  \\ddots \\\\\n0  & 0 & \\tau_{-3,-3} (a,aq)  &  \\ddots \\\\\n\\end{bmatrix}.\n$$\n\nThis has strong implications for the form of the discrete co-cycle condition (\\ref{COCYCAQ}) as well, which we come back to in section \\ref{COCYCPRF}. \n\n\\section{Combining the $d_{r,s}c_{|r|,|s|}$ Products into the $\\tau \\text{ and } \\sigma$ \\label{TAUSIGMA}}\nThe most combinatorially involved part of our proof of Theorem \\ref{ETC}  involves co-cycle condition verification. \n\nTo facilitate its formulation and the clarity of correctness of our arguments, it is\nuseful to combine the $d_{rs}c_{|r|,|s|}$ expressions into more distinctive and mnemonic expressions. As mentioned \nearlier, equations (\\ref{TAUDEF}) and (\\ref{SIGMADEF})  give the main notation we use, namely \n\\begin{align*}\n\\tau_{r,s}&=&d_{r,s}c_{|r|,|s|} & \\hspace{10mm} & \\text{if } (r \\ge 0 \\text{ and } s \\ge 0)  \\text{ or } (r <  0 \\text{ and } s < 0)\\\\\n\\sigma_{r,s}&=& d_{r,s}c_{|r|,|s|} & \\hspace{10mm} & \\text{if } (r \\ge 0 \\text{ and } s < 0)  \\text{ or } (r <  0 \\text{ and } s \\ge 0). \n\\end{align*}\nIn this way, the sign portion of the $r \\preceq s$ relation leads to the natural $2 \\times 2$ block matrix \nform of the transition matrices. (This is where the $4$ cases of the $d_{rs}$ come from.) Explicitly\nthis means we are using the notation:\n\nFor  $n \\geq 0$\n\\begin{multline}\nE_n(z; a,b,c,d|q) = \\sum_{m=0}^n \\left[ \\tau_{m,n}(a, e ; b, c, d | q) \\right] E_m(z; e,b,c,d|q)\\\\  + \\sum_{m=0}^{n-1} \\left[ \\sigma_{-(m+1),n}(a, e ; b, c, d | q) \\right] E_{-(m+1)}(z; e,b,c,d|q) \\\\\nE_{-(n+1)}(z; a,b,c,d|q) = \\sum_{m=0}^n \\left[ \\tau_{-(m+1),-(n+1)}(a, e ; b, c, d | q) \\right] E_{-(m+1)}(z; e,b,c,d|q) \\\\+ \\sum_{m=0}^n \\left[ \\sigma_{m,-(n+1)}(a, e ; b, c, d | q) \\right] E_{m}(z; e,b,c,d|q) \n\\end{multline}\nwhere for $k,n \\geq 0$\n\\begin{multline}\n\\label{ETCFRM}\n\\tau_{k,n}(a, e ; b, c, d | q) =  \\frac{ (q^{n-k+1}| q)_k (eq)^{n-k}(bc q^k, bd q^k, cd q^k, ae^{-1} | q )_{n-k}   }  { (q| q)_k (abcdq^{n+k}, bcdeq^{2k} | q)_{n-k} } \\\\\n\\sigma_{k,-(n+1)}(a, e ; b, c, d | q) =\\frac{ (q^{n-k+1}| q)_k e^{n+1-k}(bc q^k, bd  q^k, ae^{-1} | q )_{n+1-k}   (cd  q^k| q )_{n-k}   }  { (q| q)_k (abcdq^{n+k} , bcdeq^{2k} | q)_{n+1-k} } \\\\\n\\sigma_{-(k+1),n}(a, e ; b, c, d | q) =   \\frac{(q^{n-k}| q)_{k+1} (bc q^{k + 1}, bd q^{k + 1} | q )_{n-k - 1}   (cd q^{k }, ae^{-1}| q )_{n-k}  }  { (q| q)_k (abcdq^{n+k} , bcdeq^{2k+1} | q)_{n-k} } \\\\\n\\times bcd e^{n-k} q^{n + k} \\\\\n\\tau_{-(k+1),-(n+1)}(a, e ; b, c, d | q) = \\frac{(q^{n-k+1}| q)_k  e^{n-k}(bc q^{k+1}, bd q^{k+1}, cdq^k, ae^{-1}| q )_{n-k}  }  { (q| q)_k  (abcdq^{n+k+1}, bcdeq^{2k+1} | q)_{n-k} }.\n\\end{multline}\n\n\\ifEXTRAPROOFS\n\n\n\n\\begin{center} \\bf (Verification below is of $\\tau,\\sigma$ given previous definitions of $c_{rs},d_{rs}.$) \\end{center}\n\\begin{proof} For $k,n \\ge 0,$\n\\begin{enumerate}\n\\item\n\\begin{multline*}\n\\tau_{k,n}(a, e ; b, c, d | q) = d_{k,n} c_{k,n} = \\left\\{ \\frac{q^{n-k} (abcdq^{n+k-1}| q)_1}{(abcdq^{2n-1}| q)_1}  \\right\\} \\cdot \\\\\n\\left\\{ \\frac{e^{n-k} (q^{n-k+1} | q )_k(bc q^k, bd q^k, cd q^k, ae^{-1}  | q )_{n-k}   }  { (q  | q )_k (abcdq^{n+k-1}, bcdeq^{2k}  | q )_{n-k} }  \\right\\} \\\\\n=  \\frac{ (q^{n-k+1}| q)_k (eq)^{n-k}(bc q^k, bd q^k, cd q^k, ae^{-1} | q )_{n-k}   }  { (q| q)_k (abcdq^{n+k}, bcdeq^{2k} | q)_{n-k} } \n\\end{multline*}\n\\item\n\\begin{multline*}\n\\sigma_{k,-(n+1)}(a, e ; b, c, d | q) = d_{k,-(n+1)} c_{k,n+1} = \\left\\{  \\frac{ (q^{n-k+1 }|q)_{ 1 } } {( q^{n+1}, cdq^{n }|q)_{ 1 }}  \\right\\} \\cdot \\\\\n \\left\\{\\frac{e^{n-k+1} (q^{n-k+2} | q )_k(bc q^k, bd q^k, cd q^k, ae^{-1}  | q )_{n-k+1}   }  { (q  | q )_k (abcdq^{n+k}, bcdeq^{2k}  | q )_{n-k+1} }  \\right\\} \\\\\n\n = \\frac{ (q^{n-k+1}| q)_k e^{n+1-k}(bc q^k, bd  q^k, ae^{-1} | q )_{n+1-k}   (cd  q^k| q )_{n-k}   }  { (q| q)_k (abcdq^{n+k} , bcdeq^{2k} | q)_{n+1-k} } \n\\end{multline*}\n \\item\n \\begin{multline*}\n\\sigma_{-(k+1), n}(a, e ; b, c, d | q) = d_{-(k+1), n} c_{k+1,n} = \\left\\{ \\frac{bcdeq^{n+k} (q^{k+1}, cdq^k, ae^{-1}q^{n-k-1 }|q)_{1 } } { (abcdq^{2n-1}, bcdeq^{2k+1 }|q)_{ 1}} \\right\\} \\\\\n \\left\\{ \\frac{e^{n-k-1} (q^{n-k} | q )_{k+1}(bc q^{k+1}, bd q^{k+1}, cd q^{k+1}, ae^{-1}  | q )_{n-k-1}   }  { (q  | q )_{k+1} (abcdq^{n+k}, bcdeq^{2(k+1)}  | q )_{n-k-1} } \\right\\} \\\\\n= \\frac{(q^{n-k}| q)_{k+1} bcd e^{n-k} q^{n + k}(bc q^{k + 1}, bd q^{k + 1} | q )_{n-k - 1}   (cd q^{k }, ae^{-1}| q )_{n-k}  }  { (q| q)_k (abcdq^{n+k} , bcdeq^{2k+1} | q)_{n-k} } \\\\\n\\end{multline*}\n \\item\n \\begin{multline*}\n\\tau_{-(k+1),-(n+1)}(a, e ; b, c, d | q) = d_{-(k+1),-(n+1)} c_{k+1,n+1} = \\\\ \\left\\{  \\frac{ (q^{k+1}, cdq^{k}, bcdeq^{n+k+1 }|q)_{1 } } { (q^{n+1}, cdq^n, bcdeq^{2k+1 }|q)_{ 1} }  \\right\\} \\\\\n\\left\\{ \\frac{e^{n-k} (q^{n-k+1} | q )_{k+1}(bc q^{k+1}, bd q^{k+1}, cd q^{k+1}, ae^{-1}  | q )_{n-k}   }  { (q  | q )_{k+1} (abcdq^{n+k+1}, bcdeq^{2(k+1)}  | q )_{n-k} }  \\right\\} \\\\\n=  \\frac{(q^{n-k+1}| q)_k  e^{n-k}(bc q^{k+1}, bd q^{k+1}, cdq^k, ae^{-1}| q )_{n-k}  }  { (q| q)_k  (abcdq^{n+k+1}, bcdeq^{2k+1} | q)_{n-k} } \n\\end{multline*}\n\\end{enumerate}\n\\end{proof}\n\\fi\n\n\n\\section{The True $\\boldsymbol{\\mathcal{T}}(a,aq)$ Matches the $\\boldsymbol{T}(a,aq) $ of Theorem \\ref{ETC} \\label{TAQ}}\n\nHere we use the notation \n$$\n\\Delta_{ae} f = f(e) -f(a)\n$$\nfor the change in values of a function  $f$ as one moves from argument $a$ to argument $e.$\n\nSince the $\\{\\lambda_{r,s},\\mu_{r,s}\\}$ are also entries in a change of basis relationship, the true  transition functions $\\boldsymbol{\\mathcal{T}}(a,aq) $ may be expressed\nin terms of them.  However, in the low zig-zag co-degree cases, the transition matrix entries may also be conveniently determined by a \nsuccessive substitution argument.\n\n\\ifJOLT \\begin{Proposition} \\else \\begin{props} \\fi\n\\label{LCDGTR}\n\\begin{eqnarray}\n\\boldsymbol{\\mathcal{T}}^{00}_{n-1, n}(a,e) &=&\\tau_{n-1,n}= -\\Delta_{ae} \\lambda_{n-1,n} +\\left[ \\mu_{n-1,-n}(e) \\right] \\Delta_{ae} \\mu_{-n,n} \\label{DELTA00} \\\\\n \\boldsymbol{\\mathcal{T}}^{01}_{n, n}(a,e) &=&\\sigma_{n,-(n+1)}=-\\Delta_{ae}\\mu_{n,-(n+1)}  \\label{DELTA01}  \\\\\n\\boldsymbol{\\mathcal{T}}^{10}_{n, n+1}(a,e) &=&\\sigma_{-(n+1),n+1}= -\\Delta_{ae}\\mu_{-(n+1),n+1} \\label{DELTA10} \\\\\n\\boldsymbol{\\mathcal{T}}^{11}_{n-1,n}(a,e)&=&\\tau_{-n,-(n+1)} =-\\Delta_{ae} \\lambda_{-n, -(n+1)} +\\left[ \\mu_{-n,n}(e) \\right] \\Delta_{ae} \\mu_{n,-(n+1)} \n\\end{eqnarray}\n\\ifJOLT \\end{Proposition} \\else \\end{props} \\fi\n\\begin{proof}\nWe write down the proof of (\\ref{DELTA00}) and  (\\ref{DELTA10}), the other two being similar.\n\nRecall $\\lambda_{r,r}=1$ for all parameters and any sign of $r.$ Then  modulo $\\mathcal{R}_{-(n-1)}:$\n\\begin{eqnarray*}\nE_{n-1}(e) =\\left[  \\lambda_{n-1,n-1}(e)\\right] f_{n-1} & \\Rightarrow & f_{n-1}=E_{n-1}(e)\\\\\nE_{-n}(e)= \\left[ \\lambda_{-n,-n}(e) \\right] h_n + \\left[ \\mu_{n-1,-n}(e) f_{n-1}(e) \\right] &  \\Rightarrow & \\\\\n h_{n}=E_{-n}(e) - \\left[ \\mu_{n-1,-n}(e) \\right] E_{n-1}(e) \\\\\nf_n = E_n(e) -  \\left[ \\mu_{-n,n}(e) \\right] h_n - \\left[ \\lambda_{n-1,n}(e) \\right] f_{n-1} & \n\\end{eqnarray*}\n$$\\hphantom{f_n}    =E_n(e)   -  \\left[ \\mu_{-n,n}(e) \\right] \\big( E_{-n}(e) - \\left[ \\mu_{n-1,-n}(e) \\right] E_{n-1}(e)  \\big) - \\left[  \\lambda_{n-1,n}(e)\\right] E_{n-1}(e) $$\nSo\n\\begin{multline*}\nE_n(a) = f_n+ \\left[ \\mu_{-n,n}(a) \\right] h_n + \\left[  \\lambda_{n-1,n}(a)\\right] f_{n-1} \\\\\n= E_n(e)   -  \\left[ \\mu_{-n,n}(e) \\right] \\big( E_{-n}(e) - \\left[ \\mu_{n-1,-n}(e) \\right] E_{n-1}(e)  \\big) - \\left[  \\lambda_{n-1,n}(e)\\right] E_{n-1}(e) \\\\\n+ \\left[ \\mu_{-n,n}(a) \\right] \\big( E_{-n}(e) - \\left[ \\mu_{n-1,-n}(e) \\right] E_{n-1}(e) \\big) +  \\left[  \\lambda_{n-1,n}(a)\\right] E_{n-1}(e) \n\\end{multline*}\nCombining terms and comparing coefficients of $E_{-n}(e)$ (for  (\\ref{DELTA10}))  and $E_{n-1}(e)$ (for  (\\ref{DELTA00}))with the definition\n$$\nE_n(a)= \\tau_{n,n}E_n(e) + \\sigma_{-n,n} E_{-n}(e) + \\tau_{n-1,n}E_{n-1}(e) \\ \\ \\  \\text{ mod }\\mathcal{R}_{-(n-1)}\n$$\ngives the asserted formulas.\n\\end{proof}\n\nThe zig-zag co-degree $1$ formulas (\\ref{DELTA01}) and (\\ref{DELTA10}) above have an obvious linearity based on \n$$\n \\Delta_{uw} f= \\Delta_{uv} f + \\Delta_{vw} f \\text{ since } \\left( f(w) - f(u) \\right) = \\left( f(v) - f(u) \\right) + \\left( f(w) - f(v) \\right).\n$$\nThis immediately implies that the zig-zag co-degree $1$ matrix entries of $T$ satisfy what is required by the discrete co-cycle condition. We shall need\nthese special cases in the next section, so we record them in the corollary below.\n\\ifJOLT \\begin{Corollary} \\else \\begin{cors} \\fi\n\\label{CDG1LIN}\nFor any $a,e,\\text{ and } f,$\n\\begin{equation*}\n\\sigma_{n,-(n+1)}(a,e) = \\sigma_{n,-(n+1)}(f,e) +  \\sigma_{n,-(n+1)}(a,f).\n\\end{equation*}\n\\begin{equation*}\n   \\sigma_{-(n+1), n+1}(a,e) =  \\sigma_{-(n+1), n+1}(f,e) +  \\sigma_{-(n+1), n+1}(a,f). \n\\end{equation*}\n\\ifJOLT \\end{Corollary} \\else \\end{cors} \\fi\n\\ifJOLT \\begin{Proposition} \\else \\begin{props} \\fi\n\\label{CDG1AEQ}\n\nIn the special case of $e=aq,$ the zig-zag co-degree $1$ transition functions satisfy: \n\\begin{enumerate}\n\\item $\\displaystyle \\boldsymbol{\\mathcal{T}}^{01}_{n,n}(a,aq) =\\sigma_{n, -(n+1)}(a,aq) = \\frac{  aq (bc q^n, bd  q^n, q^{-1} | q )_{1}   }  {  (abcdq^{2n} | q)_{1} (abcdq^{2n+1} | q)_{1}   }$\n\\item $\\displaystyle \\boldsymbol{\\mathcal{T}}^{10}_{n, n+1}(a,aq) = \\sigma_{-(n+1), n+1}(a,aq) =\n \\frac{abcd    q^{2(n +1)}   (q^{n+1}, cd q^{n }, q^{-1}| q )_{1}  }  {  (abcdq^{2n+1} | q)_{1} (abcdq^{2(n+1)} | q)_{1}   } $\n\\end{enumerate}\n\\ifJOLT \\end{Proposition} \\else \\end{props} \\fi\n\n\\begin{proof}\n\n\\begin{enumerate}\n\\item \nBy proposition \\ref{LCDGTR}\n$$ \\boldsymbol{\\mathcal{T}}^{01}_{n, n}(a,aq) =\\sigma_{n, -(n+1)}(a,aq) =-\\Delta_{ae}\\mu_{n,-(n+1)} $$\nSo\n\\begin{multline*\n\\boldsymbol{\\mathcal{T}}^{01}_{nn}(a,aq) = - \\Big\\{ \\Big\\{ - \\frac{ \\left[ ( \\left[ aq \\right] bq^{n} | q)_1 -1 \\right] (c+d) + ( \\left[ aq \\right] +b)  }{  ( \\left[ aq \\right] bcdq^{2n} | q)_1 } \\Big\\} \\\\\n- \\Big\\{  - \\frac{ \\left[ (abq^{n} | q)_1 -1 \\right] (c+d) + (a+b)  }{  (abcdq^{2n} | q)_1 }\\Big\\} \\Big\\} \\\\%\\hspace{50mm}  \\\\\n=\\Big\\{ \\frac{1}{(abcdq^{2n} | q)_1 (abcdq^{2n+1} | q)_1  } \\Big\\}  \\cdot \\\\\n\\big\\{ (abcdq^{2n} | q)_1\\left[  - abq^{n+1} (c+d)  +(aq+b)\\right]  \\\\- (abcdq^{2n+1} | q)_1\\left[  - abq^{n} (c+d) +(a+b)\\right] \\big\\}\n\\end{multline*\nComparing with the statement of the proposition and keeping in mind that $$ q(q^{-1}| q)_1= -(q| q)_1,$$ \nwe see that we need to verify equality of the numerators, both of which are polynomials; i.e. the vanishing of\n\\begin{multline*\n  (abcdq^{2n} | q)_1\\left[  - abq^{n+1} (c+d) +(aq+b)\\right]\\\\ - (abcdq^{2n+1} | q)_1\\left[  - abq^{n}  (c+d) +(a+b)\\right] \\\\\n   +a(bc q^n| q )_{1}  (bd  q^n| q )_{1}    (q | q )_{1}  = \n\\end{multline*\nIntroducing an additional variable $u$ (intuitively, replacing $q^n$), it\nsuffices to show that the {\\em linear} in $q$  polynomial\n\\begin{multline*\n  (abcdu^{2} | q)_1\\left[  - abuq (c+d) +(aq+b)\\right] - (abcdu^2q | q)_1\\left[  - abu  (c+d) +(a+b)\\right] \\\\\n   +a(bc u| q )_{1}  (bd  u| q )_{1}    (q | q )_{1}  = 0  \n\\end{multline*\nvanishes for all $a,b,c,d,u.$ \n\\begin{enumerate}\n\\item For $q=0,$ we have \n$$ b(abcdu^{2} | q)_1- \\left[  - abu  (c+d) +(a+b)\\right] + a(bc u| q )_{1}  (bd  u| q )_{1}=0$$\n\\item For $q=1,$ we have \n\\begin{multline*}\n(abcdu^{2} | q)_1\\left[  - abu (c+d) +(a+b)\\right] \\\\- (abcdu^2 | q)_1\\left[  - abu  (c+d) +(a+b)\\right] =0\n\\end{multline*}\n\\end{enumerate}\nSo, since the $1$ variable polynomial  in $q$ with coefficients in $\\mathbb{Q}(a,b,c,d,u)$ is at most degree $1$  and vanishes at two distinct values of $q,$ it must be identically\nzero and the required identity has been established.\n\n\\item By proposition \\ref{LCDGTR}\n $$ \\boldsymbol{\\mathcal{T}}^{10}_{n, n+1}(a,aq) = \\sigma_{-(n+1), n+1}(a,aq)= -\\Delta_{ae}\\mu_{-(n+1),n+1} $$\nSo\n\\bmls\n\\boldsymbol{\\mathcal{T}}^{10}_{n,n+1}(a,aq) = - \\Big\\{ \\Big\\{   - \\frac{ (cdq^{n } | q)_1 ( q^{n+1}| q  )_1 } {  ( \\left[ aq \\right] bcdq^{2n+1} | q)_1 } \\Big\\} \n - \\Big\\{ - \\frac{ (cdq^{n} | q)_1 ( q^{n+1}| q  )_1 } {  (abcdq^{2n+1} | q)_1 }  \\Big\\} \\Big\\}\\\\% \\hspace{50mm}  \\\\\n= \\frac{ (cdq^{n} | q)_1 ( q^{n+1}| q  )_1  }{ (abcdq^{2n+1} | q)_1 (abcdq^{2(n+1)} | q)_1 }\\Big\\{(abcdq^{2n+1} | q)_1 -(abcdq^{2(n+1)} | q)_1\\Big\\}\\\\\n= \\frac{ (cdq^{n} | q)_1 ( q^{n+1}| q  )_1 \\big\\{ - abcdq^{2n+1}(q| q)_1 \\big\\} }{ (abcdq^{2n+1} | q)_1 (abcdq^{2(n+1)} | q)_1 }\\\\\n= \\frac{ (cdq^{n} | q)_1 ( q^{n+1}| q  )_1 \\big\\{  abcdq^{2(n+1)}(q^{-1}| q)_1 \\big\\} }{ (abcdq^{2n+1} | q)_1 (abcdq^{2(n+1)} | q)_1 }\n\\end{multline*}\n\n\\end{enumerate}\n\\end{proof}\n\n\\ifJOLT \\begin{Proposition} \\else \\begin{props} \\fi\n\\label{CDG2AEQ}\n\nIn the special case of $e=aq,$ the zig-zag co-degree $2$ transition functions satisfy:\n\\ \n\n\\begin{enumerate}\n\\item $\\displaystyle \\boldsymbol{\\mathcal{T}}^{00}_{n-1, n}(a,aq) =\\tau_{n-1,n}= - \\frac{   aq  (q^n,bc q^{n-1}, bd q^{n-1}, cd q^{n-1}| q )_{1}    }  {\\left[ (abcdq^{2n-1} | q)_{1}  \\right]^2  }$\n\\item $\\displaystyle \\boldsymbol{\\mathcal{T}}^{11}_{n-1,n}(a,aq) =\\tau_{-n,-(n+1)}=  -\\frac{  a(q^n, bc q^{n}, bd q^{n},cdq^{n-1}| q )_{1}  }  {\\left[  (abcdq^{2n} | q)_{1} \\right]^2 }  $\n\\end{enumerate}\n\n\\ifJOLT \\end{Proposition} \\else \\end{props} \\fi\n\n\\begin{proof}\n\\begin{enumerate}\n\\item\nBy proposition \\ref{LCDGTR}\n$$\\boldsymbol{\\mathcal{T}}^{00}_{n-1, n}(a,aq) =-\\Delta_{ae} \\lambda_{n-1,n} +\\left[ \\mu_{n-1,-n}(e) \\right] \\Delta_{ae} \\mu_{-n,n} .$$\nSo\n\\begin{multline*}\n\\boldsymbol{\\mathcal{T}}^{00}_{n-1,n}(a,aq)  = \\\\ - \\Big\\{\\Big\\{-{\\displaystyle \\frac{(q^n| q)_1}{(q| q)_1  (\\left[ aq \\right] bcdq^{2n-1}| q)_1}} \\left[(c+d)(\\left[ aq \\right] bq^{n}| q)_1 + q( \\left[ aq \\right] +b) (cdq^{n-1}| q)_1 \\right] \\Big\\}\\\\\n   - \\Big\\{ -{\\displaystyle \\frac{(q^n| q)_1}{(q| q)_1  (abcdq^{2n-1}| q)_1}} \\left[(c+d)(abq^{n}| q)_1 + q(a+b) (cdq^{n-1}| q)_1 \\right]\\Big\\}\\Big\\} \\\\\n   + \\Big\\{  - \\frac{ \\left[ (\\left[ aq \\right] bq^{n-1} | q)_1 -1 \\right] (c+d) + (  \\left[ aq \\right] +b)  }{  ( \\left[ aq \\right] bcdq^{2(n-1)} | q)_1 } \\Big\\} \\cdot \\Big\\{ \\\\\n   \\Big\\{- \\frac{ (cdq^{n - 1} | q)_1 ( q^n| q  )_1 } {  ( \\left[ aq \\right] bcdq^{2n-1} | q)_1 }  \\Big\\} \n-  \\Big\\{ - \\frac{ (cdq^{n - 1} | q)_1 ( q^n| q  )_1 } {  (abcdq^{2n-1} | q)_1 }  \\Big\\}\\Big\\}  \n\\end{multline*}\n\nUsing the notation $\\big[a,b,c,\\dots\\big]=(a| q)_1(b| q)_1(c| q)_1\\ldots,$ \nthe abbreviation variables $u=q^{n-1},y=abcd$\nand multiplying by \n$$\n \\frac{ (q,q)_1 \\left[ ( yq^{2n-1} | q)_1 \\right]^2 ( yq^{2n} | q)_1 } {(q^n| q)_1}=\\frac{ \\left[ q,yu^2q , yu^2q , yu^2q^{2}\\right] } {(uq| q)_1}\n$$ \nit suffices to show the vanishing of \n\\begin{multline*} \np_1=\\Big\\{aq \\big[q,  yu^2q^{2}, bcu, bdu, cdu \\big]    \\Big\\} \\\\\n+ \\Big\\{ \\left[ ( yu^2q| q)_1\\right]^2  \\left[(c+d)(a buq^{2}| q)_1 + q(  aq  +b) (cd u| q)_1 \\right] \\Big\\}\\\\\n- \\Big\\{   ( yu^2q | q)_1  ( yu^2q^{2} | q)_1 \\left[(c+d)(abuq| q)_1 + q(a+b) (cdu| q)_1 \\right] \\Big\\}  \\\\ \n+\\Big\\{   (q,q)_1 ( yu^2q | q)_1 \\Big[ \\left[ (abuq | q)_1 -1 \\right] (c+d) + (   aq  +b) \\Big]  (cdu | q)_1\\Big\\} \\\\\n -  \\Big\\{   (q,q)_1( yu^2q^{2} | q)_1 \\Big[ \\left[ ( abuq | q)_1 -1 \\right] (c+d) + (   aq  +b) \\Big] (cdu | q)_1 \\Big\\} \\\\\n=\\Big\\{aq \\big[q,  yu^2q^{2}, bcu, bdu, cdu \\big]    \\Big\\} \\\\\n+ ( yu^2q| q)_1 \\Big\\{(c+d) \\big\\{ ( yu^2q| q)_1 (abuq^2|q)_1 - ( yu^2q^2| q)_1 (abuq|q)_1 \\big\\}\\\\\n                                 +(cdu|q)_1 \\Big\\{ aq \\big\\{ ( yu^2q | q)_1  (q) -  ( yu^2q^{2} | q)_1 (1)  \\big\\} + bq  \\big\\{ ( yu^2q | q)_1 -  ( yu^2q^{2} | q)_1 \\big\\} \\Big\\} \\Big\\} \\\\                                 \n +  \\left[ q,cdu \\right] \\big\\{  \\left( - abuq  \\right) (c+d) + (   aq  +b)    \\big\\} \\big\\{ ( yu^2q | q)_1 -  ( yu^2q^{2} | q)_1 \\big\\}                                \n\\end{multline*}\n\nNote (keeping in mind, e.g., $(yu^2q)(q) - (yu^2q^2)(1) =0$) that\n\\begin{eqnarray*}\n( yu^2q | q)_1  (q) -  ( yu^2q^{2} | q)_1 (1)  = q-1  = -  (q| q)_1 \\\\\n( yu^2q | q)_1 -  ( yu^2q^{2} | q)_1=- yu^2q (q| q)_1 \n\\end{eqnarray*}\n\\begin{multline}\n( yu^2q | q)_1 (ab uq^2 | q)_1  - ( yu^2q^2 | q)_1  (ab u q | q)_1 \\\\\n= -yu^2q -abuq^2 +yu^2q^2 +ab u q \\\\\n=uq(q|q)_1 (ab -yu) \\\\\n= uq(q|q)_1(ab -abcdu) \\\\ \n= ab u q (q|q)_1 (cdu | q)_1 \n\\end{multline}\n\nSo $p_1$ is also equal to\n\\begin{multline*}\np_2=\\Big\\{aq \\big[q,  yu^2q^{2}, bcu, bdu, cdu \\big]    \\Big\\} \\\\\n+ ( yu^2q| q)_1 \\Big\\{(c+d) \\big\\{ ab u q (q|q)_1 (cdu | q)_1  \\big\\}\\\\\n                                 +(cdu|q)_1 \\Big\\{ aq \\big\\{  -  (q| q)_1 \\big\\} + bq  \\big\\{ - yu^2q (q| q)_1 \\big\\} \\Big\\} \\Big\\} \\\\\n +  \\left[ q,cdu \\right] \\big\\{  \\left( - abuq  \\right) (c+d) + (   aq  +b)    \\big\\} \\big\\{ - yu^2q (q| q)_1 \\big\\} \\\\\n\n\\end{multline*}\nMultiplying $p_2$ by $\\left[ q (q|q)_1\\  (cdu | q)_1 \\right]^{-1}$\nwe are reduced to showing the vanishing of\n\\begin{multline*}\np_3=\\Big\\{a \\big[  yu^2q^{2}, bcu, bdu \\big]    \\Big\\} \n+ ( yu^2q| q)_1 \\Big\\{(c+d) \\big\\{ ab u     \\big\\}\\\\\n                                 +\\Big\\{ a \\big\\{  -1   \\big\\} + b  \\big\\{ - yu^2q \\big\\} \\Big\\} \\Big\\} \n +  \\big\\{  \\left( - abuq  \\right) (c+d) + (   aq  +b)    \\big\\} \\big\\{ - yu^2 (q| q)_1 \\big\\} \\\\\n\n = a \\big[  abcdu^2q^{2}, bcu, bdu \\big]    \n+ a( abcdu^2q| q)_1 \\Big\\{b u(c+d)\n                                 -\\left( 1  +    b^2cdu^2q \\right)  \\Big\\} \\\\\n  - abcdu^2 (q| q)_1  \\big\\{  \\left( - abuq  \\right) (c+d) + (   aq  +b)    \\big\\}  \\\\\n\n = a \\big[  abcdu^2q^{2}, bcu, bdu \\big]    \n\n+ a( abcdu^2q| q)_1 \\Big\\{-(1-bcu)(1-bdu)\n                                +b^2cdu^2(1-q)   \\Big\\} \\\\\n - abcdu^2 (q| q)_1  \\big\\{ aq(1-bcu)(1-bdu)  +b(1-abcdu^2q)    \\big\\}  \\\\                                \n\n = a \\big[ bcu, bdu \\big]  \\big\\{( abcdu^2q^2| q)_1 - ( abcdu^2q| q)_1 - abcdu^2q (q| q)_1\\big\\}  \\\\\n + ab^2cdu^2 \\left[ q, abcdu^2q \\right] \\big\\{1  - 1 \\big\\}\\\\\n\n= a \\big[ bcu, bdu \\big]  \\big\\{(  abcdu^2q  (q| q)_1  - abcdu^2q (q| q)_1\\big\\} =0,\n\\end{multline*} \n thus proving the first formula.\n\\item By proposition \\ref{LCDGTR}\n$$\n\\boldsymbol{\\mathcal{T}}^{11}_{n-1,n}(a,aq) =\\\\ -\\Delta_{ae} \\lambda_{-n, -(n+1)} +\\left[ \\mu_{-n,n}(e) \\right] \\Delta_{ae} \\mu_{n,-(n+1)} .\n$$\nSo  \n\\begin{multline*}\n\\boldsymbol{\\mathcal{T}}^{11}_{n-1,n}(a,aq) =\\\\- \\Big\\{ \\Big\\{   -{\\displaystyle \\frac{(q^n| q)_1}{(q| q)_1 ( \\left[ aq \\right] bcdq^{2n}| q)_1}} \\left[(c+d)( \\left[ aq \\right] bq^{n+1}| q)_1 + q( \\left[ aq \\right] +b) (cdq^{n-1}| q)_1 \\right] \\Big\\}  \\\\\n - \\Big\\{  -{\\displaystyle \\frac{(q^n| q)_1}{(q| q)_1 (abcdq^{2n}| q)_1}} \\left[(c+d)(abq^{n+1}| q)_1 + q(a+b) (cdq^{n-1}| q)_1 \\right]   \\Big\\} \\Big\\}   \n\\end{multline*}\n \\begin{multline*}\n\\hphantom{\\boldsymbol{\\mathcal{T}}^{11}_{n-1,n}(a,aq) =} \n+\\Big\\{  - \\frac{ (cdq^{n - 1} | q)_1 ( q^n| q  )_1 } {  ( \\left[ aq \\right] bcdq^{2n-1} | q)_1 }\\Big\\} \\cdot  \\Big\\{   \\\\\n \\Big\\{   - \\frac{ \\left[ ( \\left[ aq \\right] bq^{n} | q)_1 -1 \\right] (c+d) + ( \\left[ aq \\right] +b)  }{  ( \\left[ aq \\right] bcdq^{2n} | q)_1 }\\Big\\}  \\\\\n -   \\Big\\{ - \\frac{ \\left[ (abq^{n} | q)_1 -1 \\right] (c+d) + (a+b)  }{  (abcdq^{2n} | q)_1 } \\Big\\}\\Big\\}  \n\\end{multline*}\n\nUsing the notation $\\big[a,b,c,\\dots\\big]=(a| q)_1(b| q)_1(c| q)_1\\ldots,$ \nthe abbreviation variables $u=q^{n},y=abcd$\nand multiplying by \n$$\n \\frac{ (q,q)_1  \\left[ (yu^2| q)_1 \\right]^2 (yu^2q| q)_1 } {(u| q)_1  }=\\frac{\\left[ q,yu^2, yu^2, yu^2q \\right] } { (u| q)_1},\n$$\n\nit suffices to show the vanishing of\n\\begin{multline*} \np_1=\\Big\\{ a\\left[ q, yu^2q ,bc u ,bd u \\right] (cd q^{n-1}| q )_{1}    \\Big\\} \\\\\n+ \\Big\\{ \\left[ (yu^{2},yu^2 \\right]   \\left\\{ (c+d) (a buq^{2}| q)_1 + q(aq +b) (cd q^{n-1}| q )_{1} \\right\\} \\Big\\} \\\\\n- \\Big\\{   \\left[ yu^2,yu^2q \\right] \\left\\{(c+d)(abuq| q)_1 + q(a+b)(cd q^{n-1}| q )_{1}  \\right\\}\\Big\\}  \\\\\n+ \\Big\\{  \\left[ q, yu^2\\right](cd q^{n-1}| q )_{1}\\Big\\{    -a buq  (c+d) +  aq +b)\\Big\\} \\\\\n -  \\left[ q, yu^2q \\right] (cd q^{n-1}| q )_{1}\\Big\\{   - abu  (c+d) + (a+b)  \\Big\\}  \\Big\\} \\\\\n =\\Big\\{ a\\left[ q, yu^2q ,bc u ,bd u \\right] (cd q^{n-1}| q )_{1}    \\Big\\} \\\\\n+ (yu^2 | q)_1 \\Big\\{ (c+d) \\big\\{ (yu^2 | q)_1 (a buq^{2}| q)_1 - (yu^2 q | q)_1 (a buq | q)_1 \\big\\} \\\\\n + (cd q^{n-1} | q)_1  \\Big\\{ aq \\big\\{ (yu^2 | q)_1 (q) -(yu^2 q | q)_1 (1)\\big\\} \\\\ \n +bq  \\big\\{ (yu^2 | q)_1  -(yu^2 q | q)_1 \\big\\}  \\Big\\} \\Big\\}\\\\\n  +\\left[ q, cd q^{n-1} \\right] \\Big\\{ a(1-bu(c+d)) \\big\\{ (yu^2 | q)_1 (q) -(yu^2 q | q)_1 (1)  \\big\\} \\\\ \n  +b  \\big\\{ (yu^2 | q)_1  -(yu^2 q | q)_1 \\big\\}   \\Big\\}\n\\end{multline*}\n\nNote (keeping in mind, e.g., $(yu^2)(q) - (yu^2q)(1) =0$) that\n\\begin{eqnarray*}\n( yu^2 | q)_1  (q) -  ( yu^2q | q)_1 (1)  = q-1  = -  (q| q)_1 \\\\\n( yu^2 | q)_1 -  ( yu^2q | q)_1=- yu^2 (q| q)_1\n\\end{eqnarray*}\n\\begin{multline}\n( yu^2 | q)_1 (ab uq^2 | q)_1  - ( yu^2q | q)_1  (ab u q| q)_1 \\\\\n= -yu^2 -abuq^2 +yu^2q +abuq \\\\\n=u(q|q)_1 (abq -yu) \n=abu (q|q)_1 (q-cdu) \\\\\n=abuq (q|q)_1 (1-cdq^{n-1}) \n=abuq (q|q)_1 (cdq^{n-1} | q)_1 \n\\end{multline}\n\nSo $p_1$ is also equal to\n\\begin{multline*}\np_2=\\Big\\{ a\\left[ q, yu^2q ,bc u ,bd u \\right] (cd q^{n-1}| q )_{1}    \\Big\\} \\\\\n+ (yu^2 | q)_1 \\Big\\{ (c+d) \\big\\{abu q (q|q)_1 (cduq^{n-1}|q)_1 \\big\\} \\\\\n + (cd q^{n-1} | q)_1  \\Big\\{ aq \\big\\{-  (q| q)_1   \\big\\} \n\n +bq  \\big\\{ - yu^2 (q| q)_1  \\big\\}  \\Big\\} \\Big\\}\\\\\n  +\\left[ q, cd q^{n-1} \\right] \\Big\\{ a(1-bu(c+d)) \\big\\{ -  (q| q)_1   \\big\\}  \n \n +b  \\big\\{ - yu^2 (q| q)_1  \\big\\}   \\Big\\} \n\\end{multline*}\n\n\nSo, upon multiplying by $\\left[ (q| q)_1(cdq^{n-1}| q)_1 \\right]^{-1}$ \nwe are reduced to showing the vanishing of\n\\begin{multline*}\np_3=\\Big\\{ a\\left[ yu^2q ,bc u ,bd u \\right]    \\Big\\} \n+ (yu^2 | q)_1 \\Big\\{ (c+d) \\big\\{abu q \\big\\} \\\\\n + \\Big\\{ aq \\big\\{- 1   \\big\\}  \n +bq  \\big\\{ - yu^2 \\big\\}  \\Big\\} \\Big\\}\n  +(q| q)_1\\Big\\{ a(1-bu(c+d)) \\big\\{ -  1   \\big\\}  \n +b  \\big\\{ - yu^2   \\big\\}   \\Big\\} \\\\\n =  a\\left[ abcdu^2q ,bc u ,bd u \\right]     \n+ q (abcdu^2 | q)_1 \\Big\\{ abu (c+d) \n  -a (1+b^2cdu^2)  \\Big\\} \\\\\n  +a(q| q)_1\\Big\\{- (1-bu(c+d)) \n   - b^2cdu^2     \\Big\\} \\\\\n=  a\\left[ abcdu^2q ,bc u ,bd u \\right]     \n- aq \\left[ abcdu^2, bcu, bdu \\right] \\\\\n  - a(q| q)_1 \\left[ bcu, bdu \\right] \\\\  \n=  a \\left[ bc u ,bd u \\right]  \\big\\{   (abcdu^2q |q)_1 - q (abcdu^2 |q)_1 - (q|q)_1 \\big\\} \\\\\n=  a \\left[ bc u ,bd u \\right]  \\big\\{ (q | q )_1 -(q|q)_1 \\big\\} =0 .\n\\end{multline*}\nThus the proposition proof is complete.\n\\end{enumerate}\n\n\\end{proof}\n\nPropositions \\ref{CDG1AEQ} and \\ref{CDG2AEQ} complete the proof of the second step of (\\ref{PLANA}), Proof Plan A.\n\n\\section{Proof of the Discrete Co-cycle Identity for $\\boldsymbol{T}$  \\label{COCYCPRF}}\n\nThe block form of the equation \n$$T(aq^p,aq^{p+1}) T(a,aq^p) = T(a,aq^{p+1})$$\n\nis\n\\begin{multline*}\n\\begin{bmatrix}\nT ^{00} (aq^p,aq^{p+1}& T^{01}(aq^p,aq^{p+1} \\\\\nT^{10}(aq^p,aq^{p+1} & T^{11}(aq^p,aq^{p+1} \\\\\n\\end{bmatrix}\n\\begin{bmatrix}\nT ^{00}(a,aq^p) & T^{01}(a,aq^p) \\\\\nT^{10}(a,aq^p) & T^{11}(a,aq^p) \\\\\n\\end{bmatrix} \\\\\n=\n\\begin{bmatrix}\nT ^{00}(a,aq^{p+1}) & T^{01}(a,aq^{p+1}) \\\\\nT^{10}(a,aq^{p+1}) & T^{11}(a,aq^{p+1}) \\\\\n\\end{bmatrix}\n\\end{multline*}\nSo we have the four matrix equations:\n\\begin{eqnarray*}\nT ^{00}(a,aq^{p+1})  =  \\Big[ T ^{00} (aq^p,aq^{p+1}) \\Big] \\Big[T^{00}(a,aq^p) \\Big] &  +  & \\Big[T ^{01} (aq^p,aq^{p+1}) \\Big] \\Big[T ^{10}(a,aq^p) \\Big] \\\\\nT ^{01}(a,aq^{p+1})  =  \\Big[T ^{00} (aq^p,aq^{p+1}) \\Big] \\Big[T^{01}(a,aq^p) \\Big] & +  & \\Big[T ^{01} (aq^p,aq^{p+1})  \\Big]\\Big[T ^{11}(a,aq^p) \\Big] \\\\\nT ^{10}(a,aq^{p+1})  =  \\Big[T ^{10} (aq^p,aq^{p+1}) \\Big] \\Big[T^{00}(a,aq^p) \\Big] &+  & \\Big[T ^{11} (aq^p,aq^{p+1}) \\Big] \\Big[T ^{10}(a,aq^p) \\Big]\\\\\nT ^{11}(a,aq^{p+1})  =  \\Big[T ^{10} (aq^p,aq^{p+1}) \\Big]\\Big[T^{01}(a,aq^p) \\Big] & +&  \\Big[T ^{11} (aq^p,aq^{p+1} ) \\Big] \\Big[T ^{11}(a,aq^p) \\Big]\n\\end{eqnarray*}\nKeep in mind, as described \nat then end of section \\ref{ALMSYM},\nthat the entries of each $T^{ij}(aq^p,aq^{p+1})$ are zero except (possibly) on the diagonal and the superdiagonal.\n\nIn terms of $\\tau \\text{ and } \\sigma,$ these equations are:\n\\begin{multline}\n\\label{COCYC00DSP}\n\\tau_{k,n}(a,aq^{p+1}) = \\big[ \\tau_{k,k}(aq^p,aq^{p+1})\\big] \\big[ \\tau_{k,n}(a,aq^p) \\big]+ \\big[ \\tau_{k,k+1}(aq^p,aq^{p+1})\\big] \\cdot \\big\\{ \\\\ \\big[ \\tau_{k+1, n}(a,aq^p)\\big] \\big\\}\n   +\\big[ \\sigma_{k,-(k+1)}(aq^p,aq^{p+1})\\big] \\big[ \\sigma_{-(k+1),n}(a,aq^p) \\big] \n   \\end{multline}  \n   \\begin{multline}\n   \\label{COCYC01DSP}\n\\sigma_{k,-(n+1)}(a,aq^{p+1}) = \\big[ \\tau_{k,k}(aq^p,aq^{p+1})\\big] \\big[ \\sigma_{k,-(n+1)}(a,aq^p) \\big]+ \\big[ \\tau_{k,k+1}(aq^p,aq^{p+1})\\big] \\cdot \\big\\{ \\\\\\big[ \\sigma_{k+1, -(n+1))}(a,aq^p)\\big] \\big\\}\n    +\\big[ \\sigma_{k,-(k+1)}(aq^p,aq^{p+1})\\big] \\big[ \\tau_{-(k+1),-(n+1)}(a,aq^p) \\big]\n       \\end{multline}  \n   \\begin{multline}\n   \\label{COCYC10DSP}\n  \\sigma_{-(k+1), n}(a,aq^{p+1}) = \n  \\big[ \\sigma_{-(k+1),k+1}(aq^p,aq^{p+1})\\big] \\big[ \\tau_{k+1, n}(a,aq^p)\\big] \n  \\\\  +\\big[ \\tau_{-(k+1),-(k+1)}(aq^p,aq^{p+1})\\big]\\cdot \\big\\{\n     \\big[ \\sigma_{-(k+1),n}(a,aq^p) \\big] \\big\\} \\\\\n    + \\big[ \\tau_{-(k+1),-(k+2)}(aq^p,aq^{p+1})\\big] \\big[ \\sigma_{-(k+2), n}(a,aq^p)\\big]  \n       \\end{multline}  \n   \\begin{multline}\n   \\label{COCYC11DSP}\n \\tau_{-(k+1),-(n+1)}(a,aq^{p+1}) = \n    \\big[ \\sigma_{-(k+1),k+1}(aq^p,aq^{p+1})\\big] \\big[ \\sigma_{k+1, -(n+1)}(a,aq^p)\\big] \\\\\n  + \\big[ \\tau_{-(k+1),-(k+1)}(aq^p,aq^{p+1})\\big] \n  \\big[ \\tau_{-(k+1),-(n+1)}(a,aq^p) \\big] \\\\\n  + \\big[ \\tau_{-(k+1),-(k+2)}(aq^p,aq^{p+1})\\big] \\big[ \\tau_{-(k+2), -(n+1)}(a,aq^p)\\big] \n  \\end{multline}\nIn the proofs of these identites, we will often reduce them to the vanishing of a polynomial. To further that, the following easily proven identities will often be used:\n  \n  \\ifJOLT \\begin{Lemma} \\else \\begin{lemmas} \\fi\n\\label{POCHNEG}\n\n\\ \n\n\\begin{eqnarray*}\nq^d(q^{e}| q)_1 &=& (q^{d+e}| q)_1-(q^d| q)_1 \\\\\nq^d(q^{f-g}| q)_1 &= &(q^{d+f-g}| q)_1-(q^d| q)_1 \\\\\n(q^{f-g}| q)_1 &=& q^{-g}\\left\\{ (q^{f}| q)_1-(q^g| q)_1\\right\\} \\\\\n(q^{-p}| q)_1 &=& - q^{-p}(q^p| q)_1 \\\\\n\\end{eqnarray*}\n\\ifJOLT \\end{Lemma} \\else \\end{lemmas} \\fi\n\n\n\n\n\nIt is feasible to directly check the identities (\\ref{COCYC00DSP}),(\\ref{COCYC01DSP}),(\\ref{COCYC10DSP}) and (\\ref{COCYC11DSP})\nsince there is a great deal of cancellation. However the readability of the quantities involved is enhanced by formulating some simplification lemmas\nfor ratios which may be interpreted as appearing in the identities.\n\nIt may be helpful motivationally to note that in both Lemmas   \\ref{URT} and  \\ref{VRT}, the first index of the transition coefficient in the denominator is always the\nzig-zag successor of that of the numerator.  And in Lemma \\ref{URT}, there is a further difference in the powers of $q$ between numerator and denominator; $aq^{p+1}$ in the numerator vs. $aq^{p}$ in the denominator.\n\n\\ifJOLT \\begin{Lemma} \\else \\begin{lemmas} \\fi For $k,n \\ge 0$\n\\label{URT}\n\\ \n\n\\begin{enumerate}\n\\item $\\displaystyle \\frac{ \\tau_{k,n}(a,aq^{p+1}) } { \\sigma_{-(k+1),n}(a,aq^{p}) }=\\frac{q^{n-k}(bcq^{k}, bdq^{k} , q^{-(p+1)}|q)_1 } {bcdq^{2k} (q^{n-k}, q^{n-k-p-1}|q)_1  }. $\n\n\\item $\\displaystyle  \\frac{ \\sigma_{k,-(n+1)}(a,aq^{p+1}) } { \\tau_{-(k+1),-(n+1)}(a,aq^{p}) } = \\frac{aq^{n-k+p+1} (bcq^k, bdq^k, q^{-(p+1)}|q)_1} { \n(abcdq^{n+k}, abcdq^{n+k+p+1} |q)_1}.  $\n\n\\item $\\displaystyle  \\frac{ \\sigma_{-(k+1),n}(a,aq^{p+1}) } { \\tau_{k+1,n}(a,aq^{p}) } = \\frac{abcd q^{n+k+p+1} (q^{k+1}, cdq^{k}, q^{-(p+1)}|q)_1} { (abcdq^{n+k}, abcdq^{n+k+p+1} |q)_1}.$\n\n\\item $\\displaystyle \\frac{ \\tau_{-(k+1),-(n+1)}(a,aq^{p+1}) } { \\sigma_{k+1,-(n+1)}(a,aq^{p}) } = \\frac{q^{n-k} (q^{k+1}, cdq^k, q^{-(p+1)}|q)_1} {(q^{n-k},q^{n-k-p-1}|q)_1 }. $ \n\\end{enumerate}\n\\ifJOLT \\end{Lemma} \\else \\end{lemmas} \\fi\n\n\\ifEXTRAPROOFS\n\\begin{proof}\n\\begin{enumerate}\n\\item\n\\begin{multline*}\n\\Big( \\left\\{ (q^{n-k+1}| q)_k [aq^{p+1}]^{n-k}q^{n-k}(bc q^k | q )_{n-k} \\right\\}  \\cdot \\\\\n\\left. \\left\\{\\frac{  (bd q^k| q )_{n-k}   (cd q^k| q )_{n-k}   {\\displaystyle  (q^{-(p+1)}  | q )_{n-k}  }  }  { (q| q)_k (abcdq^{n+k} | q)_{n-k}  (bcd[aq^{p+1}]q^{2k} | q)_{n-k} }\\right\\} \\right) \\cdot \\\\\n\\Big( \\Big\\{\n\\frac{1}\n{ (q^{n-k}| q)_{k+1} bcd [aq^p]^{n-k} q^{n + k}(bc q^{k + 1}| q )_{n-k - 1}} \\Big\\}  \\cdot \\\\\n\\Big\\{ \\left. \\frac\n{ (q| q)_k (abcdq^{n+k} | q)_{n-k}  (bcd[aq^p]q^{2k+1} | q)_{n-k} }\n{  (bd q^{k + 1} | q )_{n-k - 1}   (cd q^{k }| q )_{n-k}  {\\displaystyle   (q^{-p} | q )_{n-k}  } }  \n\\Big\\} \\right) \\\\\n= \\frac{q^{n-k}(bcq^{k}, bdq^{k} , q^{-(p+1)}|q)_1 } {bcdq^{2k} (q^{n-k}, q^{n-k-p-1}|q)_1  }.  \\\\\n\\end{multline*}\n\\item\n\\begin{multline*}\n\\Big( \n\\left\\{(q^{n-k+1}| q)_k  \\left[ aq^{p+1} \\right]^{n+1-k}(bc q^k| q )_{n+1-k}  \\right\\} \\cdot \\\\\n\\Big\\{ \\frac{  (bd  q^k| q )_{n+1-k}   (cd  q^k| q )_{n-k}   {(q^{-(p+1)} | q )_{n +1- k} }  }  { (q| q)_k (abcdq^{n+k} | q)_{n+1-k}  (bcd \\left[ aq^{p+1} \\right] q^{2k} | q)_{n+1-k} } \\Big\\}\n\\Big) \\\\\n\\Big(  \n\\left\\{\n\\frac{1}{(q^{n-k+1}| q)_k  \\left[ aq^{p} \\right] ^{n-k}} \n \\right\\} \\cdot \\\\\n\\Big\\{ \\frac\n{ (q| q)_k (abcdq^{n+k+1} | q)_{n-k}  (bcd \\left[ aq^{p} \\right] q^{2k+1} | q)_{n-k} }  \n{ (bc q^{k+1} | q )_{n-k}  (bd q^{k+1}| q )_{n-k}   (cd q^k| q )_{n-k}   {\\displaystyle  (q^{-p} | q )_{n-k}  }  }  \n\\Big\\}\n\\Big) \\\\\n= \\frac{aq^{n-k+p+1} (bcq^k, bdq^k, q^{-(p+1)}|q)_1} { \n(abcdq^{n+k}, abcdq^{n+k+p+1} |q)_1}. \\\\\n\\end{multline*}\n\\item\n\\begin{multline*}\n\\Big( \n\\left\\{ (q^{n-k}| q)_{k+1} bcd \\left[ aq^{p+1} \\right]^{n-k} q^{n + k}(bc q^{k + 1}| q )_{n-k - 1}  \\right\\} \\cdot \\\\\n\\Big\\{ \\frac\n{ (bd q^{k + 1} | q )_{n-k - 1}   (cd q^{k }| q )_{n-k}  {\\displaystyle   (q^{-(p+1)} | q )_{n-k}  } }  { (q| q)_k (abcdq^{n+k} | q)_{n-k}  (bcd\\left[ aq^{p+1} \\right] q^{2k+1} | q)_{n-k} }  \n\\Big\\} \\Big) \\\\\n\\Big( \n\\left\\{ \\frac{1}{ (q^{n-k}| q)_{k+1}  \\left[ aq^{p} \\right] ^{n-k-1}q^{n-k-1} } \\right\\} \\cdot \\\\\n\\Big\\{ \\frac\n{ (q| q)_{k+1} (abcdq^{n+k+1} | q)_{n-k-1}  (bcd \\left[ aq^{p} \\right] q^{2(k+1)} | q)_{n-k-1} }\n{(bc q^{k+1} | q )_{n-k-1}  (bd q^{k+1}| q )_{n-k-1}    (cd q^{k+1}| q )_{n-k-1}   {\\displaystyle  (q^{-p}| q )_{n-k-1}  }  }  \\Big\\}\n\\Big) \\\\\n= \\frac{abcd q^{n+k+p+1} (q^{k+1}, cdq^{k}, q^{-(p+1)}|q)_1} { (abcdq^{n+k}, abcdq^{n+k+p+1} |q)_1}. \\\\\n\\end{multline*}\n\\item\n\\begin{multline*}\n\\Big(  \n\\left\\{ (q^{n-k+1}| q)_k  \\left[ aq^{p+1} \\right]^{n-k} \\right\\} \\cdot \\\\\n\\Big\\{ \\frac{(bc q^{k+1}| q )_{n-k}  (bd q^{k+1}| q )_{n-k}   (cdq^k | q )_{n-k}  {\\displaystyle   (q^{-(p+1)} | q )_{n-k}  } }  { (q| q)_k  (abcdq^{n+k+1} | q)_{n-k}  (bcd\\left[ aq^{p+1} \\right] q^{2k+1} | q)_{n-k} } \n\\Big\\} \\Big) \\\\\n\\Big( \n\\Big\\{\n\\frac{1}{ (q^{n-k}| q)_{k+1}  \\left[ aq^{p} \\right] ^{n-k}}\n\\Big\\} \\cdot \\\\\n\\Big\\{ \\frac\n  { (q| q)_{k+1} (abcdq^{n+k+1} | q)_{n-k}  (bcd \\left[ aq^{p} \\right] q^{2(k+1)} | q)_{n-k} }\n{ (bc q^{k+1}| q )_{n-k}  (bd  q^{k+1}| q )_{n-k}   (cd  q^{k+1}| q )_{n-k-1}   \n  {\\displaystyle  (q^{-p} | q )_{n - k} }  }  \n  \\Big\\}\n\n  \\Big) \\\\\n= \\frac{q^{n-k} (q^{k+1}, cdq^k, q^{-(p+1)}|q)_1} {(q^{n-k},q^{n-k-p-1}|q)_1 }.\\\\\n\\end{multline*}\n\\end{enumerate}\n\n\\end{proof}\n\\fi\n\n\n\n\\ifJOLT \\begin{Lemma} \\else \\begin{lemmas} \\fi  For $k,n \\ge 0:$\n\\label{VRT}\n\\ \n\n\\begin{enumerate}\n\\item $\\displaystyle \\frac{ \\tau_{k,n}(a,aq^{p}) } { \\sigma_{-(k+1),n}(a,aq^{p}) } = \\frac{(bcq^k,bdq^k, abcdq^{n+k+p} |q)_1} {bcd q^{2k} (q^{n-k}, abcdq^{2k+p} |q)_1}.$\n\n\\item $\\displaystyle \\frac{ \\sigma_{k,-(n+1)}(a,aq^{p}) } { \\tau_{-(k+1),-(n+1)}(a,aq^{p}) } =\\frac{aq^p(bcq^k,bdq^k, q^{n-k-p}|q)_1 } {(abcdq^{n+k},abcdq^{2k+p }|q)_1 }.\n$\n\n\\item  $\\displaystyle \\frac{ \\sigma_{-(k+1),n}(a,aq^{p}) } { \\tau_{k+1,n}(a,aq^{p}) } = \\frac{abcdq^{2k+p+1}(q^{k+1},q^{n-k-p-1},cdq^k|q)_1 } {(abcdq^{n+k},abcdq^{2k+p+1 }|q)_1 }.$\n\n\n\\item $\\displaystyle \\frac{ \\tau_{-(k+1),-(n+1)}(a,aq^{p}) } { \\sigma_{k+1,-(n+1)}(a,aq^{p}) } = \\frac{ (q^{k+1}, cdq^k, abcdq^{n+k+p+1}|q)_1} {(q^{n-k}, abcdq^{2k+p+1 }|q)_1 }. $\n\n\\end{enumerate}\n\\ifJOLT \\end{Lemma} \\else \\end{lemmas} \\fi\n\n\\ifEXTRAPROOFS\n\n\\begin{proof}\n\n\\begin{enumerate}\n\\item\n\\begin{multline*}\n\\Big( \n\\left\\{ (q^{n-k+1}| q)_k [aq^p]^{n-k}q^{n-k} \\right\\} \\cdot \\\\\n\\Big\\{ \\frac{ (bc q^k | q )_{n-k}  (bd q^k| q )_{n-k}   (cd q^k| q )_{n-k}   {\\displaystyle  (q^{-p}  | q )_{n-k}  }  }  { (q| q)_k (abcdq^{n+k} | q)_{n-k}  (bcd[aq^p]q^{2k} | q)_{n-k} } \\Big\\}\n\\Big) \\\\\n\\Big( \n\\Big\\{ \\frac{1}{ (q^{n-k}| q)_{k+1} bcd [aq^p]^{n-k} q^{n + k}} \\Big\\} \\cdot \\\\ \n\\Big\\{ \\frac\n{ (q| q)_k (abcdq^{n+k} | q)_{n-k}  (bcd[aq^p]q^{2k+1} | q)_{n-k} } \n{(bc q^{k + 1}| q )_{n-k - 1}  (bd q^{k + 1} | q )_{n-k - 1}   (cd q^{k }| q )_{n-k}  {\\displaystyle   (q^{-p} | q )_{n-k}  } }  \\Big\\}\n\\Big) \\\\\n= \\frac{(bcq^k,bdq^k, abcdq^{n+k+p} |q)_1} {bcd q^{2k} (q^{n-k}, abcdq^{2k+p} |q)_1}.\\\\\n\\end{multline*}\n\\item\n\\begin{multline*}\n\\Big( \n\\left\\{ (q^{n-k+1}| q)_k  \\left[ aq^{p} \\right] ^{n+1-k} \\right\\} \\cdot \\\\\n\\frac{ (bc q^k| q )_{n+1-k}  (bd  q^k| q )_{n+1-k}   (cd  q^k| q )_{n-k}   {(q^{-p} | q )_{n +1- k} }  }  { (q| q)_k (abcdq^{n+k} | q)_{n+1-k}  (bcd \\left[ aq^{p} \\right] q^{2k} | q)_{n+1-k} }   \\Big) \\\\\n\\Big( \n\\Big\\{ \\frac{1}{ (q^{n-k+1}| q)_k  \\left[ aq^{p} \\right] ^{n-k}(bc q^{k+1} | q )_{n-k}} \\Big\\} \\cdot \\\\ \n\\Big\\{  \\frac\n{ (q| q)_k (abcdq^{n+k+1} | q)_{n-k}  (bcd \\left[ aq^{p} \\right] q^{2k+1} | q)_{n-k} }  \n{   (bd q^{k+1}| q )_{n-k}   (cd q^k| q )_{n-k}   {\\displaystyle  (q^{-p} | q )_{n-k}  }  }  \n\\Big\\}\n\\Big) \\\\\n= \\frac{aq^p(bcq^k,bdq^k, q^{n-k-p}|q)_1 } {(abcdq^{n+k},abcdq^{2k+p }|q)_1 }. \\\\\n\\end{multline*}\n\\item\n\\begin{multline*}\n\\Big( \n\\left\\{ (q^{n-k}| q)_{k+1} bcd  \\left[ aq^{p} \\right] ^{n-k} q^{n + k}\\right\\} \\cdot \\\\\n\\frac{(bc q^{k + 1}| q )_{n-k - 1}  (bd q^{k + 1} | q )_{n-k - 1}   (cd q^{k }| q )_{n-k}  {\\displaystyle   (q^{-p} | q )_{n-k}  } }  { (q| q)_k (abcdq^{n+k} | q)_{n-k}  (bcd \\left[ aq^{p} \\right] q^{2k+1} | q)_{n-k} }  \\Big) \\\\\n\\Big(\n\\Big\\{ \\frac{1}{(q^{n-k}| q)_{k+1}  \\left[ aq^{p} \\right] ^{n-k-1}q^{n-k-1} } \\Big\\} \\cdot \\\\  \n\\Big\\{  \\frac\n{ (q| q)_{k+1} (abcdq^{n+k+1} | q)_{n-k-1}  (bcd \\left[ aq^{p} \\right] q^{2(k+1)} | q)_{n-k-1} }\n{ (bc q^{k+1} | q )_{n-k-1}  (bd q^{k+1}| q )_{n-k-1}    (cd q^{k+1}| q )_{n-k-1}   {\\displaystyle  (q^{-p}| q )_{n-k-1}  }  }  \n\\Big\\}\n\\Big) \\\\\n= \\frac{abcdq^{2k+p+1}(q^{k+1},q^{n-k-p-1},cdq^k|q)_1 } {(abcdq^{n+k},abcdq^{2k+p+1 }|q)_1 }. \\\\\n\\end{multline*}\n\\item\n\\begin{multline*}\n\\Big(  \n\\left\\{ (q^{n-k+1}| q)_k   \\left[ aq^{p} \\right] ^{n-k} \\right\\} \\cdot \\\\\n\\Big\\{ \\frac{(bc q^{k+1}| q )_{n-k}  (bd q^{k+1}| q )_{n-k}   (cdq^k | q )_{n-k}  \n  {\\displaystyle   (q^{-p} | q )_{n-k}  } \n  }  { (q| q)_k  (abcdq^{n+k+1} | q)_{n-k}  (bcd \\left[ aq^{p} \\right] q^{2k+1} | q)_{n-k} } \n  \\Big\\} \\Big) \\\\\n\\Big( \n\\Big\\{ \\frac{1}{ (q^{n-k}| q)_{k+1}  \\left[ aq^{p} \\right] ^{n-k} } \\Big\\} \\cdot \\\\ \n\\frac\n  { (q| q)_{k+1} (abcdq^{n+k+1} | q)_{n-k}  (bcd \\left[ aq^{p} \\right] q^{2(k+1)} | q)_{n-k} }  \n{ (bc q^{k+1}| q )_{n-k}  (bd  q^{k+1}| q )_{n-k}   (cd  q^{k+1}| q )_{n-k-1}   \n  {\\displaystyle  (q^{-p} | q )_{n - k} }   } \n\n  \\Big) \\\\\n= \\frac{ (q^{k+1}, cdq^k, abcdq^{n+k+p+1}|q)_1} {(q^{n-k}, abcdq^{2k+p+1 }|q)_1 }.  \\\\\n\\end{multline*}\n\\end{enumerate}\n\n\\end{proof}\n\\fi\n\n\n\n\n\\begin{center} \\bf The $T^{00}$ Identity\\label{T00} \\end{center}\n\\ifJOLT \\begin{Proposition} \\else \\begin{props} \\fi\n\\label{T00PROP}\nWhen $0 \\le k \\le n-1$\n\\begin{multline*}\n\\tau_{k,n}(a,aq^{p+1}) = \\big[ \\tau_{k,k}(aq^p,aq^{p+1})\\big] \\big[ \\tau_{k,n}(a,aq^p) \\big] + \\big[ \\tau_{k,k+1}(aq^p,aq^{p+1})\\big] \\big[ \\tau_{k+1, n}(a,aq^p)\\big] \\\\\n  +\\big[ \\sigma_{k,-(k+1)}(aq^p,aq^{p+1})\\big] \\big[ \\sigma_{-(k+1),n}(a,aq^p) \\big].\n\\end{multline*}\n And \n$$  \\tau_{n,n}(a,aq^{p+1}) = \\big[ \\tau_{n,n}(aq^p,aq^{p+1})\\big] \\big[ \\tau_{n,n}(a,aq^p) \\big].$$\n\\ifJOLT \\end{Proposition} \\else \\end{props} \\fi\n\n\\begin{proof}\nThe second identity just says $1=1 \\cdot 1.$\n\n\nFor the first, upon dividing by $\\sigma_{-(k+1),n}(a,aq^p),$ we see it is sufficient to prove\n\\begin{multline*}\n\\frac{\\tau_{k,n}(a,aq^{p+1})}{\\sigma_{-(k+1),n}(a,aq^p)} = \\left\\{ \\tau_{k,k}(aq^p,aq^{p+1})\\right\\} \\left\\{ \\frac{\\tau_{k,n}(a,aq^p)}{\\sigma_{-(k+1),n}(a,aq^p)} \\right\\} \\\\ + \\left\\{ \\tau_{k,k+1}(aq^p,aq^{p+1})\\right\\} \\left\\{ \\frac{\\tau_{k+1, n}(a,aq^p)}{\\sigma_{-(k+1),n}(a,aq^p)} \\right\\} \n  +\\left\\{ \\sigma_{k,-(k+1)}(aq^p,aq^{p+1})\\right\\} .\n\\end{multline*}\nUsing Lemmas \\ref{URT} and \\ref{VRT}, this means we need to show\n\\begin{multline}\n\\label{EQN1a}\n\\left\\{ \n\\frac{q^{n-k}(bcq^{k}, bdq^{k} , q^{-(p+1)}|q)_1 } {bcdq^{2k} (q^{n-k}, q^{n-k-p-1}|q)_1  }\n\\right\\}\n  =   \\left\\{ 1 \\right\\} \n\n\\frac{(bcq^k,bdq^k, abcdq^{n+k+p} |q)_1} {bcd q^{2k} (q^{n-k}, abcdq^{2k+p} |q)_1}\n\n \\\\\n+ \\left\\{ \\frac{ (q^{2}| q)_k [aq^{p+1}]q(bc q^k | q )_{1}  (bd q^k| q )_{1}   (cd q^k| q )_{1}   (q^{-1}| q )_{1}    }  { (q| q)_k ([aq^p]bcdq^{2k+1} | q)_{1}  (bcd[aq^{p+1}]q^{2k} | q)_{1} } \\right\\}   \\cdot  \\\\ \n   \n\\left\\{ \\frac\n{(abcdq^{n+k},abcdq^{2k+p+1 }|q)_1 } \n{abcdq^{2k+p+1}(q^{k+1},q^{n-k-p-1},cdq^k|q)_1 } \n\\right\\} \\\\\n\n  +  \\left\\{\\frac{ (q| q)_k [aq^{p+1}](bc q^k| q )_{1}  (bd  q^k| q )_{1}     {\\displaystyle  (q^{-1} | q )_{1} }  }  { (q| q)_k ([aq^p]bcdq^{2k} | q)_{1}  (bcd[ aq^{p+1}]q^{2k} | q)_{1} } \\right\\} \\cdot 1 \\\\\n\\end{multline}\n\n\n\nMultiplying (\\ref{EQN1a}) by\n$$ \\frac{ bcdq^{3k+p+1} ( q, q^{n-k}, abcdq^{2k+p}, abcdq^{2k+p+1}, q^{n-k-p-1} | q )_1 } { (bcq^k, bdq^k | q)_1 }$$\nwe see it is sufficient to show the vanishing of the  polynomial $p_1$ below.\n\n\n$\\big($To arrive at the final form of $p_1,$ we use Lemma \\ref{POCHNEG} above  to simplify the following expressions:\n$$ \n(q^{-1}| q)_1,\\ \n(q^{-(p+1)}| q)_1,\\  \n(q^{n-k}| q)_1, \\ \\text{and }\n(q^{n-k-p-1}| q)_1. \\ \n\\big)$\n\n\nWe will eventually reduce this identity to the vanishing of a 1-variable polynomial in $q$ with coefficients in the field $\\mathbb{Q}(a,b,c,d,y,u,v,w)$ \nwith the property that when \n$$\ny=abcd \\hspace{5mm} u= q^n \\hspace{5mm} v= q^k\\hspace{5mm} w= q^p\n$$\nwe obtain a unit in the coefficient field times the difference between the two sides of equation (\\ref{EQN1a}) above. So, effectively, we can use the variables $y,u,v,w$\nas abbreviations for these expressions.\n\n\\begin{multline}\np_1=-\\left\\{ (yq^{2k+p} | q)_1 (yq^{2k+p+1} | q)_1  (q| q)_1   q^{n+p+1}     (q^{-(p+1)}| q )_{1}   \\right\\}   \\\\\n +   \\left\\{ 1 \\right\\}\n\n  \\left\\{  (yq^{n+k+p} | q)_1  (yq^{2k+p+1} | q)_1 q^{k+p+1}    (q| q)_1   (q^{n-k-p-1} | q )_{1}    \\right\\} \\\\\n    + \\left\\{  (yq^{n+k} | q)_1  (yq^{2k+p} | q)_1(q^{n-k}| q)_1    q^{k+p+2}       (q^{-1} | q )_{1}    \n   \n       \\right\\}  \\\\\n    +  \\left\\{ (q^{n-k}| q)_1y (q| q)_1 q^{3k+2p+2}   (q^{-1} | q )_{1}  \n     (q^{n-k-p-1} | q )_{1}    \\right\\} \n\\end{multline}\n\\begin{multline*}\n\\hphantom{p_1}=-\\left\\{ (yq^{2k+p} | q)_1 (yq^{2k+p+1} | q)_1  (q| q)_1  q^{n+p+1}   \\left[ - q^{-(p+1)}(q^{p+1}| q)_1\\right]   \\right\\} \\\\\n +   \\left\\{ 1 \\right\\}\n\n  \\left\\{  (yq^{n+k+p} | q)_1  (yq^{2k+p+1} | q)_1  (q| q)_1  q^{k+p+1}   \\left[ q^{-(k+p+1)} \\left[ (q^{n}| q)_1-(q^{k+p+1}| q)_1 \\right] \\right]    \\right\\} \\\\\n  + \\left\\{  (yq^{n+k} | q)_1  (yq^{2k+p} | q)_1  q^{k+p+2} \\left[ q^{-k} \\left[ (q^{n}| q)_1-(q^k| q)_1 \\right]\\rb       \\left[ - q^{-1}(q| q)_1 \\right]   \n   \n       \\right\\} \\\\\n +  \\left\\{  \\left[ q^{-k} \\left[ (q^{n}| q)_1-(q^k| q)_1 \\right] \\right] (q| q)_1  yq^{3k+2p+2}  \\left[- q^{-1}(q| q)_1 \\right] \\right. \\cdot \\\\ \n\\left. \\left[ q^{-(k+p+1)} \\left[ (q^{n}| q)_1-(q^{k+p+1}| q)_1 \\right] \\right]       \\right\\} \n\\end{multline*}\n\\begin{multline*}\n\\hphantom{p_1}=\\left\\{ (yq^{2k+p} | q)_1 (yq^{2k+p+1} | q)_1  (q| q)_1   q^{n}    (q^{p+1}| q)_1   \\right\\}   \\\\\n+   \\left\\{ 1 \\right\\\n\n  \\left\\{  (yq^{n+k+p} | q)_1  (yq^{2k+p+1} | q)_1  (q| q)_1   \\left[   (q^{n}| q)_1-(q^{k+p+1}| q)_1 \\right] \\    \\right\\} \\\\\n    - \\left\\{  (yq^{n+k} | q)_1  (yq^{2k+p} | q)_1  q^{p+1}   \\left[ (q^{n}| q)_1-(q^k| q)_1 \\right]       (q| q)_1    \n \n\n       \\right\\} \\hspace{40mm} \\\\\n  -  \\left\\{  \\left[ (q^{n}| q)_1-(q^k| q)_1  \\right] (q| q)_1  yq^{k+p}  \\left[ (q| q)_1 \\right]   \n\\left[ (q^{n}| q)_1-(q^{k+p+1}| q)_1 \\right]       \\right\\} \n\\end{multline*}\n\nUsing the notation $\\big[a,b,c,\\dots\\big]=(a| q)_1(b| q)_1(c| q)_1\\ldots,$ \nand multiplying by $\\left[  (q| q)_1 \\right]^{-1}$ \nit suffices to show the vanishing of \n\\begin{multline*}\np_2 = u[yv^2w, yv^2wq, wq] \n        + (vwq - u)[yuvw, yv^2wq] \\\\\n       - wq(v-u)[yuv, yv^2w]\n       -yvw(v-u)(vwq-u) (q| q)_1\n\\end{multline*}\n\nThis expression may be interpreted as a one variable polynomial of degree at most $2$ in $q$ with coefficients in the field $\\mathbb{Q}(y,u,v,w).$\n\nThe coefficient of $q^2$ is\n$$\nu (1-yv^2w) (-yv^2w) (-w) + (vw) (1-yuvw) (-yv^2w) -yvw (v-u) (vw) (-1) =0.\n$$\nSo the polynomial $p_2$ is of degree at most $1$ in $q.$\nEvaluating \n\\begin{description}\n\\item[at $q=0$ ] $p_2(0)=u(1-yv^2w)-u(1-yuvw) -yvw(v-u)(-u)=0$\n\\item[at $ q=u(vw)^{-1}$ ] Note at this point $yv^2wq=yuv$ and $wq=uv^{-1}.$ So\n$$ p_2\\big(u(vw)^{-1}\\big) = u[yv^2w, yuv](1-uv^{-1}) - uv^{-1}(v-u)[yv^2w,yuv]=0. $$\n\\end{description}\nThus the polynomial $p_2$ is $0$ and the proposition is proven.\n\n\\end{proof}\n\n\\begin{center} \\bf The $T^{01}$ Identity \\end{center}\n\n\n\\ifJOLT \\begin{Proposition} \\else \\begin{props} \\fi\n\\label{T01PROP}\nWhen $0 \\le k \\le n-1$\n\\begin{multline}\n\\sigma_{k,-(n+1)}(a,aq^{p+1}) = \\big[ \\tau_{k,k}(aq^p,aq^{p+1})\\big] \\big[ \\sigma_{k,-(n+1)}(a,aq^p) \\big] \\\\\n+ \\big[ \\tau_{k,k+1}(aq^p,aq^{p+1})\\big] \\big[ \\sigma_{k+1, -(n+1))}(a,aq^p)\\big] \\\\\n  +\\big[ \\sigma_{k,-(k+1)}(aq^p,aq^{p+1})\\big] \\big[ \\tau_{-(k+1),-(n+1)}(a,aq^p) \\big].\n\\end{multline}\nAnd\n\\begin{multline}\n\\sigma_{n,-(n+1)}(a,aq^{p+1}) = \\big[ \\tau_{n,n}(aq^p,aq^{p+1})\\big] \\big[ \\sigma_{n,-(n+1)}(a,aq^p) \\big] \\\\\n  +\\big[ \\sigma_{n,-(n+1)}(aq^p,aq^{p+1})\\big] \\big[ \\tau_{-(n+1),-(n+1)}(a,aq^p) \\big].\n\\end{multline}\n\n\n\\ifJOLT \\end{Proposition} \\else \\end{props} \\fi\n\n\\begin{proof} The second is immediate from corollary \\ref{CDG1LIN} together with the observation that $\\tau_{r,r}=1$ for any sign of $r.$\n\n\nFor the first, upon dividing by $\\tau_{-(k+1),-(n+1)}(a,aq^p),$ we see it is sufficient to prove\n\\begin{multline*}\n\\frac{\\sigma_{k,-(n+1)}(a,aq^{p+1}) } {\\tau_{-(k+1),-(n+1)}(a,aq^p)} \n= \\left\\{ \\tau_{kk}(aq^p,aq^{p+1})\\right\\} \\left\\{ \\frac{\\sigma_{k,-(n+1)}(a,aq^p)} {\\tau_{-(k+1),-(n+1)}(a,aq^p)} \\right\\} \\\\\n+ \\left\\{ \\tau_{k,k+1}(aq^p,aq^{p+1})\\right\\} \\left\\{ \\frac{\\sigma_{k+1, -(n+1))}(a,aq^p) } {\\tau_{-(k+1),-(n+1)}(a,aq^p)} \\right\\}\n  +\\left\\{ \\sigma_{k,-(k+1)}(aq^p,aq^{p+1})\\right\\} \\cdot 1.\n\\end{multline*}\nThat means we need to show\n\\begin{multline}\n\\label{EQN1b} \n\\frac{aq^{n-k+p+1} (bcq^k, bdq^k, q^{-(p+1)}|q)_1} { (abcdq^{n+k}, abcdq^{n+k+p+1} |q)_1}\n  =   \\left\\{1  \\right\\} \n \n \\left\\{  \\frac{aq^p( bcq^k,bdq^k, q^{n-k-p} |q)_1 } {(abcdq^{n+k},abcdq^{2k+p }|q)_1 } \\right\\} \\\\\n \n+ \\left\\{\\frac{ (q^{2}| q)_k \\left[ aq^{p+1} \\right] q(bc q^k | q )_{1}  (bd q^k| q )_{1}   (cd q^k| q )_{1}   {\\displaystyle  (q^{-1} | q )_{1}  }  }  { (q| q)_k ( \\left[ aq^{p} \\right] bcdq^{2k+1} | q)_{1}  (bcd\\left[ aq^{p+1} \\right] q^{2k} | q)_{1} }  \\right\\}  \\cdot \\\\\n \\left\\{ \\frac\n  {(q^{n-k}, abcdq^{2k+p+1 }|q)_1 }\n { (q^{k+1}, cdq^k, abcdq^{n+k+p+1}|q)_1} \n \\right\\} \\\\\n\n\n  +  \\left\\{ \\frac{ (q| q)_k  \\left[ aq^{p+1} \\right]  (bc q^k| q )_{1}  (bd  q^k| q )_{1}     {\\displaystyle  (q^{-1}| q )_{1} }  }  { (q| q)_k ( \\left[ aq^{p} \\right] bcdq^{2k} | q)_{1}  (bcd \\left[ aq^{p+1} \\right]  q^{2k} | q)_{1} } \\right\\}  \\cdot   \n \n  1\n\\end{multline}\n\n\nWe will eventually reduce this identity to the vanishing of a 1-variable polynomial in $y$ with coefficients in the field $\\mathbb{Q}(a,b,c,d,u,v,w,q)$ \nwith the property that when \n$$\ny=abcd \\hspace{5mm} u= q^n \\hspace{5mm} v= q^k\\hspace{5mm} w= q^p\n$$\nwe obtain a unit in the coefficient field times the difference between the two sides of equation (\\ref{EQN1b})  above. So, effectively, we can use the variables $y,u,v,w$\nas abbreviations for the above expressions.\n\nMultiplying (\\ref{EQN1b})  by  \n\\begin{equation}\n\\frac{q^k ( abcdq^{n+k}, abcdq^{n+k+p+1}, abcdq^{2k+p}, abcdq^{2k+p+1} |q)_1 } { a ( bcq^k,bdq^k | q)_1 }\n\\end{equation}\nwe see it is sufficient to show the vanishing of the  polynomial $p_1$ below. We use $y$ as an abbreviation for $abcd.$\n\n$\\big($To arrive at the final form of $p_1,$ we use Lemma \\ref{POCHNEG} above  to simplify the following expressions:\n$$\n(q^{-1}| q)_1,\\  \n(q^{-p}| q)_1,\\ \n(q^{-(p+1)}| q)_1,\\ \n(q^{n-k}| q)_1, \\ \\text{and }\n(q^{n-k-p}| q)_1. \\big)\n$$\n\\begin{multline*}\np_1 =  -\\left\\{  (yq^{2k +p} | q)_{1}   (yq^{2k +p+1} | q)_{1}  \n   q^{n+p+1}  (q^{-(p+1)}| q)_1  \\right\\}\\\\\n  +   \\left\\{1  \\right\\}   \n  \\left\\{    (yq^{n+k +p+1} | q)_{1}  (yq^{2k +p+1} | q)_{1}  q^{k+p}  \n    (q^{n-k-p}| q)_1      \\right\\} \\\\\n\n + \\left\\{\\frac{ (q^{n-k}| q)_1(yq^{n+k} | q)_{1}   (y q^{2k+p} | q)_{1}   q^{k+p+2} \n   (q^{-1}| q)_1 \n  }  { (q| q)_1 \n   }  \\right\\} \\\\\n\n\n +  \\left\\{ (y q^{n+k} | q)_{1} (y q^{n+k+p+1} | q)_{1}    q^{k+p+1}      (q^{-1}| q)_1   \\right\\}\n   \\end{multline*}      \n\\begin{multline*}\n\\hphantom{p_1 }=  -\\left\\{  (yq^{2k +p} | q)_{1}   (yq^{2k +p+1} | q)_{1}  q^{n+p+1} \\Big\\{   - q^{-(p+1)}(q^{p+1}| q)_1\\Big\\}   \\right\\}\\\\\n  +   \\left\\{1  \\right\\}   \n  \\left\\{    (yq^{n+k +p+1} | q)_{1}  (yq^{2k +p+1} | q)_{1}   q^{k+p} \n \\big\\{ q^{-(k+p)} \\left[ (q^n| q)_1 -(q^{k+p}| q)_1 \\right]  \\big\\}     \\right\\}   \\\\\n\n +\\frac{ \\left\\{ \\big\\{q^{-k} \\left[(q^n| q)_1 -(q^k| q)_1 \\right]  \\big\\} (yq^{n+k} | q)_{1}   (y q^{2k+p} | q)_{1}  q^{k+p+2} \n  \\left[ -q^{-1}(q| q)_1 \\right] \n  \\right\\}}  { (q| q)_1 } \\\\\n \n +  \\left\\{ (y q^{n+k} | q)_{1} (y q^{n+k+p} | q)_{1}    q^{k+p+1}     \\left[ -q^{-1}(q| q)_1 \\right]  \\right\\}\n  \\end{multline*}\n \\begin{multline*}\n\\hphantom{p_1 }=  \\left\\{  (yq^{2k +p} | q)_{1}   (yq^{2k +p+1} | q)_{1}   q^{n} \\Big\\{  (q^{p+1}| q)_1\\Big\\}   \\right\\}\\\\\n  +   \\left\\{1  \\right\\}   \n  \\left\\{    (yq^{n+k +p+1} | q)_{1}  (yq^{2k +p+1} | q)_{1}   \n   \\big\\{ \\left[ (q^{n}| q)_1-(q^{k+p}| q)_1\\right]    \\big\\}\n      \\right\\} \\\\\n  \n \n\n\n - \\left\\{(yq^{n+k} | q)_{1}   (y q^{2k+p} | q)_{1}   q^{p+1}\n    \\right\\}\n \\big\\{\\left[ (q^{n}| q)_1-(q^{k}| q)_1 \\right] \\big\\} \\\\\n \n -  \\left\\{ (y q^{n+k} | q)_{1} (y q^{n+k+p+1} | q)_{1}    q^{k+p}     \\left[ (q| q)_1 \\right]  \\right\\}\n  \\end{multline*}\n \n  Using the notation $\\big[a,b,c,\\dots\\big]=(a| q)_1(b| q)_1(c| q)_1\\ldots,$\nsetting $y=abcd, u=q^n,v=q^k, \\text{ and } w=q^p,$ we see it is sufficient to show the vanishing of\n\n\\begin{multline*}\np_2=u\\left[ yv^2w, yv^2wq, wq\\right] \n+(vw-u)\\left[ yuvwq, yv^2wq \\right]\\\\\n-wq(v-u) \\left[ yuv, yv^2w\\right]\n-vw \\left[ yuv, yuvwq,q\\right] \n\\end{multline*}\n \n This expression may be interpreted as a one variable polynomial of degree at most $2$ in $y$ with coefficients in the field $\\mathbb{Q}(u,v,w,q).$\n Evaluating\n \\begin{description}\n\\item [at $y=0$] $p_2(0)=u(1-wq)+(vw-u) -wq(v-u)-vw(1-q) =0.$\n\\item [at $y=(v^2wq)^{-1}$] When $y=(v^2wq)^{-1},$ note $yv^2w =q^{-1},$ $yuv= u(vwq)^{-1},$ and $yuvwq=uv^{-1}.$ So\n\\begin{multline*}\np_2\\big( (v^2wq)^{-1} \\big) = -wq(v-u) \\left[ u(vwq)^{-1}, q^{-1} \\right] -vw \\left[ u(vwq)^{-1}, uv^{-1}, q \\right]\\\\\n= (1-u(vwq)^{-1})\\big\\{  -wq(v-u)  \\left( -q^{-1} (1-q) \\right) -vw(1-uv^{-1}) (1-q) \\big\\} \\\\\n= w(1-u(vwq)^{-1})(1-q)\\big\\{ v-u -v(1-uv^{-1}) \\big\\} =0\n\\end{multline*}\n\\item [at $y=(uv)^{-1}$] When $y=(uv)^{-1},$ note $yv^2w= u^{-1}vw,$ $yv^2wq=u^{-1}vwq ,$ and $yuvwq=wq .$ So\n\\begin{multline*}\np_2\\big( (uv)^{-1} \\big) = (1-u^{-1}vwq)(1-wq) \\big\\{u(1 -  u^{-1}vw) +vw-u\\big\\}=0 \n\\end{multline*}\n\\end{description}\nThus the polynomial $p_2$ is $0$ and the proposition is proven.\n\n \n\\end{proof}\n\n\n\\begin{center} \\bf The $T^{10}$ Identity \\end{center}\n\n\\ifJOLT \\begin{Proposition} \\else \\begin{props} \\fi\n\\label{T10PROP}\nWhen $0 \\le k \\le n-2$\n\\begin{multline}\n    \\sigma_{-(k+1), n}(a,aq^{p+1}) =  \\big[ \\sigma_{-(k+1),k+1}(aq^p,aq^{p+1})\\big] \\big[ \\tau_{k+1, n}(a,aq^p)\\big] \\\\\n+\\big[ \\tau_{-(k+1),-(k+1)}(aq^p,aq^{p+1})\\big] \\big[ \\sigma_{-(k+1),n}(a,aq^p) \\big] \\\\\n  + \\big[ \\tau_{-(k+1),-(k+2)}(aq^p,aq^{p+1})\\big] \\big[ \\sigma_{-(k+2), n}(a,aq^p)\\big] \n\\end{multline}\nAnd\n\\begin{multline*}\n   \\sigma_{-n, n}(a,aq^{p+1}) =  \\big[ \\sigma_{-n,n}(aq^p,aq^{p+1})\\big] \\big[ \\tau_{n,n}(a,aq^p)\\big] \\\\\n+\\big[ \\tau_{-n,-n}(aq^p,aq^{p+1})\\big] \\big[ \\sigma_{-n,n}(a,aq^p) \\big]. \n\\end{multline*}\n\\ifJOLT \\end{Proposition} \\else \\end{props} \\fi\n\n\\begin{proof} The second is immediate from corollary \\ref{CDG1LIN} together with the observation that $\\tau_{r,r}=1$ for any sign of $r.$\n\nFor the first, upon dividing by $\\tau_{k+1,n}(a,aq^p),$ we see it is sufficient to prove\n\\begin{multline*}\n    \\frac{ \\sigma_{-(k+1), n}(a,aq^{p+1}) } { \\tau_{k+1,n}(a,aq^p)} =  \\left\\{ \\sigma_{-(k+1),k+1}(aq^p,aq^{p+1}) \\right\\} \\cdot 1 \\\\\n+\\left\\{  \\tau_{-(k+1),-(k+1)}(aq^p,aq^{p+1})  \\right\\} \\left\\{ \\frac{ \\sigma_{-(k+1),n}(a,aq^p) } { \\tau_{k+1,n}(a,aq^p) } \\right\\} \\\\\n  + \\left\\{  \\tau_{-(k+1),-(k+2)}(aq^p,aq^{p+1}) \\right\\} \\left\\{ \\frac { \\sigma_{-(k+2), n}(a,aq^p) } { \\tau_{k+1,n}(a,aq^p) } \\right\\} .\n\\end{multline*}\nThat means we need to show\n\\begin{multline}\n\\label{EQN1c}\\ \n\\frac{abcd q^{n+k+p+1} (q^{k+1}, cdq^{k}, q^{-(p+1)}|q)_1} { (abcdq^{n+k}, abcdq^{n+k+p+1} |q)_1} \\\\\n= \n   \\left\\{ \\frac{(q| q)_{k+1} bcd  \\left[ aq^{p+1} \\right]  q^{2k+1}  (cd q^{k }| q )_{1}  {\\displaystyle   (q^{-1} | q )_{1}  } }  { (q| q)_k ( \\left[ aq^{p} \\right] bcdq^{2k+1} | q)_{1}  (bcd \\left[ aq^{p+1} \\right]  q^{2k+1} | q)_{1} } \\right\\}  \\cdot \n  \n   1 \\\\\n+  \\left\\{1 \\right\\}  \\cdot \n\\left\\{ \\frac{abcdq^{2k+p+1}(q^{k+1},q^{n-k-p-1},cdq^k|q)_1 } {(abcdq^{n+k},abcdq^{2k+p+1 }|q)_1 } \\right\\} \\\\\n+  \\left\\{ \\frac{(q^{2}| q)_k   \\left[ aq^{p+1} \\right]  (bc q^{k+1}| q )_{1}  (bd q^{k+1}| q )_{1}   (cdq^k | q )_{1}  {\\displaystyle   (q^{-1}| q )_{1}  } }  { (q| q)_k  ( \\left[ aq^{p} \\right] bcdq^{2(k+1)} | q)_{1}  (bcd \\left[ aq^{p+1} \\right]  q^{2k+1} | q)_{1} } \\right\\} \\cdot \\\\   \n\\left\\{ \\frac\n{bcd q^{2(k+1)} (q^{n-k-1}, abcdq^{2k+p+2} |q)_1} \n{(bcq^{k+1},bdq^{k+1}, abcdq^{n+k+p+1} |q)_1} \\right\\} \n\\end{multline}\n\nWe will eventually reduce this identity to the vanishing of a 1-variable polynomial in $y$ with coefficients in the field $\\mathbb{Q}(a,b,c,d,u,v,w,q)$ \nwith the property that when \n$$\ny=abcd \\hspace{5mm} u= q^n \\hspace{5mm} v= q^k\\hspace{5mm} w= q^p\n$$\nwe obtain a unit in the coefficient field times the difference between the two sides of equation (\\ref{EQN1c})  above. So, effectively, we can use the variables $y,u,v,w$\nas abbreviations for the above expressions.\n\n\nMultiplying (\\ref{EQN1c})  by\n$$\n\\frac{ ( q, abcdq^{n+k}, abcdq^{n+k+p+1}, abcdq^{ 2k+p+1}, abcdq^{2k+p+2} | q )_1 } {abcd (q^{k+1}, cdq^k | q)_1 } \\\\\n$$\nwe see it is sufficient to show the vanishing of the  polynomial $p_1$ below.\n\n$\\big($To arrive at the final form of  $p_1,$ we use Lemma \\ref{POCHNEG} above  to simplify the following expressions:\n$$ \n(q^{-1}| q)_1,\\  \n  (q^{-(p+1)} | q )_{1}, \n\\text{ and }\n(q^{n-k-p-1}| q)_1. \n\\big) \n$$\n\n\n\n\n\\begin{multline*}\np_1  =  \n-\\left\\{ (yq^{2k+p+1}| q)_1(yq^{2k+p+2}| q)_1  (q| q)_1 \n q^{n + k+p+1} (q^{-(p+1)} | q )_{1}    \\right\\}  \\\\\n+  \n  \\left\\{ (yq^{n+k}| q)_1 (yq^{n+k+p+1}| q)_1  (q| q)_1  q^{2k+p+2} (q^{-1}| q)_1    \\right\\}  \\\\  \n   +  \\left\\{1 \\right\\}  \\cdot    \n\\left\\{ (yq^{n+k+p+1}| q)_1 (yq^{2k+p+2}| q)_1  (q| q)_1 \n q^{2k+p+1}  (q^{n-k-p-1}| q)_1    \\right\\} \\\\\n+  \\left\\{ (yq^{n+k}| q)_1 (yq^{2k+p+1}| q)_1(q^{n-k-1}| q)_1 q^{2k+p+3}    \n(q^{-1}| q)_1    \\right\\} \\cdot \n\\end{multline*}\n\\begin{multline*} \n\\hphantom{p_1}  =  \n-\\left\\{ (yq^{2k+p+1}| q)_1(yq^{2k+p+2}| q)_1  (q| q)_1 \n q^{n + k+p+1} \\left[ - q^{-(p+1)}(q^{p+1}| q)_1  \\right]   \\right\\}  \\\\\n+ \n     \\left\\{ (yq^{n+k}| q)_1 (yq^{n+k+p+1}| q)_1  (q| q)_1  q^{2k+p+2} \\left[  - q^{-1}(q| q)_1 \\right]   \\right\\}   \\\\  \n  +  \\left\\{1 \\right\\}  \\cdot   \n\\left\\{ (yq^{n+k+p+1}| q)_1 (yq^{2k+p+2}| q)_1(q| q)_1 \n q^{2k+p+1}  \\left[ q^{-(k+p+1)}   \\left[ (q^{n}| q)_1-(q^{k+p+1}| q)_1 \\right] \\right]  \\right\\} \\\\\n+  \\left\\{ (yq^{n+k}| q)_1 (yq^{2k+p+1}| q)_1 \\left[  q^{-(k+1)} \\left[ (q^{n}| q)_1-(q^{k+1}| q)_1 \\right]\\ \\right]  q^{2k+p+3} \n \\left[ - q^{-1}(q| q)_1 \\right]   \\right\\}   \n\\end{multline*}\n\\begin{multline*}\n\\hphantom{p_1} =  \n\\left\\{ (yq^{2k+p+1}| q)_1(yq^{2k+p+2}| q)_1 (q| q)_1 \n q^{n + k}   (q^{p+1}| q)_1     \\right\\}\\\\\n -   \n     \\left\\{ (yq^{n+k}| q)_1 (yq^{n+k+p+1}| q)_1   \\left[ (q| q)_1 \\right]^2  q^{2k+p+1}   \\right\\}   \\hspace{50mm} \\\\  \n\n  +  \\left\\{1 \\right\\}  \\cdot  \n\\left\\{ (yq^{n+k+p+1}| q)_1 (yq^{2k+p+2}| q)_1 (q| q)_1 \n q^{k}  \\left[  \\left[ (q^{n}| q)_1-(q^{k+p+1}| q)_1 \\right] \\right]  \\right\\} \\\\\n- \\left\\{ (yq^{n+k}| q)_1 (yq^{2k+p+1}| q)_1 \\left[  \\left[ (q^{n}| q)_1-(q^{k+1}| q)_1 \\right]\\ \\right] q^{k+p+1}\n (q| q)_1   \\right\\}\n\\end{multline*}\nUsing the notation $\\big[a,b,c,\\dots\\big]=(a| q)_1(b| q)_1(c| q)_1\\ldots,$ \nrecalling our abbreviation variables $u,v,\\text{ and } w,$\nand multiplying by $v^{-1} (q| q)^{-1}$ \nit suffices to show the vanishing of \n\\begin{multline*}\np_2 =  u[ yv^2wq,yv^2wq^2,wq]\n-vwq[yuv, yuvwq,q] \\\\\n +(vwq-u ) [yuvwq, yv^2wq^2 ] \n-wq(vq-u)[yuv, yv^2wq] \n\\end{multline*}\n\n\nThis expression may be interpreted as a one variable polynomial of degree at most $2$ in $y$ with coefficients in the field $\\mathbb{Q}(u,v,w,q).$\nEvaluating\n\\begin{description}\n\\item[at $y=0$ ] \n$$\np_2(0)= u(1-wq) -vwq(1-q) +(vwq-u) -wq(vq-u)=0.\n$$\n\\item[at $y=(v^2wq)^{-1}$ ] Note at this point $yv^2wq^2=q, \\ yuvwq=uv^{-1}, \\text{ and }yuv= u(vwq)^{-1}.\\ $ So\n\\begin{multline*}\np_2\\big((v^2wq)^{-1} \\big) =(1-uv^{-1}) \\big\\{  -vwq(1- u(vwq)^{-1}) (1-q) +(vwq-u)(1-q)\\big\\} \\\\\n                                         =(1-uv^{-1})(1-q)(vwq-u)  \\big\\{ -1+1 \\big\\} =0.\n\\end{multline*}\n\\item[at $y=(uvwq)^{-1}$ ] Note at this point $yv^2wq=u^{-1}v \\ yv^2wq^2= u^{-1}v q\\  \\text{ and }yuv=(wq)^{-1} $ So\n\n\\begin{multline*}\np_2\\big((uvwq)^{-1} \\big)= (1-u^{-1}v)\\big\\{ u(1-u^{-1}vq)(1-wq)-wq(vq-u)(1-(wq)^{-1} \\big\\} \\\\\n                                       = (1-u^{-1}v)(vq-u)(1-wq)\\big\\{-1 +1\\big\\} =0.\n\\end{multline*}\n\\end{description}\n\nThus $p_2$ being of degree at most  $2$ and vanishing at $3$ points implies $p_2$ is identically $0$ and the proposition proof is complete.\n\n\\end{proof}\n\n\\begin{center} \\bf The $T^{11}$ Identity \\end{center}\n\\ifJOLT \\begin{Proposition} \\else \\begin{props} \\fi\n\\label{T11PROP}\nWhen $0 \\le k \\le n-1$\n\\begin{multline}\n  \\tau_{-(k+1),-(n+1)}(a,aq^{p+1}) = \n   \\big[ \\sigma_{-(k+1),k+1}(aq^p,aq^{p+1})\\big] \\big[ \\sigma_{k+1, -(n+1)}(a,aq^p)\\big] \\\\\n+ \\big[ \\tau_{-(k+1),-(k+1)}(aq^p,aq^{p+1})\\big] \\big[ \\tau_{-(k+1),-(n+1)}(a,aq^p) \\big] \\\\\n+ \\big[ \\tau_{-(k+1),-(k+2)}(aq^p,aq^{p+1})\\big] \\big[ \\tau_{-(k+2), -(n+1)}(a,aq^p)\\big].\n\\end{multline}\nAnd \n$$  \\tau_{-(n+1),-(n+1)}(a,aq^{p+1}) = \\big[ \\tau_{-(n+1),-(n+1)}(aq^p,aq^{p+1})\\big] \\big[ \\tau_{-(n+1),-(n+1)}(a,aq^p) \\big].$$\n\\ifJOLT \\end{Proposition} \\else \\end{props} \\fi\n\n\\begin{proof}\nThe second identity just says $1=1 \\cdot 1.$\n\nFor the first, upon dividing by $ \\sigma_{k+1,-(n+1)}(a,aq^p) ,$ we see it is sufficient to prove\n\\begin{multline*}\n  \\frac{ \\tau_{-(k+1),-(n+1)}(a,aq^{p+1}) } { \\sigma_{k+1,-(n+1)}(a,aq^p) }  = \n   \\left\\{ \\sigma_{-(k+1),k+1}(aq^p,aq^{p+1}) \\right\\} \\cdot 1  \\\\\n+ \\left\\{ \\tau_{-(k+1),-(k+1)}(aq^p,aq^{p+1})\\right\\} \\left\\{  \\frac{ \\tau_{-(k+1),-(n+1)}(a,aq^p) } { \\sigma_{k+1,-(n+1)}(a,aq^p) } \\right\\} \\\\\n+ \\left\\{ \\tau_{-(k+1),-(k+2)}(aq^p,aq^{p+1})\\right\\} \\left\\{  \\frac{ \\tau_{-(k+2), -(n+1)}(a,aq^p) }{ \\sigma_{k+1,-(n+1)}(a,aq^p) } \\right\\}.\n\\end{multline*}\nThat means we need to show\n\\begin{multline}\n\\label{EQN1d}\n\\frac{q^{n-k} (q^{k+1}, cdq^k, q^{-(p+1)}|q)_1} {(q^{n-k},q^{n-k-p-1}|q)_1 } \\\\\n= \n   \\left\\{ \\frac{(q| q)_{k+1} bcd  \\left[ aq^{p+1} \\right]   q^{2k+1}  (cd q^{k }| q )_{1}  {\\displaystyle   (q^{-1} | q )_{1}  } }  { (q| q)_k ( \\left[ aq^{p} \\right] bcdq^{2k+1} | q)_{1}  (bcd \\left[ aq^{p+1} \\right]  q^{2k+1} | q)_{1} }  \\right\\}  \\cdot 1 \\\\\n  \n+  \\left\\{1 \\right\\} \n\\left\\{ \\frac{ (q^{k+1}, cdq^k, abcdq^{n+k+p+1}|q)_1} {(q^{n-k}, abcdq^{2k+p+1 }|q)_1 } \\right\\} \\\\\n+  \\left\\{ \\frac{(q^{2}| q)_k   \\left[ aq^{p+1} \\right]  (bc q^{k+1}| q )_{1}  (bd q^{k+1}| q )_{1}   (cdq^k | q )_{1}  {\\displaystyle   (q^{-1} | q )_{1}  } }  { (q| q)_k  ( \\left[ aq^{p} \\right] bcdq^{2(k+1)} | q)_{1}  (bcd \\left[ aq^{p+1} \\right]  q^{2k+1} | q)_{1} } \\right\\} \\cdot      \\\\\n   \\left\\{ \\frac\n  {(abcdq^{n+k+1},abcdq^{2k+p+2 }|q)_1 } \n  {aq^p(bcq^{k+1},bdq^{k+1}, q^{n-k-p-1}|q)_1 } \\right\\}\n \n \n\\end{multline}\n\n\nMultiplying (\\ref{EQN1d}) by\n$$\n \\frac{q^{k+p+1} ( q, q^{n-k}, q^{n-k-p-1}, abcdq^{2k+p+1},  abcdq^{2k+p+2} | q)_1} { (q^{k+1}, cdq^k | q)_1} \\\\\n$$\nwe see it is sufficient to show the vanishing of the  polynomial $p_1$ below.\n\n$\\big($To arrive at the final form of $p_1,$ we use Lemma \\ref{POCHNEG} above  to simplify the following expressions:\n$$ \n(q^{-1}| q)_1,\\ \n(q^{-p}| q)_1, \\ \n(q^{-(p+1)}| q)_1,\\  \n \\text{and }\nq^{n-k-p-1}| q)_1 \\ \n\\big) \n$$\n\nWe will eventually reduce this identity to the vanishing of a 1-variable polynomial in $y$ with coefficients in the field $\\mathbb{Q}(a,b,c,d,u,v,w,q)$ \nwith the property that when \n$$\ny=abcd \\hspace{5mm} u= q^n \\hspace{5mm} v= q^k\\hspace{5mm} w= q^p\n$$\nwe obtain a unit in the coefficient field times the difference between the two sides of equation (\\ref{EQN1d}) above. So, effectively, we can use the variables $y,u,v,w$\nas abbreviations for the above expressions.\n\n\n\\begin{multline*}\np_1=\n-\\left\\{(yq^{2k+p+1} | q)_1 (yq^{2k+p+2} | q)_1  (q| q)_1   q^{n+p+1}     (q^{-(p+1)} | q )_{1}    \\right\\} \\\\\n +\n   \\left\\{    (q^{n-k}| q)_{1}  (q| q)_1 y q^{3k+2p+3}     (q^{-1} | q )_{1}     \n  \n   (q^{n-k-p-1} | q )_{1}  \n   \\right\\} \\\\\n\n \n  +  \\left\\{1 \\right\\} \n   \\left\\{  (yq^{n+k+p+1} | q)_1  (yq^{2k+p+2} | q)_1(q| q)_1  q^{k+p+1}  (q^{n-k-p-1} | q )_{1}   \n   \\right\\} \\\\\n +  \\left\\{  (yq^{n+k+1} | q)_1  (yq^{2k+p+1} | q)_1  (q^{n-k}| q)_1   (q^{-1} | q )_{1}        \n q^{k+p+2} \n  \\right\\}  \n \\end{multline*} \n\\begin{multline*}\n\\hphantom{p_1}=\n-\\left\\{(yq^{2k+p+1} | q)_1 (yq^{2k+p+2} | q)_1  (q| q)_1   q^{n+p+1}    \\left[  - q^{-(p+1)}(q^{p+1}| q)_1 \\right]    \\right.   \\\\\n +\n   \\left\\{    \\left[  q^{-k} \\left[ (q^{n}| q)_1-(q^k| q)_1 \\right]\\rb   (q| q)_1   y  q^{3k+2p+3}     \\left[  - q^{-1}(q| q)_1  \\right]  \\right.  \\cdot \\\\\n  \n   \\left. \\left[ q^{-k-p-1} \\left[ (q^n| q)_1 - (q^{k+p+1}| q)_1 \\right]  \\right]   \\right\\} \\\\\n\n\n  +  \\left\\{1 \\right\\} \n   \\left\\{  (yq^{n+k+p+1} | q)_1  (yq^{2k+p+2} | q)_1(q| q)_1     q^{k+p+1}  \\left[ q^{-k-p-1} \\left[ (q^n| q)_1 - (q^{k+p+1}| q)_1 \\right]  \\right]         \\right\\} \\\\\n+  \\left\\{  (yq^{n+k+1} | q)_1  (yq^{2k+p+1} | q)_1   \\left[ q^{-k} \\left[ (q^{n}| q)_1-(q^k| q)_1 \\right] \\right]      \\left[  - q^{-1}(q| q)_1  \\right]     \n   q^{k+p+2} \n  \\right\\}  \n \\end{multline*}\n\\begin{multline*}\n\\hphantom{p_1}=\n\\left\\{(yq^{2k+p+1} | q)_1 (yq^{2k+p+2} | q)_1  (q| q)_1   q^{n}      (q^{p+1}| q)_1      \\right\\} \\\\\n - \\left\\{     \\left[ (q^{n}| q)_1-(q^k| q)_1 \\right]      y  q^{k+p+1}    \\left[ (q| q)_1  \\right]^2\n  \n   \\left[ (q^n| q)_1 - (q^{k+p+1}| q)_1 \\right]        \\right\\} \\\\\n\n\n+ \\left\\{1 \\right\\} \n   \\left\\{  (yq^{n+k+p+1} | q)_1  (yq^{2k+p+2} | q)_1(q| q)_1       \\left[ (q^n| q)_1 - (q^{k+p+1}| q)_1 \\right]          \\right\\} \\\\\n -  \\left\\{  (yq^{n+k+1} | q)_1  (yq^{2k+p+1} | q)_1  \\left[ (q^{n}| q)_1-(q^k| q)_1 \\right]    (q| q)_1      \n   q^{p+1} \n  \\right\\}  \n \\end{multline*}\nUsing the notation $\\big[a,b,c,\\dots\\big]=(a| q)_1(b| q)_1(c| q)_1\\ldots,$ \nrecalling our abbreviation variables $u,v,\\text{ and } w,$\nand multiplying by $\\left[ (q| q)_1 \\right]^{-1}$ \nit suffices to show the vanishing of \n\\begin{multline*}\np_2 = u[yv^2wq, yv^2wq^2, wq ] \n- yvwq(v-u)(vwq-u)(q| q)_1\\\\\n+ (vwq-u) [yuvwq,yv^2wq^2] \n -wq(v-u) [yuvq,yv^2wq] \n\\end{multline*}\nThis expression may be interpreted as a one variable polynomial of degree at most $2$ in $y$ with coefficients in the field $\\mathbb{Q}(u,v,w,q).$\nEvaluating \n\\begin{description}\n\\item[at $y=0$ ] \n$$\np_2(0)= u(1-wq) + (vwq-u)  -wq(v-u)=0.\n$$\n\\item[at $y=(v^2wq)^{-1}$ ] Note at this point $yv^2wq^2=q, \\ yvwq=v^{-1}, \\text{ and } yuvwq=uv^{-1}.$ So\n\\begin{multline*}\np_2\\big((v^2wq)^{-1} \\big)=(vwq-u) \\big\\{ -v^{-1}(v-u)(1-q) + (1-uv^{-1})(1-q)  \\big\\} \\\\\n                                         =(vwq-u) (1-uv^{-1})(1-q)  \\big\\{ -1+1 \\big\\} =0.\n\\end{multline*}\n\\item[at $y=(v^2wq^2)^{-1}$ ] Note at this point $yv^2wq=q^{-1}, \\ yvwq=(vq)^{-1}, \\text{ and } yuvq=u(vwq)^{-1}.$ So\n\\begin{multline*}\np_2\\big((v^2wq^2)^{-1} \\big)=(v-u)  \\big\\{-(vq)^{-1}(vwq-u)(1-q) - wq(1- u(vwq)^{-1})(1-q^{-1}) \\big\\} \\\\\n                                       =(v-u) (1-q)\\big\\{ -( w -u (vq)^{-1}  ) + (w- u(vq)^{-1})\\big\\}=0. \n\\end{multline*}\n\\end{description}\nThus $p_2$ being of degree $2$ and vanishing at $3$ points implies $p_2$ is identically $0$ and the proposition proof is complete.\n\n\\end{proof}\n\nThe combination of propositions \\ref{PLANA1}, \\ref{CDG1AEQ}, \\ref{CDG2AEQ}, \\ref{T00PROP}, \\ref{T01PROP}, \\ref{T10PROP}, and \\ref{T11PROP}   finishes the proof of all $3$ steps of (\\ref{PLANA}), Proof Plan A, and so completes the proof of theorem \\ref{ETC}. \n\n\\section{Reproof of the Askey-Wilson Result Using the Nonsymmetric Version \\label{REPROOF}}\n\nHaving given a direct proof of the shift-a connection coefficient formula Theorem \\ref{ETC}, we now show that it can be used to give another\nproof of the original Askey-Wilson result Theorem \\ref{PTCFRM}. \n\nFor this purpose, consistent with $x=\\cos{\\theta}, z=e^{i\\theta}$, so $2x=\\left( z +z^{-1}\\right),$ we view the $n$'th Askey-Wilson polynomial $P_n$ \nas a zig-zag monic Laurent polynomial in $z.$ So as an ordinary polynomial in $x,$ the leading coefficient would be $2^{n}.$\n\nAs earlier for the $E_r,$ we will shorten $P_n(z; a,b,c,d|q)$ to $P_n(a).$\n\nFirst note that the usual DAHA relation on $T_1$ \n$$\nT_1-T_1^{-1} = t_1^{\\frac{1}{2}} - t_1^{-\\frac{1}{2}}\n$$\ntranslates to the quadratic relation \n\\begin{equation}\n\\left( T_1+ t_1^{-\\frac{1}{2}} \\right) \\left( T_1- t_1^{\\frac{1}{2}} \\right) = 0.\n\\end{equation}\nshowing that the possible eigenvalues of $T_1$ are $ -t_1^{-\\frac{1}{2}} \\text{ and } t_1^{\\frac{1}{2}}.$\n\nDefine\n\\begin{eqnarray}\nL_1 = T_1+t_1^{-\\frac{1}{2}}& & L_2= -T_1 +t_1^{\\frac{1}{2}}.\n\\end{eqnarray}\nDefinition 1.2 in \\cite{ASKVOL} defined the Askey-Wilson $P_n$ (respectively $Q_n$) as normalized multiples of $L_1E_n$ (respectively\n$L_2 E_n$) for $n \\ge 0.$ Theorem 1.3 there (referring to \\cite{ANNALS}) pointed out that this definition of $P_n$ agrees with the usual $P_n.$\n\nIt was pointed out in Theorem 1.3 of \\cite{ASKVOL} that, up to a normalizing factor, $P_{|n|}$ is $(T_1+t_1^{-\\frac{1}{2}})E_n$ for $n \\ge 0.$\n\nA similar fact holds when $E_n$ is replaced by $E_{-n}$ for $n > 0.$ The reason for this is that by the recursion (\\ref{SRCN0}) and \n(\\ref{SRCN1}), (originally\nproved in \\cite{ASKVOL}) for $n>0,$ the $2$-dimensional subspace spanned by $E_{-n} \\text{ and } E_n$ is invariant under $T_1.$ So\n$L_1+L_2$ is a multiple of the identity on these $2$-dimensional subspaces and easy DAHA calculations show:\n\\ifJOLT \\begin{Lemma} \\else \\begin{lemmas} \\fi \n\\label{TWODPROJCD} Up to scalar factors, the operators $L_1$ and $L_2$ are algebraically orthogonal projections onto $1$-dimensional\nsubspaces of the  $2$-dimensional subspace (for $n \\ge0$)  spanned by $E_{-(n+1)} \\text{ and } E_{n+1}.$ In fact\n\\begin{enumerate}\n\\item $L_1L_2=L_2 L_1=0$\n\\item $L_1^2=( t_1^{\\frac{1}{2}}+ t_1^{-\\frac{1}{2}} )L_1$\n\\item $L_2^2=( t_1^{\\frac{1}{2}}+ t_1^{-\\frac{1}{2}} )L_2$\n\\end{enumerate}\n\\ifJOLT \\end{Lemma} \\else \\end{lemmas} \\fi \n{\\bf Remark: } By  algebraically orthogonal projections, we are referring to endomorphisms $\\pi_1$ and $\\pi_2=\\text{Identity} -\\pi_1$ satisfying\n$\\pi_1^2=\\pi_1.$ This of course also implies $\\pi_2^2=\\pi_2$ and $\\pi_1\\pi_2 =\\pi_2\\pi_1=0.$ These projections can also  be related to the natural\ninner product specified in Definition 1.4 of \\cite{ASKVOL} but we omit the details here.\n\\begin{proof}\n\\begin{enumerate}\n\\item By the basic DAHA relation\n$$ L_1= T_1+t_1^{-\\frac{1}{2}} = T_1^{-1} +t_1^{\\frac{1}{2}}.$$\nThen\n\\begin{multline*}\nL_1L_2= \\left( T_1^{-1} +t_1^{\\frac{1}{2}} \\right) \\left(  -T_1 +t_1^{\\frac{1}{2}} \\right) \\\\\n= -1 -  t_1^{\\frac{1}{2}}  \\left( T_1-T_1^{-1} \\right) +t_1 \\\\\n= -1 -  t_1^{\\frac{1}{2}}  \\left(  t_1^{\\frac{1}{2}} -  t_1^{-\\frac{1}{2}}  \\right) +t_1 =0.\n\\end{multline*}\nAnd $L_2L_1=L_1L_2.$\n\\item $\\displaystyle L_1^2=L_1(L_1+L_2)= \\left( t_1^{\\frac{1}{2}} + t_1^{-\\frac{1}{2}} \\right) L_1.$\n\\item $\\displaystyle L_2^2=(L_1+L_2)L_2= \\left( t_1^{\\frac{1}{2}} + t_1^{-\\frac{1}{2}} \\right) L_2.$\n\\end{enumerate}\n\\end{proof}\n\nConsequently, for $n>0,$ $P_n$ is also a multiple of $\\left( T_1+t_1^{-\\frac{1}{2}} \\right) E_{-n}.$ (The operator $L_1,$ up to constant factors, \nis sometimes referred to as {\\em Hecke-symmetrization}.) Translating to the operator $\\widetilde{T}_1= t_1^{\\frac{1}{2}} T_1 $ which we have mostly been using (and which\navoids square roots), we see that $P_n$ for $n>0$ is a multiple of $(\\widetilde{T}_1 +1)E_{-n}.$\n\nWith the zig-zag monic normalization conventions, Proposition \\ref{SCALE} and equation\n(\\ref{SRCNX}),\ngive us more explicitly for $n \\ge 0$ \n\n\\begin{equation}\n\\label{T1ECD} \n\\widetilde{T}_1 E_{-(n+1)} =\\left[  \\hat{c}_{n+1} \\right]^{-1} \\Big\\{E_{n+1} -  \\hat{d}_{n+1} E_{-(n+1)} \\Big\\}.\n\\end{equation}\nwhere \n$$\n \\hat{c}_{n+1} = - \\frac{1}{ab} =t_1^{-1} \n \\hspace{15mm} \n\\hat{d}_{n+1} = \\displaystyle - \\frac{( abq^{n+1} | q )_1+ab(cdq^{n}| q)_1}{ab(1-abcdq^{2n+1})} \n$$\n\n\\ifJOLT \\begin{Proposition} \\else \\begin{props} \\fi\n\\label{HECKESYMCD}\nFor $n \\ge 0,$\n$$ P_{n+1}= t_1^{-1} (\\widetilde{T}_1 +1)E_{-(n+1)}.$$\n(Also $P_0=E_0=1.)$\n\nMore explicitly\n$$\nP_{n+1}(a)  = E_{n+1} (a) + \\left[ \\gamma_{n+1} (a) \\right] E_{-(n+1)}(a)\n$$\nwhere $\\gamma_{n+1} (a)$ (really a function of $a,b,c,d,\\text{ and } q$)   is the scalar defined by\n\\begin{equation}\n\\label{GAMMACD}\n\\gamma_{n+1} (a) = \\frac{(q^{n+1},cdq^n|q)_1}{(abcdq^{2n+1}|q)_1}.\n\\end{equation}\n\\ifJOLT \\end{Proposition} \\else \\end{props} \\fi\n\\begin{proof}\nWe've already argued  $t_1^{-1} (\\widetilde{T}_1 +1)E_{-(n+1)}$ is a multiple of $P_{n+1}.$ Equation (\\ref{T1ECD}) implies $t_1^{-1} \\left[  \\hat{c}_{n+1} \\right]^{-1}=1.$ And $P_{n+1} =E_{n+1} = f_{n+1} \\text{ mod } \\mathcal{R}_{-(n+1)}$ while $E_{-(n+1)} \\in \\mathcal{R}_{-(n+1)}.$ So $t_1^{-1} (\\widetilde{T}_1 +1)E_{-(n+1)}$ gives $P_{n+1}$ exactly.\n\nRemembering $t_1\\hat{c}_{n+1}=1$ and  plugging the formulas of (\\ref{T1ECD}) into $P_{n+1}= t_1^{-1} (\\widetilde{T}_1 +1)E_{-(n+1)}$ gives us\n$$\nP_{n+1} = E_{n+1} + \\left(  \\hat{c}_{n+1} -  \\hat{d}_{n+1}  \\right) E_{-(n+1)}.\n$$\n(Since our normalization convention is that both $P_{n+1}$ and $E_{n+1}$ are zig-zag monic, the coefficient $1$ above  in front of $E_{n+1}$ was\nknown in advance.)\nOur definition of $\\gamma_{n+1} (a)$ is just a simplification of $\\hat{c}_{n+1} -  \\hat{d}_{n+1}$ as demonstrated in the easy Lemma \\ref{CMDSIMPCD} below.\n\\end{proof}\n\n\\ifJOLT \\begin{Lemma} \\else \\begin{lemmas} \\fi For $n \\ge 0$\n\\label{CMDSIMPCD}\n$$\n\\hat{c}_{n+1} -  \\hat{d}_{n+1}  = \\frac{(q^{n+1},cdq^n|q)_1}{(abcdq^{2n+1}|q)_1},\n$$\nthis quantity being the $ \\gamma_{n+1} (a)$ (really $ \\gamma_{n+1} (a,b,c,d|q)$) defined in Proposition \\ref{HECKESYMCD}.\n\\ifJOLT \\end{Lemma} \\else \\end{lemmas} \\fi \n\\begin{proof}\n\\begin{multline*}\n\\hat{c}_{n+1} -  \\hat{d}_{n+1}  =-\\frac{1}{ab} + \\frac{( abq^{n+1} | q )_1+ab(cdq^{n}| q)_1}{ab(1-abcdq^{2n+1})}  \\\\\n= -\\left(\\frac{1} {ab(1-abcdq^{2n+1})} \\right)  \\big\\{ 1 - abcdq^{2n+1} + (abq^{n+1} - 1) +ab(cdq^n-1)\\big\\}\\\\\n=\\frac{(1-q^{n+1})(1-cdq^n)} {1-abcdq^{2n+1}}.\\\\\n\\end{multline*}\n\\end{proof}\n\nNow that we know\n\\begin{eqnarray*}\nP_{n+1}(a) &=& E_{n+1}(a) + \\left[   \\gamma_{n+1}(a) \\right] \\left[ E_{-(n+1)}(a) \\right] \\\\\nP_{n+1}(e) &=& E_{n+1}(e) + \\left[   \\gamma_{n+1}(e) \\right] \\left[ E_{-(n+1)}(e) \\right], \n\\end{eqnarray*}\na natural way to obtain the Askey-Wilson connection coefficient relation Theorem \\ref{PTCFRM} is:\n\n\\begin{equation} \n\\label{PLANPCD}\n\\text{ \\em (PROOF PLAN P)  }\n\\end{equation}\n\\begin{enumerate}\n\\item Start with the top line for $P_{n+1}(a).$\n\\item Apply Theorem \\ref{ETC} to express each of $E_{\\pm(n+1)}(a)$ in terms of the $E_r(e).$ (Here $r$ can be of any sign.) \n\\item Show that the combinations of $E_{\\pm(m+1)}(e)$ (for $m\\ge 0$) which result are in fact the $c_{m+1,n+1}P_{m+1}(e)$ of the\nAskey-Wilson result. (As well as the $E_0(e)$ coefficient matching $c_{0,n+1}.$)\n\\end{enumerate}\n\nWe will now show that {\\em PROOF PLAN P} can be carried out to prove the Askey-Wilson result.\n\n\n\nRecall our generic notation for the $E$ connection coefficient relations.\nFor  $n \\geq 0:$\n\\begin{equation}\n\\label{EGENERICPLUSCD0} E_n(a) = \\sum_{m=0}^n \\left[ \\tau_{m,n} \\right] E_m(e)  + \\sum_{m=0}^{n-1} \\left[ \\sigma_{-(m+1),n} \\right] E_{-(m+1)}(e). \\end{equation}\n\\begin{equation}\n\\label{EGENERICMINUSCD0} E_{-(n+1)}(a) =  \\sum_{m=0}^n \\left[ \\tau_{-(m+1),-(n+1)}\\right] E_{-(m+1)}(e) + \\sum_{m=0}^n \\left[ \\sigma_{m,-(n+1)}\\right] E_{m}(e) . \n\\end{equation}\n\nOur original formulation of Theorem \\ref{ETC}, introduced the notation $d_{r,s}$ which is related to the (variously subscripted) $\\tau,\\sigma$ by:\n\\begin{equation}\nd_{r,s} = \\begin{cases}\n\\tau_{r,s}\/c_{|r|,|s|} & \\text{if } (r \\ge 0 \\text{ and } s \\ge 0)  \\text{ or } (r <  0 \\text{ and } s < 0)\\\\\n\\sigma_{r,s}\/c_{|r|,|s|} &  \\text{if } (r \\ge 0 \\text{ and } s < 0)  \\text{ or } (r <  0 \\text{ and } s > 0). \\\\\n\\end{cases}\n\\end{equation}\nWe return to that notation now and write:\n\\begin{equation}\n\\label{EGENERICPLUSCD} E_n(a) = \\sum_{m=0}^n \\left[ d_{m,n} c_{m,n}  \\right] E_m(e)  + \\sum_{m=0}^{n-1} \\left[ d_{-(m+1),n} c_{m+1,n}\\right] E_{-(m+1)}(e).\n\\end{equation}\n\\begin{equation}  \n\\label{EGENERICMINUSCD} E_{-(n+1)}(a) = \\sum_{m=0}^n \\left[ d_{-(m+1),-(n+1)} c_{m+1, n+1} \\right] E_{-(m+1)}(e) + \\sum_{m=0}^n \\left[ d_{m,-(n+1)} c_{m,n+1}  \\right] E_{m}(e).\n\\end{equation}\n\n\nPlugging these two into \n$$ P_{n+1}(a) = E_{n+1}(a) + \\left[   \\gamma_{n+1}(a) \\right] \\left[ E_{-(n+1)}(a) \\right] $$\nand considering the coefficients of $E_{\\pm(m+1)}(e)$ which result makes clear the relevance of the following two propositions.\n\\ifJOLT \\begin{Proposition} \\else \\begin{props} \\fi For $0 \\le m \\le n$\n\\label{EGAMMAEPLUSCD}\n\\begin{equation}\nd_{m+1,n+1} + \\gamma_{n+1}(a) d_{m+1,-(n+1)} =1.\n\\end{equation}\n\\ifJOLT \\end{Proposition} \\else \\end{props} \\fi\n\\begin{proof}\n\\begin{multline*}\nd_{m+1,n+1} + \\gamma_{n+1}(a) d_{m+1,-(n+1)} \\\\\n=\\Big\\{ \\displaystyle  \\frac{q^{n-m} (abcdq^{n+m+1}| q)_1}{(abcdq^{2n+1}| q)_1}\\Big\\} \n+\\Big\\{ \\displaystyle \\frac{(q^{n+1},cdq^n|q)_1}{(abcdq^{2n+1}|q)_1} \\Big\\} \\Big\\{  \\frac{ (q^{n-m }|q)_{ 1 } } {( q^{n+1}, cdq^{n }|q)_{ 1 }}\\Big\\} \\\\\n\\left( \\frac{1}{(abcdq^{2n+1}|q)_1 } \\right) \\big\\{q^{n-m} - abcdq^{ 2n+1} +1 - q^{n-m}\\big\\} =1.\\\\\n\\end{multline*}\n\\end{proof}\n\n\n\n\n\n\\ifJOLT \\begin{Proposition} \\else \\begin{props} \\fi\n\\label{EGAMMAEMINUSCD} For $0 \\le m \\le n$\n\\begin{equation}\nd_{-(m+1),n+1} + \\gamma_{n+1}(a) d_{-(m+1),-(n+1)} =\\gamma_{m+1}(e).\n\\end{equation}\n\\ifJOLT \\end{Proposition} \\else \\end{props} \\fi \n\\begin{proof}\n\\begin{multline*}\nd_{-(m+1),n+1} + \\gamma_{n+1}(a) d_{-(m+1),-(n+1)} \\\\\n= \\Big\\{ \\displaystyle  \\frac{bcdeq^{n+m+1} (q^{m+1}, cdq^{m}, ae^{-1}q^{n-m}|q)_{1 } } { (abcdq^{2n+1}, bcdeq^{2m+1 }|q)_{ 1}} \\Big\\} \\\\\n+ \\Big\\{ \\displaystyle \\frac{(q^{n+1},cdq^n|q)_1}{(abcdq^{2n+1}|q)_1}  \\Big\\}  \\Big\\{ \\displaystyle  \\frac{ (q^{m+1}, cdq^{m}, bcdeq^{n+m+1 }|q)_{1 } } { (q^{n+1}, cdq^{n}, bcdeq^{2m+1}|q)_{ 1} }  \\Big\\}  \\\\\n=\\Big\\{ \\frac{(q^{m+1}, cdq^{m} |q)_{1 } } { (abcdq^{2n+1}, bcdeq^{2m+1 }|q)_{ 1}} \\Big\\} \\big\\{ bcdeq^{n+m+1}(ae^{-1}q^{n-m}|q)_{1 } +(bcdeq^{n+m+1 }|q)_{ 1} \\big\\}\\\\\n=\\Big\\{ \\frac{(q^{m+1}, cdq^{m} |q)_{1 } } { (abcdq^{2n+1}, bcdeq^{2m+1 }|q)_{ 1}} \\Big\\} \\big\\{ bcdeq^{n+m+1} - abcd q^{2n+1} + 1- bcdeq^{n+m+1 } \\big\\}\\\\\n=  \\frac{(q^{m+1}, cdq^{m} |q)_{1 } } { ( bcdeq^{2m+1 }|q)_{ 1}}=\\gamma_{m+1}(e) \\\\\n\\end{multline*}\nas asserted.\n\\end{proof}\n\nNow we prove the Askey-Wilson connection coefficient result:\n\\begin{proof} (of Theorem \\ref{PTCFRM})\n\nThe case of $n=0$ checks since $P_0$ is $1,$ independent of parameters, and $c_{0,0}=1.$\n\nFor $n \\ge 0,$ using equations (\\ref{EGENERICPLUSCD}) and (\\ref{EGENERICMINUSCD}) as well as Propositions \\ref{EGAMMAEPLUSCD} and \\ref{EGAMMAEMINUSCD}\n\\begin{multline*}\nP_{n+1}(a)  = E_{n+1} (a) + \\left[ \\gamma_{n+1} (a) \\right] E_{-(n+1)}(a) \\\\\n=  \\sum_{m=0}^{n+1} \\left[ d_{m,n+1} c_{m,n+1} \\right] E_m(e)  + \\sum_{m=0}^{n} \\left[ d_{-(m+1),n+1} c_{m+1,n+1} \\right] E_{-(m+1)}(e) \\\\\n+ \\left[ \\gamma_{n+1} (a) \\right]  \\Big\\{ \\sum_{m=0}^n \\left[ d_{-(m+1),-(n+1)} c_{m+1,n+1} \\right] E_{-(m+1)}(e) + \\sum_{m=0}^n \\left[ d_{m,-(n+1)}  c_{m,n+1}\\ \\right] E_{m}(e) \\Big\\}\\\\\n\\end{multline*}\n\\begin{multline*}\n= c_{0,n+1} \\Big\\{  d_{0,n+1}  E_0(e) +  \\gamma_{n+1} (a)  d_{0,-(n+1)} \\Big\\} E_{0}(e) \\\\\n+\\sum_{m=1}^{n} c_{m,n+1}  \\big\\{  d_{m,n+1}  + \\gamma_{n+1}(a) d_{m,-(n+1)} \\big\\} E_{m}(e) + 1 \\cdot E_{n+1}(e)\\\\\n+\\sum_{m=0}^n c_{m+1,n+1} \\big\\{ d_{-(m+1),n+1}  + \\gamma_{n+1}(a) d_{-(m+1),-(n+1)}  \\big\\} E_{-(m+1)}(e)  \\\\\n\\end{multline*}\n\\begin{multline*}\n= \\text{ (by Propositions \\ref{EGAMMAEPLUSCD} and \\ref{EGAMMAEMINUSCD} }) \\\\\n= c_{0,n+1} \\cdot 1 \\cdot E_{0}(e) \n+\\sum_{m=0}^{n-1} c_{m+1,n+1} \\cdot 1 \\cdot E_{m+1}(e) + c_{n+1,n+1} E_{n+1}(e)\\\\\n+\\sum_{m=0}^{n-1} c_{m+1,n+1} \\cdot \\gamma_{m+1}(e) \\cdot E_{-(m+1)}(e) + c_{n+1,n+1} \\cdot \\gamma_{n+1}(e) \\cdot E_{-(m+1)}(e) \\\\\n= \\sum_{m=0}^{n+1} c_{m,n+1} P_m \\\\\n\\end{multline*}\ncompleting the re-proof of the Askey-Wilson Theorem \\ref{PTCFRM}.\n\\end{proof}\n\n\n\n\\noindent {\\bf Remark:} Instead of proving Lemma \\ref{TWODPROJCD} and Proposition \\ref{HECKESYMCD}, we could have based those\naspects of our proof on the discussion in Section 3 of \\cite{ZHED}. In particular, that reference clearly explains how, for $n>0,$ using the alternate \nchoice\n$$D=\\widetilde{Y}+q^{-1}abcd \\widetilde{Y}^{-1} \\hspace{5mm} \\text{(with } \\widetilde{Y}= \\widetilde{T}_1\\widetilde{T}_0) $$\nfor second order operator with Askey-Wilson polynomials $P_n$ as eigenfunctions, the corresponding eigenspace of $D$ is $4$ dimensional and spanned by\n$P_n,Q_n,E_n,\\text{ and } E_{-n}.$ Moreover $D$ commutes with both $\\widetilde{T}_1$ and $\\widetilde{T}_0,$ with $P_n$ and $Q_n$\nbeing eigenfunctions of $\\widetilde{T}_1.$ The respective eigenvalues are $t_1$ and $-1.$ Exact formulas expressing $E_{\\pm n}$\nas linear combinations of $P_n$ and the eigenvalue  $-1$ eigenfunction $Q_n^\\dagger$ of $\\widetilde{T}_0$ are also written down there.\n\n\n\n\\section{Appendix A on Change of Basis Conventions \\label{APPA} }\n\n\nLet $\\mathcal{B}=\\{e_{\\alpha}\\}_{\\alpha \\in \\mathcal{I}}$ be an ordered basis of a vector space $\\mathcal{V}$ where the index\nset $\\mathcal{I}=0,1,2,\\ldots\\ .$ For $x \\in \\mathcal{V},$ we represent the linear combination  $x=\\sum_{\\alpha \\in \\mathcal{I}} v^{\\alpha} e_{\\alpha}$\nby the column vector\n$$\n[x]_{\\mathcal{B}} = \\begin{bmatrix} v^0 \\\\ v^1 \\\\ \\vdots \\\\ \\end{bmatrix}.\n$$\n\nIf $\\overline{\\mathcal{B}}=\\{\\bar{e}_{\\beta}\\}_{\\beta \\in \\mathcal{I}}$ is another ordered basis, then \n$$\n[x]_{\\overline{\\mathcal{B}}} = \\begin{bmatrix} \\bar{v}^0 \\\\ \\bar{v}^1 \\\\ \\vdots \\\\ \\end{bmatrix}.\n$$\nis the column vector corresponding to the linear combination $x= \\sum_{\\beta \\in \\mathcal{I}} \\bar{v}^{\\beta} \\bar{e}_{\\beta}.$\n\nThen a change of basis relationship\n$$\ne_{\\alpha}=\\sum_{\\beta \\in \\mathcal{I}} T_{\\beta\\alpha}  \\bar{e}_{\\beta}\n$$\ncorresponds to\n$$\n \\begin{bmatrix} \\bar{v}^0 \\\\ \\bar{v}^1 \\\\ \\vdots \\\\ \\end{bmatrix} = T \\begin{bmatrix} v^0 \\\\ v^1 \\\\ \\vdots \\\\ \\end{bmatrix}\n$$\nwhere the row $\\beta$ column $\\alpha$ entry of $T$ is $T_{\\beta\\alpha}.$\n\nThis is easily confirmed by noting\n$$\nx=\n \\sum_{\\alpha} v^{\\alpha} e_{\\alpha} =\n \\sum_{\\alpha,\\beta} v^{\\alpha} T_{\\beta\\alpha} \\bar{e}_{\\beta}=\n \\sum_{\\beta} \\Big(\\sum_{\\alpha} T_{\\beta\\alpha} v^{\\alpha} \\Big)  \\bar{e}_{\\beta} =\n \\sum_{\\beta} \\bar{v}^{\\beta} \\bar{e}_{\\beta}=\n x.\n$$\n\nIn the case \n\\begin{multline*}\nn\\geq 0: \\  E_n(z; a,b,c,d| q) = \\sum_{m=0}^n\\tau_{m,n} E_m(z; e,b,c,d| q) \\\\+ \\sum_{m=0}^{n-1}\\sigma_{-(m+1),n} E_{-(m+1)}(z; e,b,c,d| q) \n\\end{multline*}\n\\begin{multline*}\nn\\geq0: \\  E_{-(n+1)}(z; a,b,c,d| q) = \\sum_{m=0}^{n}\\tau_{-(m+1),-(n+1)} E_{-(m+1)}(z; e,b,c,d| q) \\\\+ \\sum_{m=0}^{n}\\sigma_{m,-(n+1)} E_{m}(z; e,b,c,d| q) \n\\end{multline*}\nwe are viewing $E_0,E_1,\\ldots$ as an ordered basis for $\\mathcal{R}^0$ and $E_{-1},E_{-2}, \\ldots$ as an ordered basis for $\\mathcal{R}^1$\n\nNote that $E_{-1}$ is in the initial position (index $0$) of the second list, in conformity with viewing it as $E_{-(n+1)}$ for $n=0.$\n\nThinking of parameter $e$ as giving rise to the $\\bar{e}_{\\beta}$ basis and parameter $a$ the $e_{\\alpha}$ basis, then our generic\nchange of basis formula\n$$\ne_{\\alpha}=\\sum_{\\beta } T_{\\beta\\alpha}  \\bar{e}_{\\beta}\n$$\n becomes, in block form,\n \n $$\n T=\\begin{bmatrix}\nT ^{00} & T^{01} \\\\\nT^{10} & T^{11} \\\\\n\\end{bmatrix}\n$$\n so that {\\em for  column vectors of (blocks of length $n$ or $n+1$) components} relative to these bases\n $$\n  \\begin{bmatrix} \\bar{v}^0 \\\\ \\bar{v}^1 \\\\  \\end{bmatrix} = T \\begin{bmatrix} v^0 \\\\ v^1  \\\\ \\end{bmatrix}=\\begin{bmatrix}\nT ^{00} & T^{01} \\\\\nT^{10} & T^{11} \\\\\n\\end{bmatrix}\n\\begin{bmatrix} v^0 \\\\ v^1  \\\\ \\end{bmatrix}.\n $$\n \n Thus for $n\\ge 0$ \n \\begin{multline*}\n E_n(z; a,b,c,d| q) = \\sum_{m=0}^n\\tau_{m,n} E_m(z; e,b,c,d | q) \\\\+ \\sum_{m=0}^{n-1}\\sigma_{-(m+1),n} E_{-(m+1)}(z; e,b,c,d| q) \n\\end{multline*}\ncorresponds to \n\\begin{multline*}\nE_n(z; a,b,c,d| q) = \\sum_{m=0}^nT^{00}_{m,n} E_m(z; e,b,c,d| q) \\\\+ \\sum_{m=0}^{n-1}T^{10}_{m,n} E_{-(m+1)}(z; e,b,c,d| q) \n\\end{multline*}\n(Note the second matrix entry above really is $T^{10}_{m,n} $ (rather than $T^{01}_{m,n})$ in line with the general property (of our\nconventions) that the first  $n+1$ {\\em columns} of $T$ are expressing the decomposition of the first $n+1$ vectors $E_k(z; a,b,c,d| q) $\n$(0 \\le k \\le n)$ as linear combinations of the ordered basis  \n\\begin{multline*}\nE_0(z; e,b,c,d| q), E_1(z; e,b,c,d| q),\\ldots, \\\\ E_n(z; e,b,c,d| q), E_{-1} (z; e,b,c,d| q) ,\\ldots,  E_{-n}(z; e,b,c,d| q).\\big)\n\\end{multline*}\n\n\nSimilarly\n \\begin{multline*}\nE_{-(n+1)}(z; a,b,c,d| q) = \\sum_{m=0}^{n}\\tau_{-(m+1),-(n+1)} E_{-(m+1)}(z; e,b,c,d| q) \\\\+ \\sum_{m=0}^{n}\\sigma_{m,-(n+1)} E_{m}(z; e,b,c,d| q) \\\\\n\\end{multline*}\ncorresponds to \n\\begin{multline*}\n E_{-(n+1)}(z; a,b,c,d| q) = \\sum_{m=0}^{n}T^{11}_{m,n} E_{-(m+1)}(z; e,b,c,d| q) \\\\ + \\sum_{m=0}^{n}T^{01}_{m,n} E_{m}(z; e,b,c,d| q)\n\\end{multline*}\n\n\n\n\nThese tell us that the row index $m$ and column index $n$ entries of the four matrices are given by\n\\begin{eqnarray*}\nT^{00}_{m,n} = \\tau_{m,n} & \\hspace{15mm} & T^{01}_{m,n} = \\sigma_{m,-(n+1)} \\\\\nT^{10}_{m,n} = \\sigma_{-(m+1),n} &  \\hspace{15mm} & T^{11}_{m,n} = \\tau_{-(m+1),-(n+1)}. \\\\\n\\end{eqnarray*}\n\n\n\\section{Appendix B \\label{HAT} }\n\n\nIn this paper, we only use the zig-zag increasing cases in the following table and so have  included just the proofs of those. We mention the others, which we\nhave also proven, because they may be of interest.\n\\begin{table}[htp] \n\\caption{Summary of $\\hat{a}_m,\\hat{b}_m,\\hat{c}_m,\\hat{d}_m,$  \\ $m =n \\text{ or } -(n+1),  \\  n \\geq 0.$}\n\\begin{tabular}{|c|c|c|} \\hline\n $n \\geq 0$ {\\bf (Non-Negative Cases)} & $-(n+1) < 0$ {\\bf (Negative Cases)} \\\\ \\hline\n$ \\hat{a}_n ={\\displaystyle \\frac{q^{n}\\left[(abcdq^{2n}| q)_1\\right]^2}{(acq^{n}, bcq^{n}, adq^{n}, bdq^{n}| q)_1}}$          \n                                     & $\\displaystyle \\hat{a}_{-(n+1)} =\n                                                       - \\frac{1}{cdq^{n}}\n         $ \\\\ \\hline         \n$ \\hat{b}_n = \\left[ cdq^{2n}(abcdq^{2n}| q)_1\\right] \\cdot$  \n& $\\hat{b}_{-(n+1)}=\n                                         \\displaystyle     \\frac{(c+d)- cdq^{n }(a+b)}{cdq^{n } (abcdq^{2n}| q)_1} \n      $\\\\\n $\\displaystyle \\frac{ \\left[ ab(c+d)q^{n}  - (a +b) \\right] }{(acq^{n}, bcq^{n}, adq^{n}, bdq^{n}| q)_1}$    &  \\\\ \\hline    \n$\\displaystyle  \\hat{c}_n = \n               - \\frac{1}{ab} $\n                           & $\\displaystyle \\hat{c}_{-(n+1)} =\\frac{\\left[(abcdq^{2n+1}| q)_1\\right]^2}{(q^{n+1}, abq^{n+1}, cdq^{n}, abcdq^{n}| q)_1}$ \\\\ \\hline                              \n$\\displaystyle \\hat{d}_{n} =   \n              \\frac{( abq^{n} -1 )+ab(cdq^{n-1} -1 )}{ab(1-abcdq^{2n-1})} \n                           $& $ \\hat{d}_{-(n+1)}=\\left[ abq^{n}(abcdq^{2n+1}| q)_1 \\right] \\cdot$ \\\\ \n& $\\displaystyle \\frac{\\left[ q(abcdq^{n}| q)_1  +cd(q^{n+1}| q)_1\\right]} \n{(q^{n+1}, abq^{n+1}, cdq^{n}, abcdq^{n}| q)_1}                                                                                               \n       $ \\\\ \\hline\n $ \\widetilde{\\mu}_{n}=\n                          t_0t_1q^{n}=\n                          abcdq^{n-1}\n                          $ & $\\widetilde{\\mu}_{-(n+1)}= \n                                                   q^{-(n+1)}\n       $ \\\\ \\hline\n$ \\widetilde{\\zeta}_{0n}=\\left[ \n \\hat{a}_{n}\\right]^{-1}\\left[ \\widetilde{\\mu}_{n} - \\widetilde{\\mu}_{-(n+1)}\\right] $ \n                                                  & $ \\widetilde{\\zeta}_{0,-(n+1)}= \\left[\\hat{a}_{-(n+1)}\\right]^{-1}\\left[ \\widetilde{\\mu}_{-(n+1)} - \\widetilde{\\mu}_{n}\\right] $  \\\\\n \\ \\ \\ $={ \\displaystyle  -\\frac\n {(acq^{n}, bcq^{n}, adq^{n}, bdq^{n}| q)_1} \n {q^{2n+1} (abcdq^{2n}| q)_1 }  } $     & \\ \\ \\ $= -cdq^{-1} (abcdq^{2n}| q)_1$ \\\\ \\hline      \n$ \\widetilde{\\zeta}_{1n}=  \\left[\\hat{c}_{n}\\right]^{-1}\\left[ \\widetilde{\\mu}_{-n} - \\widetilde{\\mu}_{n}\\right] $\n                                              & $ \\widetilde{\\zeta}_{1,-(n+1)} = \\left[\\hat{c}_{-(n+1)}\\right]^{-1}\\left[ \\widetilde{\\mu}_{n+1} - \\widetilde{\\mu}_{-(n+1)}\\right] $ \\\\\n $-abq^{-n}(abcdq^{2n-1}| q)_1$                                                   & $= {\\displaystyle  -\\frac\n\n {(q^{n+1}, abq^{n+1}, cdq^{n}, abcdq^{n}| q)_1}\n  {q^{n+1} (abcdq^{2n+1}| q)_1 }  } $  \\\\ \\hline                                                    \n\\end{tabular}\n\\label{default}\n\\end{table}%\n\nFor any signs of $n,k:$\n\\begin{eqnarray*\nE_{-(n+1)} &=& \\hat{a}_{-(n+1)} \\widetilde{U}_0\\Big(E_n\\Big) +   \\hat{b}_{-(n+1)} E_n \\\\\nE_{n}& =& \\hat{a}_{n} \\widetilde{U}_0\\Big(E_{-(n+1)}\\Big) +  \\hat{b}_{n} E_{-(n+1)} \\\\\nE_{-n} &=& \\hat{c}_{-n} \\widetilde{T}_1\\Big(E_{n}\\Big) +   \\hat{d}_{-n} E_n \\\\\nE_{n} &=& \\hat{c}_{n} \\widetilde{T}_1\\Big(E_{-n}\\Big) +   \\hat{d}_{n} E_{-n} \n\\end{eqnarray*}\n\\begin{eqnarray*\n\\widetilde{U}_0\\Big(E_{k}\\Big)&=&\\left[ \\hat{a}_{-(k+1)}\\right]^{-1}    \\left[ E_{-(k+1)}  -  \\hat{b}_{-(k+1)} E_k \\right] \\\\\n\\widetilde{T}_1\\Big(E_{k}\\Big)&=&\\left[ \\hat{c}_{-k}\\right]^{-1} \\left[ E_{-k} -  \\hat{d}_{-k} E_k \\right] \\ \\  (k \\neq 0)\n\\end{eqnarray*\n\\begin{eqnarray*\n\\widetilde{U}_0\\Big(E_{-(k+1)}\\Big)&=&\\left[ \\hat{a}_{k}\\right]^{-1}    \\left[ E_{k}  -   \\hat{b}_{k} E_{-(k+1)} \\right] \\\\\n\\widetilde{T}_1\\Big(E_{-(k+1)}\\Big)&=&\\left[ \\hat{c}_{k+1}\\right]^{-1} \\left[ E_{k+1}  -   \\hat{d}_{k+1} E_{-(k+1)} \\right] \\ \\  (k \\neq -1)\n\\end{eqnarray*\n\\begin{eqnarray*\n\\widetilde{\\mathcal{S'}}_{0} E_{n} = \\widetilde{\\zeta}_{0,-(n+1)}E_{-(n+1)}  &\\hspace{10mm}  & \\widetilde{\\mathcal{S'}}_{0} E_{-(n+1)} = \\widetilde{\\zeta}_{0,n}E_{n} \\\\\n\\widetilde{\\mathcal{S}}_{1} E_{n}=\\widetilde{\\zeta}_{1, - n}E_{-n} &\\hspace{10mm} & \\widetilde{\\mathcal{S}}_{1} E_{-n} =\\widetilde{\\zeta}_{1n}E_n  \n\\end{eqnarray*\n\nSince the multiplication operator\n$$ X= t_0 \\widetilde{T}_1^{-1}  \\widetilde{ Y}  \\widetilde{U }_0,$$\nthe above imply general $4-$dimensional invariant subspaces for $X$ and $X^{-1}.$ Simplifications of the matrix entries of these rel the $\\{E_r\\}$ arise. \nThe simplified expressions are mostly products and quotients of q-Pochhammer symbols, with an occasional  monomial or sum of $2$ symbols factor.\n\n\\ifLONG\n\n\n\\section{Appendix C1: Introduction to the Shift-c Proof \\label{SHIFTC1}}\n\nThe nonsymmetric Askey-Wilson polynomials $E_n(z;a,b,c,d|q)$ are symmetric under the interchange of parameters $a$ and $b,$\nas well as interchange of $c$ and $d.$ \n\n\nIn the main body of this paper, we have presented a full proof of Theorem \\ref{ETC}. About the corresponding `shift-c' version Theorem  \\ref{ETC2},\nwe have thus far mentioned only that the `same method of proof' can be used.\nThe purpose of this addendum is to write out the details of the proof of this shift-c result. \n\nFirst, note that the connection coefficients $c_{m,n}(a, e ; b, c, d | q)$ of Theorem \\ref{PTCFRM} are symmetric in the parameters\n$c$ and $d.$  \nSo $c_{m,n}(a, e ; b, c, d | q)$is not natural to relate to the nonsymmetric connection coefficients  arising from changing the basis\n$\\cF=\\{E_r(z;a,b,c,d|q)\\}=\\{E_r\\}$  to $\\cF''= \\{E_r(z;a,b,g,d|q)\\}=\\{E_r''\\}$ as we are doing in Theorem \\ref{ETC2}.\n\nSo we start with a rewriting of equations (\\ref{AWCONSUM}) and (\\ref{AWCON}).\n\nTheorem \n\\ref{PTCFRM} here,\na result (with different normalization)  of Askey and Wilson in  \\cite{MEMOIRS}, says\n$$ P_n(z; a,b,c,d | q) =\\sum_{m \\le n}  c_{m,n}(a, e ; b, c, d | q)  P_m(z; e,b,c,d | q) $$\nwhere\n$$c_{m,n}(a, e ; b, c, d | q) =  \\frac{ (q^{n-m+1} | q )_m(bc q^m, bd q^m, cd q^m, ae^{-1}  | q )_{n-m}   }  { (q  | q )_m (abcdq^{n+m-1}, bcdeq^{2m}  | q )_{n-m} } e^{n-m}. $$\n\nReplacing $a \\to c, e \\to g, c \\to a,$ gives\n\\begin{equation} P_n(z; c,b,a,d | q) =\\sum_{m \\le n}  c_{m,n}(c, g ; b, a, d | q)  P_m(z; g,b,a,d | q) \\end{equation}\nwhere\n\\begin{equation} \\label{CRSSHIFTC} c_{m,n}( c, g ; b, a, d | q) =  \\frac{g^{n-m} (q^{n-m+1} | q )_m(ab q^m, bd q^m, ad q^m, cg^{-1}  | q )_{n-m}   }  { (q  | q )_m (abcdq^{n+m-1}, abdgq^{2m}  | q )_{n-m} }  \\end{equation}\nwhich we observe is symmetric in $a$ and $b.$\n\nKeeping in mind that the $P_n(z;a,b,c,d|q)$ are symmetric in all four parameters $a,b,c, \\text{ and }d,$ one form of the shift-c result for\nsymmetric Askey-Wilson polynomials is\n\\begin{equation} P_n(z; a,b,c,d | q) =\\sum_{m \\le n}  c_{m,n}(c, g ; a, b, d | q)  P_m(z; a,b,g,d | q) \\end{equation}\nwith equation (\\ref{CRSSHIFTC}) defining $c_{m,n}(c, g ; a, b, d | q)= c_{m,n}(c, g ; b, a, d | q).$\n\nSo the result of Theorem \\ref{ETC2} may also be expressed in the following style:\n\n\nThe expansion formula $ \\displaystyle E_s =\\sum_{r \\preceq s}\\left[  d^c_{r,s}(c, g ; a, b, d | q) \\right] \\left[ c_{|r|,|s|}(c, g ; a, b, d | q) \\right] E_r''  $ holds, where\n\\[ E_r = E_r(z; a,b,c,d|q), \\quad E_r''=E_r(z; a,b,g,d|q)\\]\nand\n\\[ \\quad d^c_{r,s}= \\begin{cases}  \n\\displaystyle  \\frac{ ( abq^s, abcdq^{r+s-1} | q)_1  }  { ( abq^r, abcdq^{2s-1} | q)_1 }  & \\text{ if }   r \\ge 0, s\\ge 0 \\\\ \n\\displaystyle  -\\frac{  abq^r ( q^{-r -s}  | q)_1 }  { ( q^{-s}, abq^r | q)_1  }   & \\text{ if }   r \\ge 0, s< 0 \\\\ \n\\displaystyle  -  \\frac{ dq^{-r-1}g (q^{-r},  abq^s, cg^{-1} q^{s+r}  | q)_1 }  { ( abcd q^{2s-1}, abdgq^{-2r-1} | q)_1 }  & \\text{ if }   r < 0, s\\ge 0 \\\\  \n\\displaystyle   \\frac{ ( q^{-r} , abdg q^{ -r-s-1} | q)_1  }  { ( q^{-s} , abdg q^{-2r-1}  | q)_1 }  & \\text{ if }   r < 0, s< 0  \n\\end{cases} \\]\n\n\n\nThus our basic working notation for the shift-c proof becomes:\n\nFor  $n \\geq 0$ and any parameter values $c,g,a,b,d,\\text{ and } q:$\n\\begin{multline*}\nE_n(a,b,c,d|q) = \\sum_{m=0}^n \\left[ \\tau^c_{m,n}(c, g ; a, b, d | q) \\right] E_m(a,b,g,d|q)\\\\  + \\sum_{m=0}^n \\left[ \\sigma^c_{-(m+1),n}(c, g ; a, b, d | q) \\right] E_{-(m+1)}(a,b,g,d|q) \\\\\nE_{-(n+1)}(a,b,c,d|q) = \\sum_{m=0}^n \\left[ \\tau^c_{-(m+1),-(n+1)}(c, g ; a, b, d | q) \\right] E_{-(m+1)}(a,b,g,d|q) \\\\+ \\sum_{m=0}^n \\left[ \\sigma^c_{m,-(n+1)}(c, g ; a, b, d | q) \\right] E_{m}(a,b,g,d|q) \n\\end{multline*}\nwhere for $k,n \\geq 0:$\n\n\\begin{multline}\n\\tau^c_{k,n}(c, g ; a, b, d | q) =  \\frac{ (q^{n-k+1} | q)_k g^{n-k}(abq^{k+1},ad q^k, bd q^k, cg^{-1}  | q )_{n-k}   }  { (q | q)_k (abcdq^{n+k}, abdgq^{2k}  | q)_{n-k} } \\\\\n\\sigma^c_{k,-(n+1)}(c, g ; a, b, d | q) = -\\frac{ (abq^{k+1} | q)_{n-k} (ad q^k, bd  q^k,cg^{-1}  | q )_{n-k+1}     }  { (q | q)_k (abcdq^{n+k} , abdgq^{2k}  | q)_{n-k+1} } \\\\\n\\times abq^k g^{n-k+1} (q^{n-k+1} | q)_k \\\\\n\\sigma^c_{-(k+1),n}(c, g ; a, b, d | q) = -\\frac{(abq^{k+1},cg^{-1} | q)_{n-k} (ad q^{k+1}, bd  q^{k+1}  | q )_{n-k-1}  }  { (q | q)_k (abcdq^{n+k} , abdgq^{2k+1}  | q)_{n-k} } \\\\\n \\times dq^k g^{n-k} (q^{n-k} | q)_{k+1}   \\\\\n\\tau^c_{-(k+1),-(n+1)}(c, g ; a, b, d | q) = \\frac{(q^{n-k+1} | q)_k  g^{n-k}(abq^{k+1},adq^{k+1}, bd q^{k+1}, cg^{-1} | q )_{n-k}  }  { (q | q)_k  (abcdq^{n+k+1}, abdgq^{2k+1}  | q)_{n-k} } .\n\\end{multline}\n\n\n\\ifEXTRAPROOFS\n\n\n\n\n\n\\begin{center} \\bf (Verification below  that the above $\\tau^c,\\sigma^c$ using the $c_{m,n}$ of equation (\\ref{CRSSHIFTC}) lead to the $d^c_{r,s}$ given in Theorem \\ref{ETC2}. \\end{center}\n\nThe basic relation is \n\\begin{align*}\n\\tau^c_{r,s}&=&d^c_{r,s}c_{|r|,|s|} & \\hspace{10mm} & \\text{if } (r \\ge 0 \\text{ and } s \\ge 0)  \\text{ or } (r <  0 \\text{ and } s < 0)\\\\\n\\sigma^c_{r,s}&= &d^c_{r,s}c_{|r|,|s|} & \\hspace{10mm} &  \\text{if } (r \\ge 0 \\text{ and } s < 0)  \\text{ or } (r <  0 \\text{ and } s > 0). \n\\end{align*}\n\n\\begin{proof} For $k,n \\ge 0,$\n\\begin{enumerate}\n\\item\n\\begin{multline*}\nd^c_{k,n} = \\frac{\\tau^c_{k,n}}{c_{k,n}} \\\\=\n\\left\\{ \\frac{ (q^{n-k+1};q)_k g^{n-k}(abq^{k+1},ad q^k, bd q^k, cg^{-1}  | q )_{n-k}   }  { (q | q)_k (abcdq^{n+k}, abdgq^{2k}  | q)_{n-k} } \\right\\} \\cdot \\\\\n\\left\\{ \n\\frac\n{ (q  | q )_k (abcdq^{n+k-1}, abdgq^{2k}  | q )_{n-k} }\n{g^{n-k} (q^{n-k+1} | q )_k(ab q^k, bd q^k, ad q^k, cg^{-1}  | q )_{n-k}   }  \n\\right\\} \\\\\n=  \\frac{ ( abq^n, abcdq^{n+k-1} | q)_1  }  { ( abq^k, abcdq^{2n-1} | q)_1 } \n\\end{multline*}\n\\item\n\\begin{multline*}\nd^c_{k,-(n+1)} = \\frac{\\sigma^c_{k,-(n+1)}}{c_{k,n+1}} \\\\=\n\\left\\{  -\\frac{ (q^{n-k+1} | q)_k abq^k g^{n-k+1}(abq^{k+1} | q)_{n-k} (ad q^k, bd  q^k,cg^{-1}  | q )_{n-k+1}     }  { (q | q)_k (abcdq^{n+k} , abdgq^{2k}  | q)_{n-k+1} }  \\right\\} \\cdot \\\\\n\\left\\{ \n\\frac\n{ (q  | q )_k (abcdq^{n+k}, abdgq^{2k}  | q )_{n-k+1} }\n{g^{n-k+1} (q^{n-k+2} | q )_k(ab q^k, bd q^k, ad q^k, cg^{-1}  | q )_{n-k+1}   }  \n\\right\\} \\\\\n=  -\\frac{  abq^k ( q^{n-k+1}  | q)_1 }  { ( q^{n+1}, abq^k | q)_1  } \n\\end{multline*}\n\\item\n\\begin{multline*}\nd^c_{-(k+1),n} = \\frac{\\sigma^c_{-(k+1),n}}{c_{k+1,n}} \\\\=\n\\left\\{- \\frac{(q^{n-k} | q)_{k+1} dq^k g^{n-k} (abq^{k+1},cg^{-1} | q)_{n-k} (ad q^{k+1}, bd  q^{k+1}  | q )_{n-k-1}  }  { (q | q)_k (abcdq^{n+k} , abdgq^{2k+1}  | q)_{n-k} } \\right\\} \\cdot \\\\\n\\left\\{ \n\\frac\n{ (q  | q )_{k+1} (abcdq^{n+k}, abdgq^{2k+2}  | q )_{n-k-1} }\n{g^{n-k-1} (q^{n-k} | q )_{k+1}(ab q^{k+1}, bd q^{k+1}, ad q^{k+1}, cg^{-1}  | q )_{n-k-1}   }  \n\\right\\} \\\\\n= -  \\frac{ dq^kg (q^{k+1},  abq^n, cg^{-1} q^{n-k-1}  | q)_1 }  { ( abcd q^{2n-1}, abdgq^{2k+1} | q)_1 } \n\\end{multline*}\n\\item\n\\begin{multline*}\nd^c_{-(k+1),-(n+1)} = \\frac{\\tau^c_{-(k+1),-(n+1)}}{c_{k+1,n+1}} \\\\=\n\\left\\{  \\frac{(q^{n-k+1} | q)_k  g^{n-k}(abq^{k+1},adq^{k+1}, bd q^{k+1}, cg^{-1} | q )_{n-k}  }  { (q | q)_k  (abcdq^{n+k+1}, abdgq^{2k+1} | q)_{n-k} }  \\right\\} \\cdot \\\\\n\\left\\{ \n\\frac\n{ (q  | q )_{k+1} (abcdq^{n+k+1}, abdgq^{2k+2}  | q )_{n-k} }\n{g^{n-k} (q^{n-k+1} | q )_{k+1}(ab q^{k+1}, bd q^{k+1}, ad q^{k+1}, cg^{-1}  | q )_{n-k}   }  \n\\right\\} \\\\\n=  \\frac{ ( q^{k+1} , abdg q^{n+k+1} | q)_1  }  { ( q^{n+1} , abdg q^{2k+1}  | q)_1 } \n\\end{multline*}\n\\end{enumerate}\n\\end{proof}\n\\fi\n\n\nThe block structure of the true transition function $\\boldsymbol{\\mathcal{T}}^c(c, g ; a, b, d | q)$ and  the claimed answer $T^c(c, g ; a, b, d | q),$\nin the theorem above are the same as in the shift-a case. We systematically use the superscript $c$ to denote the shift-c version\nof the corresponding shift-a one. And, since the parameters $a,b,d,\\text{ and } q$ are often the same, we usually omit them from \nargument lists that mostly depend on $c,g, \\text{ or both.}$\n\nOur  proof of Theorem \\ref{ETC2}  has three steps which we refer to as\n\\begin{equation} \n\\label{PLANC}\n\\text{ \\em (PROOF PLAN C)  }\n\\end{equation}\n\n\\begin{enumerate}\n\\item Show that the entries of both $T^c(c,g)$ and  $\\boldsymbol{\\mathcal{T}}^c(c,g) $ are rational functions of $g$ with coefficients in the filed $\\mathbb{Q}(a,b,c,d,q).$ {\\em (The proof is the same as in the shift-a case.)}\n\\item Show $\\boldsymbol{\\mathcal{T}}^c(c,cq)=T^c(c,cq).$\n\\item Show $T^c$ also satisfies the discrete co-cycle condition $$T^c(c,cq^{p+1}) = T^c(cq^p,cq^{p+1})T(c,cq^p)$$ for any $p \\in \\mathbb{N}.$\n\\end{enumerate}\nSince both $T^c(c,c)$ and $\\boldsymbol{\\mathcal{T}}^c(c,c)$ are the identity, and  equation \n(\\ref{COCYC}) \nsays $\\boldsymbol{\\mathcal{T}}^c$ satisfies the discrete\nco-cycle condition, it is immediate from \nparts $2$ and $3$ above\nthat  $\\boldsymbol{\\mathcal{T}}^c(c,g)$ and $T^c(c,g)$ agree whenever $g=cq^p$ for\n$p$ a non-negative integer. Now using \npart $1$ of Proof Plan C,\nwe see that each entry of the two matrices is a rational function of $g$ agreeing with the other at infinitely many points. So they must agree\n$\\big($as rational functions with coefficients in $\\mathbb{Q}(a,b,c,d,q) \\big)$ for all $g.$ \n\n\\section{Appendix C2: The True $\\boldsymbol{\\mathcal{T}}^c(c,cq)$ Matches the $\\boldsymbol{T}^c(c,cq) $ of Theorem \n\\ref{ETC2}\n\\label{SHIFTC2}\n}\n\n\\ifJOLT \\begin{Proposition} \\else \\begin{props} \\fi\n\\label{LCDGTRC}\n\\begin{eqnarray}\n\\boldsymbol{\\mathcal{T}}^{c00}_{n-1, n}(c,g) &=&\\tau^c_{n-1,n}= -\\Delta_{cg} \\lambda_{n-1,n} +\\left[ \\mu_{n-1,-n}(g) \\right] \\Delta_{ae} \\mu_{-n,n} \\label{DELTA00C} \\\\\n \\boldsymbol{\\mathcal{T}}^{c01}_{n, n}(c,g) &=&\\sigma^c_{n,-(n+1)}=-\\Delta_{cg}\\mu_{n,-(n+1)}  \\label{DELTA01C}  \\\\\n\\boldsymbol{\\mathcal{T}}^{c10}_{n, n+1}(c,g) &=&\\sigma^c_{-(n+1),n+1}= -\\Delta_{cg}\\mu_{-(n+1),n+1} \\label{DELTA10C}  \\\\\n\\boldsymbol{\\mathcal{T}}^{c11}_{n-1,n}(c,g)&=&\\tau^c_{-n,-(n+1)} =-\\Delta_{cg} \\lambda_{-n, -(n+1)} +\\left[ \\mu_{-n,n}(g) \\right] \\Delta_{cg} \\mu_{n,-(n+1)} \n\\end{eqnarray}\n\\ifJOLT \\end{Proposition} \\else \\end{props} \\fi\n\\begin{proof}\nWe write down the proof of (\\ref{DELTA00C}) and  (\\ref{DELTA10C}), the other two being similar.\n\nRecall $\\lambda_{r,r}=1$ for all parameters and any sign of $r.$ Then  modulo $\\mathcal{R}_{-(n-1)}:$\n\\begin{eqnarray*}\nE_{n-1}(g) =\\left[  \\lambda_{n-1,n-1}(g)\\right] f_{n-1} & \\Rightarrow & f_{n-1}=E_{n-1}(g)\\\\\nE_{-n}(g)= \\left[ \\lambda_{-n,-n}(g) \\right] h_n + \\left[ \\mu_{n-1,-n}(g) f_{n-1}(g) \\right] & \\Rightarrow \\\\\n  h_{n}=E_{-n}(g) - \\left[ \\mu_{n-1,-n}(g) \\right] E_{n-1}(g) \\\\\nf_n = E_n(g) -  \\left[ \\mu_{-n,n}(g) \\right] h_n - \\left[ \\lambda_{n-1,n}(g) \\right] f_{n-1} & \n\\end{eqnarray*}\n$$\\hphantom{f_n}    =E_n(g)   -  \\left[ \\mu_{-n,n}(g) \\right] \\big( E_{-n}(g) - \\left[ \\mu_{n-1,-n}(g) \\right] E_{n-1}(g)  \\big) - \\left[  \\lambda_{n-1,n}(g)\\right] E_{n-1}(g) $$\nSo\n\\begin{multline*}\nE_n(c) = f_n+ \\left[ \\mu_{-n,n}(c) \\right] h_n + \\left[  \\lambda_{n-1,n}(c)\\right] f_{n-1} \\\\\n= E_n(g)   -  \\left[ \\mu_{-n,n}(g) \\right] \\big( E_{-n}(g) - \\left[ \\mu_{n-1,-n}(g) \\right] E_{n-1}(g)  \\big) - \\left[  \\lambda_{n-1,n}(g)\\right] E_{n-1}(g) \\\\\n+ \\left[ \\mu_{-n,n}(c) \\right] \\big( E_{-n}(g) - \\left[ \\mu_{n-1,-n}(g) \\right] E_{n-1}(g) \\big) +  \\left[  \\lambda_{n-1,n}(c)\\right] E_{n-1}(g) .\n\\end{multline*}\nCombining terms and comparing coefficients of $E_{-n}(g)$ (for  (\\ref{DELTA10C}))  and $E_{n-1}(g)$ (for  (\\ref{DELTA00C}))with the definition\n$$\nE_n(c)= \\tau^c_{n,n}E_n(g) + \\sigma^c_{-n,n} E_{-n}(g) + \\tau^c_{n-1,n}E_{n-1}(g) \\ \\ \\  \\text{ mod }\\mathcal{R}_{-(n-1)}\n$$\ngives the asserted formulas.\n\\end{proof}\n\nThe zig-zag co-degree $1$ formulas (\\ref{DELTA01C}) and (\\ref{DELTA10C}) above have an obvious linearity based on \n$$\n \\Delta_{uw} f= \\Delta_{uv} f + \\Delta_{vw} f \\text{ since } \\left( f(w) - f(u) \\right) = \\left( f(v) - f(u) \\right) + \\left( f(w) - f(v) \\right).\n$$\nThis immediately implies that the zig-zag co-degree $1$ matrix entries of $T^c$ satisfy what is required by the discrete co-cycle condition. We shall need\nthese special cases in the next section, so we record them in the corollary below.\n\\ifJOLT \\begin{Corollary} \\else \\begin{cors} \\fi\n\\label{CDG1LINC}\nFor any $c,g,\\text{ and } f,$\n\\begin{eqnarray*}\n\\sigma^c_{n,-(n+1)}(c,g) &= &\\sigma^c_{n,-(n+1)}(f,g) +  \\sigma^c_{n,-(n+1)}(c,f) \\\\\n\\sigma^c_{-(n+1), n+1}(c,g) &= & \\sigma^c_{-(n+1), n+1}(f,g) +  \\sigma^c_{-(n+1), n+1}(c,f) .\n\\end{eqnarray*}\n\\ifJOLT \\end{Corollary} \\else \\end{cors} \\fi\n\n\\ifJOLT \\begin{Proposition} \\else \\begin{props} \\fi\n\\label{CDG1CGQ}\n\nIn the special case of $g=cq,$ the zig-zag co-degree $1$ transition functions satisfy\n\\begin{eqnarray*}\n\\mathcal{T}^{c01}_{n,n}(c,cq) &=&\\sigma^c_{n, -(n+1)}(c,cq) = -\\frac{  abcq^{n+1} (ad q^n, bd  q^n, q^{-1} | q )_{1}   }  {  (abcdq^{2n}, abcdq^{2n+1} | q)_{1}   } \\\\\n\\mathcal{T}^{c10}_{n, n+1}(c,cq) &=& \\sigma^c_{-(n+1), n+1}(c,cq) = \n -\\frac{cd    q^{n+1}   (q^{n+1},abq^{n+1},  q^{-1} | q )_{1}  }  { (abcdq^{2n+1}, abcdq^{2(n+1)} | q)_{1}  } \n\\end{eqnarray*}\n\\ifJOLT \\end{Proposition} \\else \\end{props} \\fi\n\\begin{proof}\n\\begin{enumerate}\n\\item \n$$  \\mathcal{T}^{c01}_{n, n}(c,cq)  =\\sigma^c_{n, -(n+1)}(c,q) =-\\Delta_{cg}\\mu_{n,-(n+1)} $$\nSo\n\\begin{multline*\n\\mathcal{T}^{c01}_{n,n}(c,cq) = - \\Big\\{ \\Big\\{ - \\frac{ \\left[ ( a bq^{n} | q)_1 -1 \\right] (cq+d) + ( a +b)  }{  ( a b\\left[ cq \\right] dq^{2n} | q)_1 } \\Big\\} \\\\\n- \\Big\\{  - \\frac{ \\left[ (abq^{n} | q)_1 -1 \\right] (c+d) + (a+b)  }{  (abcdq^{2n} | q)_1 }\\Big\\} \\Big\\} \\\\%\\hspace{50mm}  \\\\\n=\\left( \\frac{1} {(abcdq^{2n} | q)_1 (abcdq^{2n+1} | q)_1  } \\right) \\cdot \\\\\n\\Big\\{ (abcdq^{2n} | q)_1\\left[  - abq^{n} (cq+d) +(a+b)\\right] \\\\\n - (abcdq^{2n+1} | q)_1\\left[  - abq^{n} (c+d) +(a+b)\\right] \\Big\\}  \\\\\n=  \\left( \\frac{ 1 }{ (abcdq^{2n}, abcdq^{2n+1} | q)_1 } \\right) \\Big\\{  - abcq^{n+1} \\\\\n+abcq^n -abcdq^{2n}(-abdq^n+a+b)\n+abcdq^{2n+1}(-abdq^n+a+b)\\Big\\} \\\\\n=   \\frac{ abcq^n(1-q) }{ (abcdq^{2n}, abcdq^{2n+1} | q)_1 }\\Big\\{1 +abd^2q^{2n} -d(a+b)q^n\\Big\\} \\\\\n=   \\frac{ abcq^n \\left[ -q(q^{-1} | q)_1 \\right]}{ (abcdq^{2n}, abcdq^{2n+1} | q)_1 }\\Big\\{(1 -adq^n)(1 -bdq^n)\\Big\\}.\n\\end{multline*\n\\item\n $$ \\mathcal{T}^{c10}_{n, n+1}(c,cq) = \\sigma^c_{-(n+1), n+1}(c,cq)= -\\Delta_{cg}\\mu_{-(n+1),n+1} $$\n So\n\\bmls\n\\mathcal{T}^{c10}_{n,n+1}(c,cq) = - \\Big\\{ \\Big\\{   - \\frac{ (\\left[ cq \\right] dq^{n } | q)_1 ( q^{n+1} | q  )_1 } {  ( a b \\left[ cq \\right] dq^{2n+1} | q)_1 } \\Big\\} \n - \\Big\\{ - \\frac{ (cdq^{n} | q)_1 ( q^{n+1} | q  )_1 } {  (abcdq^{2n+1} | q)_1 }  \\Big\\} \\Big\\}\\\\% \\hspace{50mm}  \\\\\n = \\frac{ ( q^{n+1} | q  )_1  }{ (abcdq^{2n+1} | q)_1 (abcdq^{2(n+1)} | q)_1 }\\Big\\{(abcdq^{2n+1} | q)_1 (cdq^{n+1} | q)_1 \\\\\n -(abcdq^{2(n+1)} | q)_1 (cdq^{n} | q)_1\\Big\\}\\\\\n  = \\frac{ ( q^{n+1} | q  )_1  }{ (abcdq^{2n+1} , abcdq^{2(n+1)} | q)_1 }\\Big\\{ - abcdq^{2n+1} - cdq^{n+1} + abcdq^{2(n+1)} + cdq^{n} \\Big\\}\\\\\n    = \\frac{ ( q^{n+1} | q  )_1  }{ (abcdq^{2n+1}, abcdq^{2(n+1)} | q)_1 }\\Big\\{ - abcdq^{2n+1}(q | q)_1  + cdq^{n}(q | q)_1 \\Big\\}\\\\\n        = \\frac{cdq^n (q, q^{n+1}, abq^{n+1} | q  )_1  }{ (abcdq^{2n+1} , abcdq^{2(n+1)}  | q)_1 }\\\\\n                = \\frac{cdq^n\\left[-q (q^{-1} | q)_1 \\right] ( q^{n+1}, abq^{n+1} | q  )_1  }{ (abcdq^{2n+1} , abcdq^{2(n+1)} | q)_1 } .\n\\end{multline*}\n\\end{enumerate}\n\n\\end{proof}\n\n\\ifJOLT \\begin{Proposition} \\else \\begin{props} \\fi\n\\label{CDG2CGQ}\n\nIn the special case of $g=cq,$ the zig-zag co-degree $2$ transition functions satisfy:\n\\ \n\\begin{eqnarray*}\n\\boldsymbol{\\mathcal{T}}^{c00}_{n-1, n}(c,cq) =\\tau^c_{n-1,n}= - \\frac{   c  (q^n, abq^n,ad q^{n-1}, bd q^{n-1}| q )_{1}    }  { \\left[ (abcdq^{2n-1} | q)_{1}  \\right]^2  } \\\\\n\\boldsymbol{\\mathcal{T}}^{c11}_{n-1,n}(c,cq) =\\tau^c_{-n,-(n+1)}= - \\frac{   c  (q^n, abq^n,ad q^{n}, bd q^{n}| q )_{1}    }  { \\left[ (abcdq^{2n} | q)_{1}  \\right]^2  }\n\\end{eqnarray*}\n\n\\ifJOLT \\end{Proposition} \\else \\end{props} \\fi\n\n\\begin{proof}\n\\begin{enumerate}\n\\item\nBy proposition \\ref{LCDGTRC}\n$$\\boldsymbol{\\mathcal{T}}^{c00}_{n-1, n}(c,cq) =-\\Delta_{cg} \\lambda_{n-1,n} +\\left[ \\mu_{n-1,-n}(g) \\right] \\Delta_{cg} \\mu_{-n,n} .$$\nSo\n\\begin{multline*}\n\\boldsymbol{\\mathcal{T}}^{c00}_{n-1,n}(c,c\nq)  = \\\\ - \\Big\\{\n\\Big\\{-{\\displaystyle \\frac{(q^n| q)_1}{(q| q)_1  (ab \\left[ cq \\right] dq^{2n-1}| q)_1}} \\left[(\\left[ cq\n \\right]+d)(\\ a bq^{n}| q)_1 + q( a +b) (\\left[ cq \\right] dq^{n-1}| q)_1 \\right] \\Big\\}\\\\%\\hspace{10mm}  \\\\\n\n   - \\Big\\{ -{\\displaystyle \\frac{(q^n| q)_1}{(q| q)_1  (abcdq^{2n-1}| q)_1}} \\left[(c+d)(abq^{n}| q)_1 + q(a+b) (cdq^{n-1}| q)_1 \\right]\\Big\\}\\Big\\} \\\\%\\hspace{10mm} \\\\\n \n   + \\Big\\{  - \\frac{ \\left[ ( abq^{n-1} | q)_1 -1 \\right] (cq+d) + (   a +b)  }{  ( ab\\left[ cq \\right] dq^{2(n-1)} | q)_1 } \\Big\\} \\cdot \n   \\Big\\{ \\\\%\\hspace{30mm} \\\\\n  \n   \\Big\\{- \\frac{ (\\left[ cq \\right] dq^{n - 1} | q)_1 ( q^n| q  )_1 } {  ( ab \\left[ cq \\right] dq^{2n-1} | q)_1 }  \\Big\\} \n  \n-  \\Big\\{ - \\frac{ (cdq^{n - 1} | q)_1 ( q^n| q  )_1 } {  (abcdq^{2n-1} | q)_1 }  \\Big\\}\\Big\\} . \n\\end{multline*}\n\nUsing the notation $\\big[a,b,c,\\dots\\big]=(a| q)_1(b| q)_1(c| q)_1\\ldots,$ \nthe abbreviation variables $u=q^{n-1},y=abcd$\nand multiplying by \n$$\n \\frac{ (q,q)_1 \\left[ ( yq^{2n-1} | q)_1 \\right]^2 ( yq^{2n} | q)_1 } {(q^n| q)_1}=\\frac{ \\left[ q,yu^2q , yu^2q , yu^2q^{2}\\right] } {(uq| q)_1}\n$$\nit suffices to show the vanishing of \n\\begin{multline*} \np_1=\\Big\\{c \\big[q,  yu^2q^{2}, abuq, adu, bdu \\big]    \\Big\\} \\\\\n+ \\Big\\{ \\left[ ( yu^2q| q)_1\\right]^2  \\left[(cq+d)(a buq| q)_1 + q(  a  +b) (cd uq | q)_1 \\right] \\Big\\}\\\\\n- \\Big\\{   ( yu^2q | q)_1  ( yu^2q^{2} | q)_1 \\left[(c+d)(abuq| q)_1 + q(a+b) (cdu| q)_1 \\right] \\Big\\}  \\\\ \n+\\Big\\{   (q,q)_1 ( yu^2q | q)_1 \\Big[ \\left[ (abu | q)_1 -1 \\right] (cq+d) + (   a  +b) \\Big]  (cduq | q)_1\\Big\\} \\\\\n -  \\Big\\{   (q,q)_1( yu^2q^{2} | q)_1 \\Big[ \\left[ ( abu | q)_1 -1 \\right] (cq+d) + (   a  +b) \\Big] (cdu | q)_1 \\Big\\} \\\\\n = \\Big\\{c \\big[q,  yu^2q^{2}, abuq, adu, bdu \\big]    \\Big\\} \\\\\n + ( yu^2q| q)_1 \\Big\\{ (a buq| q)_1 \\Big\\{ c\\big\\{ ( yu^2q| q)_1  (q) -  ( yu^2q^{2} | q)_1 (1)  \\big\\} \\\\\n                                                                 + d  \\big\\{ ( yu^2q | q)_1 -  ( yu^2q^{2} | q)_1 \\big\\} \\Big\\} \\\\\n                                  +q(a+b)  \\big\\{ ( yu^2q | q)_1 (cd uq | q)_1  - ( yu^2q^2 | q)_1  (cd u | q)_1 \\big\\} \\Big\\}\\\\\n                                  +(q|q)_1\\big\\{ -abu(cq+d) +(a+b) \\big\\} \\big\\{( yu^2q | q)_1 (cd uq | q)_1  - ( yu^2q^2 | q)_1  (cd u | q)_1  \\big\\}\n\\end{multline*}\n\nNote (keeping in mind, e.g., $(yu^2q)(q) - (yu^2q^2)(1) =0$) that\n\\begin{eqnarray*}\n( yu^2q | q)_1  (q) -  ( yu^2q^{2} | q)_1 (1)  &= &q-1  = -  (q| q)_1 \\\\\n( yu^2q | q)_1 -  ( yu^2q^{2} | q)_1&=&- yu^2q (q| q)_1 \n\\end{eqnarray*}\n\\begin{multline*}\n( yu^2q | q)_1 (cd uq | q)_1  - ( yu^2q^2 | q)_1  (cd u | q)_1 \\\\\n= -yu^2q -cduq +yu^2q^2 +cdu \\\\\n=u(q|q)_1 (cd -yuq) \\\\\n= u(q|q)_1(cd -abcdq^n)  = cdu (q|q)_1 (abuq| q)_1 \n\\end{multline*} \n So $p_1$ is also equal to \n\\begin{multline*}\np_2 = \\Big\\{c \\big[q,  yu^2q^{2}, abuq, adu, bdu \\big]    \\Big\\} \\\\\n + ( yu^2q| q)_1 \\Big\\{ (a buq| q)_1 \\Big\\{c \\big\\{ -  (q| q)_1 \\big\\} \\\\\n                                                                 + d  \\big\\{ - yu^2q (q| q)_1 \\big\\} \\Big\\} \\\\\n                                  +q(a+b)  \\big\\{ cdu (q|q)_1 (abuq| q)_1  \\big\\} \\Big\\}\\\\\n\n                                  +(q|q)_1\\big\\{ -abu(cq+d) +(a+b) \\big\\} \\big\\{  cdu (q|q)_1 (abuq| q)_1  \\big\\}.\n\\end{multline*}\nMultiplying $p_2$ by $\\left[ (q|q)_1\\  (abuq| q)_1 \\right]^{-1}$\nwe are reduced to showing the vanishing of\n\\begin{multline*}\np_3= \\Big\\{c \\big[  yu^2q^{2}, adu, bdu \\big]    \\Big\\} \n + ( yu^2q| q)_1 \\Big\\{ \\Big\\{ c\\big\\{ -  1 \\big\\} \n                                                                 + d  \\big\\{ - yu^2q  \\big\\} \\Big\\} \\\\\n                                  +q(a+b)  \\big\\{ cdu   \\big\\} \\Big\\}\n                                  +(q|q)_1\\big\\{ -abu(cq+d) +(a+b) \\big\\} \\big\\{  cdu  \\big\\} \\\\ \n        =c\\big[  abcdu^2q^{2}, adu, bdu \\big] + ( abcdu^2q| q)_1\\big\\{ -c -abcd^2u^2q +cduq(a+b)\\big\\}\\\\\n        +cdu(q|q)_1\\big\\{ -abu(cq+d) +(a+b) \\big\\} .\n\\end{multline*}\nThis may be viewed as an at most degree $2$ polynomial in $q$ with coefficients in the field $\\mathbb{Q}(a,b,c,d,u).$\n\nThe coefficient of $q^2$ is\n\\begin{multline*}\n(-abcdu^2) c(1-adu)(1-bdu) + ( -abcdu^2)\\big\\{( -abcd^2u^2) \\\\ +cdu(a+b)\\big\\}  \n+(cdu)(-1) (-abcu)\\\\\n=(-abcdu^2\\big\\{  c(1-adu)(1-bdu) -abcd^2u^2 +cdu(a+b) -c \\big\\} =0.\n\\end{multline*}\nSo $p_3$ is  an at most degree $1$ polynomial in $q$ with coefficients in the field $\\mathbb{Q}(a,b,c,d,u).$\nEvaluating:\n\\begin{description}\n\\item[at $q=0$ ] \n$$\np_3(0) = c(1-adu)(1-bdu)  -c +cdu( -abdu + a +b) =0.\n$$\n\\item[at $q= (abcdu^2)^{-1} $ ] When  $q= (abcdu^2)^{-1} ,$ note   $abcdu^{2}q^2=q$ and $abc^2d u^2q=c.$ \nSo\n\\begin{multline*}\np_3\\left( (abcdu^2)^{-1} \\right)= c[q,adu,bdu] +(q|q)_1 \\big\\{ -c -abcd^2u^2 +(a+b)(cdu)\\big\\}\\\\\n=c(q|q)_1\\big\\{(1-adu)(1-bdu) -1 -abd^2u^2 + du(a+b)\\big\\}=0.\n\\end{multline*}\n\\end{description}\nSince $p_3,$ when viewed as a degree at most $1$ polynomial in $q$ with coefficients in a field, vanishes at $2$ points, it must be the\nzero polynomial, proving the first formula.\n\\item By proposition \\ref{LCDGTRC}\n$$\n\\boldsymbol{\\mathcal{T}}^{c11}_{n-1,n}(c,cq) =\\\\ -\\Delta_{cg} \\lambda_{-n, -(n+1)} +\\left[ \\mu_{-n,n}(g) \\right] \\Delta_{cg} \\mu_{n,-(n+1)} .\n$$\nSo\n\\begin{multline*}\n\\boldsymbol{\\mathcal{T}}^{c11}_{n-1,n}(a,aq) =\\\\- \\Big\\{ \n\\Big\\{   -{\\displaystyle \\frac{(q^n| q)_1}{(q| q)_1 ( ab\\left[ cq \\right] dq^{2n}| q)_1}} \\left[( \\left[ cq \\right] +d)(  a bq^{n+1}| q)_1 + q( a +b) ( \\left[ cq \\right] dq^{n-1}| q)_1 \\right] \\Big\\}  \\\\\n - \\Big\\{  -{\\displaystyle \\frac{(q^n| q)_1}{(q| q)_1 (abcdq^{2n}| q)_1}} \\left[(c+d)(abq^{n+1}| q)_1 + q(a+b) (cdq^{n-1}| q)_1 \\right]   \\Big\\} \\Big\\}   \\\\\n\n+\\Big\\{  - \\frac{ ( \\left[ cq \\right] dq^{n - 1} | q)_1 ( q^n| q  )_1 } {  ( ab\\left[ cq \\right] dq^{2n-1} | q)_1 }\\Big\\} \\cdot  \\Big\\{   \\\\\n \\Big\\{    \\frac{ a bq^{n}  (  \\left[ cq \\right] +d) - (  a +b)  }{  ( ab \\left[ cq \\right] dq^{2n} | q)_1 }\\Big\\}  \\\\\n\n -   \\Big\\{ \\frac{  abq^{n} (c+d) - (a+b)  }{  (abcdq^{2n} | q)_1 } \\Big\\}\\Big\\}  .\n\\end{multline*}  \nUsing the notation $\\big[a,b,c,\\dots\\big]=(a| q)_1(b| q)_1(c| q)_1\\ldots,$ \nthe abbreviation variables $u=q^{n-1},y=abcd$\nand multiplying by \n$$\n \\frac{ (q,q)_1  \\left[ (yu^2q^2| q)_1 \\right]^2 (yu^2q^3| q)_1 } {(u q | q)_1  }=\\frac{\\left[ q,yu^2q^2, yu^2q^2, yu^2q^3 \\right] } { (u q | q)_1},\n$$\nit suffices to show the vanishing of\n\\begin{multline*} \np_1=\\Big\\{ c\\left[ q, abuq , aduq, bduq, yu^2q^3 \\right]     \\Big\\} \\\\\n+ \\Big\\{ \\left[ (yu^{2}q^2, yu^2q^2 \\right]   \\left\\{ (cq+d)(a buq^{2}| q)_1 + q(a +b) (cd uq | q )_{1} \\right\\} \\Big\\} \\\\\n- \\Big\\{   \\left[ yu^2q^2, yu^2q^3 \\right] \\left\\{(c+d)(abuq^2| q)_1 + q(a+b)(cd u | q )_{1}  \\right\\}\\Big\\}  \\\\\n+ \\Big\\{  \\left[ q, yu^2q^2, cduq \\right] \\Big\\{    -abuq (cq+d) +  (a+b)\\Big\\} \\\\\n -  \\left[ q, yu^2q^3, cduq \\right] \\Big\\{   - abu q (c+d) + (a+b)  \\Big\\}  \\Big\\} \\\\\n = \\Big\\{ c\\left[ q, abuq , aduq, bduq, yu^2q^3 \\right]     \\Big\\} \\\\\n + (yu^{2}q^2 | q)_1 \\Big\\{  (a buq^{2}| q)_1 \\Big\\{c \\big\\{ ( yu^2q^2 | q)_1  (q) -  ( yu^2q^{3} | q)_1 (1) \\big\\} \\\\\n                      + d \\big\\{ ( yu^2q^2 | q)_1 -  ( yu^2q^{3} | q)_1 \\big\\} \\Big\\} \\\\\n                      + q(a+b)\\big\\{ ( yu^2q^2 | q)_1 (cd uq | q)_1  - ( yu^2q^3 | q)_1  (cd u | q)_1 \\big\\} \\Big\\} \\\\\n                     + [q, cduq] \\Big\\{ (-abuq) \\Big\\{ c \\big\\{ ( yu^2q^2 | q)_1  (q) -  ( yu^2q^{3} | q)_1 (1) \\big\\}  \\\\\n                     + d \\big\\{  ( yu^2q^2 | q)_1 -  ( yu^2q^{3} | q)_1 \\big\\} \\Big\\} \\\\\n                              + (a+b) \\big\\{  ( yu^2q^2 | q)_1 -  ( yu^2q^{3} | q)_1 \\big\\}       \\Big\\} .\n\\end{multline*}\nNote (keeping in mind, e.g., $(yu^2q^2)(q) - (yu^2q^3)(1) =0$) that\n\\begin{eqnarray*}\n( yu^2q^2 | q)_1  (q) -  ( yu^2q^{3} | q)_1 (1)  &=& q-1  = -  (q| q)_1 \\\\\n( yu^2q^2 | q)_1 -  ( yu^2q^{3} | q)_1&=& - yu^2q^2 (q| q)_1 \n\\end{eqnarray*}\n\\begin{multline*}\n( yu^2q^2 | q)_1 (cd uq | q)_1  - ( yu^2q^3 | q)_1  (cd u | q)_1 \\\\\n= -yu^2q^2 -cduq +yu^2q^3 +cdu \\\\\n=u(q|q)_1 (cd -yuq^2) \\\\\n= u(q|q)_1(cd -abcdq^{n+1}) \\\\ = cdu (q|q)_1 (abuq^2 | q)_1 .\n\\end{multline*} \nSo $p_1$ is also equal to \n\\begin{multline*}\np_2 =  \\Big\\{ c\\left[ q, abuq , aduq, bduq, yu^2q^3 \\right]     \\Big\\} \n + (yu^{2}q^2 | q)_1 \\Big\\{  (a buq^{2}| q)_1 \\Big\\{c \\big\\{  -  (q| q)_1 \\big\\} \\\\\n                      + d \\big\\{  - yu^2q^2 (q| q)_1 \\big\\} \\Big\\} \n                      + q(a+b)\\big\\{  cdu (q|q)_1 (abuq^2 | q)_1  \\big\\} \\Big\\} \\\\\n\n                     + [q, cduq] \\Big\\{ (-abuq) \\Big\\{ c \\big\\{  -  (q| q)_1 \\big\\}                       \n\n                     + d \\big\\{  - yu^2q^2 (q| q)_1  \\big\\} \\Big\\} \\\\\n                              + (a+b) \\big\\{  - yu^2q^2 (q| q)_1 \\big\\}       \\Big\\} .                     \n\n\\end{multline*}\nMultiplying $p_2$ by $\\left[ (q|q)_1\\   \\right]^{-1}$\nwe are reduced to showing the vanishing of\n\\begin{multline*}\np_3=  \\Big\\{ c\\left[ abuq , aduq, bduq, yu^2q^3 \\right]     \\Big\\} \\\\\n + (yu^{2}q^2 | q)_1 \\Big\\{  (a buq^{2}| q)_1 \\Big\\{c \\big\\{  -  1 \\big\\} \n                      + d \\big\\{  - yu^2q^2 \\big\\} \\Big\\} \\\\\n                      + q(a+b)\\big\\{  cdu (abuq^2 | q)_1  \\big\\} \\Big\\} \\\\\n                     + [q, cduq ] \\Big\\{ (-abuq) \\Big\\{ c \\big\\{  -  1\\big\\}                        \n                     + d \\big\\{  - yu^2q^2  \\big\\} \\Big\\} \n                              + (a+b) \\big\\{  - yu^2q^2  \\big\\}       \\Big\\} .\n\\end{multline*}\nIn $p_3$ above, every appearance of $u$ has at least one power of $q$ next to it, so it\nis natural to switch to\n$$\n\\widetilde{u} =uq=q^n.\n$$\nAlso remember $y=abcd.$ So it is sufficient to show the vanishing of the polynomial\n\\begin{multline*}\np_4=   c\\left[ ab\\widetilde{u} , ad\\widetilde{u}, bd\\widetilde{u}, abcd\\widetilde{u}^2q \\right]     \\\\\n + (abcd\\widetilde{u}^{2} | q)_1 \\Big\\{ - (a b\\widetilde{u}q | q)_1 \\Big\\{c \n                      +    abcd^2\\widetilde{u}^2  \\Big\\}\n                      +   cd(a+b)\\widetilde{u} (ab\\widetilde{u}q | q)_1   \\Big\\} \\\\\n                     + [q, cd\\widetilde{u} ] \\Big\\{ ab\\widetilde{u} \\Big\\{ c                        \n                     +    abcd^2\\widetilde{u}^2  \\Big\\} \n                              -     abcd\\widetilde{u}^2   (a+b)     \\Big\\} \n\\end{multline*}\nas an at most degree $1$ polynomial in $q$ with coefficients in the field \\\\ $\\mathbb{Q}(a,b,c,d,\\widetilde{u}).$ \nEvaluating at the two points\n\\begin{description}\n\\item[at $q=1$]\n$$p_4(1 )= \\left[ abcd\\widetilde{u}^2, ab \\widetilde{u} \\right] \\big\\{  c \\left[ ad \\widetilde{u}, bd \\widetilde{u} \\right]  -(c+abcd^2\\widetilde{u}^2) +cd\\widetilde{u}(a+b)\\big\\} =0 . $$\n\\item[at $q= (abcd\\widetilde{u}^2)^{-1} $ ] When  $q= (abcd\\widetilde{u}^2)^{-1} ,$ note   $ abcd\\widetilde{u}^{2} =q^{-1}, $ \\  $ab\\widetilde{u}q=\\left[ cd\\widetilde{u} \\right]^{-1} $ and $\\left[ ab\\widetilde{u} \\right]^{-1} = cd\\widetilde{u} q .$ \nSo\n\\begin{multline*}\np_4\\left( (abcd\\widetilde{u}^2)^{-1} \\right)=  (abcd\\widetilde{u}^{2} | q)_1 \\Big\\{ - (a b\\widetilde{u}q | q)_1 \\Big\\{c \n                      +    abcd^2\\widetilde{u}^2  \\Big\\} \\\\\n                      +   cd(a+b)\\widetilde{u} (ab\\widetilde{u}q | q)_1   \\Big\\} \\\\\n                     + [q, cd\\widetilde{u} ] \\Big\\{ ab\\widetilde{u} \\Big\\{ c                        \n                     +    abcd^2\\widetilde{u}^2  \\Big\\} \n                              -     abcd\\widetilde{u}^2   (a+b)     \\Big\\} \\\\\n=  (q^{-1}| q)_1 (a b\\widetilde{u}q | q)_1 \\Big\\{ -  \\left( c \n                      +    dq^{-1} \\right) \n                      +   cd(a+b)\\widetilde{u}  \\Big\\} \\\\\n                     + [q, ( ab\\widetilde{u}q )^{-1} ] \\Big\\{ ab\\widetilde{u} \\Big\\{ c                        \n                     +    d q^{-1} \\Big\\} \n                              -    q^{-1}  (a+b)     \\Big\\} \\\\ \n  =  (q^{-1}| q)_1 (a b\\widetilde{u}q | q)_1 \\Big\\{ -  \\left( c \n                      +   d q^{-1} \\right)\n                      +   cd(a+b)\\widetilde{u}  \\Big\\} \\\\\n                     + (q^{-1})(ab\\widetilde{u} q) [q, ( ab\\widetilde{u}q )^{-1} ] \\Big\\{  \\left(  c                        \n                     +  d  q^{-1} \\right)\n                              -    q^{-1} \\left[ ab\\widetilde{u} \\right]^{-1} (a+b)     \\Big\\} \\\\        \n  =  (q^{-1}| q)_1 (a b\\widetilde{u}q | q)_1 \\Big\\{ \\Big\\{ -  \\left( c \n                      +  d  q^{-1} \\right)\n                      +   cd(a+b)\\widetilde{u}  \\Big\\} \\\\\n                     +  \\Big\\{  \\left( c                        \n                     +  d  q^{-1} \\right) \n                              -    q^{-1} \\left( cd\\widetilde{u} q \\right)(a+b)     \\Big\\} \\Big\\}=0.                                                                                                                     \n\\end{multline*}\n\\end{description}\nThus $p_4$ being of degree at most $1$ and vanishing at $2$ points implies $p_4$ is identically $0$ and the proposition proof is complete.\n\\end{enumerate}\n\n\\end{proof}\n\nPropositions \\ref{CDG1CGQ} and \\ref{CDG2CGQ} complete the proof of the second step of (\\ref{PLANC}), Proof Plan C.\n\n\\section{Appendix C3: Proof of the Discrete Co-Cycle Identity for $\\boldsymbol{T}^c$  \\label{COCYCPRFC}}\n\nThe block form of the equation \n$$T^c(cq^p,cq^{p+1}) T^c(c,cq^{p}) = T^c(c,cq^{p+1})$$\n\nis\n\\begin{multline*}\n\\begin{bmatrix}\nT^{c00} (cq^p,cq^{p+1})& T^{c01}(cq^p,cq^{p+1} \\\\\nT^{c10}(cq^p,cq^{p+1}) & T^{c11}(cq^p,cq^{p+1} \\\\\n\\end{bmatrix}\n\\begin{bmatrix}\nT^{c00}(c,cq^p) & T^{c01}(c,cq^p) \\\\\nT^{c10}(c,cq^p) & T^{c11}(c,cq^p) \\\\\n\\end{bmatrix} \\\\\n=\n\\begin{bmatrix}\nT^{c00}(c,cq^{p+1}) & T^{c01}(c,cq^{p+1}) \\\\\nT^{c10}(c,cq^{p+1}) & T^{c11}(c,cq^{p+1}) \\\\\n\\end{bmatrix}.\n\\end{multline*}\nSo we have the four matrix equations:\n\\begin{eqnarray*}\nT^{c00}(c,cq^{p+1})  =  \\Big[ T^{c00} (cq^p,cq^{p+1}) \\Big] \\Big[T^{c00}(c,cq^p) \\Big] &  +  & \\Big[T^{c01} (cq^p,cq^{p+1}) \\Big] \\Big[T^{c10}(c,cq^p) \\Big] \\\\\nT^{c01}(c,cq^{p+1})  =  \\Big[T^{c00} (cq^p,cq^{p+1}) \\Big] \\Big[T^{c01}(c,cq^p) \\Big] & +  & \\Big[T^{c01} (cq^p,cq^{p+1})  \\Big]\\Big[T^{c11}(c,cq^p) \\Big] \\\\\nT^{c10}(c,cq^{p+1})  =  \\Big[T^{c10} (cq^p,cq^{p+1}) \\Big] \\Big[T^{c00}(c,cq^p) \\Big] &+  & \\Big[T^{c11} (cq^p,cq^{p+1}) \\Big] \\Big[T^{c10}(c,cq^p) \\Big]\\\\\nT^{c11}(c,cq^{p+1})  =  \\Big[T^{c10} (cq^p,cq^{p+1}) \\Big]\\Big[T^{c01}(c,cq^p) \\Big] & +&  \\Big[T^{c11} (cq^p,cq^{p+1} ) \\Big] \\Big[T^{c11}(c,cq^p) \\Big].\n\\end{eqnarray*}\nKeep in mind, as described in the shift-a case \nat the end of section \n\\ref{ALMSYM},\nthat the entries of each $T^{cij}(cq^p,cq^{p+1})$ are zero except (possibly) on the diagonal and the superdiagonal.\n\nIn terms of $\\tau^c \\text{ and } \\sigma^c,$ these equations are:\n\\begin{multline}\n\\label{COCYC00DSPC}\n\\tau^c_{k,n}(c,cq^{p+1}) = \\big[ \\tau^c_{k,k}(cq^p,cq^{p+1})\\big] \\big[ \\tau^c_{k,n}(c,cq^p) \\big]+ \\big[ \\tau^c_{k,k+1}(cq^p,cq^{p+1})\\big] \\cdot \\big\\{ \\\\ \\big[ \\tau^c_{k+1, n}(c,cq^p)\\big] \\big\\}\n   +\\big[ \\sigma^c_{k,-(k+1)}(cq^p,cq^{p+1})\\big] \\big[ \\sigma^c_{-(k+1),n}(c,cq^p) \\big] \n   \\end{multline}  \n   \\begin{multline}\n   \\label{COCYC01DSPC}\n\\sigma^c_{k,-(n+1)}(c,cq^{p+1}) = \\big[ \\tau^c_{k,k}(cq^p,cq^{p+1})\\big] \\big[ \\sigma^c_{k,-(n+1)}(c,cq^p) \\big]+ \\big[ \\tau^c_{k,k+1}(cq^p,cq^{p+1})\\big] \\cdot \\big\\{ \\\\\\big[ \\sigma^c_{k+1, -(n+1))}(c,cq^p)\\big] \\big\\}\n    +\\big[ \\sigma^c_{k,-(k+1)}(cq^p,cq^{p+1})\\big] \\big[ \\tau^c_{-(k+1),-(n+1)}(c,cq^p) \\big]\n       \\end{multline}  \n   \\begin{multline}\n   \\label{COCYC10DSPC}\n  \\sigma^c_{-(k+1), n}(c,cq^{p+1}) = \n  \\big[ \\sigma^c_{-(k+1),k+1}(cq^p,cq^{p+1})\\big] \\big[ \\tau^c_{k+1, n}(c,cq^p)\\big] \\\\\n    +\\big[ \\tau^c_{-(k+1),-(k+1)}(cq^p,cq^{p+1})\\big]\n  \n    \\big\\{\n     \\big[ \\sigma^c_{-(k+1),n}(c,cq^p) \\big] \\big\\} \\\\\n    + \\big[ \\tau^c_{-(k+1),-(k+2)}(cq^p,cq^{p+1})\\big] \\big[ \\sigma^c_{-(k+2), n}(c,cq^p)\\big]  \n       \\end{multline}  \n   \\begin{multline}\n   \\label{COCYC11DSPC}\n \\tau^c_{-(k+1),-(n+1)}(c,cq^{p+1}) = \n    \\big[ \\sigma^c_{-(k+1),k+1}(cq^p,cq^{p+1})\\big] \\big[ \\sigma^c_{k+1, -(n+1)}(c,cq^p)\\big] \\\\\n  + \\big[ \\tau^c_{-(k+1),-(k+1)}(cq^p,cq^{p+1})\\big] \n  \\big[ \\tau^c_{-(k+1),-(n+1)}(c,cq^p) \\big] \\\\\n  + \\big[ \\tau^c_{-(k+1),-(k+2)}(cq^p,cq^{p+1})\\big] \\big[ \\tau^c_{-(k+2), -(n+1)}(c,cq^p)\\big] \n  \\end{multline}\n\\ifJOLT \\begin{Lemma} \\else \\begin{lemmas} \\fi For $k,n \\ge 0$\n\\label{URTC}\n\\ \n\n\\begin{enumerate}\n\\item $\\displaystyle \\frac{ \\tau^c_{k,n}(c,cq^{p+1}) } { \\sigma^c_{-(k+1),n}(c,cq^{p}) }= - \\frac{ q^{n-2k} (adq^k, bdq^k, q^{-(p+1)} | q)_1} {d(q^{n-k}, q^{n-k-p-1} | q )_1 }. $\n\n\\item $\\displaystyle  \\frac{ \\sigma^c_{k,-(n+1)}(c,cq^{p+1}) } { \\tau^c_{-(k+1),-(n+1)}(c,cq^{p}) } =  - \\frac{abcq\n^{n+p+1} (adq^k, bdq^k, q^{-(p+1)}  | q )_1 } {  (abcdq^{n+k}, abcdq^{n+k+p+1} | q )_1 }.  $\n\n\\item $\\displaystyle  \\frac{ \\sigma^c_{-(k+1),n}(c,cq^{p+1}) } { \\tau^c_{k+1,n}(c,cq^{p}) } = - \\frac{ cdq^{n+p} ( q^{k+1}, abq^{k+1}, q^{-(p+1)}  | q )_1 } {  ( abcdq^{n+k}, abcdq^{n+k+p+1} | q )_1 } .$\n\n\\item $\\displaystyle \\frac{ \\tau^c_{-(k+1),-(n+1)}(c,cq^{p+1}) } { \\sigma^c_{k+1,-(n+1)}(c,cq^{p}) } =  - \\frac{q^{n-2k-1} ( q^{k+1}, abq^{k+1}, q^{-(p+1)} | q )_1 } { ab ( q^{n-k}, q^{n-k-p-1} | q )_1 } . $ \n\\end{enumerate}\n\\ifJOLT \\end{Lemma} \\else \\end{lemmas} \\fi\n\n\\ifEXTRAPROOFS\n\\begin{proof}\n\\begin{enumerate}\n\\item\n\\begin{multline*}\n\\left( \n\\frac{ (q^{n-k+1} | q)_k \\left[ cq^{p+1} \\right]^{n-k}(abq^{k+1},ad q^k, bd q^k, q^{-(p+1)}  | q )_{n-k}   }  { (q | q)_k (abcdq^{n+k}, abd \\left[ cq^{p+1} \\right] q^{2k}  | q)_{n-k} } \n\\right) \\cdot \\\\\n\\left( \n - \\frac\n { (q | q)_k (abcdq^{n+k} , abd \\left[ cq^{p} \\right] q^{2k+1}  | q)_{n-k} }\n {(q^{n-k} | q)_{k+1} dq^k \\left[ cq^{p} \\right]^{n-k} (abq^{k+1}, q^{-p} | q)_{n-k} (ad q^{k+1}, bd  q^{k+1}  | q )_{n-k-1}  }  \n\n\\right) \\\\\n= - \\frac{ q^{n-2k} (adq^k, bdq^k, q^{-(p+1)} | q)_1} {d(q^{n-k}, q^{n-k-p-1} | q )_1 } .  \\\\\n\\end{multline*}\n\\item\n\\begin{multline*}\n\\left( \n-\\frac{ (q^{n-k+1} | q)_k abq^k  \\left[ cq^{p+1} \\right] ^{n-k+1}(abq^{k+1}| q)_{n-k} (ad q^k, bd  q^k,q^{-(p+1)}  | q )_{n-k+1}     }  { (q | q)_k (abcdq^{n+k} , abd \\left[ cq^{p+1} \\right] q^{2k}  | q)_{n-k+1} }  \n\\right) \\cdot \\\\\n\\left(  \n\\frac\n{ (q | q)_k  (abcdq^{n+k+1}, abd \\left[ cq^p \\right] q^{2k+1}  | q)_{n-k} }\n{(q^{n-k+1} | q)_k   \\left[ cq^p \\right] ^{n-k}(abq^{k+1},adq^{k+1}, bd q^{k+1}, q^{-p} | q )_{n-k}  }  \n\\right) \\\\\n= - \\frac{abcq\n^{n+p+1} (adq^k, bdq^k, q^{-(p+1)}  | q )_1 } {  (abcdq^{n+k}, abcdq^{n+k+p+1} | q )_1 }. \\\\\n\\end{multline*}\n\\item\n\\begin{multline*}\n\\Big(\n- \\big\\{ (q^{n-k} | q)_{k+1} dq^k \\left[ cq^{p+1} \\right]^{n-k} \\big\\} \\cdot \\\\\n \\frac\n {(abq^{k+1}, q^{-(p+1)}  | q)_{n-k} (ad q^{k+1}, bd  q^{k+1}  | q )_{n-k-1}  } \n { (q | q)_k (abcdq^{n+k} , abd \\left[ cq^{p+1} \\right] q^{2k+1}  | q)_{n-k} } \n\\Big)\n\\Big(\n\\big\\{ (q | q)_{k+1} \\big\\} \\cdot \\\\\n\\Big\\{ \\frac\n{  (abcdq^{n+k+1}, abd \\left[ cq^{p} \\right] q^{2k+2}  | q)_{n-k-1} }\n{ (q^{n-k} | q)_{k+1} \\left[ cq^{p} \\right]^{n-k-1}(abq^{k+2},ad q^{k+1}, bd q^{k+1}, q^{-p}  | q )_{n-k-1}   }  \\Big\\}\n\\Big) \\\\\n= - \\frac{ cdq^{n+p} ( q^{k+1}, abq^{k+1}, q^{-(p+1)}  | q )_1 } {  ( abcdq^{n+k}, abcdq^{n+k+p+1} | q )_1 }. \\\\\n\\end{multline*}\n\\item\n\\begin{multline*}\n\\left(\n\\frac{(q^{n-k+1} | q)_k   \\left[ cq^{p+1} \\right]^{n-k}(abq^{k+1},adq^{k+1}, bd q^{k+1}, q^{-(p+1)} | q )_{n-k}  }  { (q | q)_k  (abcdq^{n+k+1}, abd \\left[ cq^{p+1} \\right] q^{2k+1}  | q)_{n-k} }   \n\\right) \\cdot \\\\\n\\left( \n-\\frac\n{ (q | q)_{k+1} (abcdq^{n+k+1} , abd  \\left[ cq^{p} \\right] q^{2k+2}  | q)_{n-k} }\n{ (q^{n-k} | q)_{k+1} abq^{k+1}  \\left[ cq^{p} \\right]^{n-k}(abq^{k+2} | q)_{n-k-1} (ad q^{k+1}, bd  q^{k+1} , q^{-p}  | q )_{n-k}     }  \n\\right)\n\\\\\n= - \\frac{q^{n-2k-1} ( q^{k+1}, abq^{k+1}, q^{-(p+1)} | q )_1 } { ab ( q^{n-k}, q^{n-k-p-1} | q )_1 }.\\\\\n\\end{multline*}\n\\end{enumerate}\n\n\\end{proof}\n\\fi\n\n\\ifJOLT \\begin{Lemma} \\else \\begin{lemmas} \\fi  For $k,n \\ge 0:$\n\\label{VRTC}\n\\ \n\n\\begin{enumerate}\n\\item $\\displaystyle \\frac{ \\tau^c_{k,n}(c,cq^{p}) } { \\sigma^c_{-(k+1),n}(c,cq^{p}) } = - \\frac {(adq^k,bdq^k,abcdq^{n+k+p} | q)_1 } { dq^k ( q^{n-k}, abcd q^{2k+p} | q)_1} .$\n\n\\item $\\displaystyle \\frac{ \\sigma^c_{k,-(n+1)}(c,cq^{p}) } { \\tau^c_{-(k+1),-(n+1)}(c,cq^{p}) } = - \\frac{abc q^{k+p}  ( adq^k, bdq^k, q^{n-k-p}  | q )_1 } {  (abcdq^{n+k}, abcdq^{2k+p}  | q )_1 }.$\n\n\\item  $\\displaystyle \\frac{ \\sigma^c_{-(k+1),n}(c,cq^{p}) } { \\tau^c_{k+1,n}(c,cq^{p}) } = - \\frac {cdq^{k+p}(q^{k+1},q^{n-k-p-1}, abq^{k+1} | q )_1 } { (abcdq^{n+k}, abcd q^{2k+p+1} | q )_1}.$\n\n\\item $\\displaystyle \\frac{ \\tau^c_{-(k+1),-(n+1)}(c,cq^{p}) } { \\sigma^c_{k+1,-(n+1)}(c,cq^{p}) } =  - \\frac{ ( q^{k+1}, abq^{k+1}, abcdq^{n+k+p+1}  | q )_1 } { abq^{k+1}  ( q^{n-k}, abcdq^{2k+p+1}  | q )_1 }  . $\n\n\\end{enumerate}\n\\ifJOLT \\end{Lemma} \\else \\end{lemmas} \\fi\n\n\\ifEXTRAPROOFS\n\n\\begin{proof}\n\n\\begin{enumerate}\n\\item\n\\begin{multline*}\n\\left( \n\\frac{ (q^{n-k+1} | q)_k \\left[ cq^{p} \\right]^{n-k}(abq^{k+1},ad q^k, bd q^k, q^{-p}  | q )_{n-k}   }  { (q | q)_k (abcdq^{n+k}, abd \\left[ cq^{p} \\right] q^{2k}  | q)_{n-k} }\n\\right) \\cdot \\\\\n\\left( \n- \\frac\n { (q | q)_k (abcdq^{n+k} , abd \\left[ cq^{p} \\right] q^{2k+1}  | q)_{n-k} }\n {(q^{n-k} | q)_{k+1} dq^k \\left[ cq^{p} \\right]^{n-k} (abq^{k+1}, q^{-p}  | q)_{n-k} (ad q^{k+1}, bd  q^{k+1}  | q )_{n-k-1}  }  \n\n\\right) \\\\\n= - \\frac {(adq^k,bdq^k,abcdq^{n+k+p} | q)_1 } { dq^k ( q^{n-k}, abcd q^{2k+p} | q)_1}  .\\\\\n\\end{multline*}\n\\item\n\\begin{multline*}\n\\left(  \n-\\frac{ (q^{n-k+1} | q)_k abq^k  \\left[ cq^p \\right] ^{n-k+1}(abq^{k+1} | q)_{n-k} (ad q^k, bd  q^k,q^{-p}  | q )_{n-k+1}     }  { (q | q)_k (abcdq^{n+k} , abd \\left[ cq^p \\right] q^{2k}  | q)_{n-k+1} } \n\\right) \\cdot \\\\\n\\left(\n\\frac\n{ (q | q)_k  (abcdq^{n+k+1}, abd \\left[ cq^p \\right] q^{2k+1}  | q)_{n-k} }\n{(q^{n-k+1} | q)_k   \\left[ cq^p \\right] ^{n-k}(abq^{k+1},adq^{k+1}, bd q^{k+1}, q^{-p} | q )_{n-k}  }  \n\\right) \\\\\n= - \\frac{abc q^{k+p}  ( adq^k, bdq^k, q^{n-k-p}  | q )_1 } {  (abcdq^{n+k}, abcdq^{2k+p}  | q )_1 }. \\\\\n\\end{multline*}\n\\item\n\\begin{multline*}\n\\left( \n - \\frac{(q^{n-k} | q)_{k+1} dq^k \\left[ cq^{p} \\right] ^{n-k} (abq^{k+1}, q^{-p}  | q)_{n-k} (ad q^{k+1}, bd  q^{k+1}  | q )_{n-k-1}  }  { (q | q)_k (abcdq^{n+k} , abd \\left[ cq^{p} \\right] q^{2k+1}  | q)_{n-k} } \n\\right) \\\\\n\\left( \n\\frac\n{ (q | q)_{k+1} (abcdq^{n+k+1}, abd \\left[ cq^{p} \\right] q^{2k+2}  | q)_{n-k-1} } \n{ (q^{n-k} | q)_{k+1} \\left[ cq^{p} \\right] ^{n-k-1}(abq^{k+2},ad q^{k+1}, bd q^{k+1},  q^{-p}   | q )_{n-k-1}   }  \n\\right) \\\\\n= - \\frac {cdq^{k+p}(q^{k+1},q^{n-k-p-1}, abq^{k+1} | q )_1 } { (abcdq^{n+k}, abcd q^{2k+p+1} | q )_1}. \\\\\n\\end{multline*}\n\\item\n\\begin{multline*}\n\\left(\n\\frac{(q^{n-k+1} | q)_k   \\left[ cq^{p} \\right] ^{n-k}(abq^{k+1},adq^{k+1}, bd q^{k+1}, q^{-p} | q )_{n-k}  }  { (q | q)_k  (abcdq^{n+k+1}, abd \\left[ cq^{p} \\right] q^{2k+1}  | q)_{n-k} }  \n  \\right) \\cdot \\\\\n\\left( \n-\\frac\n{ (q | q)_{k+1} (abcdq^{n+k+1} , abd  \\left[ cq^{p} \\right] q^{2k+2}  | q)_{n-k} }\n{ (q^{n-k} | q)_{k+1} abq^{k+1}  \\left[ cq^{p} \\right]^{n-k}(abq^{k+2} | q)_{n-k-1} (ad q^{k+1}, bd  q^{k+1} , q^{-p}  | q )_{n-k}     }  \n  \\right) \\\\\n= - \\frac{ ( q^{k+1}, abq^{k+1}, abcdq^{n+k+p+1}  | q )_1 } { abq^{k+1}  ( q^{n-k}, abcdq^{2k+p+1}  | q )_1 } .  \\\\\n\\end{multline*}\n\\end{enumerate}\n\n\\end{proof}\n\\fi\n\n\n\\begin{center} \\bf The $T^{c00}$ Identity\\label{T00C} \\end{center}\n\\ifJOLT \\begin{Proposition} \\else \\begin{props} \\fi\n\\label{T00PROPC}\nWhen $0 \\le k \\le n-1$\n\\begin{multline*}\n\\tau^c_{k,n}(c,cq^{p+1}) = \\big[ \\tau^c_{k,k}(cq^p,cq^{p+1})\\big] \\big[ \\tau^c_{k,n}(c,cq^p) \\big] + \\big[ \\tau^c_{k,k+1}(cq^p,cq^{p+1})\\big] \\big[ \\tau^c_{k+1, n}(c,cq^p)\\big] \\\\\n  +\\big[ \\sigma^c_{k,-(k+1)}(cq^p,cq^{p+1})\\big] \\big[ \\sigma^c_{-(k+1),n}(c,cq^p) \\big].\n\\end{multline*}\n And \n$$  \\tau^c_{n,n}(c,cq^{p+1}) = \\big[ \\tau^c_{n,n}(cq^p,cq^{p+1})\\big] \\big[ \\tau^c_{n,n}(c,cq^p) \\big].$$\n\\ifJOLT \\end{Proposition} \\else \\end{props} \\fi\n\n\\begin{proof}\nThe second identity just says $1=1 \\cdot 1.$\n\n\nFor the first, upon dividing by $\\sigma^c_{-(k+1),n}(c,cq^p),$ we see it is sufficient to prove\n\\begin{multline*}\n\\frac{\\tau^c_{k,n}(c,cq^{p+1})}{\\sigma^c_{-(k+1),n}(c,cq^p)} = \\left\\{ \\tau^c_{k,k}(cq^p,cq^{p+1})\\right\\} \\left\\{ \\frac{\\tau^c_{k,n}(c,cq^p)}{\\sigma^c_{-(k+1),n}(c,cq^p)} \\right\\} \\\\ + \\left\\{ \\tau^c_{k,k+1}(cq^p,cq^{p+1})\\right\\} \\left\\{ \\frac{\\tau^c_{k+1, n}(c,cq^p)}{\\sigma^c_{-(k+1),n}(c,cq^p)} \\right\\} \n  +\\left\\{ \\sigma^c_{k,-(k+1)}(cq^p,cq^{p+1})\\right\\} .\n\\end{multline*}\nUsing Lemmas \\ref{URTC} and \\ref{VRTC}, this means we need to show\n\\begin{multline}\n\\label{EQN1ac}\n\\left\\{ \n- \\frac{ q^{n-2k} (adq^k, bdq^k, q^{-(p+1)} | q)_1} {d(q^{n-k}, q^{n-k-p-1} | q )_1 }\n\\right\\} \n  =   \\left\\{ 1 \\right\\} \\cdot\n\n \\left\\{  - \\frac {(adq^k,bdq^k,abcdq^{n+k+p} | q)_1 } { dq^k ( q^{n-k}, abcd q^{2k+p} | q)_1}\\right\\} \\\\\n\n+  \n\\left\\{ \\frac{ (q^{2} | q)_k \\left[ cq^{p+1} \\right] (abq^{k+1},ad q^k, bd q^k, q^{-1}  | q )_{1}   }  { (q | q)_k (ab \\left[ cq^p \\right] dq^{2k+1}, abd \\left[ cq^{p+1} \\right] q^{2k}  | q)_{1} }  \\right\\} \\cdot \\\\\n\\left\\{ \n- \\frac \n{ (abcdq^{n+k}, abcd q^{2k+p+1} | q )_1}\n{cdq^{k+p}(q^{k+1},q^{n-k-p-1}, abq^{k+1} | q )_1 } \n\\right\\} \\\\\n  + \\left\\{ \n   -\\frac{ (q | q)_k abq^k  \\left[ cq^{p+1} \\right] (ad q^k, bd  q^k,q^{-1}  | q )_{1}     }  { (q | q)_k (ab \\left[ cq^p \\right] dq^{2k} , abd \\left[ cq^{p+1} \\right] q^{2k}  | q)_{1} } \\right\\} \n\\cdot 1 \\\\\n\\end{multline}\n\n\n\nMultiplying (\\ref{EQN1ac}) by\n$$\n\\frac{ dq^{2k+p+1} (q, q^{n-k}, q^{n-k-p-1}, abcdq^{2k+p}, abcdq^{2k+p+1} | q )_1 } { (adq^k, bdq^k | q)_1 }\n$$\nwe see it is sufficient to show the vanishing of the  polynomial $p_1$ below.\n\nWe will eventually reduce this identity to the vanishing of a 1-variable polynomial in $q$ with coefficients in the field $\\mathbb{Q}(a,b,c,d,y,u,v,w)$ \nwith the property that when \n$$\ny=abcd \\hspace{5mm} u= q^n \\hspace{5mm} v= q^k\\hspace{5mm} w= q^p\n$$\nwe obtain a unit in the coefficient field times the difference between the two sides of equation (\\ref{EQN1ac}) above. So, effectively, we can use the variables $y,u,v,w$\nas abbreviations for these expressions.\n\n\\begin{multline}\np_1= \\left\\{q^{n+p+1} (yq^{2k+p} | q)_1 (yq^{2k+p+1} | q)_1  (q| q)_1       (q^{-(p+1)}| q )_{1}   \\right\\}   \\\\\n\n\n  -\\left\\{ q^{k+p+1} (yq^{n+k+p} | q)_1  (yq^{2k+p+1} | q)_1     (q| q)_1   (q^{n-k-p-1} | q )_{1}    \\right\\} \\\\\n    -\\left\\{ q^{k+p+2} (q^{n-k}|q)_1 (yq^{n+k} | q)_1  (yq^{2k+p} | q)_1      (q^{-1} | q )_{1}    \n   \n       \\right\\}  \\\\\n    - \\left\\{q^{3k+2p+2} (q^{n-k}| q)_1y (q| q)_1  (q^{n-k-p-1}|q)_1   (q^{-1} | q )_{1}  \n      \\right\\} \n\\end{multline}\n\\begin{multline*}\n\\hphantom{p_1}=\\left\\{ q^{n+p+1} (yq^{2k+p} | q)_1 (yq^{2k+p+1} | q)_1  (q| q)_1  \\left[ - q^{-(p+1)}(q^{p+1}| q)_1\\right]   \\right\\} \\\\\n\n-  \\left\\{  q^{k+p+1} (yq^{n+k+p} | q)_1  (yq^{2k+p+1} | q)_1     (q| q)_1  \\left[ q^{-(k+p+1)} \\left[ (q^{n}| q)_1-(q^{k+p+1}| q)_1 \\right] \\right]    \\right\\} \\\\\n  - \\left\\{ q^{k+p+2}  (yq^{n+k} | q)_1  (yq^{2k+p} | q)_1   \\left[ q^{-k} \\left[ (q^{n}| q)_1-(q^k| q)_1 \\right] \\right]       \\left[ - q^{-1}(q| q)_1 \\right]   \n   \n       \\right\\} \\\\\n - \\left\\{  yq^{3k+2p+2}(q| q)_1   \\left[ q^{-k} \\left[ (q^{n}| q)_1-(q^k| q)_1 \\right] \\right]   \\right. \\cdot \\\\\n\\left.  \\left[ q^{-(k+p+1)} \\left[ (q^{n}| q)_1-(q^{k+p+1}| q)_1 \\right] \\right]      \\left[- q^{-1}(q| q)_1 \\right]   \\right\\} \n\\end{multline*}\n\\begin{multline*}\n\\hphantom{p_1}=- \\left\\{ q^{n} (yq^{2k+p} | q)_1 (yq^{2k+p+1} | q)_1  (q| q)_1  \\left[ (q^{p+1}| q)_1\\right]   \\right\\} \\\\\n\n-  \\left\\{   (yq^{n+k+p} | q)_1  (yq^{2k+p+1} | q)_1     (q| q)_1   \\left[ (q^{n}| q)_1-(q^{k+p+1}| q)_1 \\right]     \\right\\} \\\\\n  + \\left\\{ q^{p+1}  (yq^{n+k} | q)_1  (yq^{2k+p} | q)_1     \\left[ (q^{n}| q)_1-(q^k| q)_1 \\right]        \\left[ (q| q)_1 \\right]   \n   \n       \\right\\} \\\\\n + \\left\\{  yq^{k+p}(q| q)_1     \\left[ (q^{n}| q)_1-(q^k| q)_1 \\right]     \n  \\left[ (q^{n}| q)_1-(q^{k+p+1}| q)_1 \\right]       \\left[ (q| q)_1  \\right]  \\right\\} \n\\end{multline*}\nUsing the notation $\\big[a,b,c,\\dots\\big]=(a| q)_1(b| q)_1(c| q)_1\\ldots,$ \nand multiplying by $\\left[  (q| q)_1 \\right]^{-1}$ \nit suffices to show the vanishing of \n\\begin{multline*}\np_2 = -u[yv^2w, yv^2wq, wq] \n        -(vwq - u)[yuvw, yv^2wq] \\\\\n       +wq(v-u)[yuv, yv^2w]\n       +yvw(v-u)(vwq-u) (q| q)_1.\n\\end{multline*}\n\nThis expression may be interpreted as a one variable polynomial of degree at most $2$ in $q$ with coefficients in the field $\\mathbb{Q}(y,u,v,w).$\n\nThe coefficient of $q^2$ is\n\\begin{multline*}\n-u (1-yv^2w) (-yv^2w) (-w) - (vw) (1-yuvw) (-yv^2w) +yvw (v-u) (vw) (-1) \\\\\n=yv^2w^2 \\big\\{ -u (1-yv^2w) - (v) (1-yuvw)(-1)+(v-u)(-1)   \\big\\} \n=0.\n\\end{multline*}\nSo the polynomial $p_2$ is of degree at most $1$ in $q.$\nEvaluating \n\\begin{description}\n\\item[at $q=0$ ] $p_2(0)=-u(1-yv^2w)+u(1-yuvw) +yvw(v-u)(-u)=0$\n\\item[at $ q=u(vw)^{-1}$ ] Note at this point $yv^2wq=yuv$ and $wq=uv^{-1}.$ So\n$$ p_2\\big(u(vw)^{-1}\\big) =- u[yv^2w, yuv](1-uv^{-1}) +uv^{-1}(v-u)[yv^2w,yuv]=0. $$\n\\end{description}\n\nThus the polynomial $p_2$ is $0$ and the proposition is proven.\n\\end{proof}\n\n\n\n\\begin{center} \\bf The $T^{c01}$ Identity \\end{center}\n\n\n\\ifJOLT \\begin{Proposition} \\else \\begin{props} \\fi\n\\label{T01PROPC}\nWhen $0 \\le k \\le n-1$\n\\begin{multline}\n\\sigma^c_{k,-(n+1)}(c,cq^{p+1}) = \\big[ \\tau^c_{k,k}(cq^p,cq^{p+1})\\big] \\big[ \\sigma^c_{k,-(n+1)}(c,cq^{p}) \\big] \\\\\n+ \\big[ \\tau^c_{k,k+1}(cq^p,cq^{p+1})\\big] \\big[ \\sigma^c_{k+1, -(n+1))}(c,cq^{p})\\big] \\\\\n  +\\big[ \\sigma^c_{k,-(k+1)}(cq^p,cq^{p+1})\\big] \\big[ \\tau^c_{-(k+1),-(n+1)}(c,cq^{p}) \\big].\n\\end{multline}\nAnd\n\\begin{multline}\n\\sigma^c_{n,-(n+1)}(c,cq^{p+1}) = \\big[ \\tau^c_{n,n}(cq^p,cq^{p+1})\\big] \\big[ \\sigma^c_{n,-(n+1)}(c,cq^{p}) \\big] \\\\\n  +\\big[ \\sigma^c_{n,-(n+1)}(cq^p,cq^{p+1})\\big] \\big[ \\tau^c_{-(n+1),-(n+1)}(c,cq^{p}) \\big].\n\\end{multline}\n\n\n\\ifJOLT \\end{Proposition} \\else \\end{props} \\fi\n\n\\begin{proof} The second is immediate from corollary \\ref{CDG1LINC} together with the observation that $\\tau^c_{r,r}=1$ for any sign of $r.$\n\n\nFor the first, upon dividing by $\\tau^c_{-(k+1),-(n+1)}(c,cq^{p}),$ we see it is sufficient to prove\n\\begin{multline*}\n\\frac{\\sigma^c_{k,-(n+1)}(c,cq^{p+1}) } {\\tau^c_{-(k+1),-(n+1)}(c,cq^{p})} \n= \\left\\{ \\tau^c_{k,k}(cq^p,cq^{p+1})\\right\\} \\left\\{ \\frac{\\sigma^c_{k,-(n+1)}(c,cq^{p})} {\\tau^c_{-(k+1),-(n+1)}(c,cq^{p})} \\right\\} \\\\\n+ \\left\\{ \\tau^c_{k,k+1}(cq^p,cq^{p+1})\\right\\} \\left\\{ \\frac{\\sigma^c_{k+1, -(n+1))}(c,cq^{p}) } {\\tau^c_{-(k+1),-(n+1)}(c,cq^{p})} \\right\\}\n  +\\left\\{ \\sigma^c_{k,-(k+1)}(cq^p,cq^{p+1})\\right\\} \\cdot 1.\n\\end{multline*}\nThat means we need to show\n\\begin{multline}\n\\label{EQN1bc}\n\\left\\{\n- \\frac{abcq\n^{n+p+1} (adq^k, bdq^k, q^{-(p+1)}  | q )_1 } {  (abcdq^{n+k}, abcdq^{n+k+p+1} | q )_1 }\n \\right\\}  \n  =   \\left\\{1  \\right\\} \n \n  \\left\\{  \n\n  - \\frac{abc q^{k+p}  ( adq^k, bdq^k, q^{n-k-p}  | q )_1 } {  (abcdq^{n+k}, abcdq^{2k+p}  | q )_1 }\n \\right\\} \n   \\\\\n \n  + \\left\\{\n \n  \\frac{ (q^{2} | q)_k \\left[ cq^{p+1} \\right](abq^{k+1},ad q^k, bd q^k, q^{-1}  | q )_{1}   }  { (q | q)_k (ab \\left[ cq^{p} \\right] dq^{2k+1}, abd \\left[ cq^{p+1} \\right] q^{2k}  | q)_{1} } \n \\right\\} \n \n   \\cdot \\\\\n \\left\\{  \n - \\frac\n { abq^{k+1}  ( q^{n-k}, abcdq^{2k+p+1}  | q )_1 }\n { ( q^{k+1}, abq^{k+1}, abcdq^{n+k+p+1}  | q )_1 } \n\n  \\right\\} \\\\\n\n + \\left\\{ \n\n  -\\frac{ (q | q)_k abq^k  \\left[ cq^{p+1} \\right]  (ad q^k, bd  q^k,q^{-1}  | q )_{1}     }  { (q | q)_k (ab \\left[ cq^{p} \\right] dq^{2k} , abd \\left[ cq^{p+1} \\right] q^{2k}  | q)_{1} } \n \\right\\} \n\n   \\cdot   \n \n  1\n\\end{multline}\n\n\n\nWe will eventually reduce this identity to the vanishing of a 1-variable polynomial in $y$ with coefficients in the field $\\mathbb{Q}(a,b,c,d,u,v,w,q)$ \nwith the property that when \n$$\ny=abcd \\hspace{5mm} u= q^n \\hspace{5mm} v= q^k\\hspace{5mm} w= q^p\n$$\nwe obtain a unit in the coefficient field times the difference between the two sides of equation (\\ref{EQN1bc})  above. So, effectively, we can use the variables $y,u,v,w$\nas abbreviations for the above expressions.\n\nMultiplying (\\ref{EQN1bc})  by  \n\\begin{equation}\n-\\frac{ (yq^{n+k}, yq^{n+k+p+1}, yq^{2k+p}, yq^{2k+p+1}  |q)_1 } { abc (adq^k, bdq^k  | q)_1 }\n\\end{equation}\nwe see it is sufficient to show the vanishing of the  polynomial $p_1$ below. We use $y$ as an abbreviation for $abcd.$\n\\begin{center} (We chose the $-1$ factor above to make $p_1$ a perfect match for the $p_1$ in the shift-a case.)\\end{center}\n\\begin{multline*}\np_1 =  -\\left\\{  (yq^{2k +p} | q)_{1}   (yq^{2k +p+1} | q)_{1}  \n   q^{n+p+1}  (q^{-(p+1)}| q)_1  \\right\\}\\\\\n  +   \\left\\{1  \\right\\}   \n  \\left\\{    (yq^{n+k +p+1} | q)_{1}  (yq^{2k +p+1} | q)_{1}  q^{k+p}  \n    (q^{n-k-p}| q)_1      \\right\\} \\\\\n\n + \\left\\{\\frac{ (q^{n-k}| q)_1(yq^{n+k} | q)_{1}   (y q^{2k+p} | q)_{1}   q^{k+p+2} \n   (q^{-1}| q)_1 \n  }  { (q| q)_1 \n   }  \\right\\} \\\\\n\n\n +  \\left\\{ (y q^{n+k} | q)_{1} (y q^{n+k+p+1} | q)_{1}    q^{k+p+1}      (q^{-1}| q)_1   \\right\\}\n   \\end{multline*}      \n\\begin{multline*}\n\\hphantom{p_1 }=  -\\left\\{  (yq^{2k +p} | q)_{1}   (yq^{2k +p+1} | q)_{1}  q^{n+p+1} \\left[   - q^{-(p+1)}(q^{p+1}| q)_1 \\right]   \\right\\}\\\\\n  +   \\left\\{1  \\right\\}   \n  \\left\\{    (yq^{n+k +p+1} | q)_{1}  (yq^{2k +p+1} | q)_{1}   q^{k+p} \n \\left[ q^{-(k+p)} \\left[ (q^n| q)_1 -(q^{k+p}| q)_1 \\right]  \\right]     \\right\\}   \\\\\n\n +\\frac{ \\left\\{ \\left[ q^{-k} \\left[(q^n| q)_1 -(q^k| q)_1 \\right]  \\right] (yq^{n+k} | q)_{1}   (y q^{2k+p} | q)_{1}  q^{k+p+2} \n  \\left[ -q^{-1}(q| q)_1 \\right] \n  \\right\\}}  { (q| q)_1 } \\\\\n \n +  \\left\\{ (y q^{n+k} | q)_{1} (y q^{n+k+p+1} | q)_{1}    q^{k+p+1}     \\left[ -q^{-1}(q| q)_1 \\right]  \\right\\}\n  \\end{multline*}\n \\begin{multline*}\n\\hphantom{p_1 }=  \\left\\{  (yq^{2k +p} | q)_{1}   (yq^{2k +p+1} | q)_{1}   q^{n} \\left[ (q^{p+1}| q)_1 \\right]   \\right\\}\\\\\n  +   \\left\\{1  \\right\\}   \n  \\left\\{    (yq^{n+k +p+1} | q)_{1}  (yq^{2k +p+1} | q)_{1}   \n   \\left[ \\left[ (q^{n}| q)_1-(q^{k+p}| q)_1\\right]    \\right]\n      \\right\\} \\\\\n\n - \\left\\{  q^{p+1} (yq^{n+k} | q)_{1}   (y q^{2k+p} | q)_{1}  \n    \\right\\}\n \\left[ \\left[ (q^{n}| q)_1-(q^{k}| q)_1 \\right] \\right] \\\\\n \n -  \\left\\{ (y q^{n+k} | q)_{1} (y q^{n+k+p+1} | q)_{1}    q^{k+p}     \\left[ (q| q)_1 \\right]  \\right\\}.\n  \\end{multline*}\n \n  Using the notation $\\big[a,b,c,\\dots\\big]=(a| q)_1(b| q)_1(c| q)_1\\ldots,$\nsetting $y=abcd, u=q^n,v=q^k, \\text{ and } w=q^p,$ we see it is sufficient to show the vanishing of\n\n\\begin{multline*}\np_2=u\\left[ yv^2w, yv^2wq, wq\\right] \n+(vw-u)\\left[ yuvwq, yv^2wq \\right]\\\\\n-wq(v-u) \\left[ yuv, yv^2w\\right]\n-vw \\left[ yuv, yuvwq,q\\right]. \n\\end{multline*}\n \n This expression may be interpreted as a one variable polynomial of degree at most $2$ in $y$ with coefficients in the field $\\mathbb{Q}(u,v,w,q).$\n Evaluating\n \\begin{description}\n\\item [at $y=0$] $p_2(0)=u(1-wq)+(vw-u) -wq(v-u)-vw(1-q) =0.$\n\\item [at $y=(v^2wq)^{-1}$] When $y=(v^2wq)^{-1},$ note $yv^2w =q^{-1},$ $yuv= u(vwq)^{-1},$ and $yuvwq=uv^{-1}.$ So\n\\begin{multline*}\np_2\\big( (v^2wq)^{-1} \\big) = -wq(v-u) \\left[ u(vwq)^{-1}, q^{-1} \\right] -vw \\left[ u(vwq)^{-1}, uv^{-1}, q \\right]\\\\\n= (1-u(vwq)^{-1})\\big\\{  -wq(v-u)  \\left( -q^{-1} (1-q) \\right) -vw(1-uv^{-1}) (1-q) \\big\\} \\\\\n= w(1-u(vwq)^{-1})(1-q)\\big\\{ v-u -v(1-uv^{-1}) \\big\\} =0.\n\\end{multline*}\n\\item [at $y=(uv)^{-1}$] When $y=(uv)^{-1},$ note $yv^2w= u^{-1}vw,$ $yv^2wq=u^{-1}vwq ,$ and $yuvwq=wq .$ So\n\\begin{multline*}\np_2\\big( (uv)^{-1} \\big) = (1-u^{-1}vwq)(1-wq) \\big\\{u(1 -  u^{-1}vw) +vw-u\\big\\}=0. \n\\end{multline*}\n\\end{description}\nThus the polynomial $p_2$ is $0$ and the proposition is proven.\n\n\\end{proof}\n\n\n\\begin{center} \\bf The $T^{c10}$ Identity \\end{center}\n\n\\ifJOLT \\begin{Proposition} \\else \\begin{props} \\fi\n\\label{T10PROPC}\nWhen $0 \\le k \\le n-2$\n\\begin{multline}\n    \\sigma^c_{-(k+1), n}(c,cq^{p+1}) =  \\big[ \\sigma^c_{-(k+1),k+1}(cq^p,cq^{p+1})\\big] \\big[ \\tau^c_{k+1, n}(c,cq^{p})\\big] \\\\\n+\\big[ \\tau^c_{-(k+1),-(k+1)}(cq^p,cq^{p+1})\\big] \\big[ \\sigma^c_{-(k+1),n}(c,cq^{p}) \\big] \\\\\n  + \\big[ \\tau^c_{-(k+1),-(k+2)}(cq^p,cq^{p+1})\\big] \\big[ \\sigma^c_{-(k+2), n}(c,cq^{p})\\big] \n\\end{multline}\nAnd\n\\begin{multline*}\n   \\sigma^c_{-n, n}(c,cq^{p+1}) =  \\big[ \\sigma^c_{-n,n}(cq^p,cq^{p+1})\\big] \\big[ \\tau^c_{n,n}(c,cq^{p})\\big] \\\\\n+\\big[ \\tau^c_{-n,-n}(cq^p,cq^{p+1})\\big] \\big[ \\sigma^c_{-n,n}(c,cq^{p}) \\big]. \n\\end{multline*}\n\\ifJOLT \\end{Proposition} \\else \\end{props} \\fi\n\n\\begin{proof} The second is immediate from corollary \\ref{CDG1LINC} together with the observation that $\\tau^c_{rr}=1$ for any sign of $r.$\n\nFor the first, upon dividing by $\\tau^c_{k+1,n}(c,cq^{p}),$ we see it is sufficient to prove\n\\begin{multline*}\n    \\frac{ \\sigma^c_{-(k+1), n}(c,cq^{p+1}) } { \\tau^c_{k+1,n}(c,cq^{p})} =  \\left\\{ \\sigma^c_{-(k+1),k+1}(cq^p,cq^{p+1}) \\right\\} \\cdot 1 \\\\\n+\\left\\{  \\tau^c_{-(k+1),-(k+1)}(cq^p,cq^{p+1})  \\right\\} \\left\\{ \\frac{ \\sigma^c_{-(k+1),n}(c,cq^{p}) } { \\tau^c_{k+1,n}(c,cq^{p}) } \\right\\} \\\\\n  + \\left\\{  \\tau^c_{-(k+1),-(k+2)}(cq^p,cq^{p+1}) \\right\\} \\left\\{ \\frac { \\sigma^c_{-(k+2), n}(c,cq^{p}) } { \\tau^c_{k+1,n}(c,cq^{p}) } \\right\\} .\n\\end{multline*}\nThat means we need to show\n\\begin{multline}\n\\label{EQN1cc}\\ \n\\left\\{\n- \\frac{ cdq^{n+p} ( q^{k+1}, abq^{k+1}, q^{-(p+1)}  | q )_1 } {  ( abcdq^{n+k}, abcdq^{n+k+p+1} | q )_1 }\n\\right\\}\n\\\\\n= \n\\left\\{\n- \\frac{(q | q)_{k+1} dq^k  \\left[ cq^{p+1} \\right]  (abq^{k+1},q^{-1} | q)_{1}   }  { (q | q)_k (ab \\left[ cq^{p} \\right] dq^{2k+1} , abd \\left[ cq^{p+1} \\right] q^{2k+1}  | q)_{1} } \n\\right\\}\n\\cdot 1 \\\\\n+  \\left\\{1 \\right\\}  \\cdot \n\\left\\{\n- \\frac {cdq^{k+p}(q^{k+1},q^{n-k-p-1}, abq^{k+1} | q )_1 } { (abcdq^{n+k}, abcd q^{2k+p+1} | q )_1}\n\\right\\}\n \\\\\n+ \n\\left\\{\n\\frac{(q^{2} | q)_k   \\left[ cq^{p+1} \\right] (abq^{k+1},adq^{k+1}, bd q^{k+1}, q^{-1} | q )_{1}  }  { (q | q)_k  (ab \\left[ cq^{p} \\right] dq^{2k+2}, abd \\left[ cq^{p+1} \\right] q^{2k+1}  | q)_{1} } \n\\right\\} \n\\cdot \\\\   \n\\left\\{  - \\frac \n{ dq^{k+1} ( q^{n-k-1}, abcd q^{2k+p+2} | q)_1}\n{(adq^{k+1},bdq^{k+1},abcdq^{n+k+p+1} | q)_1 } \n\\right\\} \\\\\n\\end{multline}\n\nWe will eventually reduce this identity to the vanishing of a 1-variable polynomial in $y$ with coefficients in the field $\\mathbb{Q}(a,b,c,d,u,v,w,q)$ \nwith the property that when \n$$\ny=abcd \\hspace{5mm} u= q^n \\hspace{5mm} v= q^k\\hspace{5mm} w= q^p\n$$\nwe obtain a unit in the coefficient field times the difference between the two sides of equation (\\ref{EQN1cc})  above. So, effectively, we can use the variables $y,u,v,w$\nas abbreviations for the above expressions.\n\n\nMultiplying (\\ref{EQN1cc})  by\n$$ \n- \\frac{ q( q, yq^{n+k}, yq^{n+k+p+1}, yq^{2k+p+1}, yq^{2k+p+2}  | q )_1 } {cd (q^{k+1}, abq^{k+1}  | q)_1 } \\\\\n$$\nwe see it is sufficient to show the vanishing of the  polynomial $p_1$ below.\n\n\n\n\n\n\n\\begin{multline*}\np_1  =  \n- \\left\\{ q^{n +p+1}  (q| q)_1  (yq^{2k+p+1}| q)_1(yq^{2k+p+2}| q)_1 \n (q^{-(p+1)} | q )_{1}    \\right\\}  \\\\\n+ \n  \\left\\{ q^{k+p+2}  (q| q)_1  (yq^{n+k}| q)_1 (yq^{n+k+p+1}| q)_1  (q^{-1}| q)_1    \\right\\}  \\\\  \n   + \\left\\{1 \\right\\}  \\cdot    \n\\left\\{ q^{k+p+1} (q| q)_1  (q^{n-k-p-1}| q)_1  (yq^{n+k+p+1}| q)_1 (yq^{2k+p+2}| q)_1   \n    \\right\\} \\\\\n+ \\left\\{ q^{k+p+3} (q^{n-k-1}| q)_1  (yq^{n+k}| q)_1 (yq^{2k+p+1}| q)_1    \n(q^{-1}| q)_1    \\right\\} \\cdot \n\\end{multline*}\n\\begin{multline*} \n\\hphantom{p_1}  =  \n-\\left\\{q^{n +p+1}(q| q)_1  (yq^{2k+p+1}| q)_1(yq^{2k+p+2}| q)_1  \n  \\left[ - q^{-(p+1)}(q^{p+1}| q)_1  \\right]   \\right\\}  \\\\\n+ \n     \\left\\{ q^{k+p+2}  (q| q)_1  (yq^{n+k}| q)_1 (yq^{n+k+p+1}| q)_1    \\left[  - q^{-1}(q| q)_1 \\right]   \\right\\}   \\\\  \n  +  \\left\\{1 \\right\\}  \\cdot   \n\\left\\{ q^{k+p+1}  (q| q)_1 (yq^{n+k+p+1}| q)_1 (yq^{2k+p+2}| q)_1 \n  \\left[ q^{-(k+p+1)}   \\left[ (q^{n}| q)_1-(q^{k+p+1}| q)_1 \\right] \\right]  \\right\\} \\\\\n+  \\left\\{ q^{k+p+3}  (yq^{n+k}| q)_1 (yq^{2k+p+1}| q)_1 \\left[  q^{-(k+1)} \\left[ (q^{n}| q)_1-(q^{k+1}| q)_1 \\right]\\ \\right]  \n \\left[ - q^{-1}(q| q)_1 \\right]   \\right\\}   \n\\end{multline*}\n\\begin{multline*}\n\\hphantom{p_1} =  \n\\left\\{ q^{n }   (yq^{2k+p+1}| q)_1(yq^{2k+p+2}| q)_1 (q| q)_1 \n  (q^{p+1}| q)_1     \\right\\}\\\\\n -   \n     \\left\\{  q^{k+p+1}   (yq^{n+k}| q)_1 (yq^{n+k+p+1}| q)_1   \\left[ (q| q)_1 \\right]^2  \\right\\}   \\hspace{50mm} \\\\  \n\n  +  \\left\\{1 \\right\\}  \\cdot  \n\\left\\{ (yq^{n+k+p+1}| q)_1 (yq^{2k+p+2}| q)_1 (q| q)_1 \n   \\left[  \\left[ (q^{n}| q)_1-(q^{k+p+1}| q)_1 \\right] \\right]  \\right\\} \\\\\n- \\left\\{q^{p+1} (yq^{n+k}| q)_1 (yq^{2k+p+1}| q)_1 \\left[  \\left[ (q^{n}| q)_1-(q^{k+1}| q)_1 \\right]\\ \\right] \n (q| q)_1   \\right\\} \n\\end{multline*}\nUsing the notation $\\big[a,b,c,\\dots\\big]=(a| q)_1(b| q)_1(c| q)_1\\ldots,$ \nrecalling our abbreviation variables $u,v,\\text{ and } w,$\nand multiplying by $ (q| q)^{-1}$ \nit suffices to show the vanishing of \n\\begin{multline*}\np_2 =  u[ yv^2wq,yv^2wq^2,wq]\n-vwq[yuv, yuvwq,q] \\\\\n +(vwq-u ) [yuvwq, yv^2wq^2 ] \n-wq(vq-u)[yuv, yv^2wq] .\n\\end{multline*}\n\n\nThis expression may be interpreted as a one variable polynomial of degree at most $2$ in $y$ with coefficients in the field $\\mathbb{Q}(u,v,w,q).$\nEvaluating\n\\begin{description}\n\\item[at $y=0$ ] \n$$\np_2(0)= u(1-wq) -vwq(1-q) +(vwq-u) -wq(vq-u)=0.\n$$\n\\item[at $y=(v^2wq)^{-1}$ ] Note at this point $yv^2wq^2=q, \\ yuvwq=uv^{-1}, \\text{ and }yuv= u(vwq)^{-1}.\\ $ So\n\\begin{multline*}\np_2\\big((v^2wq)^{-1} \\big) =(1-uv^{-1}) \\big\\{  -vwq(1- u(vwq)^{-1}) (1-q) +(vwq-u)(1-q)\\big\\} \\\\\n                                         =(1-uv^{-1})(1-q)(vwq-u)  \\big\\{ -1+1 \\big\\} =0.\n\\end{multline*}\n\\item[at $y=(uvwq)^{-1}$ ] Note at this point $yv^2wq=u^{-1}v, \\ yv^2wq^2= u^{-1}v q\\  \\text{ and }yuv=(wq)^{-1} $ So\n\\begin{multline*}\np_2\\big((uvwq)^{-1} \\big)= (1-u^{-1}v)\\big\\{ u(1-u^{-1}vq)(1-wq)-wq(vq-u)(1-(wq)^{-1} \\big\\} \\\\\n                                       = (1-u^{-1}v)(vq-u)(1-wq)\\big\\{-1 +1\\big\\} =0.\n\\end{multline*}\n\\end{description}\n\nThus $p_2$ being of degree at most  $2$ and vanishing at $3$ points implies $p_2$ is identically $0$ and the proposition proof is complete.\n\n\\end{proof}\n\n\n\\begin{center} \\bf The $T^{c11}$ Identity \\end{center}\n\\ifJOLT \\begin{Proposition} \\else \\begin{props} \\fi\n\\label{T11PROPC}\nWhen $0 \\le k \\le n-1$\n\\begin{multline}\n  \\tau^c_{-(k+1),-(n+1)}(c,cq^{p+1}) = \n   \\big[ \\sigma^c_{-(k+1),k+1}(cq^p,cq^{p+1})\\big] \\big[ \\sigma^c_{k+1, -(n+1)}(c,cq^{p})\\big] \\\\\n+ \\big[ \\tau^c_{-(k+1),-(k+1)}(cq^p,cq^{p+1})\\big] \\big[ \\tau^c_{-(k+1),-(n+1)}(c,cq^{p}) \\big] \\\\\n+ \\big[ \\tau^c_{-(k+1),-(k+2)}(cq^p,cq^{p+1})\\big] \\big[ \\tau^c_{-(k+2), -(n+1)}(c,cq^{p})\\big].\n\\end{multline}\nAnd \n$$  \\tau^c_{-(n+1),-(n+1)}(c,cq^{p+1}) = \\big[ \\tau^c_{-(n+1),-(n+1)}(cq^p,cq^{p+1})\\big] \\big[ \\tau^c_{-(n+1),-(n+1)}(c,cq^{p}) \\big].$$\n\\ifJOLT \\end{Proposition} \\else \\end{props} \\fi\n\n\\begin{proof}\nThe second identity just says $1=1 \\cdot 1.$\n\nFor the first, upon dividing by $ \\sigma^c_{k+1,-(n+1)}(c,cq^{p}) ,$ we see it is sufficient to prove\n\\begin{multline*}\n  \\frac{ \\tau^c_{-(k+1),-(n+1)}(c,cq^{p+1}) } { \\sigma^c_{k+1,-(n+1)}(c,cq^{p}) }  = \n   \\left\\{ \\sigma^c_{-(k+1),k+1}(cq^p,cq^{p+1}) \\right\\} \\cdot 1  \\\\\n+ \\left\\{ \\tau^c_{-(k+1),-(k+1)}(cq^p,cq^{p+1})\\right\\} \\left\\{  \\frac{ \\tau^c_{-(k+1),-(n+1)}(c,cq^{p}) } { \\sigma^c_{k+1,-(n+1)}(c,cq^{p}) } \\right\\} \\\\\n+ \\left\\{ \\tau^c_{-(k+1),-(k+2)}(cq^p,cq^{p+1})\\right\\} \\left\\{  \\frac{ \\tau^c_{-(k+2), -(n+1)}(c,cq^{p}) }{ \\sigma^c_{k+1,-(n+1)}(c,cq^{p}) } \\right\\}.\n\\end{multline*}\nThat means we need to show\n\\begin{multline}\n\\label{EQN1dc}\n\\left\\{\n- \\frac{q^{n-2k-1} ( q^{k+1}, abq^{k+1}, q^{-(p+1)} | q )_1 } { ab ( q^{n-k}, q^{n-k-p-1} | q )_1 } \n\\right\\}\n\\\\\n= \n\\left\\{\n- \\frac{(q | q)_{k+1} dq^k  \\left[ cq^{p+1} \\right] (abq^{k+1}, q^{-1} | q)_{1}  }  { (q | q)_k (ab \\left[ cq^{p} \\right] dq^{2k+1} , abd \\left[ cq^{p+1} \\right] q^{2k+1}  | q)_{1} } \n\\right\\} \n  \\cdot 1  \\\\\n+\\left\\{ 1\n\\right\\}\n\\cdot\n\\left\\{\n- \\frac{ ( q^{k+1}, abq^{k+1}, abcdq^{n+k+p+1}  | q )_1 } { abq^{k+1}  ( q^{n-k}, abcdq^{2k+p+1}  | q )_1 }  \n\\right\\}\n\\\\\n+\\left\\{\n\\frac{(q^{2} | q)_k   \\left[ cq^{p+1} \\right] (abq^{k+1},adq^{k+1}, bd q^{k+1}, q^{-1} | q )_{1}  }  { (q | q)_k  (ab \\left[ cq^{p} \\right] dq^{2k+2}, abd \\left[ cq^{p+1} \\right] q^{2k+1}  | q)_{1} } \n\\right\\} \n\\cdot \\\\\n   \n    \\left\\{\n    - \\frac\n    {  (abcdq^{n+k+1}, abcdq^{2k+p+2}  | q )_1 }\n    {abc q^{k+p+1}  ( adq^{k+1}, bdq^{k+1}, q^{n-k-p-1}  | q )_1 } \n   \n    \\right\\}\n\n\\end{multline}\n\n\nMultiplying (\\ref{EQN1dc}) by\n\n$$\n- \\frac{abq^{2k+p+2} (q, q^{n-k}, q^{n-k-p-1}, yq^{2k+p+1}, yq^{2k+p+2}  | q)_1} { ( q^{k+1}, abq^{k+1} | q)_1} \\\\\n$$\nwe see it is sufficient to show the vanishing of the  polynomial $p_1$ below.\n\n\nWe will eventually reduce this identity to the vanishing of a 1-variable polynomial in $y$ with coefficients in the field $\\mathbb{Q}(a,b,c,d,u,v,w,q)$ \nwith the property that when \n$$\ny=abcd \\hspace{5mm} u= q^n \\hspace{5mm} v= q^k\\hspace{5mm} w= q^p\n$$\nwe obtain a unit in the coefficient field times the difference between the two sides of equation (\\ref{EQN1dc}) above. So, effectively, we can use the variables $y,u,v,w$\nas abbreviations for the above expressions.\n\n\n\\begin{multline*}\np_1=\n-\\left\\{ q^{n+p+1} (q| q)_1   (yq^{2k+p+1} | q)_1 (yq^{2k+p+2} | q)_1        (q^{-(p+1)} | q )_{1}    \\right\\} \\\\\n +\n   \\left\\{ yq ^{3k+2p+3}  (q| q)_1     (q^{n-k}| q)_{1}    (q^{n-k-p-1} | q )_{1}      (q^{-1} | q )_{1}     \n  \n   \\right\\} \\\\\n\n \n  +  \\left\\{1 \\right\\} \n   \\left\\{ q^{k+p+1} (q| q)_1    (q^{n-k-p-1} | q )_{1}  (yq^{n+k+p+1} | q)_1  (yq^{2k+p+2} | q)_1   \n   \\right\\} \\\\\n +  \\left\\{ q^{k+p+2}  (q^{n-k}| q)_1  (yq^{n+k+1} | q)_1  (yq^{2k+p+1} | q)_1   (q^{-1} | q )_{1}        \n  \\right\\}  \n \\end{multline*} \n\\begin{multline*}\n\\hphantom{p_1}=\n-\\left\\{q^{n+p+1}  (q| q)_1  (yq^{2k+p+1} | q)_1 (yq^{2k+p+2} | q)_1       \\left[  - q^{-(p+1)}(q^{p+1}| q)_1 \\right]    \\right.   \\\\\n +\n   \\Big\\{   y  q^{3k+2p+3}    (q| q)_1  \\left[  q^{-k} \\left[ (q^{n}| q)_1-(q^k| q)_1 \\right]\\rb    \\cdot \\\\\n  \n   \\left[ q^{-k-p-1} \\left[ (q^n| q)_1 - (q^{k+p+1}| q)_1 \\right]  \\right]   \\left[  - q^{-1}(q| q)_1  \\right]      \\Big\\} \\\\\n  +  \\left\\{1 \\right\\} \n   \\left\\{   q^{k+p+1}  (q| q)_1   \\left[ q^{-k-p-1} \\left[ (q^n| q)_1 - (q^{k+p+1}| q)_1 \\right]  \\right]   (yq^{n+k+p+1} | q)_1   (yq^{2k+p+2} | q)_1        \\right\\} \\\\\n+  \\left\\{  q^{k+p+2}  \\left[ q^{-k} \\left[ (q^{n}| q)_1-(q^k| q)_1 \\right] \\right]    (yq^{n+k+1} | q)_1     (yq^{2k+p+1} | q)_1   \\left[  - q^{-1}(q| q)_1  \\right]    \n  \\right\\}  \n \\end{multline*}\n\\begin{multline*}\n\\hphantom{p_1}=\n\\left\\{ q^{n}  (q| q)_1   (yq^{2k+p+1} | q)_1 (yq^{2k+p+2} | q)_1        (q^{p+1}| q)_1      \\right\\} \\\\\n - \\left\\{   y  q^{k+p+1}   \\left[ (q| q)_1  \\right]^2    \\left[ (q^{n}| q)_1-(q^k| q)_1 \\right]     \n  \n   \\left[ (q^n| q)_1 - (q^{k+p+1}| q)_1 \\right]        \\right\\} \\\\\n+ \\left\\{1 \\right\\} \n   \\left\\{  (q| q)_1    \\left[ (q^n| q)_1 - (q^{k+p+1}| q)_1 \\right]          (yq^{n+k+p+1} | q)_1  (yq^{2k+p+2} | q)_1    \\right\\} \\\\\n -  \\left\\{ q^{p+1}   (q| q)_1  \\left[ (q^{n}| q)_1-(q^k| q)_1 \\right]   (yq^{n+k+1} | q)_1  (yq^{2k+p+1} | q)_1      \n  \\right\\}  \n \\end{multline*}\nUsing the notation $\\big[a,b,c,\\dots\\big]=(a| q)_1(b| q)_1(c| q)_1\\ldots,$ \nrecalling our abbreviation variables $u,v,\\text{ and } w,$\nand multiplying by $\\left[ (q| q)_1 \\right]^{-1}$ \nit suffices to show the vanishing of \n\\begin{multline*}\np_2 = u[yv^2wq, yv^2wq^2, wq ] \n- yvwq(v-u)(vwq-u)(q| q)_1\\\\\n+ (vwq-u) [yuvwq,yv^2wq^2] \n -wq(v-u) [yuvq,yv^2wq] \n\\end{multline*}\nThis expression may be interpreted as a one variable polynomial of degree at most $2$ in $y$ with coefficients in the field $\\mathbb{Q}(u,v,w,q).$\nEvaluating \n\\begin{description}\n\\item[at $y=0$ ] \n$$\np_2(0)= u(1-wq) + (vwq-u)  -wq(v-u)=0.\n$$\n\\item[at $y=(v^2wq)^{-1}$ ] Note at this point $yv^2wq^2=q, \\ yvwq=v^{-1}, \\text{ and } yuvwq=uv^{-1}.$ So\n\\begin{multline*}\np_2\\big((v^2wq)^{-1} \\big)=(vwq-u) \\big\\{ -v^{-1}(v-u)(1-q) + (1-uv^{-1})(1-q)  \\big\\} \\\\\n                                         =(vwq-u) (1-uv^{-1})(1-q)  \\big\\{ -1+1 \\big\\} =0.\n\\end{multline*}\n\\item[at $y=(v^2wq^2)^{-1}$ ] Note at this point $yv^2wq=q^{-1}, \\ yvwq=(vq)^{-1}, \\text{ and } yuvq=u(vwq)^{-1}.$ So\n\\begin{multline*}\np_2\\big((v^2wq^2)^{-1} \\big)=(v-u)  \\big\\{-(vq)^{-1}(vwq-u)(1-q) - wq(1- u(vwq)^{-1})(1-q^{-1}) \\big\\} \\\\\n                                       =(v-u) (1-q)\\big\\{ -( w -u (vq)^{-1}  ) + (w- u(vq)^{-1})\\big\\}=0. \n\\end{multline*}\n\\end{description}\nThus $p_2$ being of degree $2$ and vanishing at $3$ points implies $p_2$ is identically $0$ and the proposition proof is complete.\n\n\\end{proof}\n\nThe combination of Propositions \\ref{PLANA1} (the $\\boldsymbol{\\mathcal{T}}(c, g ; a, b, d | q) $ version),  \\ref{CDG1CGQ}, \\ref{CDG2CGQ}, \\ref{T00PROPC}, \\ref{T01PROPC}, \\ref{T10PROPC}, and \\ref{T11PROPC}   finishes the proof of all $3$ steps of (\\ref{PLANC}), Proof Plan C, and so completes the proof of Theorem \\ref{ETC2}. \n\n\n\n\\fi\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}