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+{"text":"\\section{Introduction}\n\nNGC 5548 is a bright, low-redshift (z $=$ 0.0172) Seyfert 1 galaxy that has \nreceived a considerable amount of attention over the past decade. In particular, \nNGC 5548 has been the subject of a number of intensive spectroscopic monitoring \ncampaigns in the optical and UV (Korista et al. 1995). \nThese efforts have yielded important results on the nature of the continuum \nsource and the size, geometry, and kinematics of the broad-line region (BLR), \nfor which the responsivity peaks at about 20 light days from the nucleus\n(Peterson et al. 1994). Little attention has been paid to the narrow-line region \n(NLR) in NGC 5548, however, although it could lead to a better understanding of \nthe circumnuclear environment at much greater distances from the nucleus of this\notherwise well-studied active galaxy.\n\nIn general, studies of the NLR in active galaxies are important for \nunderstanding the nature of the NLR clouds and the interaction of the central \ncontinuum source with the surrounding galaxy on large scales. Emission-line \nstudies and detailed photoionization modeling are particularly useful \nfor determining the range of physical conditions and reddening amongst the NLRs \nin active galaxies. A comparison of the NLR properties in Seyfert \n1 and Seyfert 2 galaxies should be helpful in testing unified theories, which \npostulate that the two types are the same object viewed from \ndifferent perspectives, such that the continuum source and BLR are ``hidden'' in \nSeyfert 2 galaxies (Miller \\& Goodrich 1990; Antonucci 1993). If this basic \nhypothesis is correct, then the intrinsic properties of the NLRs in Seyfert 1 \nand Seyfert 2 galaxies should {\\it not} show large systematic differences.\n\nThe narrow lines in Seyfert 1 galaxies are more difficult to measure \nthan in Seyfert 2 galaxies, due to blending with the broad lines. However, given \nspectra with sufficient signal-to-noise ratio and spectral resolution, these \ncomponents can be isolated and measured reasonably well\n(Crenshaw \\& Peterson 1986). Measurements of the narrow lines in the optical are \ngiven by Cohen (1983) for a large number of Seyfert 1 galaxies. In the UV, these \nmeasurements were difficult with the low spectral resolution of the \n{\\it International Ultraviolet Explorer} ({\\it IUE}), but are possible with \ninstruments on \nthe {\\it Hubble Space Telescope} ({\\it HST}), such as the Faint Object \nSpectrograph (FOS).\n\nThe FOS UV spectra of NGC 5548 presented in a previous paper (Crenshaw, Boggess, \n\\& Wu 1993; hereafter Paper I) provide a good starting point for detailed \nstudies of the NLR in a Seyfert 1 galaxy. These observations happened to occur\nat a time when the broad emission lines (and continuum fluxes) were at an \nhistoric low in the UV, and the contrast between the broad and narrow \ncomponents is thereby enhanced. Paper I gives the UV spectrum and measurements \nof the broad and narrow lines in NGC 5548. Relative to the other narrow lines, \nC~IV $\\lambda$1549 is much stronger in NGC 5548 than in Seyfert 2 galaxies, \nindicating a higher ionization parameter and\/or harder continuum in the NLR of \nNGC 5548. Narrow Mg II $\\lambda$2800 emission is very weak or absent in NGC \n5548, and Paper I presents two possible explanations: 1) the NLR clouds lack the \npresence of a partially-ionized zone (i.e., they are optically thin to ionizing \nradiation), and\/or 2) dust grains are present in the NLR clouds, and the Mg II \nflux is weak due to depletion and\/or destruction from multiple scatterings and \neventual absorption of the photons by dust (Kraemer \\& Harrington 1986; Ferland \n1992).\n\nWe now have the opportunity to investigate the preliminary results from Paper I \nin more detail, by including ground-based optical spectra and photoionization \nmodels. From the ground-based monitoring campaigns, we have selected \nspectra that cover the full optical range (3000 -- 10,000 \\AA) and were\nobserved around the same time period as the UV data, when the broad-line fluxes \nwere very low. The combination of optical and UV lines provides a wide range of \nemission-line diagnostics, as well as an opportunity to deredden the lines using \nthe He~II recombination lines. We can then use multicomponent \nphotoionization models to match the dereddened line \nratios and probe the physical conditions in the NLR of NGC 5548.\n\n\\section{Observations and Data Analysis}\n\n\\subsection{UV and Optical Spectra}\n\nWe obtained the FOS UV spectra of NGC 5548 through a 1\\arcsecpoint0 circular \naperture on 1992 July 5 UT. Paper I gives the details of the observations and \nmeasurements, along with the UV spectrum and emission-line fluxes (with \nassociated errors). The observations were made prior to the installation of \nCOSTAR on {\\it HST}, so near-simultaneous {\\it IUE} spectra were used to adjust \nthe absolute flux levels of the FOS spectra. As noted in Paper I, \nthe scale factors needed to bring the FOS continuum fluxes up to the {\\it IUE} \nlevels are around 1.4 -- 1.5, which are somewhat higher than the \nvalues of 1.1 -- 1.3 for our other Seyfert observations. We concluded that the \nSeyfert nucleus may not have been accurately centered in the aperture. Koratkar \net al. (1996) suggest that another possible explanation for the discrepancy in \nabsolute fluxes is nonlinearity in the {\\it IUE} detectors. However, we have \nseen no evidence for this possibility in other observations at these flux \nlevels, so we have no reason to distrust the {\\it IUE} fluxes. In addition, \nreprocessed versions of these spectra that we obtained from the {\\it HST} and \n{\\it IUE} archives have not changed the original fluxes by more than 10\\%, so we \ncontinue to use the values from Paper I. Some of the emission lines in Paper I \nhave only a single number quoted for the flux (as opposed to separate values for \nthe broad and narrow components); a single value represents the narrow-line \ncontribution, since the broad component is either not present or too weak to be \ndetected in these cases.\n\nWe selected two optical spectra obtained during a four-year monitoring campaign \non NGC 5548 (Peterson et al. 1994), from a time interval of $\\sim$30 days when \nthe H$\\beta$ and continuum light curves were at their lowest levels to date.\nThe spectra were chosen on the basis of their large wavelength coverage\n(3000 - 10000 \\AA), high signal-to-noise ratio ($\\geq$50 per resolution \nelement in the continuum at 5200 \\AA), and acceptable resolution ($\\sim$8 \n\\AA). The spectra were obtained through a 4\\arcsecpoint0 x \n10\\arcsecpoint0 aperture with the 3.0-m Shane telescope $+$ Kast spectrograph on \n1992 April 21 and 1992 May 23 UT. Additional details on the observations are \ngiven by Peterson et al. (1994). The absolute flux levels were \nadjusted by scaling the optical spectra so that the [O~III] $\\lambda$5007 flux \nis 5.58 x 10$^{-13}$ ergs s$^{-1}$ cm$^{-2}$, a value determined from \nobservations through large apertures on spectrophotometric nights (Peterson et \nal. 1991). The scale factors we used are 1.36 for the 1992 April 24 spectrum and \n1.01 for the 1992 May 23 spectrum.\n\nPlots of the optical spectra are shown in Figure 1 (the UV spectrum is shown in \nPaper I). The contrast between the broad and narrow components of the permitted \nemission lines is most clearly seen in H$\\beta$. The 1992 April 21 spectrum was \nobtained at an historic low level, with a continuum flux of  \nF$_{\\lambda}$(5100~\\AA) $=$ 5.5 x 10$^{-15}$ ergs s$^{-1}$ \ncm$^{-2}$ \\AA$^{-1}$ and total H$\\beta$ flux of F(H$\\beta$) $=$ 3.2 x 10$^{-13}$ \nergs s$^{-1}$ cm$^{-2}$. The flux levels are a little higher for the 1992 May 23 \nspectrum with F$_{\\lambda}$(5100 \\AA) $=$ 6.0 x 10$^{-15}$ ergs s$^{-1}$ \ncm$^{-2}$ \\AA$^{-1}$ and F(H$\\beta$) $=$ 3.7 x 10$^{-13}$ ergs s$^{-1}$ \ncm$^{-2}$.  At the time of the FOS \nobservations on 1992 July 5, the continuum and H$\\beta$ fluxes were close to the \nsame levels as those from the second optical spectrum, according to the light \ncurves of Peterson et al. (1994).\n\nAlthough the optical aperture is much larger than the one used for the UV \nobservations, we have substantial evidence that it does not contain much \nadditional NLR flux. Unfortunately, there are no {\\it HST} narrow-band images in \n[O~III] or other strong lines that could be used to directly determine the \ndistribution of narrow-line emission close to the nucleus. However, there is \nsignificant evidence that the apparent size of the NLR is very small in NGC \n5548. Peterson et al. (1995) find from a ground-based image that the [O~III] \nemission is pointlike, given a point-spread function that is characterized by a \nwidth of 2\\arcsecpoint0 (FWHM). In addition, Wilson \\& Ulvestad (1982) show that \nin an aperture that is 4\\arcsecpoint2 in diameter, the [O~III] $\\lambda$5007 \nfluxes at positions offset from the nucleus by 4\\arcsecpoint5 -- 6$''$\nare about 100 times weaker than the nuclear flux.\nMore importantly, in Paper I we found that the strongest UV lines in the {\\it \nIUE} 20$''$ x 10$''$ aperture have fluxes that are only $\\sim$ 20\\% higher than \nthose in the FOS 1\\arcsecpoint0 aperture. Thus, the observed UV to optical line \nratios that we quote are at most 20\\%  too low, which has little effect on our \ncomparisons with the model results.\n\nIn order to measure the flux of each narrow optical line, we used\na local baseline determined by linear interpolation between adjacent \ncontinuum regions or broad profile wings (in the case of profiles \nconsisting of broad and narrow components). For severely blended lines like \nH$\\alpha$ and [N~II] $\\lambda\\lambda$6548, 6584, we used the [O~III] \n$\\lambda$5007 profile as a template to deblend the lines (see Crenshaw \\& \nPeterson 1986). The adopted flux for each narrow component is the average of the \nvalues from each of the two spectra.\n\nWe determined the reddening of the narrow emission lines from the He~II \n$\\lambda$1640\/$\\lambda$4686 ratio and the Galactic reddening curve of Savage \\& \nMathis (1979). For the temperatures and densities typical of the NLR, the He II \nlines are due to recombination, and this particular ratio only varies from 6.3 \nto 7.6 (Seaton 1978); we adopt an intrinsic value of 7.2, consistent with our \nmodel values (Section 3). The observed He~II $\\lambda$1640\/$\\lambda$4686 ratio \nis 5.5 $\\pm$ 1.6, which yields a reddening of E$_{B-V}$ $=$ 0.07 mag \n$^{+0.09}_{-0.06}$. The portion of the reddening that is due to our own Galaxy \nis E$_{B-V}$ $=$ 0.03 mag, determined from a neutral hydrogen column density of \nN$_{HI}$ $=$ 1.6 x 10$^{20}$ cm$^{-2}$ (Murphy et al. 1996) and the relationship \nE$_{B-V}$ $=$N$_{HI}$\/5.2 x 10$^{21}$ cm$^{-2}$ (Shull \\& Van Steenburg 1985).\nWe note that the intrinsic reddening of the narrow emission lines in this \nSeyfert 1 galaxy, E$_{B-V}$ $\\approx$ 0.04 mag, is much smaller than typical \nvalues of 0.2 -- 0.4 mag obtained for Seyfert 2 galaxies (MacAlpine 1988; \nFerland \\& Osterbrock 1986; Kraemer et al. 1994). \n\nWe determined errors in the dereddened ratios from the sum in quadrature of the \nerrors from three sources: photon noise, different reasonable continuum \nplacements, and reddening. Errors in the optical ratios are dominated by \ncontinuum placement, whereas errors in the UV to optical ratios are due to both \ncontinuum placement and uncertainties in the reddening correction. Errors in the \nweak lines in both regions also have a significant contribution from photon \nnoise. As we discussed earlier in this section, there are some possible sources \nof systematic error in the UV to optical line ratios, on which we placed upper \nlimits of $\\sim$20\\%.\n\nTable 1 gives the observed and dereddened narrow-line ratios relative to \nH$\\beta$, and errors in the dereddened ratios. Cohen (1983) gives the next \nmost comprehensive list of optical line ratios; in general, Cohen's observed \nratios agree with ours to within the errors. A number of investigators have \nindependently determined the narrow H$\\beta$\/[O~III] $\\lambda$5007 ratio in NGC \n5548 (Cohen 1983; Crenshaw \\& Peterson 1986; Peterson 1987; Wamsteker et al. \n1990; Rosenblatt et al. 1992; Wanders \\& Peterson 1996): these values range from \n0.10 to 0.15, compared to our value of 0.12 $\\pm$ 0.01.\n\n\\subsection{The Ionizing Continuum}\n\nEstimates of the ionizing continuum are needed as input values for the \nphotoionization models of the NLR. We choose the continuum data points given by \nKrolik et al. (1991), since they represent the historic mean levels for this \nobject. As always, the greatest uncertainty is the shape of the extreme \nultraviolet (EUV) continuum. Figure 2 \ngives the UV continuum point closest to the EUV region, at 1340 \\AA, and the \nX-ray continuum points from Krolik et al. (cf. Turner \n\\& Pounds 1989; Clavel et al. 1991) in terms of luminosity (ergs \ns$^{-1}$ Hz$^{-1}$), which we have adjusted for a Hubble constant of H$_{0}$ $=$ \n75 km s$^{-1}$ Mpc$^{-1}$. The dotted line in Figure 1 gives Krolik et al.'s \ncontinuum fit in the EUV, which is a power law determined from the UV data along \nwith an exponential cutoff designed to meet the first X-ray point. We prefer a \nfit with two power laws (in the form L$_{\\nu}$ $=$ K$\\nu^{\\alpha}$), given the \nevidence for an upturn in the spectrum at energies smaller than 1 -- 2 keV \n(i.e., a soft X-ray excess). A fit to these data yields  $\\alpha$ $=$ \n$-$1.40$\\pm$0.03 in the EUV and soft X-rays, and $\\alpha$ $=$ $-$0.40$\\pm$0.03 \nin the hard X-rays; the break point is at $\\nu$ $=$ 10$^{17.1}$ Hz$^{-1}$ (1.3 \nkeV).\n\nNGC 5548 was monitored by the {\\it Extreme Ultraviolet Explorer} ({\\it EUVE}) \nover a two month period during 1993 March -- May \n(Marshall et al. 1997).  During this time, the EUV flux varied by a factor of \nfour from peak to minimum, and the average flux (corrected for Galactic \nneutral hydrogen absorption) was 135 $\\mu$Jy at $\\sim$76 \\AA.\nThese observations provide an important constraint on the EUV ionizing \ncontinuum, but were not used directly in our continuum fit for two reasons. \nFirst, the neutral hydrogen absorption due to our Galaxy is well known (see the \nprevious section), but there could be additional absorption along the \nline of sight. Second, the {\\it EUVE} flux, averaged over two months, may not \nbe representative of the average flux over many years. Given these caveats, we \nplot the {\\it EUVE} continuum point in Figure 2 for comparison with our adopted \ncontinuum; the error bar was determined from an estimate of $\\pm$10\\% \nuncertainty in \nGalactic N$_{HI}$ (see Murphy et al. 1996). The {\\it EUVE} point is slightly \nhigher than \nthe continuum fits in Figure 2, but appears to be consistent with our and Krolik \net al.'s \nadoption of a relatively steep continuum. If we use Krolik et al.'s continuum or \na continuum formed by joining the UV, {\\it EUVE}, and X-ray points with line \nsegments \n(in log space), the total EUV flux increases by factors of only 1.19 and 1.30, \nrespectively, and the flux at the frequency of the {\\it EUVE} observation \nincreases by \nfactors of 1.40 and 1.84, respectively. The effects of adopting these other \ncontinua are small, and will be discussed later in the paper.\n\n\\section{Photoionization Models}\n\nIn modeling the narrow-line emission of NGC 5548, we have adhered to our basic \nphilosophy of keeping the number of free parameters to a minimum, by using the \navailable observational constraints and the simplest assumptions possible.\nThe parameters are varied until the best agreement is obtained with the \nobserved line ratios, and additional input parameters are only included if \nthey are needed to provide a reasonable match to the majority of the  \nlines. Discrepancies between the model predictions and specific lines are then \ninvestigated to provide further insight into the physical conditions. In some \ncases we generate variations on the standard model using additional parameters \n(such as dust) or nonstandard values of the initial parameters (such as nonsolar \nabundances) to illustrate our ideas for resolving the discrepancies.\n\n\\subsection{Methodology}\n\nThe basic modeling methodology that we employ is described in \nKraemer et al. (1994) and the details of the photoionization code are given in \nKraemer (1985). To review the major points, we assume plane parallel \ngeometry, which is reasonable if the gas is ionized by radiation from a central \nsource at a distance that is large compared to the extent of the cloud. The gas \nis assumed to be atomic (i.e., there is no molecular component). For radiation \nbounded models, we stop the integration into the slab when the electron \ntemperature falls below 5000 K and there is no longer any significant line \nemission. The emission line photon escape is through the ionized face of the \nslab. Details of the treatment of dust in the models are described in Kraemer \n(1985). Since the work of Kraemer et al. (1994), we have added iron to the \nelements modeled in the code. The atomic data that we used can be found in \nPradhan \\& Peng (1994), and references contained therein, as well as through \nFerland's ``Cloudy and Associates'' World Wide Web site \n(http:\/\/www.pa.uky.edu\/$\\sim$gary\/cloudy).\nThe final output of these models is an emission line spectrum. The line\nstrengths are tabulated relative to H$\\beta$. In addition, the model\ngives the volume emissivity of H$\\beta$, from which we can determine\nthe mass of gas required to produce the observed line emission, the\nefficiency of production of H$\\beta$ photons, and an estimate of the covering \nfactor.\n\nIn order to keep the input parameters to a minimum, we kept two of them fixed \nfor our standard model: the shape of the ionizing continuum and the abundances. \nWe used the simplest possible ionizing continuum consistent with the \nobservations, as described in Section 2.2. In addition, we have assumed \nsolar elemental abundances for the standard model as follows (see Lambert \\& \nLuck 1978): He = 0.1, C = 3.4 x 10$^{-4}$, O = 6.8 x 10$^{-4}$, N = 1.2 x \n10$^{-4}$, Ne = 1.1 x 10$^{-4}$, S = 1.5 x 10$^{-4}$, Si = 3.1 x 10$^{-5}$, Mg = \n3.3 x 10$^{-5}$, Fe = 4 x 10$^{-5}$ (relative to hydrogen by number).\n\nOur photoionization models are parameterized in terms of the density of\natomic hydrogen (N$_{H}$) and the dimensionless ionization parameter at the \nilluminated face of the cloud:\n\n\\begin{equation}\nU = \\int^{\\infty}_{\\nu_0} ~\\frac{L_\\nu}{h\\nu}~d\\nu ~\/~ (4\\pi~D^2~N_{H}~c),\n\\end{equation}\n\n\\noindent where {\\it L$_{\\nu}$} is the frequency-dependent luminosity of \nthe ionizing continuum, {\\it D} is the distance between the cloud and the \nionizing source, and h$\\nu_{0}$ = 13.6 eV. \n\nWe show that we must add two enhancements to our standard model to obtain an \nacceptable match to the observations. First, we need two components of gas,  \ncharacterized by different ionization parameters and densities. Second, we show \nthat the inner component must be optically thin (i.e., radiation bounded) at the \nLyman edge (13.6 eV). We are then able to vary the ionization parameter and \ndensity of each component to match the observations. Of course, the resulting \nstandard model is an oversimplification, since it is likely that the NLR clouds \nare characterized by a number of different ionization parameters, densities, and \noptical depths. We have effectively averaged the initial conditions for each of \nthese components to fit the largest selection of line ratios. Given this \nsimplification, the two-component model gives a surprisingly good fit \nto the observations.\n\nThe narrow-line spectrum of NGC 5548 is dominated by high ionization lines,\nsuch as C~IV $\\lambda$1549, N V $\\lambda$1240, and [Ne V] $\\lambda\\lambda$\n3346, 3426, as well as the coronal lines of [Fe~VII] and [Fe~X]. This indicates \nthat there is a high ionization component relatively near the central source, \nbut presumably outside the BLR, since these lines are much narrower\n($\\leq$~500~km~s$^{-1}$ FWHM) than the broad lines ($\\sim$ 5000 km s$^{-1}$ \nFWHM).\nAlso present in the spectrum are relatively strong lines of [N II] \n$\\lambda\\lambda$6584, 6548 and [O II] $\\lambda$3727. These are typical of the \nnarrow emission line regions of many Seyfert 2 galaxies (Koski 1978; Shuder\n\\& Osterbrock 1981), and presumably arise in a component of relatively low \nionization and low density.\n\nThe choice of density for these models is based on the relative strengths\nof certain forbidden emission lines in the spectrum of NGC 5548. \nAn emission line will not be an important coolant in gas with sufficiently \nlarge electron density that collisional de-excitation of the line will dominate\nover radiative transition (see Osterbrock 1989).  For example, the observed \n[Ne~V]~$\\lambda$3426\/H$\\beta$\nratio indicates that the density of the component in which that emission\noriginates must be less than $\\sim$5~x~10$^{7}$ cm$^{-3}$ (DeRobertis \\& \nOsterbrock 1984). Given the hardness of the ionizing continuum, one would expect \nhigh temperatures in this highly ionized gas.\nThe lower limit on the density of this component can be estimated from the\nrelative strength of the [O~III] $\\lambda$5007 emission. The ratio of [O III] \n$\\lambda$5007\/ [Ne V] $\\lambda$3426\nis less than 5, which indicates either low density, highly ionized gas, as\nis often seen in the extended NLR of Seyfert 2 galaxies (Storchi-\nBergmann et al. 1996), or density greater than 10$^{6}$ cm$^{-3}$ and \nsome collisional supression of the $\\lambda$5007 line. The ratio of [O~III] \n$\\lambda$4363\/$\\lambda$5007 is very high ($\\sim$0.09). In the low density \nlimit, this ratio is\n$\\sim$7 x 10$^{-3}$ at a temperature of 10$^{4}$~K (Osterbrock 1989). \nAt this low density, it is unlikely that \ntemperatures consistent with photoionization equilibrium can increase this\nratio to the observed value, and probable that the large \n$\\lambda$4363\/$\\lambda$5007\nratio is due to the fact that some of this emission arises in gas of high\ndensity (i.e., $>$ 10$^{6}$ cm$^{-3}$). \nAt this density and level of ionization, the \n[N~II]~ $\\lambda\\lambda$6548, 6584 emission from this component will be \nnegligible, so there must be a lower ionization and lower density component \npresent. In order for the [N~II] lines to be among the principal coolants from \nthis component, its density must be less than 1 x 10$^{5}$ cm$^{-3}$. Other \nstudies (cf. Filippenko \\& Halpern 1984; Filippenko 1985; Kraemer et al. 1994) \nhave shown that a range of densities in the NLR of Seyfert galaxies is likely, \nso it is not surprising that this condition exists in NGC 5548.\n\nTo summarize, the range in density, along with the presence of emission lines \nfrom a wide range of ionization states, indicates that more than one model \ncomponent is needed to fit the NLR spectrum of NGC 5548. \nThe existence of strong high ionization lines such as C IV \n$\\lambda$1549, N V $\\lambda$1240, and [Ne V] $\\lambda\\lambda$3346, 3426, and our \nevidence for high densities in the region that they are produced, requires a \ncomponent of gas relatively close to the central source. The weakness of Mg II \n$\\lambda$2800 and [O I] $\\lambda\\lambda$6300, 6364 indicate that these gas \nclouds lack a significant partially ionized zone, and therefore must be\n{\\it optically thin} to the ionizing radiation (or ``matter bounded''). \nWe further investigate the conditions in these two components below.\n\n\\subsection {Model Results and Comparison to Observations}\n\nOur approach in modeling NGC 5548 was to fit the high ionization\ncomponent first and then add components as needed to fit the lower\nionization lines (in the end, only one additional component was needed). Given \nthe constraints and assumptions described in the\nprevious section, we arrived at values of N$_{H}$ $=$ 1 x 10$^{7}$ cm$^{-3}$ and\nU $=$ 10$^{-1.5}$ for the high ionization component. Substantially lower \ndensities would result in \n[O~III]~$\\lambda$5007 being too strong, and higher densities would quench the \n[Ne V] emission. A higher ionization parameter is possible but, given our EUV \ncontinuum, would not increase the relative strengths of any of the high \nionization lines other than [Fe X] $\\lambda$6374, at the expense of putting this \ncomponent at distances much closer than $\\sim$1 pc from the continuum source \n(see Section 4). Models were run to varying optical depth at the Lyman limit \n$\\tau_{0}$, with the constraint that the Mg II and [O I] lines could not become \ntoo strong. After comparing the results of models run with $\\tau_{0}$ $=$ 1.5 to \n10, we found that $\\tau_{0}$ $=$ 2.5 gave the best fit. The emission line \nspectrum from this model, INNER, is given in Table 2.\n\nA second component, OUTER, was needed to fit the lower ionization lines. \nWe found that U $=$ 10$^{-2.5}$ and N$_{H}$ $=$ 2 x 10$^{4}$ cm$^{-3}$ gave a \ngood simultaneous fit to the [O~III]~$\\lambda$5007\/[O~II]~$\\lambda$3727 and \n[N~II] $\\lambda$6584\/H$\\beta$ ratios. Unlike our models for Mrk 3 and I~Zw~92 \n(Kraemer \\& Harrington 1986; Kraemer et al. 1994), there was no need to add a \nthird component for NGC 5548, since there is no obvious contribution from a \ncomponent of very low density ($<$ 10$^{3}$ cm$^{-3}$) low \nionization gas, such as very strong [O II] $\\lambda$3727 and [N~I] $\\lambda$5200 \nlines. The resulting emission line spectrum from OUTER is also included in \nTable 2. (We will discuss two variations on INNER and OUTER in the next \nsection.)\n\nIn order to fit the observed (and dereddened) narrow-line spectrum of NGC 5548, \nwe combined the output spectrum of the two standard components INNER and OUTER. \nIn previous studies, we attempted to weigh the contributions from each component \nto fit specific emission line ratios. For the model of NGC 5548, we simply took \nan equal contribution from INNER and OUTER. The relative simplicity\nof the narrow-line spectrum and lack of a strong contribution from very \nlow-ionization gas makes such a simple fit possible. The combined spectrum\nis given, along with the dereddened observed spectrum for comparison, in\nTable 3 (horizontal lines in the table indicate that the models do not predict \nthe strengths of these emission lines). \n\nComparison of the model predictions to the dereddened observed spectrum in Table \n3 shows agreement, to within the errors, for most of the lines. In particular, \nthese include C~IV~$\\lambda$1549, He II \n$\\lambda$1640, [O~II] $\\lambda$3727, [Ne III] $\\lambda$3869, [N II] \n$\\lambda$6584, and the Balmer decrement.\nIn radiation bounded gas,\nthe ratio of the He II lines to H$\\beta$ is strongly dependent on the shape of \nthe ionizing continuum, because neutral hydrogen is the \ndominant absorber of ionizing radiation between 13.6 eV and 54.4 eV, while above \n54.4 eV, singly ionized helium dominates.\nIf there is a component of matter bounded gas, He II\/H$\\beta$ is less easily \npredicted. The accuracy of our fit to this ratio indicates that the relative \ncontributions of the matter and radiation bounded components are approximately \ncorrect, given the observational constraints on the ionizing continuum.\nThe fact that we have a reasonable fit \nfor lines that span a wide range of ionization and critical densities supports \nour values for density and ionization parameter. Most of the \ndiscrepancies between the observations and models are in the lowest and highest \nionization lines, which we will address below. \n\n\\subsection{Discrepancies and Possible Explanations}\n\nFirst, we address differences between the predicted and observed ratios for the \nlow ionization lines. The Mg II $\\lambda$2800 and [O I] $\\lambda\\lambda$6300, \n6364 lines are still predicted to be too strong by our standard model, by \nfactors $\\geq$ 6 and 2.5, respectively. Nearly all this emission is coming from \nOUTER. Two factors determine the strength of the [O I] lines: the hardness of \nthe ionizing continuum and the physical depth of the emission line clouds. Since \nit appears that we have a good fit for the ionizing continuum, the weakness in \nthe observed [O~I] lines gives a limit on the depth of the clouds. Truncating \nthe integration of OUTER at $\\tau_{0}$ $\\approx$ 1000 would give a better fit to \nthe [O I] without affecting the other important line ratios. This results in a \ncloud depth of $\\sim$2.5 x 10$^{15}$ cm.\nThe overprediction of the Mg II $\\lambda$2800 line strength presents a somewhat\ndifferent problem. In order to reduce the contribution of this line from INNER, \nwe assumed a matter bounded model for this component. To provide a better match \nto the observation of little or no Mg II emission, we can reduce the model \ncontribution by modifying OUTER. The Mg$^{+}$ emissivity is greatest near the \nH$^{+}$\/H$^{0}$ transition zone in OUTER, so a \nsimple truncation at much lower optical depths is not feasible, as it would have\na much greater effect on the other line ratios. \n\nAn obvious explanation for the weak observed Mg II is depletion of the magnesium \ninto dust grains, along with suppression of the resonance photons by multiple \nscatterings and eventual absorption by dust. This was suggested in Paper I (cf. \nKraemer \\& Harrington 1986; Ferland 1992). For comparison with our standard \nmodel, we generated a version of OUTER that includes dust, assuming \na dust to gas ratio that is 30\\% of that found in the Galactic interstellar \nmedium, with equal amounts of graphite and silicate grains and accompanying \ndepletions. These assumptions were made to avoid biasing our results by simply \nhaving all of the Mg depleted into dust grains. We assumed relative element \ndepletions as calculated by Seab \\& \nShull (1983), and the grain size distributions determined by Mathis et al. \n(1977) and Draine \\& Lee (1984); details of the treatment of dust in the code \nare given by Kraemer (1985). The results of the model are given in Table 2, and \nnot only show a significant drop in the relative strength of Mg II \n$\\lambda$2800, but also a drop in the Ly$\\alpha$ strength, as is expected due to \nthe preferential dust absorption of multiply scattered UV resonance lines. The \nlower Ly$\\alpha$\/H$\\beta$ ratio is a better fit to the observations. A \nsubstantially larger dust-to-gas ratio than we assumed would result in a \nLy$\\alpha$\/H$\\beta$ ratio that is lower than observed. Therefore, \nit is likely that there is some dust mixed in with the low-ionization gas, \nalthough with a lower dust to gas ratio than found in the ISM, and that \ndepletion coupled with the resonance line suppression explain the weak Mg II. \n\nSecond, we address discrepancies in the high ionization lines.\nSpecifically, the lines of N~V, [Ne~V], [Fe~VII], and [Fe~X] are too weak by \nfactors of 2 to 4 compared to the observations.\nAs we mentioned earlier, the density is well constrained, so increasing the \nionization parameter brings the gas well within 1 pc, into the realm of \nthe BLR. However, these lines are relatively narrow (FWHM $\\leq$ 500 km \ns$^{-1}$, see Moore et al. 1996) and they are not likely to arise very close to \nthe BLR. The strengths of the high ionization lines can also be enhanced \nrelative to H$\\beta$ by truncating the integration of INNER at a lower optical \ndepth. However, this has the problem of enhancing the He II emission relative to \nH$\\beta$ in the model. More likely solutions to the problem of underpredicting \nthe high ionization lines include 1) shock ionization, 2) a large ``blue \nbump'' in the EUV continuum, or 3) supersolar abundances.\n\nPredicting the strengths of the coronal lines has always been a problem with \nsimple photoionization models, as Viegas-Aldrovandi \\& Contini (1989) discuss in \nsome detail for the Fe lines. They suggest that there may be shocked gas mixed \nin with the photoionized clouds and that these high ionization lines may arise \nthere. Although this is certainly a possible factor, there may be other \nplausible explanations which avoid adding another level of complexity.\n\nAn obvious way in which the coronal lines might be enhanced is if there\nwere a component of ionizing radiation that contributed significantly\nat energies between 100 and 500 eV. Although there has been some\nspeculation about the presence of a ``blue bump'' in the EUV, recent work\nby Zheng et al. (1997) on low-redshift quasars shows that the {\\it near} EUV \ncontinuum is likely to be much steeper than previously \nsupposed. In NGC 5548, the {\\it EUVE} continuum point in Figure 2 is further \nconfirmation that a large blue bump is not present in the spectrum \nof NGC 5548. Another possibility may be diffuse radiation from the intercloud \nmedium. Tran (1995) has shown that there may be a contribution to the continuum \nradiation in some Seyfert 2 galaxies from thermal emission from the intercloud\nmedium responsible for the scattering of the hidden BLR emission into\nthe observer's line of sight. If a similar medium with temperatures\n$\\approx$ 5~x~10$^{5}$~K exists in the NLR of NGC 5548, it is possible that \nfree-free radiation and line emission that arise within it may contribute to the \nionization of the inner narrow-line gas. Although this component would be weak \ncompared to the continuum radiation emitted by the central source, it could have\na significant local contribution to the ionization balance of clouds existing\nwithin this inner region. However, recent observations suggest that the extended \nUV continuum seen in some Seyferts may be due to starbursts (Heckman et al. \n1997), which would not contribute to the high ionization lines.\n\nIn studies of medium redshift QSOs, Ferland et al. (1996) found\nevidence of supersolar abundances. There is no direct evidence of\nelemental enhancements in Seyfert galaxies, but it is certainly not\nimplausible, particularly near the nucleus, where the most intense activity\noccurs. As Oliva (1996) points out, the coronal line emission will be enhanced \nproportionally to the abundance of the atomic species.\nAs a comparison, we ran a version of INNER with a heavy element\nabundance that is twice solar, and the results are shown in Table 3. In \nparticular,\nthe relative strengths of the [Fe VII] and [Ne V] lines have increased,\nwhile many lines, such as C IV $\\lambda$1549 and C III] $\\lambda$1909, show \nlittle change. It is possible, then, that the observed strength of \nsome of these lines is in part due to enhanced abundances. Note that we have not \nattempted to adjust the increase in abundances to fit assumptions about the\ntype of star formation that might be expected.\n\n\\section{Discussion}\n\nFrom our standard model, we can estimate several global properties of the NLR in \nNGC 5548, including the covering factor and physical size, in addition to more \nlocal properties, including optical depth and presence of dust.\nWe determine the covering factor of the NLR gas from the observed and model \nvalues of the ``conversion efficiency'', $\\eta$ , which is the ratio of H$\\beta$ \nphotons to ionizing photons. The covering factor is given by\nC~$=$~$\\eta$(observed)\/$\\eta$(model). A value of C $>$ 1 would indicate that the \nionizing radiation is anisotropic, which we would not expect to be the case \nfor a Seyfert 1 galaxy, since the central source is seen directly in such \nobjects. Assuming Ho= 75 km s$^{-1}$ Mpc$^{-1}$, the observed H$\\beta$ \nflux corresponds to a luminosity of 3.8 x 10$^{40}$ ergs s$^{-1}$, or 9.3 x \n10$^{51}$  H$\\beta$ photons s$^{-1}$. From the continuum observations described \nin section 2.2, we calculate a total luminosity of ionizing photons of 1.09 x \n10$^{54}$ s$^{-1}$. This yields an observed $\\eta$ = 0.009. Our fit to the \nobserved emission line spectrum assumed that each component in our model \ncontributed 50\\% of the H$\\beta$ emission. The resulting values of $\\eta$ were \n0.06 for INNER and 0.11 for OUTER, and the covering factors are 0.07 and 0.04, \nrespectively, so C(NLR) = 0.11. The value is small compared to those found for \nSeyfert 2 galaxies, which are often $>$ 1 (Kinney et al. 1991; Kraemer et al. \n1994), and there is no evidence for anisotropic radiation. Note that this \nestimate of covering factor does not include the BLR clouds, which contribute at \nleast 50\\% of the flux in many of the strong lines, even when NGC 5548 is in its \nlowest state (Paper I). \n\nGiven the ionization parameters and densities of the two components from our \nstandard model, as well as the ionizing luminosity, the \ncharacteristic sizes (i.e., radii) for the two emitting regions are 1 pc for \nINNER, and 70 pc for OUTER. \n(Using the higher continuum luminosity given by the {\\it EUVE} point in Figure 2 \nwould increase these values by a factor of only $\\sqrt{1.30}$, or 1.14.)\nThus, the NLR of NGC 5548 is {\\it physically} \ncompact, since the size of OUTER is much smaller than typical values of 200 -- \n1000 pc determined for Seyfert 2 galaxies, using the same methods that we have \ndescribed in this paper (Kraemer \\& Harrington 1986; Kraemer et al. 1994).\nAlthough there have been reports of extended emission from NGC 5548 (Wilson et \nal. 1989), the contribution to the integrated emission line spectrum from this \nregion is small (Peterson et al. 1995); we estimated the contribution to the \nnarrow UV lines outside of a 1$''$ aperture (330 pc for H$_{0}$ $=$ 75 km \ns$^{-1}$ Mpc$^{-1}$) to be only $\\sim$20\\%. This is consistent with our finding \nthat the majority of the narrow emission must arise in a region with a \n``diameter'' of 140 pc.\n\nPogge (1989) and Schmitt \\& Kinney (1996) claim that the\n{\\it apparent} size of the NLR in a Seyfert 1 galaxy is typically much \nsmaller than that of a Seyfert 2 galaxy, although this may be due to a selection \neffect, since most of the Seyfert 2s in the {it HST} archive were selected on \nthe basis of their extended emission (Wilson 1997).\nNevertheless, it is clear that the majority of the Seyfert 1 galaxies in these \nstudies are apparently compact. This cannot be explained \nby viewing angle alone, since the opening angles \nof the presumed ionization cones are large, and in the simplest version of the \nunified model, the apparent extent of most Seyfert galaxies should be much \nlarger, even if viewed ``pole-on'' (Schmitt \\& Kinney 1996).\nOur results show that a possible explanation for the small apparent size of the \nNLRs in many Seyfert 1 galaxies is that they are truly (i.e., physically) \ncompact. Schmitt \\& Kinney explain this phenomenon as a result of\nthe orientation of the obscuring torus with respect\nto the plane of the galaxy. However, their model does not explain the dominance\nof the high ionization lines in NGC 5548 and many other Seyfert 1 galaxies, a \nfeature not generally seen in Seyfert 2 galaxies \n(Koski 1978; Shuder \\& Osterbrock 1981). Thus, if the unified model applies we \nmight expect that the narrow emission line spectrum of this Seyfert 1 galaxy \nwould resemble that of Seyfert 2 galaxies, and, further, we would expect \nto see a noticeable contribution to the spectrum from gas hundreds of parsecs \nfrom the central source. It is possible that the high ionization region in \nSeyfert 2 galaxies is obscured by a torus, but this does not explain the absence \nof a low ionization component in NGC 5548. Not only do our models show that the \nnarrow-line spectrum can be well fitted without such a component, but it is \nclear that the NLR of NGC 5548 is dominated by high ionization gas that must \nbe located close to the central source.\n\nA few Seyfert 1 galaxies do indeed show significant NLR emission at\nlarge distances from the central source. For example, {\\it HST} observations of \nNGC 4151 (Evans et al. 1993; Hutchings et al. 1997) reveal an array of knots and \nfilaments out to a kiloparsec that are almost certainly ionized by the central \ncontinuum. It may be that some Seyfert 1 galaxies, such as NGC 4151, have \nextended emission line regions and narrow lines in their spectra that closely\nresemble Seyfert 2 galaxies. Cohen (1983) studied the optical narrow\nline spectra of a group of Seyfert 1 galaxies and found some resemblance\nalthough, as a group, they appeared to be of somewhat higher ionization.\nCertainly, some of the galaxies in his study have spectra that are\nindistinguishable from those of type 2 Seyferts, but others, like NGC 5548, \nappear to be dominated by high ionization lines, a condition that appears to be \nrare among Seyfert 2 galaxies. It would be extremely interesting if one could \ndetermine if these high ionization objects are as compact as NGC 5548 appears to \nbe. Unfortunately, optical spectra alone are insufficient for detailed \nmodeling of the NLR, and thus far, accurate measurements of the narrow-line \nstrengths in the UV have only been obtained for NGC 5548\nand NGC 4151 (Ferland \\& Mushotzky 1982).\n\nAs we stated above, our finding that the inner component of gas is\noptically thin at the Lyman limit is based both on the absence of strong\nMg II emission and the relative strength of the high ionization lines.\nIf this is indeed true, not only for NGC 5548 but for other Seyfert 1\ngalaxies, it may be a clue to the origin of the inner narrow-line\ngas. Thin filaments or knots of reasonably high density ($\\sim$ 10$^{7}$ \ncm$^{-3}$) could be the result of outflow from the BLR, either as condensations\nin an expanding intercloud medium or as ``tails'' of BLR clouds, driven\nout by radiation pressure. If so, the small physical depths of the clouds\ninferred from our modeling ($\\sim$ 10$^{14}$ cm) may constrain the sizes of \ntheir BLR progenitors. Note that there has been some recent success in \nmodels of the BLR using a component of\noptically thin clouds (Shields, Ferland, \\& Peterson 1995).\nThere have been other studies suggesting that the \nclouds in the NLR are matter bounded or that some mix of radiation bounded and \nmatter bounded clouds exist (Wilson et al. 1997; Viegas-Aldrovandi 1988). Such a \nmix might give rise to the filamentary structure seen in some [O III] images of \nSeyfert galaxies rather than the typical molecular clouds found in spiral \ngalaxies, and may give a clue about the origin of the NLR gas.\n\nOne other point regarding the NLR gas is that it appears that there\nmay be some dust present within the outer components of the emitting\nregion. Dust is certainly able to exist in clouds at this proximity\nto the central source, with dust temperatures reaching a few hundred\nK (Kraemer \\& Harrington 1986). Nevertheless, the history of the dust\nin this gas is unknown. Our finding that the outer clouds\nmay not be purely radiation bounded would indicate that they are not\nsimply interstellar molecular clouds that have an outer shell ionized\nby the central source. Our finding that there is probably dust mixed in with\nthis gas will be important in determining the origin of this component.\n\nFinally, Moore et al. (1996) find a correlation of ionization potential of the \nnarrow emission lines with velocity width in NGC 5548. Combined with our finding \nthat, to first order, the ionization level decreases with distance, this \nindicates that the radial velocities decrease with increasing distance. This \ntrend is also seen in a more direct fashion in spatially resolved spectra of the \ninner NLR in NGC 4151 (Hutchings et al. 1997).\n\n\\section{Conclusions}\n\nWe have analyzed UV and optical spectra of the Seyfert 1 galaxy NGC 5548 that \nwere obtained when it was at an historical minimum. We were able to \nisolate the narrow emission line components (which do not vary in flux on short \ntime scales), due to the relative weakness of the more rapidly variable broad \nemission lines that are usually blended with the narrow components.\nWe have constructed photoionization models of the narrow-line region of this \ngalaxy, and are able to successfully match the observed dereddened ratios of a \nlarge number of emission lines to within the errors, with the \nexceptions noted in Section 3.3. Since we used the best direct observational \nevidence of the shape of the ionizing continuum, rather than making adjustments \nbased on fitting emission line ratios, the quality of this fit is particularly \nsatisfying. The fact that a good fit was obtained for the permitted emission \nlines, such as C~IV~$\\lambda$1549, and the forbidden lines, such as [Ne III] \n$\\lambda$3869, [O III] $\\lambda$5007, [O II] $\\lambda$3727, [N II] \n$\\lambda$6584, etc., indicates that the range of physical conditions assumed \nin these models is approximately correct.\n\nFrom our analysis and modeling of these spectra, we can make several statements \nregarding the physical conditions in the NLR of NGC 5548. First, it is clear \nthat the principal source of ionization in the NLR of NGC 5548 is the central \ncontinuum source. This conclusion is borne out by the quality of the fit to the \nemission line spectrum. The NLR covering factor is reasonably small (C $=$ \n0.11), so the NLR gas does not need to intercept much of the ionizing continuum \nto produce the observed emission lines fluxes. A second conclusion is that the \nhighly ionized gas in the inner part of the NLR appears to be optically thin at \nthe Lyman limit ($\\tau_{0}$ $\\approx$ 2.5), which yields constraints on the \nphysical depths of these clouds and may provide a clue to their origin. We have \nalso presented evidence for supersolar abundances in the inner portion of the \nNLR, and dust in the outer portion. \n\nThe most important conclusion that we have reached in this study is\nthat the NLR of NGC 5548 is physically compact, with the majority of emission \ncoming from a distance $\\leq$~70 pc from the nucleus. By contrast, the unified \nmodel of Seyfert galaxies suggests that the {\\it physical} dimensions of the NLR \nin Seyfert 1 and 2 galaxies should be similar. Additional studies of the type \nthat we have presented in this paper, particularly of other Seyfert 1 galaxies, \nare important for testing this aspect of the unified model.\n\n\\acknowledgments\n\nWe thank the referee, Andrew Wilson, for helpful comments and suggestions.\nWe are grateful to Fred Bruhweiler and Pat Harrington for helpful discussions on \nthe physical properties of iron and the availability of atomic data.\nS.B.K. and D.M.C. acknowledge support from NASA grant NAG~5-4103.\nA.V.F acknowledges support from NASA grant NAG~5-3556, and B.M.P. \nacknowledges support from NSF grant AST-9420080.\n\n\\clearpage\t\t\t       \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{SecIntro}\nA crucial step in the history of General Relativity (GR) was Einstein's adoption of the principle of general covariance which states that the form of our physical laws should be independent of any choice of coordinate systems. The conceptual benefits of writing a theory in a coordinate-free way are immense. A generally covariant formulation of a theory has at least two major benefits: 1) it more clearly exposes the theory's geometric background structure, and 2) it thereby helps clarify our understanding of the theory's symmetries (i.e., its structure\/solution preserving transformations). It does both of these by disentangling the theory's substantive content from representational artifacts which arise in particular coordinate representations~\\cite{Pooley2015,Norton1993,EarmanJohn1989Weas}. Thus, general covariance is an indispensable tool for a modern understanding of spacetime theories. \n\nMotivated by quantum gravity, one may wish to extend these notions to quantum spacetime theories (whatever those are). Relatedly, one might want to extend these notions to discrete spacetime theories (i.e., lattice theories\\footnote{Given the results of this paper and of \\cite{DiscreteGenCovPart1}, calling these ``lattice theories'' can be misleading. This would be analogous to referring to continuum spacetime theories as ``coordinate theories''. As I will discuss, in both cases the coordinate systems\/lattice structure are merely representational artifacts and so do not deserve ``first billing'' so to speak. All lattice theories are best thought of as lattice-representable theories. Similarly, the term ``discrete spacetime theories'' ought to be here read as ``discretely-representable spacetime theories''. As discussed here (and in \\cite{DiscreteGenCovPart1}), the defining feature of such theories is that they have a finite density of degrees of freedom, see the work of Achim Kempf~\\cite{UnsharpKempf,Kempf2003,Kempf2004a,Kempf2004b,Kempf2006}}.). In \\cite{DiscreteGenCovPart1} I developed two analogs of general covariance for such discrete spacetime theories in a non-Lorentzian setting. The aim of this paper is to extend these results to a Lorentzian setting. Indeed, the analysis provided here is nearly identical to the one carried out in \\cite{DiscreteGenCovPart1}, although each paper is self-contained.\n\nIn either setting these discrete analogs of general covariance reveal that lattice structure is rather less like a fixed background structure or a fundamental part of some underlying manifold and rather more like a coordinate system, i.e., merely a representational artifact. Indeed, these discrete analogs are built upon a rich analogy between the lattice structures appearing in our discrete spacetime theories and the coordinate systems appearing in our continuum spacetime theories. \n\nThis paper is largely inspired by the brilliant work of mathematical physicist Achim Kempf~\\cite{Kempf_1997,UnsharpKempf,Kempf2000b,Kempf2003,Kempf2004a,Kempf2004b,Kempf2006,Martin2008,Kempf_2010,Kempf2013,Pye2015,Kempf2018} among others~\\cite{PyeThesis,Pye2022,BEH_2020}. A key feature present both here and in Kempf's work is the sampling property of bandlimited function revealed by the Nyquist-Shannon sampling theory~\\cite{GARCIA200263,SamplingTutorial,UnserM2000SyaS}. I review sampling theory in more detail in Sec.~\\ref{SecSamplingTheory}, but let me overview here. Bandlimited functions are those with have a limited extent in Fourier space (i.e., compact support). Bandlimited functions have the following sampling property: they can be exactly reconstructed knowing only the values that they take on any sufficiently dense sample lattice. What ``sufficiently dense'' means is fixed in terms of the size of the function's support in Fourier space.\n\nNyquist-Shannon sampling theory was first discovered in the context of information processing as a way of converting between analog and digital signals (i.e., between continuous and discrete information). Sampling theory found its first application in fundamental spacetime physics with Kempf's \\cite{Kempf_1997,UnsharpKempf}, ultimately leading to his thesis that ``Spacetime could be simultaneously continuous and discrete, in the same way that information can be'' \\cite{Kempf_2010}. Kempf's thoughts on these topics is the primary inspiration for this paper and deserves wider appreciation by the philosophy of physics community. For an overview of Kempf's works on this topic see~\\cite{Kempf2018}.\n\nMy thesis in \\cite{DiscreteGenCovPart1} is in broad agreement with Kempf's with one crucial alteration. I stress that the sampling property of bandlimited functions indicates that bandlimited physics can be simultaneously \\textit{represented as} continuous and discrete, (i.e., on a continuous or discrete spacetime). However, I further argue (both here and in \\cite{DiscreteGenCovPart1}) that when one investigates these two representations one finds substantial issue with taking the discrete representation as fundamental. These issues stem from the rich analogy between the lattice structures and coordinate systems mentioned above.\n\nThis analogy is supported here (and in \\cite{DiscreteGenCovPart1}) by the three lessons each of which tell against an intuitions one is likely to have regarding lattice structure. To motivate these (wrong) intuitions, consider the following situation.\n\nSuppose that after substantial empirical investigation of our micro-physical reality we find what appear to be ``lattice artifacts''. For instance, we may find ourselves restricted to only quarter rotation, or one-sixth rotation symmetries. Intuitively, this would suggest that the world is fundamentally set on a lattice of the kind shown in Fig.~\\ref{FigLat}, i.e., a square or hexagonal lattice.\n\n\\begin{figure}[t!]\n\\includegraphics[width=0.4\\textwidth]{PaperFigures\/FigLat.pdf}\n\\caption{Three of the lattice structures in space considered throughout this paper. The arrows indicate the indexing conventions for the lattice sites. Repeated in time, these give us something like a square 2D lattice, a cubic 3D lattice, and a hexagonal 3D lattice respectively. [\\textit{Reproduced with permission from} \\cite{DiscreteGenCovPart1}.]}\\label{FigLat}\n\\end{figure}\n\nSuppose that we have great predictive success when modeling the world as being set on (for instance) a square lattice with next-to-nearest neighbor interactions. Would this in any way prove that the world is fundamentally set on such a lattice? No, all this would prove is that the world can be \\textit{faithfully represented} on such a lattice with such interactions, at least empirically. Anything can be faithfully represented in any number of ways, this is just mathematics. Some extra-empirical work must be done to know if we should take the lattice structure appearing in this representation seriously. That is, we must ask which parts of the theory are substantive and which parts are merely representational? The discrete analogs of general covariance developed here (and in \\cite{DiscreteGenCovPart1}) answer this question: lattice structures are coordinate-like representational artifacts and so ultimately have no physical content.\n\nTo flesh out the contrary received position however, let us proceed without this analogy for the moment. We can ask: beyond merely appearing in our hypothetical empirically successful theory, what reason do we have to take the lattice structures which appear in this theory seriously? Well, intuitively the lattice structures appear to play a very substantial role in these theories, not merely a representational one. One likely has the following three interconnected first intuitions regarding the role that the lattice and lattice structure play in discrete spacetime theories:\n\\begin{itemize}\n\\item[1.] They restrict our possible symmetries. Taking the lattice structure to be a part of the theory's fixed background structure, our possible symmetries are limited to those which preserve this fixed structure. Intuitively a theory set on a square lattice can only have the symmetries of that lattice. Similarly for a hexagonal lattice, or even an unstructured lattice.\n\\item[2.] Differing lattice structures distinguishes our theories. Two theories with different lattice structures (e.g., square, hexagonal, irregular, etc.) cannot be identical. As suggested above they have different fixed background structures and as therefore have different symmetries.\n\\item[3.] The lattice is fundamentally ``baked-into'' the theory. Firstly, it is what the fundamental fields are defined over: they map lattice sites (and in \\cite{DiscreteGenCovPart1} times) into some value space. Secondly, the bare lattice is what the lattice structure structures. Thirdly, it is what limits us to discrete permutation symmetries in advance of further limitations from the lattice structure.\n\\end{itemize}\nThese intuitions will be fleshed out and made more concrete in Sec.~\\ref{SecKlein1}. However, as this paper demonstrates, each of the above intuitions are doubly wrong and overhasty.\n\nWhat goes wrong with the above intuitions is that we attempted to directly transplant our notions of background structure and symmetry from continuous to discrete spacetime theories. This is an incautious way to proceed and is apt to lead us astray. Recall that, as discussed above, our notions of background structure and symmetry are best understood in light of general covariance. It is only once we understand what is substantial and what is merely representational in our theories that we have any hope of properly understanding them. Therefore, we ought to instead first transplant a notion of general covariance into our discrete spacetime theories and then see what conclusions we are led to regarding the role that the lattice and lattice structure play in our discrete spacetime theories. This transplant has been done in a non-Lorentzian setting in \\cite{DiscreteGenCovPart1}. Here I extend these results to a Lorentzian setting.\n\nThis paper will teach us three lessons each of which negates one of the above intuitions about the role that lattice structure plays in discrete spacetime theories.\n\nFirstly, as I will show, taking any lattice structure seriously as a fixed background structure systematically under predicts the symmetries that discrete theories can and do have. Indeed, as I will show neither the bare lattice itself nor its lattice structure in any way restrict a theory's possible symmetries. In \\cite{DiscreteGenCovPart1}, for non-Lorentzian theories I have shown that there is no conceptual barrier to having a theory with continuous translation and rotation symmetries formulated on a discrete lattice. Indeed, in \\cite{DiscreteGenCovPart1} I presented a perfectly rotation invariant lattice theory. As I discuss in \\cite{DiscreteGenCovPart1}, this is analogous to the familiar fact that there is no conceptual barrier to having a continuum theory with rotational symmetry formulated on a Cartesian coordinate system. Here, I repeat this analysis in a Lorentzian context. In Sec.~\\ref{PerfectLorentz}, I present a perfectly Lorentzian lattice theory. \n\nSecondly, as I will show, discrete theories which are initially presented to us with very different lattice structures (i.e., square vs. hexagonal) may nonetheless turn out to be completely equivalent theories or to be overlapping parts of some larger theory. Moreover, given any discrete theory with some lattice structure we can always re-describe it using a different lattice structure. As I will discuss, this is analogous to the familiar fact that our continuum theories can be described in different coordinates, and moreover we can switch between these coordinate systems freely.\n\nThirdly, as I will show, in addition to being able to switch between lattice structures, we can also reformulate any discrete theory in such a way that it has no lattice structure whatsoever. Indeed, we can always do away with the lattice altogether. As I will discuss, this is analogous to the familiar fact that any continuum theory can be written in a generally covariant (i.e., coordinate-free) way.\n\nThese three lessons combine to give us a rich analogy between lattice structures and coordinate systems. It is from this rich analogy that the central claims of this paper follow. Namely, from this analogy it follows that the lattice structure supposedly underlying any discrete ``lattice'' theory has the same level of physical import as coordinates do, i.e., none at all. Thus, as I argued in \\cite{DiscreteGenCovPart1}, the world cannot be ``fundamentally set on a square lattice'' (or any other lattice) any more than it could be ``fundamentally set in a certain coordinate system''. Like coordinate systems, lattice structures are just not the sort of thing that can be fundamental; they are both thoroughly merely representational. Spacetime cannot be a lattice (even when it might be representable as such). Specifically, I claimed that properly understood, there are no such things as lattice-fundamental theories, rather there are only lattice-representable theories. This paper extends these conclusions to a Lorentzian context.\n\nOnce one begins thinking of lattices as a merely representational structure, a path opens for perfectly Lorentzian lattice theories. As proponents of causal set theory correctly point out, no single fixed spacetime lattice is Poincar\\'e invariant. This (apparently) spells big trouble for any lattice-based Lorentzian theories. They, however, avoid this issue by considering instead a random Poisson sprinkling of lattice points which does not pick out any preferred direction and hence does not explicitly break Poincar\\'e symmetry, at least on average. However, given the deflationary position this paper takes towards lattices, I claim there is no issue to be avoided. Like coordinate systems, lattice structures are just a representational tool for helping us express our theory. There is no need for the symmetries of our representational tools to latch onto the symmetries of the thing being represented. Cartesian coordinates are distorted under Lorentz boosts, but we can still use them to describe our Lorentzian theories without issue. The same is true of lattices. Indeed, in Sec.~\\ref{PerfectLorentz} I will present a perfectly Lorentzian lattice theory.\n\n\\subsection{Outline of the Paper}\nIn Sec.~\\ref{SecSevenKG}, I will introduce seven discrete Klein Gordon equations in an interpretation-neutral way and solve their dynamics. Then, in Sec.~\\ref{SecKlein1}, I will make a first attempt at interpreting these theories. I will (ultimately wrongly) identify their underlying manifold, locality properties, and symmetries. Among other issues, a central problem with this first attempt is that it takes the lattice itself to be the underlying spacetime manifold and thereby unequivocally cannot support continuous translation and rotation symmetries. This systematically under predicts the symmetries that these theories can and do have.\n\nIn Sec.~\\ref{SecKlein2}, I will provide a second attempt at interpreting these theories which fixes this issue (albeit in a slightly unsatisfying way). In particular, in this second attempt I deny that the lattice is the underlying spacetime manifold. Instead, I ``internalize'' it into the theory's value space. Fruitfully, this second interpretation does allow for continuous translation and rotation symmetries and even a (limited) Lorentz boost symmetry. However, the key move here of ``internalization'' has several unsatisfying consequences. For instance, the continuous symmetries we find here are all classified as internal (i.e, associated with the value space) whereas intuitively they ought to be external (i.e, associated with the manifold).\n\nWe thus will need a third attempt at interpreting these theories which externalizes these symmetries. Sec.~\\ref{SecExtPart1} - Sec.~\\ref{SecExtPart2} lay the groundwork for this third interpretation. In particular, they describe a principled way of 1) inventing a continuous spacetime manifold for our formerly discrete theories to live on and 2) embedding our theory's states\/dynamics onto this manifold as a new dynamical field. In the middle of this, in Sec.~\\ref{SecSamplingTheory}, I will provide an informal overview of the primary mathematical tools used in the latter half of this paper. Namely, I will review the basics of Nyquist-Shannon sampling theory and bandlimited functions. \n\nWith this groundwork complete, in Sec.~\\ref{SecKlein3} and Sec.~\\ref{SecKlein3Extra} I will provide a third attempt at interpreting these seven theories which fixes all issues arising in the previous two interpretations. For instance, like in my second attempt, this third interpretation can support continuous translation and rotation symmetries as well as a (limited) Lorentz boost symmetry. However, unlike the second attempt it realizes them as external symmetries (i.e., associated with the underlying manifold, not the theory's value space).\n\nIn Sec.~\\ref{SecDisGenCov}, I will review the lessons learned in these three attempts at interpretation. As I will discuss, the lessons learned combine to give us a rich analogy between lattice structures and coordinate systems. As I will discuss, there are actually two ways of fleshing out this analogy: one internal and one external. This section spells out these analogies in detail, each of which gives us a discrete analog of general covariance. I find reason to prefer the external notion, but either is likely to be fruitful for further investigation\/use. Sec.~\\ref{PerfectLorentz} provides us with a perfectly Lorentzian lattice theory as promised.\n\nFinally, in Sec. \\ref{SecConclusion} I will summarize the results of this paper, discuss its implications, and provide an outlook of future work.\n\nFor comments on what this means for the the dynamical vs geometrical spacetime debate~\\cite{EarmanJohn1989Weas,TwiceOver,BelotGordon2000GaM,Menon2019,BrownPooley1999,Nonentity,HuggettNick2006TRAo,StevensSyman2014Tdat,DoratoMauro2007RTbS,Norton2008,Pooley2013} see \\cite{DiscreteGenCovPart1}. Here I will focus on the implications this work has for quantum gravity especially causal set theory~\\cite{Surya2019}.\n\n\\section{Seven Discrete Klein Gordon Equations}\\label{SecSevenKG}\nIn this section I will introduce seven discrete Klein Gordon equations (KG1-KG7) in an interpretation-neutral way and solve their dynamics. These theories are all describable as being set on a lattice in both space and time. In each of these theories the lattice in space will simply repeat itself in time. I consider the following three cases for the lattice in space: a uniform 1D lattice, a square 2D lattice, and a hexagonal 2D lattice, see Fig.~\\ref{FigLat}. Repeated in time, these give us something like a square 2D lattice, a cubic 3D lattice, and a hexagonal 3D lattice respectively.\n\nAs harmful as these choices seem to be to Lorentz invariance (as well as continuous translation and rotation invariance) as I will show they ultimately pose no barrier to our theories having these symmetries. As I will argue, these choices of lattice are ultimately merely choices of representation which have absolutely nothing to do with the thing being represented. In particular, there is no need for our representational structure to have the same symmetries as the thing being represented. There is no issue with using Cartesian coordinates to describe a rotationally invariant state\/dynamics. I claim that analogously there is no issue with using a lattice to describe a state\/dynamics with continuous translation and rotation invariance and even Lorentz invariance. This claim has already been demonstrated in \\cite{DiscreteGenCovPart1} for states\/dynamics with continuous translation and rotation invariance. This paper extends this claim about lattice structures to Lorentz invariance as well.\n\n\\subsection{Introducing KG1-KG7}\nTo begin, let us consider the continuum Klein Gordon equations in $1+1$ and $2+1$ dimensions:\n\\begin{align}\\label{KG00}\n&\\text{\\bf Continuum Klein Gordon Eq. (KG00):}\\\\\n\\nonumber\n&\\partial_{t}^2\\varphi(t,x) = (\\partial_x^2-M^2) \\, \\varphi(t,x)\\\\\n\\label{KG0}\n&\\text{\\bf Continuum Klein Gordon Eq. (KG0):}\\\\\n\\nonumber\n&\\partial_{t}^2\\varphi(t,x,y) = (\\partial_x^2+\\partial_y^2-M^2) \\, \\varphi(t,x,y)\n\\end{align}\nwith some mass $M\\geq0$. For a generally covariant (i.e., coordinate-free) view of these theories, see Sec.~\\ref{SecFullGenCov}. \n\nFor our first three discrete Klein Gordon theories, let us consider the theories with only nearest-neighbor (N.N.) interactions on the above discussed lattices which best approximate KG0 and KG00. Namely:\n\\begin{align}\\label{KG1Long}\n&\\text{\\bf 1D N.N. Klein Gordon Eq. (KG1):}\\\\\n\\nonumber\n&[\\phi_{j-1,n}-2\\phi_{j,n}+\\phi_{j+1,n}]\\\\\n\\nonumber\n&\\quad=[\\phi_{j,n-1}-2\\phi_{j,n}+\\phi_{j,n+1}]\n-\\mu^2 \\phi_{j,n}\\\\\n&\\text{\\bf Square N.N. Klein Gordon Eq. (KG4):}\\label{KG4Long}\\\\\n\\nonumber\n&[\\phi_{j-1,n,m}-2\\phi_{j,n,m}+\\phi_{j+1,n,m}]\\\\\n\\nonumber\n&\\quad = [ \\ \\phi_{j,n-1,m}-2\\phi_{j,n,m}+\\phi_{j,n+1,m}\\\\\n\\nonumber\n&\\quad \\ \\ +\\phi_{j,n,m-1}-2\\phi_{j,n,m}+\\phi_{j,n,m+1}]\n-\\mu^2 \\phi_{j,n,m}\\\\\n&\\text{\\bf Hexagonal N.N. Klein Gordon Eq. (KG5):}\\label{KG5Long}\\\\\n\\nonumber\n&[\\phi_{j-1,n,m}-2\\phi_{j,n,m}+\\phi_{j+1,n,m}]\\\\\n\\nonumber\n&\\quad = \\frac{2}{3}[ \\ \\phi_{j,n-1,m}-2\\phi_{j,n,m}+\\phi_{j,n+1,m}\\\\\n\\nonumber\n&\\qquad \\quad +\\phi_{j,n,m-1}\\!-\\!2\\phi_{j,n,m}\\!+\\!\\phi_{j,n,m+1}\\\\\n\\nonumber\n&\\qquad \\quad +\\phi_{j,n+1,m-1}-2\\phi_{j,n,m}+\\phi_{j,n-1,m+1}]\n-\\mu^2 \\phi_{j,n,m}\n\\end{align}\nwith $j\\in\\mathbb{Z}$ indexing time and $n\\in\\mathbb{Z}$ and $m\\in\\mathbb{Z}$ indexing space. See Fig.~\\ref{FigLat} for the indexing convention. Here $\\mu\\in\\mathbb{R}$ is a dimensionless number playing the role of the field's mass. The terms in square brackets in the above expressions are the best possible approximations of the second derivative on each lattice which make use of only nearest neighbor interactions. These theories are named KG1, KG4, and KG5 in correspondence with the discrete heat equations considered in \\cite{DiscreteGenCovPart1} and in anticipation of their further treatment later in this section. \n\nThis section has promised to introduce these theories in an interpretation neutral way. As such, some of the above discussion needs to be hedged. In particular, in introducing these theories I have made casual comparison between parts of these theories' dynamical equations and various approximations of the second derivative. While, as I will discuss, such comparisons can be made, to do so immediately is unearned. It comes dangerously close to imagining the spacetime lattices discussed above as being embedded in a continuous manifold. This may be something we want to do later (see Sec.~\\ref{SecExtPart1}), but it is a non-trivial interpretational move which ought not be done so casually. \n\nCrucially, in this paper I will begin by analyzing these theories as discrete-native theories. As such, it's important to think of these discrete spacetime theories as self-sufficient theories in their own right. We must not begin by thinking of them as various discretizations or bandlimitations of the continuum theories. While, as I will discuss, these discrete theories have some notable relationships to various continuum theories it is important to resist any temptation to see these continuum theories as ``where they came from''. Rather, let us pretend these theories ``came from nowhere'' and let us see what sense we can make of them.\n\nAnother bit of hedging: in introducing the above three theories I casually associated them with the lattice structures shown in Fig.~\\ref{FigLat} (each repeated in time). Making such associations ab initio is unwarranted. While we may eventually associate these theories with those lattice structures we cannot do so immediately. Such an association would need to be made following careful consideration of the dynamics. (Such an exercise is carried out in Sec.~\\ref{SecKlein1}.) Beginning here in an interpretation-neutral way these theories ought to be seen as being defined over a completely unstructured lattice. \n\nI will reflect this concern in my notation as follows. The labels for the lattice sites are presently too structured (e.g., $(j,n)\\in\\mathbb{Z}^2$ and $(j,n,m)\\in\\mathbb{Z}^3$). Instead we ought to think of the lattice sites as having labels $\\ell\\in L$ for some set $L$. Crucially, at this point the set of labels for the lattice sites, $L$, is just that, an unstructured set.\n\nUp to isomorphism (here, generic bijections, i.e. generic relabelings), sets are uniquely specified by their cardinality. The set of labels for the lattice sites is here countable, $\\ell\\in L\\cong\\mathbb{Z}\\cong\\mathbb{Z}^2\\cong\\mathbb{Z}^3$. Reframed this way the above discussed theories each consider the same discrete variables $\\phi_\\ell\\in\\mathbb{R}$. In particular, KG1 considers variables $\\phi_\\ell$ which under some convenient relabeling of the lattice sites, $\\ell\\in L\\mapsto (j,n)\\in\\mathbb{Z}^2$, satisfies Eq.~\\eqref{KG1Long}. Similarly, KG4 and KG5 consider variables $\\phi_\\ell$ which under some convenient relabeling of the lattice sites, $\\ell\\in L\\mapsto (j,n,m)\\in\\mathbb{Z}^3$, satisfy Eq.~\\eqref{KG4Long} and Eq.~\\eqref{KG5Long} respectively.\n\nIt's important to stress that the mere existence of these convenient relabelings by itself has no interpretative force. The fact that our labels $(j,n)\\in\\mathbb{Z}^2$ and $(j,n,m)\\in\\mathbb{Z}^3$ in some sense form a square 2D lattice and cubic 3D lattice in no way forces us to think of $L$ as being structured in this way (indeed, we might later like to think of $L$ as a hexagonal 3D lattice). In particular, the fact that these labels are in a sense equidistant from each other does not force us to think of the lattice sites as being equidistant from each other. Nor are we forced to think that ``the distance between lattice sites'' to be meaningful at all. Dynamical considerations may later push us in this direction, but the mere convenience of this labeling should not.\n\nI have above introduced three out of seven discrete Klein Gordon theories. In order to introduce the other four theories, it is convenient (but not necessary) to first reformulate things. In particular, let us reorganize the $\\phi_\\ell$ variables into a vector, namely,\n\\begin{align}\\label{PhiDef}\n\\bm{\\Phi}= \\sum_{\\ell\\in L} \\phi_\\ell \\, \\bm{b}_\\ell. \n\\end{align}\nwhere $\\bm{b}_\\ell$ is a linearly-independent basis vector for each $\\ell\\in L$ and \\mbox{$\\bm{\\Phi}$} is a vector in the vector space \\mbox{$\\mathbb{R}^L\\coloneqq\\text{span}(\\{\\bm{b}_\\ell\\}_{\\ell\\in L})$}. For later reference, it should be noted that $\\phi_\\ell$ is also a vector in a vector space: namely, $F_L$ the space of functions $f:L\\to\\mathbb{R}$. Note that Eq.~\\eqref{PhiDef} is an vector space isomorphism between these vector spaces, $\\mathbb{R}^L\\cong F_L$. Everything which follows concerning $\\bm{\\Phi}\\in\\mathbb{R}^L$ has an isomorphic description in terms of $\\phi_\\ell\\in F_L$.\n\nRecall that for KG1 the lattice sites $\\ell\\in L$ have a convenient relabeling in terms of two integer indices, \\mbox{$\\ell\\in L\\mapsto (j,n)\\in\\mathbb{Z}^2$}. We can use this relabeling to grant the vector space a tensor product structure as \\mbox{$\\mathbb{R}^L\\mapsto\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$} by taking \\mbox{$\\bm{b}_\\ell\\mapsto \\bm{e}_j\\otimes\\bm{e}_n$} where \n\\begin{align}\n\\bm{e}_m = (\\dots,0,0,1,0,0,\\dots)^\\intercal\\in\\mathbb{R}^\\mathbb{Z}\n\\end{align}\nwith the 1 in the $m^\\text{th}$ position. Under this restructuring of KG1 we have, \n\\begin{align}\\label{PhiVec1}\n\\bm{\\Phi}= \\sum_{j,n\\in\\mathbb{Z}} \\phi_{j,n} \\, \\bm{e}_j\\otimes\\bm{e}_n. \n\\end{align}\nIn these terms the dynamics of KG1 is given by,\n\\begin{align}\n\\label{DKG1}\n&\\text{\\bf Klein Gordon Equation 1 (KG1):}\\\\\n\\nonumber\n&\\Delta_{(1),\\text{j}}^2\\,\\bm{\\Phi}=\\Delta_{(1),\\text{n}}^2 \\, \\bm{\\Phi}\n-\\mu^2\\bm{\\Phi}\n\\end{align}\nwhere the notation $A_\\text{j}\\coloneqq A\\otimes\\openone$ and $A_\\text{n}\\coloneqq\\openone\\otimes A$ mean $A$ acts only on the first or second tensor space respectively. The linear operator $\\Delta_{(1)}^2$ appearing twice in the above expression is the following bi-infinite Toeplitz matrix:\n\\begin{align}\\label{Delta12}\n\\Delta_{(1)}^2=\\{\\Delta^+,\\Delta^-\\}\n&=\\text{Toeplitz}(1,\\,-2,\\,1)\\\\\n\\nonumber\n\\Delta^+&=\\text{Toeplitz}(0,-1,\\,1)\\\\\n\\nonumber\n\\Delta^-&=\\text{Toeplitz}(-1,\\,1,\\,0)\n\\end{align}\nwhere the curly brackets indicate the anticommutator, $\\{A,B\\}= \\frac{1}{2}(A\\,B + B\\,A)$. Recall that Toeplitz matrices are so called diagonal-constant matrices with \\mbox{$[A]_{i,j}=[A]_{i+1,j+1}$}. Thus, the values in the above expression give the matrix's values on either side of the main diagonal.\n\nAlthough above I warned about thinking in terms of derivative approximations prematurely, a few comments are here warranted. Note that $\\Delta^+$ is associated with the forward derivative approximation, $\\Delta^-$ is be associated with the backwards derivative approximation, and $\\Delta_{(1)}^2$ is associated with the nearest neighbor second derivative approximation,\n\\begin{align}\n\\nonumber\n\\Delta_{(1)}^2\/\\epsilon^2:\\ \\partial_x^2 f(x)\n&\\approx\\frac{f(x+\\epsilon)-2 f(x)+f(x-\\epsilon)}{\\epsilon^2}.\n\\end{align}\nAs stressed above, we ought to be cautious not to lean too heavily on these relationships when interpreting these discrete theories. \n\nIn addition to KG1, I will also consider two more theories with ``improved derivative approximations''. Namely,\n\\begin{align}\n\\label{DKG2}\n&\\text{\\bf Klein Gordon Equation 2 (KG2):}\\\\\n\\nonumber\n&\\Delta_{(2),\\text{j}}^2\\,\\bm{\\Phi}=\\Delta_{(2),\\text{n}}^2 \\ \\bm{\\Phi}-\\mu^2 \\, \\bm{\\Phi}\\\\\n\\label{DKG3}\n&\\text{\\bf Klein Gordon Equation 3 (KG3):}\\\\\n\\nonumber\n&D_{\\text{j}}^2\\,\\bm{\\Phi}=D_{\\text{n}}^2 \\ \\bm{\\Phi}-\\mu^2 \\, \\bm{\\Phi}\n\\end{align}\nwhere\n\\begin{align}\\label{BigToeplitz}\n\\Delta_{(2)}^2&=\\text{Toeplitz}(\\frac{-1}{12},\\,\\frac{4}{3},\\,\\frac{-5}{2},\\,\\frac{4}{3},\\,\\frac{-1}{12})\\\\\n\\nonumber\nD&=\\text{Toeplitz}(\\dots,\\!\\frac{-1}{5},\\!\\frac{1}{4},\\!\\frac{-1}{3},\\!\\frac{1}{2},\\!-1,\\!0,\\!1,\\!\\frac{-1}{2},\\!\\frac{1}{3},\\!\\frac{-1}{4},\\!\\frac{1}{5},\\!\\dots)\\\\\n\\nonumber\nD^2&=\\text{Toeplitz}(\\dots,\\!\\frac{-2}{16},\\!\\frac{2}{9},\\!\\frac{-2}{4},\\!\\frac{2}{1},\\!\\frac{-2\\pi^2}{6},\\!\\frac{2}{1},\\!\\frac{-2}{4},\\!\\frac{2}{9},\\!\\frac{-2}{16},\\!\\dots).\n\\end{align}\nNote that $\\Delta_{(2)}^2$ is related to the next-to-nearest-neighbor approximation to the second derivative. Obviously, the longer range we make our derivative approximations the more accurate they can be. The infinite-range operator $D$ (and its square $D^2$) in some sense are the best discrete approximations to the derivative (and second derivative) possible. The defining property of $D$ is that it is diagonal in the (discrete) Fourier basis with spectrum,\n\\begin{align}\\label{LambdaD}\n\\lambda_D(k)=-\\mathrm{i}\\,\\underline{k}    \n\\end{align}\nwhere $\\underline{k}=k$ for $k\\in[-\\pi,\\pi]$ repeating itself cyclically with period $2\\pi$ outside of this region. This is in tight connection with the continuum derivative operator $\\partial_x$ which is diagonal in the (continuum) Fourier basis with spectrum \\mbox{$\\lambda_{\\partial_x}(k)=-\\mathrm{i} \\, k$} for $k\\in[-\\infty,\\infty]$. \n\nAlternatively, one can construct $D^2$ in the following way: generalize $\\Delta_{(1)}^2$ and $\\Delta_{(2)}^2$ to $\\Delta_{(n)}^2$ namely the best second derivative approximation which considers up to $n^\\text{th}$ neighbors to either side. Taking the limit $n\\to\\infty$ gives $D^2=\\lim_{n\\to\\infty}\\Delta_{(n)}^2$. Other aspects of $D$ will be discussed in Sec.~\\ref{SecSamplingTheory} (including its related derivative approximation Eq.~\\eqref{ExactDerivative}) but enough has been said for now.\n\nWhile these connections to derivative approximations allow us to export some intuitions from the continuum theories into these discrete theories, we must resist this (at least for now). In particular, I should stress again that we should not  be thinking of any of KG1, KG2 and KG3 as coming from the continuum theory under some approximation of the derivative.\n\nLet's next reformulate KG4 and KG5 in terms of \\mbox{$\\bm{\\Phi}\\in\\mathbb{R}^L$}. In these cases we have a convenient relabeling of the lattice sites in terms of three integer indices, $\\ell\\mapsto (j,n,m)$. As before we can use this relabeling to grant the vector space a tensor product structure as $\\mathbb{R}^L\\mapsto\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$ by taking by taking \\mbox{$\\bm{b}_\\ell\\mapsto \\bm{e}_j\\otimes\\bm{e}_n\\otimes\\bm{e}_m$}. Under this restructuring we have,\n\\begin{align}\\label{PhiVec2}\n\\bm{\\Phi}= \\sum_{j,n,m\\in\\mathbb{Z}} \\phi_{j,n,m} \\, \\bm{e}_j\\otimes\\bm{e}_n\\otimes\\bm{e}_m. \n\\end{align}\nIn these terms the dynamics of KG4 given above (namely, Eq.~\\eqref{KG4Long}) is now given by,\n\\begin{align}\\label{DKG4}\n&\\text{\\bf Klein Gordon Equation 4 (KG4):}\\\\\n\\nonumber\n&\\Delta_{(1),\\text{j}}^2\\,\\bm{\\Phi}=(\\Delta_{(1),\\text{n}}^2+\\Delta_{(1),\\text{m}}^2) \\ \\bm{\\Phi}-\\mu^2 \\, \\bm{\\Phi}\n\\end{align}\nA similar treatment of the dynamics of KG5 (namely, Eq.~\\eqref{KG5Long}) gives us,\n\\begin{align}\\label{DKG5}\n&\\text{\\bf Klein Gordon Equation 5 (KG5):}\\\\\n\\nonumber\n&\\Delta_{(1),\\text{j}}^2\\,\\bm{\\Phi} =\\frac{2}{3} \\, \\Big[\\Delta^2_{(1),\\text{n}}+ \\Delta^2_{(1),\\text{m}}\\\\\n\\nonumber\n&\\qquad\\qquad\\quad \\ +\\big\\{\\Delta^+_\\text{m}-\\Delta^+_\\text{n},\\Delta^-_\\text{m}-\\Delta^-_\\text{n}\\big\\}\\Big] \\, \\bm{\\Phi}-\\mu^2 \\, \\bm{\\Phi}\n\\end{align}\nWhile the third term in the square brackets looks complicated, it is just the analog of $\\Delta^2_{(1),\\text{n}}$ and $\\Delta^2_{(1),\\text{m}}$ but in the $m-n$ direction. See Eq.~\\eqref{Delta12}.\n\nFinally, in addition to KG4 and KG5 I consider the following two theories:\n\\begin{align}\n\\label{DKG6}\n&\\text{\\bf Klein Gordon Equation 6 (KG6):}\\\\\n\\nonumber\n&D_{\\text{j}}^2\\,\\bm{\\Phi}=(D_{\\text{n}}^2+D_{\\text{m}}^2) \\ \\bm{\\Phi}-\\mu^2 \\, \\bm{\\Phi}\\\\\n\\label{DKG7}\n&\\text{\\bf Klein Gordon Equation 7 (KG7):}\\\\\n\\nonumber\n&D_{\\text{j}}^2\\,\\bm{\\Phi}= \\frac{2}{3}\\left(D^2_\\text{n}+D^2_\\text{m}+(D_\\text{m}-D_\\text{n})^2\\right) \\, \\bm{\\Phi}-\\mu^2 \\, \\bm{\\Phi}\n\\end{align}\nwhich resemble KG4 and KG5 but which make use of an infinite range coupling between lattice sites. Having introduced these seven theories, let us next solve their dynamics.\n\n\\subsection{Solving Their Dynamics}\nConveniently, each of KG1-KG7 admit planewave solutions. Moreover, in each case these planewave solutions form a complete basis of solutions.\n\nConsidering first KG1-KG3 we have solutions of the form,\n\\begin{align}\\label{PlaneWave123}\n\\phi_{j,n}(\\omega,k)=e^{-\\mathrm{i}\\, \\omega \\, j - \\mathrm{i}\\, k \\, n}.\n\\end{align}\nwith $\\omega,k\\in\\mathbb{R}^2$. It should be noted however, that outside of the range $\\omega,k\\in[-\\pi,\\pi]$ these planewaves repeat themselves with period $2\\pi$ due to Euler's identity, $\\exp(2\\pi\\mathrm{i})=1$. In terms of $\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$ these planewaves are:\n\\begin{align}\n\\bm{\\Phi}(\\omega,k)=\\sum_{j,n\\in\\mathbb{Z}} \\phi_{j,n}(\\omega,k) \\, \\bm{e}_j\\otimes\\bm{e}_n. \n\\end{align}\nFrom this planewave basis we can recover the $\\bm{e}_j\\otimes\\bm{e}_n$ basis as:\n\\begin{align}\n\\bm{e}_j\\otimes\\bm{e}_n\n&=\\frac{1}{(2\\pi)^2}\\int\\!\\!\\!\\!\\int_{-\\pi}^\\pi e^{\\mathrm{i}\\,\\omega \\,j+\\mathrm{i}\\, k\\, n}\\, \\bm{\\Phi}(\\omega,k)\\,\\d\\omega\\d k. \n\\end{align}\nThese planewaves are only a solution if $\\omega$ and $k$ satisfy the theory's dispersion relation which can be straight-forwardly calculated from the theory's dynamics:\n\\begin{align}\n\\text{KG1:}& \\quad \\!\\! 2-2\\,\\cos(\\omega)= \\mu^2+2-2\\,\\cos(k)\\\\\n\\text{KG2:}& \\quad \\!\\! \\frac{1}{6}\\,(\\cos(2\\,\\omega)-16\\,\\cos(\\omega)+15)\\\\\n\\nonumber\n&= \\mu^2+\\frac{1}{6}\\,(\\cos(2\\,k)-16\\,\\cos(k)+15)\\\\\n\\text{KG3:}& \\quad \\!\\! \\underline{\\omega}^2= \\mu^2+\\underline{k}^2\n\\end{align}\nwhere $\\underline{\\omega}=\\omega$ and $\\underline{k}=k$ for $\\omega,k\\in[-\\pi,\\pi]$ repeating themselves cyclically with period $2\\pi$ outside of this region. Note that the dispersion relation for KG3 follows from Eq.~\\eqref{LambdaD}, essentially from the definition of $D$.\n\n\\begin{figure}[t!]\n\\includegraphics[width=0.45\\textwidth]{PaperFigures\/FigKleinDisp.pdf}\n\\caption{The dispersion relations for the planewave solutions to the discrete Klein Gordon equations are plotted as a function of wavenumber for KG1, KG2 and KG3 with $\\mu=1.25$.}\\label{FigKleinDisp}\n\\end{figure}\n\nFig.~\\ref{FigKleinDisp} shows these dispersion relations restricted to the region $\\omega,k\\in[-\\pi,\\pi]$ with a field mass of $\\mu=1.25$. Qualitatively, KG1-KG3 all seem to agree with each other at low wavenumbers. They appear to mostly differ with respects to the rate at which high wavenumber planewaves oscillate. Let's investigate how these theories behave for planewaves with periods and wavelengths which span many lattice sites, that is with $\\vert\\omega\\vert,\\vert k\\vert\\ll\\pi$.\n\\begin{align}\n\\nonumber\n\\text{KG1:}& \\quad \\!\\! \\omega^2= \\mu^2+k^2+\\frac{\\omega^4-k^4}{12}+\\mathcal{O}(\\omega^6)+\\mathcal{O}(k^6)\\\\\n\\nonumber\n\\text{KG2:}& \\quad \\!\\! \\omega^2= \\mu^2+k^2-\\frac{\\omega^6-k^6}{90}+\\mathcal{O}(\\omega^8)+\\mathcal{O}(k^8)\\\\\n\\text{KG3:}& \\quad \\!\\! \\omega^2 = \\mu^2+k^2.\n\\end{align}\nNote that the dispersion relation for KG3 exactly matches that of the continuum theory, not only within this regime but for all $\\omega,k\\in[-\\pi,\\pi]$. In the $\\vert\\omega\\vert,\\vert k\\vert\\ll\\pi$ regime, KG2 gives a better approximation of the continuum theory than KG1 does. This is due to its longer range coupling giving a better approximation of the derivative. \n\nIf we consider only solutions with all or most of their planewave support with $\\vert\\omega\\vert,\\vert k\\vert\\ll\\pi$, we have an approximate one-to-one correspondence between the solutions to these theories. This is roughly why each of these theories have the same continuum limit, namely KG00 defined above. In terms of the rate at which these theories converge to the continuum theory in the continuum limit, one can expect KG3 to outpace KG2 which outpaces KG1. (As I discussed in \\cite{DiscreteGenCovPart1}, this is in a way counter-intuitive: why does the most non-local discrete theory give the best approximation of our perfectly local continuum theory?)\n\nHowever, while interesting in their own right, these relationships with the continuum theory are not directly helpful in helping us understand KG1-KG3 in their own terms as discrete-native theories.\n\nMoving on to KG4-KG7, their planewave solutions are of the form, \n\\begin{align}\n\\phi_{j,n,m}(\\omega,k_1,k_2)=e^{-\\mathrm{i}\\,\\omega \\,j-\\mathrm{i}\\, k_1\\, n-\\mathrm{i}\\, k_2\\, m}\n\\end{align}\nwith $\\omega,k_1,k_2\\in\\mathbb{R}^2$. Again, it should be noted however, that outside of the range \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$} these planewaves repeat themselves with period $2\\pi$ due to Euler's identity, $\\exp(2\\pi\\mathrm{i})=1$. In terms of \\mbox{$\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$} these planewaves are:\n\\begin{align}\n\\nonumber\n\\bm{\\Phi}(\\omega,k_1,k_2)=\\sum_{j,n,m\\in\\mathbb{Z}} \\phi_{j,n,m}(\\omega,k_1,k_2) \\, \\bm{e}_j\\otimes\\bm{e}_n\\otimes\\bm{e}_m. \n\\end{align}\nFrom this planewave basis we can recover the $\\bm{e}_j\\otimes\\bm{e}_n\\otimes\\bm{e}_m$ basis as:\n\\begin{align}\n&\\bm{e}_j\\otimes\\bm{e}_n\\otimes\\bm{e}_m\\\\\n\\nonumber\n&=\\frac{1}{(2\\pi)^3}\\int\\!\\!\\!\\!\\int\\!\\!\\!\\!\\int_{-\\pi}^\\pi e^{\\mathrm{i}\\,\\omega \\,j+\\mathrm{i}\\, k_1 n+\\mathrm{i}\\, k_2 m}\\,\\bm{\\Phi}(\\omega,k_1,k_2)\\,\\d\\omega\\d k_1 \\d k_2. \n\\end{align}\n\nThe dispersion relation for each of these theories is given by:\n\\begin{align}\n\\text{KG4:}& \\ 2-2\\cos(\\omega)= \\mu^2\\!+\\!4\\!-\\!2\\cos(k_1)\\!-\\!2\\cos(k_2)\\\\\n\\nonumber\n\\text{KG5:}& \\  2-2\\,\\cos(\\omega)= \\mu^2 \\\\\n\\nonumber\n&+\\frac{4}{3}\\big[3-\\cos(k_1)-\\cos(k_1)-\\cos(k_2-k_1)\\big]\\\\\n\\nonumber\n\\text{KG6:}& \\ \\underline{\\omega}^2= \\mu^2 +\\underline{k_1}^2 +\\underline{k_2}^2\\\\\n\\nonumber\n\\text{KG7:}& \\ \\underline{\\omega}^2= \\mu^2 +\\frac{2}{3}\\big[\\underline{k_1}^2 +\\underline{k_2}^2 +(\\underline{k_2}-\\underline{k_1})^2\\big].\n\\end{align}\nNote that the dispersion relation for KG6 and KG7 follow from Eq.~\\eqref{LambdaD}, essentially from the definition of $D$.\n\nUnlike KG1-KG3, these theories do not all agree with each other in the small $\\omega,k_1,k_2$ regime. KG4 and KG6 agree that for $\\vert\\omega\\vert, \\vert k_1\\vert, \\vert k_2\\vert \\ll \\pi$ we have \\mbox{$\\omega^2= \\mu^2 +k_1^2 +k_2^2$}. Moreover, KG5 and KG7 agree with each other in this regime, but not with KG4 and KG6. Do we have two different results in the continuum limit here?\n\nCloser examination reveals that we do not. The key to realizing this is to note that under the transformation, \n\\begin{align}\\label{SkewKG7KG6}\n\\omega\\mapsto \\omega,\\quad\nk_1\\mapsto k_1,\\quad\nk_2\\mapsto \\frac{1}{2}\\, k_1+\\frac{\\sqrt{3}}{2}\\,k_2.\n\\end{align}\nwe have the dispersion relation for KG7 mapping exactly onto the one for KG6. The inverse of this map is\n\\begin{align}\\label{SkewKG6KG7}\n\\omega\\mapsto \\omega,\\quad\nk_1\\mapsto k_1,\\quad\nk_2\\mapsto \\frac{2k_2-k_1}{\\sqrt{3}}.\n\\end{align}\nTechnically, when acting on the planewaves $\\bm{\\Phi}(\\omega,k_1,k_2)$ these transformations are only each other's inverses when we have \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$} both before and after the transformation. This is due to the $2\\pi$ periodicity of these planewaves. Fortunately however, all of KG7's planewave solutions with \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$} remain in this region after applying Eq.~\\eqref{SkewKG7KG6}. The same is true of KG6: its planewave solutions with \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$} also remain in this region after applying Eq.~\\eqref{SkewKG6KG7}. As I will soon discuss, this means we have an exact one-to-one correspondence between KG6 and KG7's solutions (much ado will be made about this later.) Applying this transformation to KG5 does not map it onto KG4, but it does bring their $\\vert\\omega\\vert,\\vert k_1\\vert,\\vert k_2\\vert\\ll\\pi$ behavior into agreement. \n\nThus, if we consider only solutions with all or most of their planewave support with $\\vert\\omega\\vert,\\vert k_1\\vert,\\vert k_2\\vert\\ll\\pi$ (or the appropriately transformed regime for KG5 and KG7) then we have an approximate one-to-one correspondence between the solutions to these theories. Within this regime we can define their common continuum limit, KG0. Repeating our analysis of the convergence rates of KG1-KG3 here, we expect KG6 and KG7 to converge in the continuum limit faster than KG4 and KG5 do.\n\n\\vspace{0.25cm}\n\nThis paper will make three attempts at interpreting these seven discrete theories. Allow me to identify in advance three important points of comparison between these interpretations. \n\nThe first important point of comparison is what sense they make of these different convergence rates in the continuum limit. As discussed above, in terms of this convergence rate we expect \\mbox{$\\text{KG3}>\\text{KG2}>\\text{KG1}$} and similarly \\mbox{$\\text{KG6}, \\text{KG7} > \\text{KG4}, \\text{KG5}$} with higher rated theories converging more quickly. This is in tension with our intuitive sense of locality for these theories: judging locality by the number of lattice sites coupled together we have\n\\mbox{$\\text{KG1} > \\text{KG2} > \\text{KG3}$} with higher rated theories being more local and similarly\n\\mbox{$\\text{KG4},\\text{KG5} > \\text{KG6},\\text{KG7}$}. How is it that our most non-local theories are somehow the nearest to our perfectly local continuum theory?\n\nRegarding how these three interpretations deal with this tension, not much changes between the discrete heat equations considered in \\cite{DiscreteGenCovPart1} and the discrete Klein Gordon equations considered here. As such, I will leave any detailed discussion of this issue to \\cite{DiscreteGenCovPart1} and direct the interested reader there. Roughly, the second and third interpretations deal with this tension by negating or reversing all of the above intuitive locality judgements.\n\nA second important point of comparison between these three interpretations will be what sense they make of the  above-noted exact one-to-one correspondence between KG6 and KG7's solutions. (More will be said about this in Sec.~\\ref{SecKlein2}.) It is important to note that the mere existence of such a one-to-one correspondence does not automatically mean that these theories are identical or even equivalent; All it means technically is that their solution spaces have the same cardinality. As I will discuss, some of the coming interpretations recognize KG6 and KG7 as being equivalent whereas others do not.\n\nA third important point of comparison between the coming interpretations will be what sense they make of these theories having continuous symmetries. For instance, the dispersion relation for KG6 appears to be in some sense rotation invariant and even Lorentz invariant (at least in Fourier space and staying inside of the region \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$}). In a sense, KG7 might have these symmetries too: given the above-noted one-to-one correspondence between the solutions of KG6 and KG7, there may be some (skewed) sense in which KG7 is rotation invariant and Lorentz invariant as well. All of this will be made precise later on. As I will discuss, some interpretations consider KG6 and KG7 to have a rotation symmetry and even (limited) Lorentz boost symmetries whereas others do not. As I will discuss in Sec.~\\ref{PerfectLorentz}, KG6 and KG7 can be seen as representationally-limited parts of a larger perfectly Lorentzian lattice theory.\n\nHaving introduced these theories and solved their dynamics in an interpretation-neutral way. We can now make a first (ultimately misled) attempt at interpreting them.\n\n\\section{A First Attempt at Interpreting KG1-KG7}\\label{SecKlein1}\nNow that we have introduced these seven discrete theories and solved their dynamics, let's get on to interpreting them. Let us begin by following our first intuitions and analyze these seven discrete theories concerning their underlying manifold, locality properties and symmetries. Ultimately however, as I will discuss later, much of the following is misled and will need to be revisited and revised later. Luckily, retracing where we went wrong here will be instructive later.\n\nLet's start by taking the initial formulation of the above theories in terms of $\\phi_\\ell$ seriously, i.e. Eq.~\\eqref{KG1Long}, Eq.~\\eqref{KG4Long} and Eq.~\\eqref{KG5Long}. Taken literally as written, what are these theories about? Intuitively these theories are about a field $\\phi_\\ell$ which maps lattice sites ($\\ell\\in L\\cong\\mathbb{Z}\\cong\\mathbb{Z}^2\\cong\\mathbb{Z}^3$) into field amplitudes ($\\phi_\\ell\\in\\mathbb{R}$). That is a field $\\phi:Q\\to \\mathcal{V}$ with a discrete manifold $Q=L$ and value space $\\mathcal{V}=\\mathbb{R}$. Thus, taking $\\phi:Q\\to \\mathcal{V}$ seriously as a fundamental field leads us to thinking of $Q=L$ as the theory's underlying manifold and $\\mathcal{V}=\\mathbb{R}$ as the theory's value space. It is important to note that here, $Q$ is the entire manifold, it is not being thought of as embedded in some larger manifold. (However, a view like this will be considered in Sec.~\\ref{SecExtPart1}.) \n\nTaking $Q$ to be these theories' underlying manifold has consequences for our understanding of the locality of these theories. In a highly intuitive sense, theory KG1 is the most local in that it couples together the fewest lattice sites (only nearest neighbors). Following this KG2 is the next most local in the same sense: it couples only next-to-nearest neighbors. Finally, in this sense KG3 is the least local, it has an infinite range coupling. As mentioned above, there is some tension however with the rate we expect each of these theories to converge at in the continuum limit. How is it that our most non-local theories are somehow the nearest to our perfectly local continuum theory? This first interpretation can do little to resolve this tension, I refer the interested reader to \\cite{DiscreteGenCovPart1} for further discussion.\n\n\\subsection{Intuitive Symmetries}\nWith this manifold $Q=L$ and value space $\\mathcal{V}=\\mathbb{R}$ picked out, what can we expect of these theories' symmetries? For any spacetime theory there are roughly three kinds of symmetries: 1) external symmetries associated with automorphisms of the manifold, here $\\text{Auto}(Q)$, 2) internal symmetries associated with automorphisms of the value space, here $\\text{Auto}(\\mathcal{V})$, and gauge symmetries which result from allowing these internal symmetries to vary smoothly across the manifold. But what are the relevant notions of automorphism here?\n\nAnswering this question for $\\text{Auto}(Q)$ will require us to distinguish what structures are ``built into'' $Q$ and what are ``built on top of'' $Q$. The analogous distinction in the continuum case is that we generally take the manifold's differentiable structure to be built into it while the Minkowski metric, for instance, is something additional built on top of the manifold. In this paper, I am officially agnostic on where we draw this line in the discrete case. However, for didactic purposes I will here be as conservative as possible giving $Q$ as little structure as is sensible. Note that the less structure we associate with $Q$ the larger the class of relevant automorphisms $\\text{Auto}(Q)$ will be. Thus, I am taking $\\text{Auto}(Q)$ to be as large as it can reasonably be. \n\nHere the minimal structure we can reasonably associate with $Q=L$ is that of a set. As such the largest $\\text{Auto}(Q)$ could reasonably be is permutations of the lattice sites, \\mbox{$\\text{Auto}(Q)=\\text{Perm}(L)$}. \n\nIn addition to $\\text{Auto}(Q)$ we might also have internal symmetries $\\text{Auto}(\\mathcal{V})$ and gauge symmetries. While in general there may be abundant internal or gauge symmetries, for the present cases there are not many. In particular, for all of the above-mentioned theories we only have $\\mathcal{V}=\\mathbb{R}$. As mentioned following Eq.~\\eqref{PhiDef}, our (potentially off-shell) discrete fields are themselves vectors $\\phi_\\ell\\in F_L$. Namely, they are closed under addition and scalar multiplication and hence form a vector space. This addition and scalar multiplication is carried out lattice-site-by-lattice-site. Thus, the field's value space $\\mathcal{V}=\\mathbb{R}$ is also structured like a vector space. \n\nThe value space $\\mathcal{V}=\\mathbb{R}$ may additionally have more structure than this. However, as above, for didactic purposes I will here minimize the assumed structure in order to maximize possible symmetries. We can even drop the zero vector from our consideration taking $\\mathcal{V}=\\mathbb{R}$ to be an affine vector space.\nTherefore, I will take \\mbox{$\\text{Auto}(\\mathcal{V})=\\text{Aff}(\\mathbb{R})$} such that our internal symmetries are linear-affine rescalings of $\\phi_\\ell$, namely $\\phi_\\ell\\mapsto c_1\\phi_\\ell+c_2$. We can find the theory's gauge symmetries by letting \\mbox{$c_1,c_2\\in\\mathbb{R}$} vary smoothly across $Q$. That is, $\\phi_\\ell\\mapsto c_{\\ell,1}\\phi_\\ell+c_{\\ell,2}$.\n\nThus, in total, for KG1-KG7 the widest scope of symmetry transformations available to us (at least on this interpretation) are:\n\\begin{align}\\label{PermutationLong}\ns:\\quad \\phi_\\ell\\mapsto c_{P(\\ell),1} \\phi_{P(\\ell)}+c_{P(\\ell),2}\n\\end{align}\nfor some permutation $P:L\\to L$.\n\nFor later reference it will be convenient to translate these potential symmetry transformations in terms of the vector, $\\bm{\\Phi}\\in\\mathbb{R}^L$, as \n\\begin{align}\\label{Permutation}\ns:\\quad\\bm{\\Phi}\\mapsto C_1\\,P\\,\\bm{\\Phi}+\\bm{c}_2,\n\\end{align}\nfor some permutation matrix, $P$, a diagonal matrix $C_1$ and a vector $\\bm{c}_2$. Here $P$ captures the theory's possible external symmetries:  the possibility of permuting lattice sites. The diagonal matrix $C_1$ and the vector $\\bm{c}_2$ capture the theory's possible gauge symmetries: the possibility of linear-affine rescalings of $\\phi_\\ell$ which vary smoothly across $Q$.\n\nI will next discuss which transformations of this form preserve the dynamics of KG1-KG7. It should be clear from the outset however that (at least on this interpretation) these theories cannot have continuous spacial translation and rotation, let alone Lorentzian boost symmetries. Indeed, I have been charitable considering the lattice sites structured only as a set (perhaps artificially) increasing the size of $\\text{Auto}(Q)$. Given this, it would be highly surprising if we found KG1-KG7 to have symmetries outside of this set. (Such a surprise is coming in the Sec.~\\ref{SecKlein2}.) \n\nAs I will show in Sec.~\\ref{SecKlein2}, this first interpretation of these theories systematically under predicts the symmetries that discrete spacetime theories can and do have. Fixing this issue will lead one to a discrete analog of general covariance. We here under-predict symmetries because we are taking these theories' lattice structures too seriously. Properly understood, they are merely a coordinate-like representational artifact and so do not limit our symmetries. Before that however, let's see the symmetries these theories have on this interpretation.\n\n\\subsubsection*{Symmetries of KG1-KG7: First Attempt}\nWhat then are the symmetries of KG1-KG7 according to this interpretation? A technical investigation of the symmetries of KG1-KG7 on this interpretation is carried out in Appendix~\\ref{AppA}, but the results are the following. For KG1-KG3 the dynamical symmetries of the form Eq.~\\eqref{PermutationLong} are:\n\\begin{flushleft}\\begin{enumerate}\n    \\item[1)] discrete shifts which map lattice site \\mbox{$(j,n)\\mapsto (j-d_1,n-d_2)$} for some integers $d_1,d_2\\in\\mathbb{Z}$,\n    \\item[2)] two negation symmetries which map lattice site $(j,n)\\mapsto (-j,n)$ and $(j,n)\\mapsto (j,-n)$ respectively,\n    \\item[3)] global linear rescaling which maps \\mbox{$\\phi_\\ell\\mapsto c_1\\phi_\\ell$} for some $c_1\\in\\mathbb{R}$,\n    \\item[4)] local affine rescaling which maps \\mbox{$\\phi_\\ell\\mapsto \\phi_\\ell + c_{2,\\ell}(t)$} for some $c_{2,\\ell}(t)$ which is also a solution of the dynamics.\n\\end{enumerate}\\end{flushleft}\nThese are the symmetries of a uniform 1D lattices in space  and a uniform 1D lattice in time, \\mbox{$z_{j,n}=(j,n)\\in\\mathbb{R}^2$} (plus linear-affine rescalings). These are two independent 1D lattices (rather than a single square 2D lattice) because we do not have quarter rotations between space and time among our symmetries. Previously I had warned against prematurely interpreting the lattice sites underlying KG1-KG3 as being organized into a square lattice. As it turns out, having investigated these theories' dynamical symmetries this warning was warranted.\n\nWhat about KG4 and KG6? For KG4 and KG6 the dynamical symmetries of the form Eq.~\\eqref{PermutationLong} are:\n\\begin{flushleft}\\begin{enumerate}\n    \\item[1)] discrete shifts which map lattice site \\mbox{$(j,n,m)\\mapsto (j-d_1,n-d_2,m-d_3)$} for some integers $d_1,d_2,d_3\\in\\mathbb{Z}$,\n    \\item[2)] three negation symmetries which map lattice site $(j,n,m)\\mapsto (-j,n,m)$ and $(j,n,m)\\mapsto (j,-n,m)$ and $(j,n,m)\\mapsto (j,n,-m)$ respectively,\n    \\item[3)] a 4-fold symmetry which maps lattice site \\mbox{$(j,n,m)\\mapsto (j,m,-n)$}, \n    \\item[4)] global linear rescaling which maps \\mbox{$\\phi_\\ell\\mapsto c_1\\phi_\\ell$} for some $c_1\\in\\mathbb{R}$,\n    \\item[5)] local affine rescaling which maps \\mbox{$\\phi_\\ell\\mapsto \\phi_\\ell + c_{2,\\ell}(t)$} for some $c_{2,\\ell}(t)$ which is also a solution of the dynamics.\n\\end{enumerate}\\end{flushleft}\nThese are the symmetries of a square 2D lattice in space and a uniform 1D lattice in time, \\mbox{$z_{j,n,m}=(j,n,m)\\in\\mathbb{R}^3$} (plus linear-affine rescalings). The above 4-fold symmetry corresponds to quarter rotation in space. These are two independent lattices (rather than a single cubic 3D lattice) because we do not have quarter rotations between space and time among our symmetries. Previously I had warned against prematurely interpreting the lattice sites underlying KG4 and KG6 as being organized into a cubic lattice. As it turns out, having investigated these theories' dynamical symmetries this warning was warranted.\n\nWhat about KG5 and KG7? For KG5 and KG7 the dynamical symmetries of the form Eq.~\\eqref{PermutationLong} are:\n\\begin{flushleft}\\begin{enumerate}\n    \\item[1)] discrete shifts which map lattice site \\mbox{$(j,n,m)\\mapsto (j-d_1,n-d_2,m-d_3)$} for some integers $d_1,d_2,d_3\\in\\mathbb{Z}$,\n    \\item[2)] an exchange symmetry which maps lattice site $(j,n,m)\\mapsto (j,m,n)$,\n    \\item[3)] a 6-fold symmetry which maps lattice site \\mbox{$(j,n,m)\\mapsto (j,-m,n+m)$}. (Roughly, this permutes the three terms in the right hand side of Eq.~\\eqref{DKG5} for KG5 and Eq.~\\eqref{DKG7} for KG7.),\n    \\item[4)] global linear rescaling which maps \\mbox{$\\phi_\\ell\\mapsto c_1\\phi_\\ell$} for some $c_1\\in\\mathbb{R}$,\n    \\item[5)] local affine rescaling which maps \\mbox{$\\phi_\\ell\\mapsto \\phi_\\ell + c_{2,\\ell}(t)$} for some $c_{2,\\ell}(t)$ which is also a solution of the dynamics.\n\\end{enumerate}\\end{flushleft}\nThese are the symmetries of a hexagonal 2D lattice in space and a uniform 1D lattice in time, \\mbox{$z_{j,n,m}=(j,n+m\/2,\\sqrt{3}m\/2)\\in\\mathbb{R}^3$} (plus linear-affine rescalings). The above 6-fold symmetry corresponds to one-sixth rotation in space. Previously I had warned against prematurely interpreting the lattice sites underlying KG5 and KG7 as being organized into a cubic 3D lattice, prompted by the convenient relabeling $\\ell\\mapsto (j,n,m)$. As it turns out, having investigated these theories' dynamical symmetries this warning was well warranted.\n\nThus, by investigating these theories' dynamical symmetries we were able to find what sort of lattice structure the assumed-to-be unstructured lattice $L$ actually has for each theory (e.g. in space a uniform 1D lattice, a square lattice, and a hexagonal lattice each together with a uniform 1D lattice in time).\n\n\\vspace{0.25cm}\n\nFinally, in this interpretation what sense can be made of KG6 and KG7 having a nice one-to-one correspondence between their solutions discussed at the end of Sec.~\\ref{SecSevenKG}? While this correspondence between solutions certainly exists, little sense can be made of it here in support of the equivalence of these theories. As the above discussion has revealed this interpretation associates very different symmetries to KG6 and KG7 and correspondingly very different lattice structures. While there is nothing technically wrong per se with this assessment our later interpretations will make better sense of this correspondence.\n\n\\vspace{0.25cm}\n\nTo summarize, this interpretation has the benefit of being highly intuitive. Taking the fields given to us, \\mbox{$\\phi:Q\\to\\mathbb{R}$}, seriously we identified the underlying manifold as $Q=L$. From this we got some intuitive notions of locality. Moreover, by finding these theories' dynamical symmetries we were able to grant some more structure to their lattice sites. By and large, the interpretation seems to validate all of the first intuitions laid out in Sec.~\\ref{SecIntro}. On this interpretation, the lattice seems to play a substantive role in the theory: it seems to restrict our symmetries, it seems to distinguish our theories from one another, be essentially ``baked-into'' the formalism. (As I will discuss in the next section, none of this is right.)\n\nHowever, there are three major issues with this interpretation which will become clear in light of our later interpretations. Firstly, our locality assessments are in tension with the rates at which these theories converge to the (perfectly local) continuum theory in the continuum limit, see \\cite{DiscreteGenCovPart1} for further discussion. Secondly, despite the niceness of the one-to-one correspondence between the solutions to KG6 and KG7, this interpretation regards them as significantly different theories: with different lattice structures and (here consequently) different symmetries. The final issue (which will become clear in the next section) is that this interpretation drastically under predicts the kinds of symmetries which KG1-KG7 can and do have. In fact, each of KG1-KG7 have a hidden continuous translation symmetry. Moreover, KG6 and KG7 have a hidden continuous rotation symmetry. Moreover still, KG3, KG6, and KG7 all have a hidden (limited) Lorentzian boost symmetry.\n\nAs I will discuss, the root of all of these issues is taking the theory's lattice structure too seriously. As I will argue, when properly understood, they are merely a coordinate-like representational artifact. Indeed, as I will show in the next section they do not limit our theory's symmetries. Moreover, theories appearing initially with different lattice structures may nonetheless be equivalent. Finally, these theories can always be reformulated to refer to no lattice structure at all. These three points establish a strong analogy between the lattice structures appearing in our discrete spacetime theories and the coordinate systems appearing in our continuum theories. Ultimately, spelling out this analogy in detail in Sec.~\\ref{SecDisGenCov} will give us a discrete analog of general covariance (now extended to a Lorentzian context). Indeed, this will lead us to a perfectly Lorentzian lattice theory in Sec.~\\ref{PerfectLorentz}.\n\n\\section{A Second Attempt at Interpreting KG1-KG7}\\label{SecKlein2}\nIn the previous section, I claimed that KG1-KG7 have hidden continuous translation and rotation symmetries and even (limited) Lorentzian boost symmetries. But how can this be? How can discrete spacetime theories have such continuous symmetries? As I discussed in the previous section, if we take our underlying manifold to be \\mbox{$Q=L$} then these theories clearly cannot support continuous translation and rotation symmetries let alone Lorentzian boosts.\n\nIn order to avoid this conclusion we must deny the premise, $Q$ must not be the underlying manifold. What led us to believe $Q$ was the underlying manifold? We arrived at this conclusion by focusing on $\\phi_\\ell\\in F_L$ and thereby taking the real scalar field $\\phi:Q\\to\\mathcal{V}$ to be fundamental. $Q$ is the underlying manifold because it is where our fundamental field maps from. In order to avoid this conclusion we must deny the premise, the field $\\phi:Q\\to\\mathcal{V}$ must not be fundamental. \n\nBut if $\\phi:Q\\to\\mathcal{V}$ is not fundamental then what is? Fortunately, our above discussion has already provided us with another object which we might take as fundamental. Namely, $\\bm{\\Phi}$ defined in Eq.~\\eqref{PhiDef}. These vectors $\\bm{\\Phi}\\in\\mathbb{R}^L$ are in a one-to-one correspondence with the discrete fields $\\phi_\\ell\\in F_L$, Moreover, these vector spaces are isomorphic $\\mathbb{R}^L\\cong F_L$ with Eq.~\\eqref{PhiDef} being a vector space isomorphism between them.\n\nOn this second interpretation I will be taking the formulations of KG1-KG7 in terms of $\\bm{\\Phi}$ seriously: namely Eq.~\\eqref{DKG1}, Eq.~\\eqref{DKG2}, Eq.~\\eqref{DKG3}, and Eqs.~\\eqref{DKG4}-\\eqref{DKG7}. Taken literally as written, what are these theories about? These theories are intuitively about an infinite dimensional vector ($\\bm{\\Phi}\\in\\mathbb{R}^L$) which satisfies some dynamical constraint. \n\nThere are two ways in which one might try to make sense of $\\bm{\\Phi}$ as a field $\\bm{\\Phi}:\\mathcal{M}\\to\\mathcal{V}$ with some manifold $\\mathcal{M}$ and some value space $\\mathcal{V}$. Firstly, one might consider a one-point manifold $\\mathcal{M}=p$ with $\\bm{\\Phi}$ simply being the field value there $\\bm{\\Phi}\\coloneqq \\bm{\\Phi}(p)$. Alternatively, one might try to make sense of $\\bm{\\Phi}$ as a field ``from nowhere'' with an empty manifold $\\mathcal{M}=\\emptyset$. On this view $\\bm{\\Phi}$ is a vector field which takes in no input and returns the vector $\\bm{\\Phi}$.\n\nIn any case, on this interpretation $Q$ is no longer the underlying manifold for KG1-KG7. Indeed, on this interpretation the lattice sites, $L$, are no longer our spacetime manifold (there may not even be a manifold). Rather, they have been ``internalized'' into the value space $\\mathcal{V}=\\mathbb{R}^L$. In particular, in defining this vector space we have associated with each lattice site $\\ell\\in L$ a basis vector $\\bm{b}_\\ell$. See Eq.~\\eqref{PhiDef}. However, as I will discuss, these particular basis vectors play no special role in these theories. Indeed, looking back at the dynamics for each of KG1-KG7 written in terms of $\\bm{\\Phi}$, one can see that in each case it can be made basis-independent.\n\nLet's see what consequences this interpretive stance has for these theories' locality and symmetry. To preview: this second interpretation either dissolves or resolves all of our issues with the first interpretation. Firstly, the tension is dissolved between our theories' differences in locality and their differences in convergence rate in the continuum limit. With no spacetime manifold we no longer have access to any spaciotemporal notion of locality. There simply are no differences in locality anymore. I refer the interested reader to \\cite{DiscreteGenCovPart1} for further discussion.\n\nSecondly, KG6 and KG7 are seen to be equivalent in a stronger sense. And thirdly, perhaps most importantly, this interpretation reveals KG1-KG7's hidden continuous translation and rotation symmetries and even (limited) Lorentzian boost symmetries. However, as I will discuss, this interpretation has some issues of its own which will ultimately require us to make a third attempt at interpreting these theories in Sec.~\\ref{SecKlein3}.\n\n\\subsection{Internalized Symmetries}\nHow does this internalization move affect our theory's capacity for symmetry? How can we now have continuous translation and rotation symmetries as well as a Lorentzian boost symmetry? At first glance, this may appear to have made things worse. Without a manifold (or even if the manifold is a single point) we no longer have any possibility for external symmetries. However, while there are certainly less possible external symmetries, we are now open to a wider range of internal symmetries. It is among these internal symmetries that we will find KG1-KG7's hidden continuous translation and rotation symmetries and even (limited) Lorentzian boost symmetries. As I will argue these symmetries can reasonably be given these names despite being internal symmetries. (In Sec.~\\ref{SecKlein3} I will present a third attempt at interpreting these theories which ``externalizes'' these symmetries, making them genuinely spacial translations, rotations, and Lorentzian boosts.)\n\nWith our focus now on $\\bm{\\Phi}\\in\\mathbb{R}^L$, let us consider its possibilities for symmetries. As discussed above, we have no possibility of external symmetries associated with the manifold. However, we do have possible internal symmetries associated with the value space (i.e., an infinite dimensional vector space) we now have the full range of invertible linear-affine transformations over $\\mathbb{R}^L$, namely $\\text{Auto}(\\mathcal{V})=\\text{Aff}(\\mathbb{R}^L)$. There are no gauge symmetries here as there is no longer any manifold for them to smoothly vary across. Thus, in total the possibly symmetries for our theories under this interpretation are,\n\\begin{align}\\label{GaugeVR}\ns:\\quad\\bm{\\Phi}\\mapsto \\Lambda\\,\\bm{\\Phi}+\\bm{c}\n\\end{align}\nfor some invertible linear transformation $\\Lambda\\in\\text{GL}(\\mathbb{R}^L)$ and some vector $\\bm{c}\\in \\mathbb{R}^L$. \n\nContrast this with the symmetries available to us on our first interpretation, namely Eq.~\\eqref{Permutation}. We can role this new class of possible symmetries back onto our first interpretation as follows: Note that because $\\mathbb{R}^L \\cong F_L$ we have $\\text{Aff}(\\mathbb{R}^L)\\cong \\text{Aff}(F_L)$. The set of transformations $\\text{Aff}(F_L)$ acting on $\\phi_\\ell\\in F_L$ is much larger than what we previously considered: namely, \\mbox{$\\text{Auto}(\\mathbb{V})=\\text{Aff}(\\mathbb{R})$} varying smoothly over \\mbox{$\\text{Auto}(Q)=\\text{Perm}(L)$}. Indeed, the present interpretation has a significantly wider class of symmetries than before. \n\nMoving back to our second interpretation, our previous class of transformations (i.e, \\mbox{$\\text{Auto}(\\mathbb{V})=\\text{Aff}(\\mathbb{R})$} varying smoothly over \\mbox{$\\text{Auto}(Q)=\\text{Perm}(L)$}) corresponds to only a subset of our present consideration: $\\text{Aff}(\\mathbb{R}^L)$. The difference is that before we could only apply a permutation matrix $P$ followed by a diagonal matrix $C_1$ whereas now we are allowed a general linear transformation $\\Lambda$. \n\nNote that permutation and diagonal matrices are basis-dependent notions. Our first interpretation took the lattice sites $\\ell\\in L$ seriously as a part of the manifold $Q=L$ and this is reflected in its conception of symmetries. Converted into $\\mathbb{R}^L$ this first conception of these theories' possible symmetries gives special treatment to the basis associated with the lattice sites, namely $\\{\\bm{b}_\\ell\\}_{\\ell\\in L}$. In particular, on our first interpretation, our possible symmetries are those of the form Eq.~\\eqref{GaugeVR} which \\textit{additionally} preserve this basis (up to rescaling, and reordering).\n\nThis basis receives no special treatment on this second interpretation. While it is true that $\\{\\bm{b}_\\ell\\}_{\\ell\\in L}$ were used in the initial construction of $\\bm{\\Phi}$, after this they no longer play any special role. We are always free to redescribe $\\bm{\\Phi}$ in a different basis if we wish. Indeed, here any change of basis transformation is of the form Eq.~\\eqref{GaugeVR} and hence a symmetry.\n\nWith this attachment to the basis $\\{\\bm{b}_\\ell\\}_{\\ell\\in L}$ dropped, we now have a wider class of symmetries. Indeed, everything which was previously considered a symmetry will be here as well and possibly more. Perhaps among this more general class of symmetries we may find new continuous symmetries. Let's see.\n\n\\subsubsection*{Symmetries of KG1-KG7: Second Attempt}\nWhich of the above transformations are symmetries for KG1-KG7? A non-exhaustive  investigation of the symmetries of KG1-KG7 on this interpretation is carried out in Appendix~\\ref{AppA}, but the results are the following. For KG1 and KG2 the dynamical symmetries of the form Eq.~\\eqref{GaugeVR} include:\n\\begin{flushleft}\\begin{enumerate}\n    \\item[1)]  action by $T^\\epsilon_\\text{j}$ sending $\\bm{\\Phi}\\mapsto T^\\epsilon_\\text{j} \\bm{\\Phi}$ where \\mbox{$T^\\epsilon_\\text{j}=T^\\epsilon\\otimes\\openone$} with $T^\\epsilon$ defined below. Similarly for $T^\\epsilon_\\text{n}=\\openone\\otimes T^\\epsilon$,\n    \\item[2)] two negation symmetries which map basis vectors as  $\\bm{e}_j\\otimes\\bm{e}_n\\mapsto \\bm{e}_{-j}\\otimes\\bm{e}_n$ and $\\bm{e}_j\\otimes\\bm{e}_n\\mapsto \\bm{e}_j\\otimes\\bm{e}_{-n}$ respectively,\n    \\item[3)] a local Fourier rescaling symmetry which maps \\mbox{$\\bm{\\Phi}(\\omega,k)\\mapsto \\tilde{f}(\\omega,k)\\,\\bm{\\Phi}(\\omega,k)$} for some non-zero complex function $\\tilde{f}(\\omega,k)\\in\\mathbb{C}$ with $\\omega,k\\in[-\\pi,\\pi]$,\n    \\item[4)] local affine rescaling which maps \\mbox{$\\bm{\\Phi}\\mapsto \\bm{\\Phi} + \\bm{c}_2$} for some $\\bm{c}_2$ which is also a solution of the dynamics.\n\\end{enumerate}\\end{flushleft}\nThese are exactly the same symmetries that we found on the previous interpretation with two differences: Firstly, global rescaling $\\phi_\\ell\\mapsto c_1 \\phi_\\ell$ has been refined to a local Fourier rescaling. Note that the discrete Fourier transform itself is in $\\text{Aff}(\\mathbb{R}^L)$ and so is in the class of potential symmetries considered here.\n\nSecondly, discrete shifts have been replaced with action by \n\\begin{align}\\label{TDef}\nT^\\epsilon\\coloneqq\\text{exp}(-\\epsilon D)\n\\end{align}\nwith $\\epsilon\\in\\mathbb{R}$ acting on each tensor factor. A straight-forward calculation shows that $T^\\epsilon$ acts on the planewave basis \\mbox{$\\bm{\\Phi}(k)\\coloneqq \\sum_{n\\in\\mathbb{Z}}e^{-\\mathrm{i} k n}\\bm{e}_n$} with $k\\in[-\\pi,\\pi]$ as\n\\begin{align}\n\\nonumber\nT^\\epsilon:\\bm{\\Phi}(k)\\mapsto \\exp(\\mathrm{i} k \\epsilon)\\,\\bm{\\Phi}(k).\n\\end{align}\nUsing \\mbox{$\\bm{e}_m\n=\\frac{1}{2\\pi}\\int_{-\\pi}^\\pi e^{\\mathrm{i}\\, k\\, m}\\, \\bm{\\Phi}(k)\\,\\d k$} we can recover how $T^\\epsilon$ acts on the basis $\\bm{e}_m$ as\n\\begin{align}\nT^\\epsilon: \\bm{e}_m \\mapsto \\sum_{b\\in\\mathbb{Z}} S_m(b+\\epsilon) \\, \\bm{e}_b\n\\end{align}\nwhere \n\\begin{align}\\label{SincDef}\nS(y)=\\frac{\\sin(\\pi y)}{\\pi y}, \\quad\\text{and}\\quad\nS_m(y)=S(y-m),       \n\\end{align}\nare the normalized and shifted sinc functions. Note that $S_n(m)=\\delta_{nm}$ for integers $n$ and $m$.\n\nAs I will now discuss, $T^\\epsilon$ can be thought of as a continuous translation operator for three reasons despite it being here classified as an internal symmetry. Note that none of these reasons depend on $T^\\epsilon$ being a symmetry of the dynamics.\n\nFirst note that $T^\\epsilon$ is a generalization of the discrete shift operation in the sense that taking $\\epsilon=d_1\\in\\mathbb{Z}$ reduces action by $T^\\epsilon$ to the map $T^{d_1}:\\bm{e}_m\\mapsto \\bm{e}_{m-d_1}$ on basis vectors and relatedly the map $m\\mapsto m-d_1$ on lattice sites. \n\nSecond note that $T^\\epsilon$ is additive in the sense that $T^{\\epsilon_1}\\,T^{\\epsilon_2}=T^{\\epsilon_1+\\epsilon_2}$. In the language or representation theory $T^\\epsilon$ is a representation of the translation group on the vector space $\\mathbb{R}^\\mathbb{Z}\\cong\\mathbb{R}^L$. In particular, this means \\mbox{$T^{1\/2}\\,T^{1\/2}=T^1$}: there is something we can do twice to move one space forward. The same is true for all fractions adding to one. \n\nThird, recall from the discussion following Eq.~\\eqref{LambdaD} that $D$ is closely related to the continuum derivative operator $\\partial_x$, exactly matching its spectrum for $k\\in[-\\pi,\\pi]$. Recall also that the derivative is the generator of translation, i.e. $h(x-\\epsilon)=\\text{exp}(-\\epsilon\\, \\partial_x) h(x)$. In this sense also \\mbox{$T^\\epsilon=\\text{exp}(-\\epsilon D)$} is a translation operator. More will be said about $T^\\epsilon$ in Sec.~\\ref{SecSamplingTheory}.\n\nThus we have our first big lesson: despite the fact that KG1-KG2 can be represented on a lattice, they nonetheless have a continuous translation symmetry. This continuous translation symmetry was hidden from us on our first interpretation because we there took the lattice to be hard-wired in as a part of the manifold. Here, we do not take the lattice structure so seriously. We have internalized it into the value space where it then disappears from view as just one basis among many.\n\nBefore KG3 had the same symmetries as KG1 and KG2, now it does not. However, allow me to skip over KG3 temporarily. In the previous interpretation the symmetries of KG4 and KG6 matched each other, both being associated with a 2D square lattice. Moreover, the symmetries of KG5 and KG7 matched each other, both being associated with a hexagonal 2D lattice. Here however, these pairings are broken up and a new matching pair is formed between KG6 and KG7. More will be said about this momentarily.\n\nLet's consider KG4 first. For KG4 the dynamical symmetries of the form Eq.~\\eqref{GaugeVR} include:\n\\begin{flushleft}\\begin{enumerate}\n    \\item[1)] action by $T^\\epsilon_\\text{j}$ sending $\\bm{\\Phi}\\mapsto T^\\epsilon_\\text{j} \\bm{\\Phi}$. Similarly for $T^\\epsilon_\\text{n}$ and $T^\\epsilon_\\text{m}$,\n    \\item[2)] three negation symmetries which map basis vectors as \\mbox{$\\bm{e}_{j}\\otimes\\bm{e}_{n}\\otimes\\bm{e}_{m}\\mapsto \\bm{e}_{-j}\\otimes\\bm{e}_{n}\\otimes\\bm{e}_{m}$}, and \\mbox{$\\bm{e}_{j}\\otimes\\bm{e}_{n}\\otimes\\bm{e}_{m}\\mapsto \\bm{e}_{j}\\otimes\\bm{e}_{-n}\\otimes\\bm{e}_{m}$}, and \\mbox{$\\bm{e}_{j}\\otimes\\bm{e}_{n}\\otimes\\bm{e}_{m}\\mapsto \\bm{e}_{j}\\otimes\\bm{e}_{n}\\otimes\\bm{e}_{-m}$} respectively,\n    \\item[3)] a 4-fold symmetry which maps basis vectors as \\mbox{$\\bm{e}_{j}\\otimes\\bm{e}_{n}\\otimes\\bm{e}_{m}\\mapsto \\bm{e}_{j}\\otimes\\bm{e}_{m}\\otimes\\bm{e}_{-n}$},\n    \\item[4)] a local Fourier rescaling symmetry which maps \\mbox{$\\bm{\\Phi}(\\omega,k_1,k_2)\\mapsto \\tilde{f}(\\omega,k_1,k_2)\\,\\bm{\\Phi}(\\omega,k_1,k_2)$} for some non-zero complex function $\\tilde{f}(\\omega,k_1,k_2)\\in\\mathbb{C}$ with $\\omega,k_1,k_2\\in[-\\pi,\\pi]$,\n    \\item[5)] local affine rescaling which maps \\mbox{$\\bm{\\Phi}\\mapsto \\bm{\\Phi} + \\bm{c}_2$} for some $\\bm{c}_2$ which is also a solution of the dynamics.\n\\end{enumerate}\\end{flushleft}\nThese are exactly the symmetries which we found on our first interpretation (plus local Fourier rescaling) but with action by $T^\\epsilon_\\text{j}$, $T^\\epsilon_\\text{n}$ and $T^\\epsilon_\\text{m}$ replacing the discrete shifts. The same discussion following Eq.~\\eqref{TDef} applies here, justifying us calling these continuous translation operations. Thus, KG4 has three continuous translation symmetries despite being initially represented on a lattice, Eq.~\\eqref{KG4Long}.\n\nLet's next consider KG5. For KG5 the dynamical symmetries of the form Eq.~\\eqref{GaugeVR} include:\n\\begin{flushleft}\\begin{enumerate}\n    \\item[1)] action by $T^\\epsilon_\\text{j}$ sending $\\bm{\\Phi}\\mapsto T^\\epsilon_\\text{j} \\bm{\\Phi}$. Similarly for $T^\\epsilon_\\text{n}$ and $T^\\epsilon_\\text{m}$,\n    \\item[2)] a negation symmetry which maps basis vectors as \\mbox{$\\bm{e}_{j}\\otimes\\bm{e}_{n}\\otimes\\bm{e}_{m}\\mapsto \\bm{e}_{-j}\\otimes\\bm{e}_{n}\\otimes\\bm{e}_{m}$} and an exchange symmetry which maps basis vectors as \\mbox{$\\bm{e}_{j}\\otimes\\bm{e}_{n}\\otimes\\bm{e}_{m}\\mapsto \\bm{e}_{j}\\otimes\\bm{e}_{m}\\otimes\\bm{e}_{n}$},\n    \\item[3)] a 6-fold symmetry which maps basis vectors as \\mbox{$\\bm{e}_{j}\\otimes\\bm{e}_{n}\\otimes\\bm{e}_{m}\\mapsto \\bm{e}_{j}\\otimes\\bm{e}_{-m}\\otimes\\bm{e}_{n+m}$}. (Roughly, this permutes the three terms in Eq.~\\eqref{DKG5}),\n    \\item[4)] a local Fourier rescaling symmetry which maps \\mbox{$\\bm{\\Phi}(\\omega,k_1,k_2)\\mapsto \\tilde{f}(\\omega,k_1,k_2)\\,\\bm{\\Phi}(\\omega,k_1,k_2)$} for some non-zero complex function $\\tilde{f}(\\omega,k_1,k_2)\\in\\mathbb{C}$ with $t\\in\\mathbb{R}$ and $\\omega,k_1,k_2\\in[-\\pi,\\pi]$,\n    \\item[5)] local affine rescaling which maps \\mbox{$\\bm{\\Phi}\\mapsto \\bm{\\Phi} + \\bm{c}_2$} for some $\\bm{c}_2$ which is also a solution of the dynamics.\n\\end{enumerate}\\end{flushleft}\nThese are exactly the symmetries which we found on our first interpretation (plus local Fourier rescaling) but with action by $T^\\epsilon_\\text{j}$, $T^\\epsilon_\\text{n}$ and $T^\\epsilon_\\text{m}$ replacing the discrete shifts. The same discussion following Eq.~\\eqref{TDef} applies here, justifying us calling these continuous translation operations. Thus, KG5 has three continuous translation symmetries despite being initially represented on a lattice, Eq.~\\eqref{KG5Long}.\n\nBefore moving on to analyze the symmetries of KG6 and KG7, let's first see what this interpretation has to say about them being equivalent to one another. As noted at the end of Sec.~\\ref{SecSevenKG}, KG6 and KG7 have a nice one-to-one correspondence between their solutions. Allow me to spell this out in detail now.\n\nBefore this, however, it is worth briefly noting a rather weak sense in which each of KG4-KG7 are equivalent to each other. As noted following Eq.~\\eqref{SkewKG7KG6} and Eq.~\\eqref{SkewKG6KG7} there is an approximate one-to-one correspondence between each of these theories in the $\\vert \\omega\\vert,\\vert k_1\\vert,\\vert k_2\\vert\\ll\\pi$ regime as they approach their common continuum limit, KG0. By contrast,  as I will show, KG6 and KG7 have an \\textit{exact} one-to-one correspondence over \\textit{the whole of} $\\sqrt{k_1^2+k_2^2}<\\pi$ and indeed more. This includes all of their solutions but not all of $\\omega,k_1,k_2\\in[-\\pi,\\pi]$.\n\nThis one-to-one correspondence is mediated by the transformations Eq.~\\eqref{SkewKG7KG6} and Eq.~\\eqref{SkewKG6KG7}. Let's first rewrite these in terms of \\mbox{$\\bm{\\Phi}\\in\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$} as follows. Consider first the transformation which maps the dispersion relation for KG7 onto the one for KG6 (namely, Eq.~\\eqref{SkewKG7KG6}). Consider its action on the planewave basis \n$\\bm{\\Phi}(\\omega,k_1,k_2)$ with \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$}, namely\n\\begin{align}\n\\nonumber\n\\Lambda_{\\text{KG7}\\to\\text{KG6}}:\\bm{\\Phi}(\\omega,k_1,k_2)\\mapsto \\bm{\\Phi}\\left(\\omega,k_1,\\frac{1}{2}k_1+\\frac{\\sqrt{3}}{2}k_2\\right)  \n\\end{align}\nA straight-forward calculation shows this acts on the \\mbox{$\\bm{e}_j\\otimes\\bm{e}_n\\otimes\\bm{e}_m$} basis as:\n\\begin{align}\\label{SkewKG7KG6Basis}\n&\\Lambda_{\\text{KG7}\\to\\text{KG6}}:\\bm{e}_j\\otimes\\bm{e}_n\\otimes\\bm{e}_m\\mapsto \\\\\n\\nonumber\n&\\bm{e}_j\\otimes\\left(\\sum_{b_1,b_2\\in\\mathbb{Z}}S_n(b_1+b_2\/2) \\,S_m(\\sqrt{3}\\, b_2\/2)\\, \\bm{e}_{b_1}\\otimes\\bm{e}_{b_2}\\right).  \n\\end{align}\nConsider also the transformation which maps the dispersion relation for KG6 onto the one for KG7 (namely, Eq.~\\eqref{SkewKG6KG7}). Consider its action on the planewave basis \n$\\bm{\\Phi}(\\omega,k_1,k_2)$ with \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$}, namely as\n\\begin{align}\n\\nonumber\n\\Lambda_{\\text{KG6}\\to\\text{KG7}}:\\bm{\\Phi}(\\omega,k_1,k_2)\\mapsto \\bm{\\Phi}\\left(\\omega,k_1,\\frac{2k_2-k_1}{\\sqrt{3}}\\right)  \n\\end{align}\nA straight-forward calculation shows this acts on the \\mbox{$\\bm{e}_j\\otimes\\bm{e}_n\\otimes\\bm{e}_m$} basis as:\n\\begin{align}\n&\\Lambda_{\\text{KG6}\\to\\text{KG7}}:\\bm{e}_j\\otimes\\bm{e}_n\\otimes\\bm{e}_m\\mapsto \\\\\n\\nonumber\n&\\bm{e}_j\\otimes\\left(\\sum_{b_1,b_2\\in\\mathbb{Z}}S_n(b_1-b_2\/\\sqrt{3}) \\,S_m(2\\, b_2\/\\sqrt{3})\\, \\bm{e}_{b_1}\\otimes\\bm{e}_{b_2}\\right).  \n\\end{align}\n\nIt should be noted however that despite the fact that Eq.~\\eqref{SkewKG7KG6} and Eq.~\\eqref{SkewKG6KG7} are each other's inverses, $\\Lambda_{\\text{KG7}\\to\\text{KG6}}$ and $\\Lambda_{\\text{KG6}\\to\\text{KG7}}$ are not each other's inverses (at least not on the whole of \\mbox{$\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$}). This is due to the $2\\pi$ periodic identification of the planewaves $\\bm{\\Phi}(\\omega,k_1,k_2)$. Indeed, when viewed as acting on $\\mathcal{V}=\\mathbb{R}^L$, the transformation $\\Lambda_{\\text{KG7}\\to\\text{KG6}}$ is not even invertible. They are only each other's inverses when we have \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$} both before and after these transformations.\n\nFor these reasons we need to consider the following two subspaces of \\mbox{$\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$}: \n\\begin{align}\n\\mathbb{R}^L_\\text{KG7}\\!\\coloneqq\\text{span}(&\\bm{\\Phi}(\\omega,k_1,k_2)\\vert\\text{with }\\omega,k_1,k_2\\in[-\\pi,\\pi]\\\\\n\\nonumber\n&\\text{ before and after applying Eq.~\\eqref{SkewKG7KG6}})\\\\\n\\mathbb{R}^L_\\text{KG6}\\!\\coloneqq\\text{span}(&\\bm{\\Phi}(\\omega,k_1,k_2)\\vert\\text{with }\\omega,k_1,k_2\\in[-\\pi,\\pi]\\\\\n\\nonumber\n&\\text{ before and after applying Eq.~\\eqref{SkewKG6KG7}}).\n\\end{align}\nFor later reference it should be noted that the rotation invariant subspace,\n\\begin{align}\\label{RLrotinv}\n\\mathbb{R}^L_\\text{rot.inv}\\coloneqq\\text{span}\\left(\\bm{\\Phi}(\\omega,k_1,k_2)\\Big\\vert\\,\\sqrt{k_1^2+k_2^2}<\\pi\\right)\n\\end{align}\nis a subspace of $\\mathbb{R}^L_\\text{KG6}$, that is $\\mathbb{R}^L_\\text{rot.inv}\\subset \\mathbb{R}^L_\\text{KG6}$.\n\nRestricted to $\\mathbb{R}^L_\\text{KG6}$ and $\\mathbb{R}^L_\\text{KG7}$ these transformations are invertible and indeed are each other's inverses. Fortunately, all of KG6's solutions are in $\\mathbb{R}^L_\\text{KG6}$ (and moreover they are in $\\mathbb{R}^L_\\text{rot.inv}$ as well). Similarly, all of KG7's solutions are in $\\mathbb{R}^L_\\text{KG7}$. Thus, $\\Lambda_{\\text{KG7}\\to\\text{KG6}}$ maps generic solutions to KG7 onto generic solutions for KG6 in an invertible way. Therefore, $\\Lambda_{\\text{KG6}\\to\\text{KG7}}$ and $\\Lambda_{\\text{KG7}\\to\\text{KG6}}$ give us not only a one-to-one correspondence between the solutions to KG6 and KG7 but a solution-preserving vector-space isomorphism between KG6 and KG7. One can gloss this situation saying: the KPMs of KG6 and KG7 are not isomorphic, but their DPMs are.\n\nThe fact that this is solution-preserving vector-space isomorphism rather than merely a one-to-one correspondence has substantial consequences for these theories' symmetries. Namely, this forces KG6 and KG7 to have the same symmetries. This is because these transformations are both of the form Eq.~\\eqref{GaugeVR} (but notably not of the form Eq.~\\eqref{Permutation}) for any symmetry transformation for KG6 there is a corresponding symmetry transformation for KG7 and vice versa. \n\nThis is in strong contrast to the results of our previous analysis in Sec.~\\ref{SecKlein1}. There KG6 was seen to have symmetries associated with a square 2D lattice and KG7 was seen to have the symmetries associated with a hexagonal 2D lattice. By contrast, in the present interpretation KG6 and KG7 are thoroughly equivalent: We have a solution-preserving vector-space isomorphism between them. Thus, on this interpretation the only difference between KG6 and KG7 is a change of basis.\n\nThus we have our second big lesson: despite the fact that KG6 and KG7 can be represented with very different lattice structures (i.e., a square lattice versus a hexagonal lattice) they have nonetheless turned out to be completely equivalent to one another. This equivalence was hidden from us on our first interpretation because we there took the lattice too seriously. As I will now discuss, this reduced their continuous rotation symmetries down to quarter rotations and one-sixth rotations respectively and thereby made them inequivalent. Here, we do not take the lattice structure so seriously. We have here internalized it into the value space where it subsequently disappears from view as just one basis among many.\n\nIn addition to switching between lattice structures, in this interpretation we can also do away with them altogether. In this interpretation, a lattice structure is associated with a choice of basis for $\\mathcal{V}=\\mathbb{R}^L$. A choice of basis (like a choice of coordinates) is ultimately a merely representational choice, reflecting no physics. We can always choose, if we like, to work within a basis-free formulation of these theories. That is, ultimately, a lattice-free formulation of these theories. Thus we have our third big lesson: given a discrete spacetime theory with some lattice structure we can always reformulate it in such a way that it has no lattice structure whatsoever.\n\nIn the rest of this subsection I will only discuss the symmetries KG6; analogous conclusions are true for KG7 after applying $\\Lambda_{\\text{KG6}\\to\\text{KG7}}$. For KG6 the dynamical symmetries of the form Eq.~\\eqref{GaugeVR} include:\n\\begin{flushleft}\\begin{enumerate}\n    \\item[1)] action by $T^\\epsilon_\\text{j}$ sending $\\bm{\\Phi}\\mapsto T^\\epsilon_\\text{j} \\bm{\\Phi}$. Similarly for $T^\\epsilon_\\text{n}$ and $T^\\epsilon_\\text{m}$,\n    \\item[2)] three negation symmetries which map basis vectors as \\mbox{$\\bm{e}_{j}\\otimes\\bm{e}_{n}\\otimes\\bm{e}_{m}\\mapsto \\bm{e}_{-j}\\otimes\\bm{e}_{n}\\otimes\\bm{e}_{m}$}, and \\mbox{$\\bm{e}_{j}\\otimes\\bm{e}_{n}\\otimes\\bm{e}_{m}\\mapsto \\bm{e}_{j}\\otimes\\bm{e}_{-n}\\otimes\\bm{e}_{m}$}, and \\mbox{$\\bm{e}_{j}\\otimes\\bm{e}_{n}\\otimes\\bm{e}_{m}\\mapsto \\bm{e}_{j}\\otimes\\bm{e}_{n}\\otimes\\bm{e}_{-m}$} respectively,\n    \\item[3)] action by $R^\\theta$  sending $\\bm{\\Phi}\\mapsto R^\\theta \\bm{\\Phi}$ with $R^\\theta$ defined below. (This being a symmetry requires some qualification as I will discuss below.),\n    \\item[4)] action by $\\Lambda^w_\\text{j,n}$ sending $\\bm{\\Phi}\\mapsto \\Lambda^w_\\text{j,n} \\bm{\\Phi}$ with $\\Lambda^w_\\text{j,n}$ defined below. Similarly for $\\Lambda^w_\\text{j,m}$ which is defined below as well. (This being a symmetry requires some qualification as I will discuss below.),\n    \\item[5)] a local Fourier rescaling symmetry which maps \\mbox{$\\bm{\\Phi}(\\omega,k_1,k_2)\\mapsto \\tilde{f}(\\omega,k_1,k_2)\\,\\bm{\\Phi}(\\omega,k_1,k_2)$} for some non-zero complex function $\\tilde{f}(\\omega,k_1,k_2)\\in\\mathbb{C}$ with $\\omega,k_1,k_2\\in[-\\pi,\\pi]$,\n    \\item[6)] local affine rescaling which maps \\mbox{$\\bm{\\Phi}\\mapsto \\bm{\\Phi} + \\bm{c}_2$} for some $\\bm{c}_2$ which is also a solution of the dynamics.\n\\end{enumerate}\\end{flushleft}\nAs with KG4 and KG5, we have here gained local Fourier rescaling and action by $T^\\epsilon_\\text{j}$, $T^\\epsilon_\\text{n}$ and $T^\\epsilon_\\text{m}$ has replaced the discrete shifts from before. The same discussion following Eq.~\\eqref{TDef} applies here, justifying us calling these continuous translation operations. Thus, KG6 (and KG7) have three continuous translation symmetries despite being initially represented on a lattice.\n\nAdditionally, we have the quarter rotation symmetry from our first interpretation replaced with action by $R^\\theta$, which as I will argue is essentially a continuous rotation transformation. Before that, it is worth noting a rather weak sense in which each of KG4-KG7 are rotation invariant. Each of these theories is approximately rotation invariant in the $\\vert \\omega\\vert,\\vert k_1\\vert,\\vert k_2\\vert\\ll\\pi$ regime as they approach the continuum limit. By contrast, as I will show, KG6 is \\textit{exactly} rotation invariant over \\textit{the whole of} $\\sqrt{k_1^2+k_2^2}<\\pi$, that is the whole of $\\mathbb{R}^L_\\text{rot.inv.}$. Since all of KG6's solutions lie inside of $\\mathbb{R}^L_\\text{rot.inv.}$, $R^\\theta$ will always map its solutions to solutions in an invertible way, and will hence be a symmetry.\n\nThis alleged continuous rotation transformation $R^\\theta$ is given by \n\\begin{align}\\label{RthetaDef}\nR^\\theta \\coloneqq \\exp(-\\theta (N_\\text{n} D_\\text{m}-N_\\text{m} D_\\text{n}))\n\\end{align}\nwith $\\theta\\in\\mathbb{R}$ and where $N$ is the ``position operator'' which acts on the basis $\\bm{e}_m$ as $N\\bm{e}_m=m\\,\\bm{e}_m$ for $m\\in\\mathbb{Z}$. Thus \\mbox{$N_\\text{n}=\\openone\\otimes N\\otimes\\openone$} and \\mbox{$N_\\text{m}=\\openone\\otimes \\openone\\otimes N$} return the second and third index respectively when acting on $\\bm{e}_j\\otimes \\bm{e}_n\\otimes \\bm{e}_m$.\n\nA straight-forward calculation shows that $R^\\theta$ acts on the planewave basis $\\bm{\\Phi}(\\omega,k_1,k_2)$ with \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$} as\n\\begin{align}\nR^\\theta:\\,&\\bm{\\Phi}(\\omega,k_1,k_2) \\mapsto\\\\\n\\nonumber\n&\\bm{\\Phi}(\\omega,\\cos(\\theta)k_1-\\sin(\\theta)k_2,\\sin(\\theta)k_1+\\cos(\\theta)k_2).\n\\end{align}\nand acts on the basis $\\bm{e}_j\\otimes \\bm{e}_n\\otimes \\bm{e}_m$ as\n\\begin{align}\n&R^\\theta: \\bm{e}_j\\!\\otimes\\bm{e}_n\\!\\otimes\\bm{e}_m \\mapsto\\bm{e}_{j}\\otimes\\!\\!\\!\\!\\sum_{b_1,b_2\\in\\mathbb{Z}} R_{nm}^{b_1 b_2}(\\theta)\\,\n\\bm{e}_{b_1}\\!\\!\\otimes\\bm{e}_{b_2}\\\\\n\\nonumber\n&R_{nm}^{b_1 b_2}\\!(\\theta)\\!=\\!S_n(\\cos(\\theta) b_1 \\!-\\! \\sin(\\theta) b_2) S_m(\\sin(\\theta) b_1\\!+\\! \\cos(\\theta)b_2).\n\\end{align}\n\nIt should be noted that $R^\\theta$ is not invertible (at least not on the whole of \\mbox{$\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$}). As I will soon discuss, this is due to the $2\\pi$ periodic identification of the planewaves $\\bm{\\Phi}(\\omega,k_1,k_2)$. However, $R^\\theta$ is invertible over $\\mathbb{R}^L_\\text{rot.inv.}$ which contains all of KG6's solutions. Thus, for KG6 $R^\\theta$ always maps solutions to solutions in an invertible way, and is hence a symmetry.\n\nTo see why $R^\\theta$ is not invertible over all of $\\mathcal{V}=\\mathbb{R}^L$ note that $R^{\\pi\/4}$ maps two different planewaves to the same place: Firstly note,\n\\begin{align}\nR^{\\pi\/4}\\bm{\\Phi}(\\omega,\\pi,\\pi)&=\\bm{\\Phi}(\\omega,0,\\sqrt{2}\\pi)\\\\\n\\nonumber\n&=\\bm{\\Phi}(\\omega,0,\\sqrt{2}\\pi-2\\pi)\n\\end{align}\nsince the planewaves repeat themselves with period $2\\pi$. Secondly note, \n\\begin{align}\n\\nonumber\nR^{\\pi\/4}\\bm{\\Phi}(\\omega,\\pi-\\sqrt{2}\\pi,\\pi-\\sqrt{2}\\pi)&=\\bm{\\Phi}(\\omega,0,\\sqrt{2}\\pi-2\\pi).\n\\end{align}\nSuch issues do not arise when $\\sqrt{k_1^2+k_2^2}<\\pi$. Thus, when we restrict our attention to $\\mathbb{R}^L_\\text{rot.inv}$ (which contains all of KG6's solutions) then $R^\\theta$ is invertible and indeed a symmetry. One can gloss this situation saying: rotation does not map the KPMs of KG6 onto themselves in an invertible way, but it does for the DPMs of KG6. If we cut the KPMs of KG6 down to $\\mathbb{R}^L_\\text{rot.inv}$ this minor issue is fixed.\n\nAs I will now discuss, $R^\\theta$ can be thought of as a continuous rotation operator for three reasons despite it being here an internal symmetry. First note that $R^\\theta$ is a generalization of quarter rotation operation in the sense that taking $\\theta=\\pi\/2$ reduces action by $R^\\theta$ to the map \\mbox{$R^{\\pi\/2}:\\bm{e}_{j}\\otimes\\bm{e}_{n}\\otimes\\bm{e}_{m}\\mapsto \\bm{e}_{j}\\otimes\\bm{e}_{m}\\otimes\\bm{e}_{-n}$} on basis vectors and relatedly the map $(j,n,m)\\mapsto (j,m,-n)$ on lattice sites.\n\nSecond, note that restricted to $\\mathbb{R}^L_\\text{rot.inv.}$, $R^\\theta$ is cyclically additive in the sense that $R^{\\theta_1}\\,R^{\\theta_2}=R^{\\theta_1+\\theta_2}$ with $R^{2\\pi}=\\openone$. In the language of representation theory, $R^\\theta$ is a representation of the rotation group on the vector space $\\mathbb{R}^L_\\text{rot.inv.}$. In particular, this means \\mbox{$R^{\\pi\/4}\\,R^{\\pi\/4}=R^{\\pi\/2}$}.  There is something we can do twice to make a quarter rotation. Similarly for all fractional rotations. Moreover, note that together with the above discussed translations, these constitute a representation of the Euclidean group over $\\mathbb{R}^L_\\text{rot.inv.}$.\n\nThird, recall from the discussion following Eq.~\\eqref{LambdaD} that $D$ is closely related to the continuum derivative operator $\\partial_x$, exactly matching its spectrum for $k\\in[-\\pi,\\pi]$. Recall also that rotations are generated through the derivative as \\mbox{$h(R^\\theta(x,y))= \\exp(-\\theta (x \\partial_y-y \\partial_x))h(x,y)$}. In this sense also $R^\\theta$ is a rotation operator. More will be said about $R^\\theta$ in Appendix~\\ref{AppA}.\n\nThis adds to our first big lesson: despite the fact that KG6 and KG7 can be represented on a cubic 3D lattice and a hexagonal 3D lattice respectively, they nonetheless both have a continuous rotation symmetry. This, in addition to their continuous translation symmetries. These continuous translation and rotations symmetries were hidden from us on our first interpretation because we there took the lattice representations too seriously. Here, we do not take the lattice structure so seriously. Instead, we have internalized it into the value space where it then disappears from view as just one basis among many.\n\nLet's next discuss KG6's (limited) Lorentzian boost symmetry. In addition to generalizing the 4-fold symmetry into to a continuous rotation symmetry, KG6 also has a brand new symmetry on this interpretation, namely ``action by $\\Lambda^w_\\text{j,n}$ and\/or $\\Lambda^w_\\text{j,m}$''. As I will argue these are essentially Lorentz boost transformations. Before that, it is worth noting a rather weak sense in which each of KG4-KG7 are Lorentz invariant. Each of these theories is approximately Lorentz invariant as they approach the continuum limit regime $\\vert \\omega\\vert,\\vert k_1\\vert,\\vert k_2\\vert\\ll\\pi$ at least boosts parameters $w$ which keep them in this regime. By contrast, as I will show, KG6 is \\textit{exactly} Lorentz invariant over \\textit{a finite-sized region} around $\\omega,k_1,k_2=0$ and boost parameter $w=0$.\n\nThis alleged Lorentz boost transformations are given by\n\\begin{align}\\label{LambdaWDef}\n\\Lambda^w_\\text{j,n}\\coloneqq\\exp(-w (&N_\\text{j} D_\\text{n} + N_\\text{n} D_\\text{j})),\\\\\n\\nonumber\n\\Lambda^w_\\text{j,m}\\coloneqq\\exp(-w (&N_\\text{j} D_\\text{m} + N_\\text{m} D_\\text{j}))\n\\end{align}\nwith $w\\in\\mathbb{R}$. Note that $\\Lambda^w_\\text{j,n}$ acts only on the first and second tensor factor, whereas $\\Lambda^w_\\text{j,m}$ acts only on the first and third factors. In what follows I will focus on $\\Lambda^w_\\text{j,n}$, with similar results following for $\\Lambda^w_\\text{j,m}$.\n\nA straight-forward calculation shows that $\\Lambda^w_\\text{j,n}$ acts on the planewave basis $\\bm{\\Phi}(\\omega,k_1,k_2)$ with \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$} as\n\\begin{align}\n\\Lambda^w_\\text{j,n}:\\,&\\bm{\\Phi}(\\omega,k_1,k_2) \\mapsto\\\\\n\\nonumber\n&\\bm{\\Phi}(\\cosh(w)\\omega+\\sinh(w)k_1,\\sinh(w)\\omega+\\cosh(w)k_1,k_2).\n\\end{align}\nThis is a Lorentz boost in discrete Fourier space. It follows from this that $\\Lambda^w_\\text{j,n}$ acts on the basis $\\bm{e}_j\\otimes \\bm{e}_n\\otimes\\bm{e}_m$ as\n\\begin{align}\n\\nonumber\n&\\Lambda^w_\\text{j,n}: \\bm{e}_j\\otimes\\!\\bm{e}_n\\otimes\\!\\bm{e}_m\\mapsto\\!\\!\\!\\!\\sum_{b_1,b_2\\in\\mathbb{Z}} \\Lambda_{jn}^{b_1 b_2}(w)\\,\n\\bm{e}_{b_1}\\otimes\\!\\bm{e}_{b_2}\\otimes\\!\\bm{e}_m\\\\\n&\\Lambda_{jn}^{b_1 b_2}(w)=S_j(\\cosh(w) b_1 + \\sinh(w) b_2)\\\\ \n\\nonumber\n&\\qquad\\qquad\\times S_n(\\sinh(w) b_1+\\cosh(w) b_2).\n\\end{align}\nLike with $R^\\theta$ discussed above, $\\Lambda^w_\\text{j,n}$ is a symmetries of the dynamics in a qualified sense: namely, $\\Lambda^w_\\text{j,n}$ is not invertible (at least not on the whole of \\mbox{$\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$}). Indeed, each of these transformations is only invertible for a subspace of \\mbox{$\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$}. As before, this is due to the $2\\pi$ periodic identification of the planewaves $\\bm{\\Phi}(\\omega,k_1,k_2)$. To see this note that $\\Lambda^{\\ln(2)}_\\text{j,n}$ maps two different planewaves to the same place: Firstly note,\n\\begin{align}\n\\Lambda^{\\ln(2)}_\\text{j,n}\\bm{\\Phi}(\\pi,\\pi,k_2)&=\\bm{\\Phi}(2\\pi,2\\pi,k_2)\\\\\n\\nonumber\n&=\\bm{\\Phi}(0,0,k_2)\n\\end{align}\nsince the planewaves repeat themselves with period $2\\pi$. Secondly note, \n\\begin{align}\n\\nonumber\n\\Lambda^{\\ln(2)}_\\text{j,n}\\bm{\\Phi}(0,0,k_2)&=\\bm{\\Phi}(0,0,k_2)\n\\end{align}\nSuch issues do not arise when we have \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$} both before and after the Lorentz transformation. \n\nThus, $\\Lambda^{w}_\\text{j,n}$ and $\\Lambda^{w}_\\text{j,m}$ are invertible over the portion of \\mbox{$\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$} spanned by planewaves $\\bm{\\Phi}(\\omega,k_1,k_2)$ which satisfy the following two conditions:\n\\begin{align}\\label{BoostCond}\n\\vert\\cosh(w)\\,\\omega+\\sinh(w)\\, k_1\\,\\vert<&\\pi\\\\\n\\nonumber\n\\vert\\sinh(w)\\,\\omega+\\cosh(w)\\, k_2\\,\\vert<&\\pi.\n\\end{align}\n\nUnfortunately, unlike with $R^\\theta$ this issue cannot be so easily contained: the region where both $\\Lambda^{w}_\\text{j,n}$ and $\\Lambda^{w}_\\text{j,m}$ are invertible depends on $w$ and indeed, shrinks to nothing as $w\\to\\pm\\infty$. When boosted enough, any planewave except $\\omega=k_1=k_2=0$ will leave the region \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$}. Moreover, for any $w$ there is some planewave in \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$} which when boosted by $w$ leaves this region. However, despite this, for every planewave in $\\omega,k_1,k_2\\in(-\\pi,\\pi)$ there is some small enough boost $w\\neq0$ which keeps it in \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$}. Moreover, for every \\mbox{$w$} there is some neighborhood around \\mbox{$\\omega,k_1,k_2=0$} where boosting by $w$ leaves us in \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$}. When restricted to acting on the span of these planewaves $\\Lambda^{w}_\\text{j,n}$ and $\\Lambda^{w}_\\text{j,n}$ are both invertible.\n\nIt is in the following sense that $\\Lambda^{w}_\\text{j,n}$ and $\\Lambda^{w}_\\text{j,m}$ are symmetries of KG6. Consider solutions to KG6 which are supported only over planewaves in some finite neighborhood of $\\omega,k_1,k_2=0$. Consider $\\Lambda^{w}_\\text{j,n}$ and $\\Lambda^{w}_\\text{j,m}$ with $w$ in some finite neighborhood of $w=0$. Choose the neighborhoods such that Eq.~\\eqref{BoostCond} is satisfied throughout. Restricting our attention to this regime, all of these transformations map exactly map solutions to solutions in an invertible way. As I claimed above, KG6 is \\textit{exactly} Lorentz invariant over \\textit{a finite-sized region} around $\\omega,k_1,k_2=0$ and boost parameter $w=0$.\n\nRecall as mentioned above that for all planewaves in $\\omega,k_1,k_2\\in(-\\pi,\\pi)$ there is some small enough boost \\mbox{$w\\neq0$} which keeps it in \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$}. Thus, KG6 has a differential Lorentz boost invariance over all of \\mbox{$\\omega,k_1,k_2\\in(-\\pi,\\pi)$}. Note that the same is true of translations and rotations. Including translations and rotations, KG6 has a differential Poincar\\'e invariance over $\\omega,k_1,k_2\\in(-\\pi,\\pi)$. Concretely, taken all together $\\Lambda^w_\\text{j,n}$, $\\Lambda^w_\\text{j,m}$, $R^\\theta$, $T^\\epsilon_\\text{j}$, $T^\\epsilon_\\text{n}$ and $T^\\epsilon_\\text{m}$ form a representation of the Poincar\\'e algebra over the space spanned by \\mbox{$\\omega,k_1,k_2\\in(-\\pi,\\pi)$}. In particular, this means that for every algebraic fact about the differential Poincar\\'e transformations there is an analogous fact here with the group action being replaced by matrix multiplication. Exponentiating this representation of the Poincar\\'e algebra we recover a finite part of the Poincar\\'e group. Namely, the finite-sized collection of states and transformations satisfying:\n\\begin{align}\\label{PoincareCond}\n\\vert\\cosh(w)\\,\\omega+\\sinh(w)\\,\\vert k\\vert\\,\\vert<&\\pi\\\\\n\\nonumber\n\\vert\\sinh(w)\\,\\omega+\\cosh(w)\\,\\vert k\\vert\\,\\vert<&\\pi\\\\\n\\nonumber\n\\vert\\cos(\\theta)\\,k_1-\\sin(\\theta) k_2\\vert<&\\pi\\\\\n\\nonumber\n\\vert\\sin(\\theta)\\,k_1+\\cos(\\theta) k_2\\vert<&\\pi.\n\\end{align}\nIndeed, KG6 is \\textit{exactly} Poincar\\'e invariant over \\textit{a finite-sized region} around $\\omega,k_1,k_2=0$ and $w=\\theta=0$.\n\nAs I will now discuss, in this regime $\\Lambda^w_\\text{j,m}$ and $\\Lambda^w_\\text{j,n}$ can be thought of as implementing Lorentzian boosts for two reasons despite being here categorized as an internal symmetry. Firstly, recall the close relationship noted above between $D$ and $\\partial_x$. Recall also that Lorentz boosts are generated through the derivative as \\mbox{$h(\\Lambda^w(t,x))= \\exp(-w (x \\partial_t + t \\partial_x))h(t,x)$}. \n\nSecondly, as I have already mentioned, together with our above discussed translations and rotations,  $\\Lambda^w_\\text{j,m}$ and $\\Lambda^w_\\text{j,n}$ give us a representation of the Poincar\\'e algebra and even a finite portion of the Poincar\\'e group.\n\nThus we have yet another addendum to our first big lesson: despite the fact that KG6 and KG7 can be represented on a cubic 3D lattice and a hexagonal 3D lattice respectively, they nonetheless both have a Lorentzian boost symmetries (in a finite but limited regime). This, in addition to their continuous translation and rotation symmetries. As impressive as this admittedly limited Lorentz symmetry is, we can do better: In Sec.~\\ref{PerfectLorentz} I will provide a perfectly Lorentzian lattice theory.\n\nFinally, let's consider KG3. For KG3 the dynamical symmetries of the form Eq.~\\eqref{GaugeVR} include:\n\\begin{flushleft}\\begin{enumerate}\n    \\item[1)] action by $T^\\epsilon_\\text{j}$ sending $\\bm{\\Phi}\\mapsto T^\\epsilon_\\text{j} \\bm{\\Phi}$. Similarly for $T^\\epsilon_\\text{n}$.\n    \\item[2)] two negation symmetries which map basis vectors as \\mbox{$\\bm{e}_{j}\\otimes\\bm{e}_{n}\\mapsto \\bm{e}_{-j}\\otimes\\bm{e}_{n}$} and \\mbox{$\\bm{e}_{j}\\otimes\\bm{e}_{n}\\mapsto \\bm{e}_{j}\\otimes\\bm{e}_{-n}$} respectively,\n    \\item[3)] action by $\\Lambda^w_\\text{jn}$ sending $\\bm{\\Phi}\\mapsto \\Lambda^w_\\text{jn} \\bm{\\Phi}$ where \\mbox{$\\Lambda^w_\\text{jn}$} is defined above,\n    \\item[4)] a local Fourier rescaling symmetry which maps \\mbox{$\\bm{\\Phi}(\\omega,k)\\mapsto \\tilde{f}(\\omega,k)\\,\\bm{\\Phi}(\\omega,k)$} for some non-zero complex function $\\tilde{f}(\\omega,k)\\in\\mathbb{C}$ with $\\omega,k\\in[-\\pi,\\pi]$,\n    \\item[5)] local affine rescaling which maps \\mbox{$\\bm{\\Phi}\\mapsto \\bm{\\Phi} + \\bm{c}_2$} for some $\\bm{c}_2$ which is also a solution of the dynamics.\n\\end{enumerate}\\end{flushleft}\nAs with KG1 and KG2 we have gained local Fourier rescaling and discrete shifts have been replaced with action by $T^\\epsilon$. As discussed above we are justified in calling $T^\\epsilon$ a continuous translation operation. For KG3 we also have a new symmetry, namely action by $\\Lambda^w_\\text{jn}$. For the reasons discussed above, this can be thought of as a (limited) Lorentzian boost symmetry despite it being here classified as an internal symmetry.\n\n\\vspace{0.25cm}\nTo summarize: this second attempt at interpreting KG1-KG7 has fixed all of the issues with our previous interpretation. Firstly, there is no longer any tension between these theories' differing locality properties and the rates at which they converge to the (perfectly local) continuum theory in the continuum limit. (There are no longer any differences in locality.) See \\cite{DiscreteGenCovPart1} for further discussion. Secondly, the fact that we have a nice one-to-one correspondence between the solutions to KG6 and KG7 is now more satisfyingly reflected in their matching symmetries. Finally, this interpretation has exposed the fact that KG1-KG7 have hidden continuous translation and rotation symmetries as well as a (limited) Lorentzian boost symmetries.\n\nBy and large, the interpretation invalidates all of the first intuitions laid out in Sec.~\\ref{SecIntro}. As this interpretation has revealed, the lattice seems to play a merely representational role in the theory: it does not restrict our symmetries. Moreover, theories initially appearing with different lattice structures may nonetheless turn out to be completely equivalent. The process for switching between lattice structures is here a change of basis in the value space. Indeed, we have a third lesson: there is no sense in which these lattice structures are essentially ``baked-into'' these theories; on this interpretation our theories make no reference to any lattice structure if we work in a basis-independent way. No basis is dynamically favored. \n\nAs I discussed in \\cite{DiscreteGenCovPart1} these three lessons lay the foundation for a strong analogy between the lattice structures which appear in our discrete spacetime theories and the coordinate systems which appear in our continuum spacetime theories. This analogy here extended to Lorentzian theories.\n\nThese are substantial lessons, but ultimately this interpretation has a few issues of its own. Firstly, the way that the tension is dissolved between locality and convergence in the continuum limit is unsatisfying. Intuitively, we ought to be able to extract intuitions about locality from the lattice sites. \n\nMoreover, while this interpretation has indeed exposed KG1-KG7's hidden symmetries, the way it classifies them seems wrong. They are here classified as internal symmetries (i.e., symmetries on the value space) whereas intuitively they should be external symmetries (i.e., symmetries on the manifold).\n\nThe root of all of these issues is taking the theory's lattice structure to be internalized into the theory's value space. Our third attempt at interpreting these theories will fix this by externalizing these symmetries. As I will discuss, this gives us access to a perspective within which we can find a perfectly Lorentzian lattice theory (i.e., not limited as the above theories are).\n\n\\section{Externalizing these theories - Part 1}\\label{SecExtPart1}\nIn the previous section it was revealed that KG1-KG7 have hidden continuous symmetry transformations which intuitively correspond to spacial translations and rotation and even (limited) Lorentzian boosts. In our first attempt at interpreting KG1-KG7 the possibility of such symmetries were outright denied, see Sec.~\\ref{SecKlein1}. In our second attempt, these hidden symmetries were exposed, but they were classified (unintuitively) as internal symmetries, see Sec.~\\ref{SecKlein2}. This is due to an ``internalization'' move made in our second interpretation. This move also had the unfortunate consequence of undercutting our ability to use the lattice sites to reason about locality.\n\nIn this section I will show how we can externalize these symmetries by in a principled way 1) inventing a continuous manifold for our formerly discrete theories to live on and 2) embedding our theory's states\/dynamics onto this manifold as a new dynamical field.\n\n\\subsection{A Principled Choice of Spacetime Manifold}\\label{SecKlein3A}\nIf we are going to externalize these symmetries then we need to have a big enough manifold on which to do the job. What spacetime manifold $\\mathcal{M}$ might be up to the task? The first thing we must do is pick out which of our theory's symmetries we would like to externalize (there may be some symmetries we want to keep internal). For KG1-KG7 we want to externalize the following symmetries: continuous translations, mirror reflections, as well as  discrete rotations for KG4 and KG5 and continuous rotations for KG6 and KG7. For each theory we can collect these dynamical symmetries together in a group $G^\\text{dym}_\\text{to-be-ext}$. Clearly, our choice of spacetime manifold $\\mathcal{M}$ needs to be big enough to have $G^\\text{dym}_\\text{to-be-ext}$ as a subgroup of $\\text{Diff}(\\mathcal{M})$. Let us call this the symmetry-fitting constraint\\footnote{As I will soon discuss, there may be transformations which we want to externalize even when they are not symmetries.}. \n\nOf course, symmetry-fitting alone doesn't uniquely specify which manifold we ought to use. If $\\mathcal{M}$ works, then so does any larger $\\mathcal{M}'$ with $\\mathcal{M}$ as a sub-manifold. For standard Occamistic reasons, it is natural to go with the smallest manifold which gets the job done. The larger the gap between the groups $G^\\text{dym}_\\text{to-be-ext}$ and $\\text{Diff}(\\mathcal{M})$ the more fixed spacetime structures will need to be introduced later on (see Sec.~\\ref{SecFullGenCov}).\n\nIn principle, we are free to pick any large-enough manifold which we like to embed KG1-KG7 onto. However, perhaps surprisingly, if we make natural choices about how the translation operations we have already identified (see Eq.~\\eqref{TDef}) fit onto the new spacetime manifold then our choice of $\\mathcal{M}$ is actually fixed up to diffeomorphism. In particular, I demand the following: Certain translation operations on $\\mathbb{R}^L$ are to correspond (perhaps in a complicated way) to parallel transport on the new spacetime manifold $\\mathcal{M}$. For KG1-KG3 these to-be-externalized translation operations are $T^{\\epsilon_1}_\\text{j}$ and $T^{\\epsilon_2}_\\text{n}$. For KG4-KG7 these to-be-externalized translation operations are $T^{\\epsilon_1}_\\text{j}$, $T^{\\epsilon_2}_\\text{n}$, and \n$T^{\\epsilon_3}_\\text{m}$. Let us call this the translation-matching constraint. As I will soon show, this constraint fixes the new spacetime manifold $\\mathcal{M}$ up to diffeomorphism.\n\nBefore fleshing this out, however, it's worth reflecting on two questions focusing on KG4-KG7: What exactly makes $T^{\\epsilon_1}_\\text{j}$, $T^{\\epsilon_2}_\\text{n}$, and $T^{\\epsilon_3}_\\text{m}$ translation operations? Moreover, what motivation do we have to externalize these particular translation operations? To answer: firstly, these can be thought of as translation operations for the reasons discussed following Eq.~\\eqref{TDef}. Note these reasons are unrelated to the fact that these are dynamical symmetries of KG4-KG7. Suppose the dynamics of KG4 given by Eq.~\\eqref{KG4Long} was modified to have explicit dependence on the index $n$. In this case, $T^{\\epsilon_2}_\\text{n}$ would no longer be a dynamical symmetry but it would still be a translation operation (and moreover, one worth externalizing).\n\nSecondly, why externalize these translation operations in particular? As I will now discuss, any motivation for externalizing these particular translation operations must come from the dynamics. Forgoing any dynamical considerations, all we can say about KG1-KG7 is that they concern vectors $\\bm{\\Phi}\\in\\mathbb{R}^L$ (or alternatively functions \\mbox{$\\phi:L\\to\\mathbb{R}$} in $F_L$). Recall that pre-dynamics the set of labels for lattice sites $L$ is uncountable but otherwise unstructured; We might index it using any number of indices we like, \\mbox{$L\\cong\\mathbb{Z}\\cong\\mathbb{Z}^2\\cong\\dots\\cong\\mathbb{Z}^{17}\\cong\\dots$}. Using each of these re-indexings we can grant $\\mathbb{R}^L$ different tensor product structures, \\mbox{$\\mathbb{R}^L\\cong\\mathbb{R}^Z\\cong\\mathbb{R}^Z\\otimes\\mathbb{R}^Z\\cong\\dots$}. In each tensor factor we can define a translation operation $T^\\epsilon$ as in Eq.~\\eqref{TDef}. Thus, the vector space $\\mathbb{R}^L$ supports representations of: the 1D translation group, the 2D translation group, $\\dots$, the 17D translation group, etc. Given all these possibilities, why externalize $T^{\\epsilon_1}_\\text{j}$, $T^{\\epsilon_2}_\\text{n}$, and $T^{\\epsilon_3}_\\text{m}$ in particular? The answer must come from the dynamics.\n\nOne good reason to consider $T^{\\epsilon_1}_\\text{j}$, $T^{\\epsilon_2}_\\text{n}$, and $T^{\\epsilon_3}_\\text{m}$ worthy of externalization is that they are dynamical symmetries of KG4-KG7. However, as mentioned above, something might be a translation operation (and moreover, one worthy of externalizing) even if it's not a dynamical symmetry. Unfortunately, I don't have a perfect rule for how to identify such cases. The best I can offer is to check whether its associated derivative (or some function thereof) appears in the dynamical equations.\n\nRegardless, its sufficiently clear for KG4-KG7 that $T^{\\epsilon_1}_\\text{j}$, $T^{\\epsilon_2}_\\text{n}$, and \n$T^{\\epsilon_3}_\\text{m}$ are among the translation operations worthy of externalizing. Our translation-matching constraint suggests that these ought to correspond (perhaps in a complicated way) to parallel transport on the new spacetime manifold $\\mathcal{M}$. I claim that (at least in this case) this constraint fixes the spacetime manifold $\\mathcal{M}$ up to diffeomorphism.\n\nTo show this I will first pick out at each point \\mbox{$p\\in\\mathcal{M}$} on the manifold three independent directions in the tangent space at $p$. To realize the translation-matching constraint, I demand that differential translation by $T^{\\epsilon_1}_\\text{j}$, $T^{\\epsilon_2}_\\text{n}$, and $T^{\\epsilon_3}_\\text{m}$ is then to be carried out on the manifold as parallel transport in these three directions. Note that this already implies that the dimension of $\\mathcal{M}$ is at least three.\n\nIndeed, in this case we have good reason to take $\\mathcal{M}$ to be exactly three dimensional. To see this, note that \n\\begin{align}\\label{RLCover}\n\\mathbb{R}^L&\\cong \\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\\\\n&=\\text{span}(T^{\\epsilon_1}_\\text{j}  T^{\\epsilon_2}_\\text{n}  T^{\\epsilon_3}_\\text{m} \\, \\bm{e}_0\\!\\otimes\\bm{e}_0\\!\\otimes\\bm{e}_0\\vert (\\epsilon_1,\\epsilon_2,\\epsilon_3)\\in\\mathbb{R}^3)\n\\end{align}\nThat is, beginning from $\\bm{e}_0\\otimes\\bm{e}_0\\otimes\\bm{e}_0$ these translations cover all of $\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$. That is, they cover all of our state space in the second interpretation. The three translations $T^{\\epsilon_1}_\\text{j}$, $T^{\\epsilon_2}_\\text{n}$, and $T^{\\epsilon_3}_\\text{m}$ are thus already enough to cover every kinematic possibility. We need no further dimensions and so by Occam's razor we may then take the spacetime manifold $\\mathcal{M}$ to be three dimensional. \n\nBut how does the translation-matching constraint (which I have yet to state technically) fix $\\mathcal{M}$ up to diffeomorphism? As I will now discuss, (at least in this case) $\\mathcal{M}$ it is fixed up to diffeomorphism by the group theoretic properties of $T^{\\epsilon_1}_\\text{j}$, $T^{\\epsilon_2}_\\text{n}$, and $T^{\\epsilon_3}_\\text{m}$. Let $G_\\text{trans.}$ be the group formed by all compositions of $T^{\\epsilon_1}_\\text{j}$, $T^{\\epsilon_2}_\\text{n}$, and $T^{\\epsilon_3}_\\text{m}$. Note that the group $G_\\text{trans.}$ is a Lie group, and hence is also a differentiable manifold.\n\nRecall that so far the translation-matching constraint associated to each of our differential translations, a direction in the tangent space of each point $p\\in\\mathcal{M}$. Since $G_\\text{trans.}$ contains nothing but these translations, this gives us a one-to-one correspondence between the algebra $g_\\text{trans.}$ of $G_\\text{trans.}$ and (at least a subspace of) the of the tangent space of each point on the manifold. Let's take these tangent vectors to very smoothly across the manifold. Let $\\mathcal{M}_\\text{trans.}(p)$ be the submanifold of points on the spacetime manifold reachable from $p\\in\\mathcal{M}$ by following these tangent vectors. The algebra $g_\\text{trans.}$ is related to the group $G_\\text{trans.}$ in the same way that this tangent space it related to $\\mathcal{M}_\\text{trans.}(p)$, namely by repeated application of the exponential map. Hence, I take translation-matching to demand that $\\mathcal{M}_\\text{trans.}(p)\\cong G_\\text{trans.}$ as differentiable manifolds for every $p\\in\\mathcal{M}$. \n\nAs discussed above, we have reason to take $\\mathcal{M}$ to be three dimensional. Hence (assuming that $\\mathcal{M}$ is connected) any three dimensional submanifold (e.g. $\\mathcal{M}_\\text{trans.}(p)$) must in fact be the entire manifold. Therefore we have $\\mathcal{M}\\cong G_\\text{trans.}$ as differentiable manifolds.\n\nBut what can we say about this translation group $G_\\text{trans.}$ viewed as a manifold? Notice that these three continuous translation operations comprising $G_\\text{trans.}$ all commute with each other. As such for any combination of them $T_\\text{generic}\\in G_\\text{trans.}$ we always have a unique factorization of the form,\n\\begin{align}\\label{Tgeneric}\nT_\\text{generic} = T^{\\epsilon_1}_\\text{j} \\ T^{\\epsilon_2}_\\text{n} \\ T^{\\epsilon_3}_\\text{m}\n\\end{align}\nMoreover, this factorization represents all of $G_\\text{trans.}$ without redundancy. Indeed, we can smoothly parameterize all of $G_\\text{trans.}$ using parameters \\mbox{$(\\epsilon_1,\\epsilon_2,\\epsilon_3)\\in\\mathbb{R}^3$} in a one-to-one way. That is, $G_\\text{trans.}\\cong\\mathbb{R}^3$ as differentiable manifolds. Therefore, we have $\\mathcal{M}\\cong\\mathbb{R}^3$ as well. Note this means that we have access to a global coordinate system for $\\mathcal{M}$.\n\nThus, from the translation-matching constraint alone we are forced to take $\\mathcal{M}\\cong\\mathbb{R}^3$ for KG4-KG7. For KG1-KG3 the same translation-matching constraint forces us to take $\\mathcal{M}\\cong\\mathbb{R}^2$. Note that in both cases these manifolds satisfy the symmetry-fitting constraint: $\\text{Diff}(\\mathcal{M})$ is large enough to contain $G^\\text{dyn}_\\text{to-be-ext}$.\n\n\\subsection{A Principled Choice of Embedding}\nNow that we have a spacetime manifold selected, we need to somehow embed $\\bm{\\Phi}$ (or equivalently $\\phi_\\ell$) into it. Here I will begin with $\\bm{\\Phi}$ with the approach from $\\phi_\\ell$ being addressed in Sec.~\\ref{SecExtPart2}. The goal in either case is to construct in a principled way from either $\\bm{\\Phi}$ or $\\phi_\\ell$ a new field \\mbox{$\\phi:\\mathcal{M}\\to\\mathbb{R}$} defined over this manifold. \n\nBefore this, however, recall that our move from the first to the second interpretation was mediated by means of a vector-space isomorphism $F_L\\cong\\mathbb{R}^L$ namely Eq.~\\eqref{PhiDef}. Here too, our reinterpretation will be mediated by a vector-space isomorphism.\n\nLet $F(\\mathcal{M})$ be the set of all real scalar functions \\mbox{$f:\\mathcal{M}\\to\\mathbb{R}$}. Notice that this is a vector space as it is closed under addition and scalar multiplication. Our goal here is to in a principle way construct an isomorphism between \\mbox{$\\mathbb{R}^L$} and some vector subspace of \\mbox{$F(\\mathcal{M})$}. Indeed, $F(\\mathcal{M})$ is much larger than $\\mathbb{R}^L$ (having an uncountably infinite dimension). As such we will only ever map $\\mathbb{R}^L$ onto some subset of it, $F\\subset F(\\mathcal{M})$. In order for us to have a vector space isomorphism between $\\mathbb{R}^L$ and $F$ the following conditions must be satisfied. $F$ must be: 1) closed under addition and scalar multiplication and hence a vector space, and 2) countably infinite dimensional and hence isomorphic to $\\mathbb{R}^L$, i.e., \\mbox{$F\\cong\\mathbb{R}^\\mathbb{Z}\\cong\\mathbb{R}^L$}. With the target subset characterized, the embedding of $\\bm{\\Phi}$ onto $\\mathcal{M}$ is then accomplished by picking a vector-space isomorphism $E:\\mathbb{R}^L\\to F$. The embedded field is then gives to us as $\\phi\\coloneqq E(\\bm{\\Phi})$.\n\nIt appears we have a great deal of freedom here. Suppose that, in line with the translation-matching constraint discussed above, we take $\\mathcal{M}\\cong\\mathbb{R}^2$ for KG1-KG3 and $\\mathcal{M}\\cong\\mathbb{R}^3$ for KG4-KG7. In this case, we still have great freedom in picking both the target vector space $F\\subset F(\\mathcal{M})$ and the isomorphism $E:\\mathbb{R}^L\\to F$. However, as I will now discuss, translation-matching also drastically limits these choices as well. Allow me to demonstrate with KG4-KG7.\n\nFirst, allow me to explicitly define a coordinate system for $\\mathcal{M}\\cong G_\\text{trans.}\\cong\\mathbb{R}^3$. As discussed above we already have an explicit smooth parametrization of $G_\\text{trans.}$ via \\mbox{$(\\epsilon_1,\\epsilon_2,\\epsilon_3)\\in\\mathbb{R}^3$}, see Eq.~\\eqref{Tgeneric}. In addition to this, concretely realizing our translation-matching constraint requires us to fix a smooth map from $G_\\text{trans.}$ to $\\mathcal{M}$. Combined these two maps give us a global coordinate system $(\\epsilon_1,\\epsilon_2,\\epsilon_3)\\in\\mathbb{R}^3$ for $\\mathcal{M}$. We can then rescale these to give new coordinates $(t,x,y)\\in\\mathbb{R}^3$ by adding in a length scale $a>0$ with $t=\\epsilon_1 a$, $x=\\epsilon_2 a$, and $y=\\epsilon_3 a$ for some fixed length scale $a$. \n\nIn total this picks out a diffeomorphism \\mbox{$d_\\text{trans.}:\\mathbb{R}^3\\to\\mathcal{M}$} which assigns global coordinates $(t,x,y)\\in\\mathbb{R}^3$ to $\\mathcal{M}$. For much of the next sections I will work with $\\phi$ in these coordinates. Namely I will work with the pull back, $\\phi\\circ d_\\text{trans.}:\\mathbb{R}^3\\to\\mathbb{R}$. One may worry that this choice of coordinates is arbitrary. Indeed, if one changes the smooth map from $G_\\text{trans.}$ to $\\mathcal{M}$ referenced above we end up with a different coordinate system $d_\\text{trans.}$ related to the original by some diffeomorphism. While this is true, no matter how one realizes the translation-matching constraint there will be some coordinate system would result from the above construction. In what follows, nothing depends on how the translation-matching constraint is realized.\n\nA word of warning. The coordinates used above might end up lying on the manifold in a curvilinear way. Crucially, the naive distance and metric structure associated with these coordinates do not automatically have any spaciotemporal significance. At present, our manifold $\\mathcal{M}$ still has no metric. When a metric arises later, it will be due to dynamical considerations, not from any choice of coordinates. A coordinate-independent view of these theories will be given in Sec.~\\ref{SecFullGenCov}.\n\nWith this said, by construction we have the following in correspondences in these coordinate:\n\\begin{flushleft}\\begin{enumerate}\n    \\item Action by $T_\\text{j}^\\epsilon$ on $\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$ acts on the manifold $\\mathcal{M}$ in these coordinates as $(t,x,y)\\mapsto(t-\\epsilon\\,a,x,y)$,\n    \\item Action by $T_\\text{n}^\\epsilon$ on $\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$ acts on the manifold $\\mathcal{M}$ in these coordinates as $(t,x,y)\\mapsto(t,x-\\epsilon\\,a,y)$,\n    \\item Action by $T_\\text{m}^\\epsilon$ on $\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$ acts on the manifold $\\mathcal{M}$ in these coordinates as $(t,x,y)\\mapsto(t,x,y-\\epsilon\\,a)$.\n\\end{enumerate}\\end{flushleft}\nIn particular, matching up differential translations on $\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$ with those on $\\mathcal{M}$ requires\n\\begin{align}\nE(D_\\text{j}\\,\\bm{\\Phi})&=a\\,\\partial_t E(\\bm{\\Phi})\\\\\n\\nonumber\nE(D_\\text{n}\\bm{\\Phi})&=a\\,\\partial_x E(\\bm{\\Phi})\\\\\n\\nonumber\nE(D_\\text{m}\\bm{\\Phi})&=a\\,\\partial_y E(\\bm{\\Phi})\n\\end{align}\nin these coordinates for all $\\bm{\\Phi}\\in\\mathbb{R}^L$.\n\nEvaluating these conditions in the planewave basis for $\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$ (namely $\\bm{\\Phi}(\\omega,k_1,k_2)$ for \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$}) we see that $E(\\bm{\\Phi}(\\omega,k_1,k_2))$ must be simultaneously an eigenvector of $\\partial_t$, $\\partial_x$, and $\\partial_y$ with eigenvalues $-\\mathrm{i} \\omega\/a$, $-\\mathrm{i} k_1\/a$, and $-\\mathrm{i} k_2\/a$ respectively. This uniquely picks out the continuum planewaves:\n\\begin{align}\\label{PlaneWaveCont}\n\\text{KG1-KG3:}\\quad&\\phi(t,x;\\omega,k)\n\\coloneqq e^{-\\mathrm{i} \\omega t -\\mathrm{i} k x}\\\\\n\\nonumber\n\\text{KG4-KG7:}\\quad&\\phi(t,x,y;\\omega,k_1,k_2)\n\\coloneqq e^{-\\mathrm{i} \\omega t -\\mathrm{i} k_1 x-\\mathrm{i} k_2 y}.\n\\end{align}\nIn particular, we are forced to take\n\\begin{align}\\label{Eftilde}\nE:\\,&\\bm{\\Phi}(\\omega,k_1,k_2)\\\\\n\\nonumber\n&\\mapsto\\tilde{f}(\\omega,k_1,k_2)\\, \\phi(t,x,y;\\omega\/a,k_1\/a,k_2\/a)\n\\end{align}\nfor some complex function $\\tilde{f}(\\omega,k_1,k_2)\\in\\mathbb{C}$. That is, each discrete planewave must map onto something proportional to the corresponding continuous planewaves (rescaled by $a$)\n\nNote that up to a local rescaling of Fourier space (which recall is a symmetry of the dynamics) our embedding is uniquely fixed by translation-matching. Even without this consideration however, our translation-matching constraint still fixes the vector space $F\\subset F(\\mathcal{M})$ we can embed \\mbox{$\\bm{\\Phi}\\in\\mathbb{R}^L$} onto, note the following. We must have \\mbox{$\\tilde{f}(\\omega,k_1,k_2)\\neq0$} for all \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$} otherwise $E$ will not be invertible an hence not an isomorphism. Therefore $F$ is fixed as,\n\\begin{align}\n&F=\\text{span}(E(\\bm{\\Phi}(\\omega,k_1,k_2))\\vert \\omega,k_1,k_2\\in[-\\pi,\\pi])\\\\\n\\nonumber\n&=\\!\\text{span}(\\tilde{f}(\\omega,k_1,k_2)\\,\\phi(t,x,y;\\frac{\\omega}{a},\\frac{k_1}{a},\\frac{k_2}{a})\\vert \\omega,k_1,k_2\\in\\![-\\pi,\\pi])\\\\\n\\nonumber\n&=\\text{span}(\\phi(t,x,y;\\frac{\\omega}{a},\\frac{k_1}{a},\\frac{k_2}{a})\\vert \\omega,k_1,k_2\\in[-\\pi,\\pi])\\\\\n\\nonumber\n&=\\text{span}(\\phi(t,x,y;\\omega,k_1,k_2)\\vert \\omega,k_1,k_2\\in[-\\pi\/a,\\pi\/a]).\n\\end{align}\nIn light of this, let us define\n\\begin{align}\n&\\text{KG1-KG3:}\\\\\n\\nonumber\n&F^K\\coloneqq\\text{span}(\\phi(t,x;\\omega,k)\\vert \\omega,k\\in[-K,K])\\\\\n&\\text{KG4-KG7:}\\\\\n\\nonumber\n&F^K\\coloneqq\\text{span}(\\phi(t,x,y;\\omega,k_1,k_2)\\vert \\omega,k_1,k_2\\in[-K,K])\n\\end{align}\nwhere $K=\\pi\/a$. These are the spaces of bandlimited functions with bandwidth $\\omega,k\\in[-K,K]$ and \\mbox{$\\omega,k_1,k_2\\in[-K,K]$} respectively. Thus, demanding translation-matching we are forced to map $\\bm{\\Phi}\\in\\mathbb{R}^L$ onto some bandlimited $\\phi(t,x,y)\\in F=F^K$ (at least in the coordinates $d_\\text{trans.}$).\n\nThe next section will discuss in detail some remarkable properties of bandlimited functions, namely their sampling property. Before that however, let's find an interpretation for the $\\tilde{f}(\\omega,k_1,k_2)$ function appearing in Eq.~\\eqref{Eftilde}. First note that applying $E$ to the basis vector $\\bm{e}_0\\otimes\\bm{e}_0\\otimes\\bm{e}_0$ we have\n\\begin{align}\\label{Embedf}\nE(\\bm{e}_0\\otimes\\bm{e}_0\\otimes\\bm{e}_0)=f\\!\\left(\\frac{t}{a},\\frac{x}{a},\\frac{y}{a}\\right)\n\\end{align}\nwhere $f(t,x,y)$ is the inverse Fourier transform of $\\tilde{f}(\\omega,k_1,k_2)$. Next note that by applying $T_\\text{j}^\\epsilon$, $T_\\text{n}^\\epsilon$, and $T_\\text{m}^\\epsilon$ with integer arguments we can get from $\\bm{e}_0\\otimes\\bm{e}_0\\otimes\\bm{e}_0$ to any other basis vector, $\\bm{e}_j\\otimes\\bm{e}_n\\otimes\\bm{e}_m$. In these coordinates this means,\n\\begin{align}\n&E(\\bm{e}_j\\otimes\\bm{e}_n\\otimes\\bm{e}_m)=f\\left(\\frac{t}{a}-j,\\frac{x}{a}-n,\\frac{y}{a}-m\\right)\n\\end{align}\nThus, we can understand a choice of $\\tilde{f}$ as picking a profile $f(t,x,y)\\in F^\\pi$. A translated and rescaled copy of this profile is then associated with each basis vector.\n\nTo review: Our translation-matching considerations have greatly constrained our choice of both the spacetime manifold $\\mathcal{M}$ and the embedding of $\\bm{\\Phi}$ onto $F(\\mathcal{M})$. In particular, $\\mathcal{M}$ was forced to be diffeomorphic to $\\mathbb{R}^2$ for KG1-KG3 and to $\\mathbb{R}^3$ for KG4-KG7. Moreover, the vector space $F\\subset F(\\mathcal{M})$ which we map $\\bm{\\Phi}\\in\\mathbb{R}^L$ into is forced to be the space of bandlimited functions with some bandwidth (i.e., bandlimited in coordinates $d_\\text{trans.}$ with $\\omega,k_1,k_2\\in[-K,K]$).\n\nBeyond this, our only freedom left is in picking a profile $f$ to associate with each basis vector $\\bm{e}_j\\otimes\\bm{e}_n\\otimes\\bm{e}_m$. In Sec.~\\ref{SecExtPart2} I will motivate a principled choice for $f$. Before that however, let's discuss bandlimited function in some detail.\n\n\\section{Brief Review of Bandlimited Functions and Nyquist-Shannon Sampling Theory}\\label{SecSamplingTheory}\n The previous section has given us reason to care about bandlimited functions. A bandlimited function is one whose Fourier transform has compact support. The bandwidth of such a function is the extent of its support in Fourier space. As I will now discuss, such functions have a remarkable sampling property: they can be exactly reconstructed knowing only their values at a (sufficiently dense) set of sample points. The study of such functions constitutes Nyquist-Shannon sampling theory. For a selection of introductory texts on sampling theory see ~\\cite{GARCIA200263,SamplingTutorial,UnserM2000SyaS}.\n\nTo introduce the topic I will at first restrict our attention to the one-dimensional case with uniform sample lattice before generalizing to higher dimensions and non-uniform samplings later on.\n\n\\subsection{One Dimension Uniform Sample Lattices}\\label{Sec1DUniform}\nConsider a generic bandlimited function, $f_\\text{B}(x)$, with a bandwidth of $K$. That is, a function $f_\\text{B}(x)$ such that its Fourier transform,\n\\begin{align}\n\\mathcal{F}[f_\\text{B}(x)](k)\\coloneqq\\int_{-\\infty}^\\infty f_\\text{B}(x) \\, e^{-\\mathrm{i} k x} \\d x,\n\\end{align}\nhas support only for wavenumbers $\\vert k\\vert< K$.\n\nSuppose that we know the value of $f_\\text{B}(x)$ only at the regularly spaced sample points, $x_n=n\\,a+b$, with some spacing, \\mbox{$0\\leq a\\leq a^*\\coloneqq\\pi\/K$}, and offset, $b\\in\\mathbb{R}$. Let \\mbox{$f_n=f_\\text{B}(x_n)$} be these sample values. Having only the discrete sample data, $\\{(x_n,f_n)\\}_{n\\in\\mathbb{Z}}$, how well can we approximate the function? \n\nThe Nyquist-Shannon sampling theorem~\\cite{ShannonOriginal} tells us that from this data we can reconstruct $f_\\text{B}$ exactly everywhere! That is, from this discrete data, $\\{(x_n,f_n)\\}_{n\\in\\mathbb{Z}}$, we can determine everything about the function $f_\\text{B}$ everywhere. In particular, the following reconstruction is exact, \n\\begin{align}\\label{SincRecon}\nf_\\text{B}(x) \n= S_n\\!\\left(\\frac{x-b}{a}\\right) f_n,\n\\end{align}\nwhere\n\\begin{align}\nS(y)=\\frac{\\sin(\\pi y)}{\\pi y}, \\quad\\text{and}\\quad\nS_n(y)=S(y-n), \n\\end{align}\nare the normalized and shifted sinc functions. Note that $S_n(m)=\\delta_{nm}$ for integers $n$ and $m$. Moreover, note that each $S_n(x)$ is both $L_1$ and $L_2$ normalized and that taken together the set $\\{S_n(x)\\}_{n\\in\\mathbb{Z}}$ forms an orthonormal basis with respects to the $L_2$ inner product. The fact that any bandlimited function can be reconstructed in this way is equivalent to the fact that this orthonormal basis spans the space of bandlimited functions with bandwidth of $K=\\pi$.\n\n\\begin{figure}\n\\includegraphics[width=0.4\\textwidth]{PaperFigures\/Fig1DSamplesa.pdf}\n\\includegraphics[width=0.4\\textwidth]{PaperFigures\/Fig1DSamplesb.pdf}\n\\includegraphics[width=0.4\\textwidth]{PaperFigures\/Fig1DSamplesc.pdf}\n\\includegraphics[width=0.4\\textwidth]{PaperFigures\/Fig1DSamplesd.pdf}\n\\includegraphics[width=0.4\\textwidth]{PaperFigures\/Fig1DSamplese.pdf}\n\\caption{Several different (but completely equivalent) graphical representations of the bandlimited function \\mbox{$f_\\text{B}(x)=1+S(x-1\/2)+x\\,S(x\/2)^2$} with bandwidth of $K=\\pi$ and consequently a critical spacing of $a^*=\\pi\/K=1$. Subfigure a) shows the function values for all $x$. b) shows the values of $f_\\text{B}$ at $x_n=n\/2$. Since $1\/2<a^*=1$ this is an instance of oversampling. c) shows the values of $f_\\text{B}$ at $x_n=n$. This is an instance of critical sampling. d) shows the values of $f_\\text{B}$ at $x_n=n+1\/3$. This too is an instance of critical sampling. e) shows a non-uniform sampling of $f_\\text{B}$. From any of these samplings we can recover the function $f_\\text{B}$ exactly everywhere. [\\textit{Reproduced with permission from} \\cite{DiscreteGenCovPart1}.]}\\label{Fig1DSamples}\n\\end{figure}\n\nAs a concrete example, let us consider the function $f_\\text{B}(x)=1+S(x-1\/2)+x\\,S(x\/2)^2$, shown in Fig.~\\ref{Fig1DSamples}a). This function has a bandwidth of $K=\\pi$ and so has a critical sample spacing of $a^*=\\pi\/K=1$. Thus, we can fully reconstruct $f_\\text{B}(x)$ knowing only its values at $x_n=n\\,a+b$ for any spacing $a\\leq a*=1$. In particular the sample values at $x_n=n\/2$ are sufficient to exactly reconstruct the function, see Fig.~\\ref{Fig1DSamples}b). So too are the sample values at the integers $x_n=n$ and at $x_n=n+1\/3$, see Fig.~\\ref{Fig1DSamples}c) and Fig.~\\ref{Fig1DSamples}d). In each of these cases the reconstruction is given by Eq.~\\eqref{SincRecon}.\n\nEverything about this function can be reconstructed from any uniform sample lattice with $a\\leq a^*=1$. In particular, the value of $f_\\text{B}$ two third's of the way between sample point, $f_\\text{B}(2\/3)$, is fixed by $\\{(n,f_\\text{B}(n))\\}_{n\\in\\mathbb{Z}}$ even though we have no sample at or even near $x=2\/3$. The derivative of $f_\\text{B}$ at zero, $f_\\text{B}'(0)$, is fixed by $\\{(n,f_\\text{B}(n))\\}_{n\\in\\mathbb{Z}}$ even though the only sample point we have in this neighborhood is $f_\\text{B}(0)$. Moreover, the derivative at $x=2\/3$, namely $f_\\text{B}'(2\/3)$, is fixed by $\\{(n,f_\\text{B}(n))\\}_{n\\in\\mathbb{Z}}$ even we have no sample points in the neighborhood.\n\nOn first exposure this may be shocking: how can a function's behavior everywhere be fixed by its value at a discrete set of points? When $f_\\text{B}$ is represented discretely, where has all of the information gone? Where is the information about the derivative at $x=2\/3$ stored in the discrete representation? \n\nTo see this, it is convenient (but not necessary) to organize our sample values $f_n=f(x_n)$ into a vector as,\n\\begin{align}\n\\bm{f}&=(\\dots,f_{-1},f_0,f_1,f_2,\\dots)^\\intercal\n= \\sum_{n\\in\\mathbb{Z}} f_{n} \\, \\bm{w}_n\n\\end{align}\nwhere $\\bm{w}_n = (\\dots,0,0,1,0,0,\\dots)^\\intercal\\in\\mathbb{R}^\\mathbb{Z}$ with the 1 in the $n^\\text{th}$ position. \n\nThe values that $f_\\text{B}$ takes at the sample points $x_n$ can be recovered from $\\bm{f}$ as\n\\begin{align}\n\\nonumber\nf_\\text{B}(x_n) \n&= f_n\n= \\bm{w}_n^\\intercal\\bm{f},\n\\end{align}\nNext notice that translating a bandlimited function preserves its bandwidth. As such both $f_\\text{B}(x)$ and $f_\\text{B}(x+\\epsilon a)$ can be represented as Eq.~\\eqref{SincRecon}. Using this fact, we can recover the values that $f_\\text{B}$ takes away from the sample points (i.e., at \\mbox{$x=x_n+\\epsilon\\,a$} for $\\epsilon\\in\\mathbb{R}$) as\n\\begin{align}\\label{TBdef}\nf_\\text{B}(x_n+\\epsilon\\,a) \n&= \\sum_{m=-\\infty}^\\infty S_m(n+\\epsilon) \\ f_m\n=\\bm{w}_n^\\intercal \\, T_\\text{B}^\\epsilon \\, \\bm{f}\n\\end{align}\nwhere the entries of the matrix $T^\\epsilon_\\text{B}$ are $[T^\\epsilon_\\text{B}]_{i,j}=S_i(j+\\epsilon)$. Note $T_\\text{B}^\\epsilon$ acts as the translation operator for this representation of bandlimited functions. If $\\bm{f}$ represents $f_\\text{B}(x)$ then $T^\\epsilon_\\text{B}\\bm{f}$ represents $f_\\text{B}(x+\\epsilon \\,a)$.\n\nFrom this translation operator we can identify the derivative operator for bandlimited functions, $D_\\text{B}$, as \n\\begin{align}\n\\nonumber\nD_\\text{B}\\coloneqq\\lim_{\\epsilon\\to0} \\frac{T^\\epsilon_\\text{B}-\\openone}{\\epsilon}. \\end{align}\nIt should be noted that $D_\\text{B}$ and $T_\\text{B}^\\epsilon$ commute and moreover we have the usual relationship between derivatives and translations, $T_\\text{B}^\\epsilon=\\exp(\\epsilon\\, D_\\text{B})$.\n\nFrom the above definition of $D_\\text{B}$ one can easily work out its matrix entries as \\mbox{$[D_\\text{B}]_{i,j}=(-1)^{i-j}\/(i-j)$} when $i\\neq j$ and 0 when $i=j$. That is, \n\\begin{align}\\label{DBMatrix}\nD_\\text{B}&=\\text{Toeplitz}(\\dots,\\!\\frac{1}{4},\\!\\frac{-1}{3},\\!\\frac{1}{2},\\!-1,\\!0,\\!1,\\!\\frac{-1}{2},\\!\\frac{1}{3},\\!\\frac{-1}{4},\\!\\dots)\n\\end{align}\nNote that $D_\\text{B}$ acts as the derivative operator for this representation of bandlimited functions. If $\\bm{f}$ represents $f_\\text{B}(x)$ then $\\frac{1}{a}D_\\text{B}\\bm{f}$ represents $f_\\text{B}'(x)$.\n\nComparing this with the $D$ operator introduced in Eq.~\\eqref{BigToeplitz} we see that they are numerically identical. Indeed, $D_\\text{B}=D$ and moreover $T_\\text{B}^\\epsilon=T^\\epsilon$. If we were to extend our discussion to two-dimensional functions we could find a discrete representation of the rotation operator for bandlimited functions, $R_\\text{B}^\\theta$. This would come out numerically equal to the $R^\\theta$ operator introduced earlier in Eq.~\\eqref{RthetaDef}, namely $R_\\text{B}^\\theta=R^\\theta$. See Appendix~\\ref{AppA} for further discussion. Thus, the discrete notions of derivative, translation, and rotation that we have been using up until now are intimately connected with bandlimited functions.\n\nIt should be noted that $D=D_\\text{B}$ gives us the following remarkable derivative approximation (which is exact for bandlimited functions):\n\\begin{align}\\label{ExactDerivative}\n\\partial_x f(x)\n&\\approx2\\sum_{m=1}^\\infty (-1)^{m+1} \\frac{f(x+m\\,a)-f(x-m\\,a)}{2\\,m\\,a}.\n\\end{align}\nRelatedly, we have the second derivative approximation (which is exact for bandlimited functions):\n\\begin{align}\\label{ExactDerivative2}\n\\partial_x^2 f(x)\n&\\approx2\\sum_{m=1}^\\infty (-1)^{m+1} \\frac{f(x+m\\,a)+f(x-m\\,a)}{m^2\\,a^2}\\\\\n\\nonumber\n&-\\frac{\\pi^2}{3}f(x).\n\\end{align}\nNamely, when $f$ is bandlimited with bandwidth of $K$ and $a\\leq\\pi\/K$ then these formulas are exact. Moreover, if the Fourier transform of $f$ is mostly supported in $[-K,K]$ with thin tails (e.g, Gaussian tails) outside this region, then these are very good derivative approximations. \n\nUltimately we can compute any derivative of $f_\\text{B}$ anywhere from our sample data as,\n\\begin{align}\n\\nonumber\n\\partial_x^r\\,f_\\text{B}(x_n+\\epsilon\\,a) \n=\\frac{1}{a^r}\\,\\bm{w}_n^\\intercal \\, D_\\text{B}^r T_\\text{B}^\\epsilon \\, \\, \\bm{f}.\n\\end{align}\nThus, we can recover any value or derivative of $f_\\text{B}$ from its values on any sufficiently dense uniform sample lattice.\n\nNote that each $f_\\text{B}$ is representable in this way on a wide number of sample lattices with differing spacings $a<\\pi\/K$ and differing offsets $b$. Translating between these different descriptions of $f_\\text{B}$ is equivalent to a change of basis on the vector of sample values $\\bm{f}$.\n\n\\subsection{Non-Uniform Sample Lattices}\\label{Sec2DSampling}\nThe previous subsection showed how any value or derivative of $f_\\text{B}$ can be recovered from its values on any sufficiently dense uniform sample lattice. Moreover, it showed how changing between representing $f_\\text{B}$ with different uniform sample lattices is ultimately just a change of basis on $\\bm{f}$. \n\nAs I will now discuss, we can also represent $\\bm{f}$ on any sufficiently dense non-uniform lattice. To motivate this, consider first an oversampling of $f_\\text{B}$. For example, figure Fig.~\\ref{Fig1DSamples}b) shows $f_\\text{B}$ sampled at twice the necessary frequency. This is a representation of $\\bm{f}$ in an overcomplete basis. Imagine oversampling by a factor of ten with a spacing of $a=a^*\/10$. Intuitively, this sample lattice has ten times the information needed to recover the function exactly. If we were to delete all but every tenth data point we would still be able to recover the function. But what if we just half of the sample points, but did so randomly? This would result in a non-uniform sample lattice. See for instance Fig.~\\ref{Fig1DSamples}e). Hopefully, the reader has some intuition that at least some non-uniform sample lattices are sufficient to exactly reconstruct $f_\\text{B}$.\n\nThe scope of such non-uniform samplings is established by various non-uniform sampling theorems~\\cite{GARCIA200263,SamplingTutorial}. The details of these theorems are not important here; They can all be summarized as saying that reconstruction is possible when our non-uniform sample points are ``sufficiently dense'' in some technical sense. The sampling shown in Fig. \\ref{Fig1DSamples}e) is sufficiently dense. The reconstruction in the non-uniform case is significantly more complicated than it is in the uniform case. Rather than Eq.~\\eqref{SincRecon}, in the non-uniform case our reconstruction is of the form,\n\\begin{align}\\label{GenRecon}\nf_\\text{B}(x)=\\sum_{m=-\\infty}^\\infty G_m(x;\\{z_n\\}_{n\\in\\mathbb{Z}}) \\, f_\\text{B}(z_m) \n\\end{align}\nfor some reconstruction functions, $G_m$, which depend in a complicated way on the location of all of the other sample points, $\\{z_n\\}_{n\\in\\mathbb{Z}}$.\n\n\n\\subsection{Higher Dimensional Sampling}\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=0.95\\textwidth]{PaperFigures\/Fig2DSamples.pdf}\n\\caption{Several different (but completely equivalent) graphical representations of the bandlimited function given by Eq.~\\eqref{J1}. This function has a bandwidth of $\\sqrt{k_x^2+k_y^2}<K=\\pi$ and so has a critical spacing of $a^*=\\pi\/K=1$ in every direction. The scale of each subfigure is 5x5. In each subfigure, the colored regions are the Voronoi cells around the sample points (black). Subfigure a) shows the function values for all $x$. b) shows $f_\\text{B}$ sampled on a square lattice with $z_{n,m}=(n\/2,m\/2)$. Since $1\/2<a^*=1$ this is an instance of oversampling. c) shows $f_\\text{B}$ sampled on a square lattice with $z_{n,m}=(n,m)$. This is an instance of critical sampling since $a=a^*=1$. d) shows $f_\\text{B}$ sampled on a square lattice with $z_{n,m}=(n+m,n-m)\/\\sqrt{2}$. e) $f_\\text{B}$ sampled on a hexagonal lattice of with \\mbox{$z_{n,m}=(n+m\/2,\\sqrt{3}m\/2)\\in\\mathbb{R}^2$}. f) shows $f_\\text{B}$ sampled on an irregular lattice. [\\textit{Reproduced with permission from} \\cite{DiscreteGenCovPart1}.]}\n\\label{Fig2DSamples}\n\\end{figure*}\n\nThe same story about bandlimited functions is largely true in higher dimensions as well. A two-dimensional function $f_\\text{B}(x,y)$ is bandlimited if is Fourier transform $\\mathcal{F}[f_\\text{B}(x,y)](k_x,k_y)$ is compactly supported in the \\mbox{($k_x$, $k_y$)-plane}. Specifying the value of the bandwidth is less straightforward in the high dimensional case as the Fourier transform's support may have different extents in different directions. However, any compact region can be bounded in a square. We can thus always imagine $f_\\text{B}(x,y)$ as being bandlimited with \\mbox{$k_x,k_y\\in[-K,K]$} for some $K>0$. As such, we can represent $f_\\text{B}(x,y)$ with a (sufficiently dense) uniform sample lattice in both the $x$ and $y$ directions. That is, we can represent any bandlimited $f_\\text{B}(x,y)$ in terms of its sample values on a sufficiently dense square lattice. Once we have such a uniform sampling, the reasoning carried out above applies unchanged. We can represent $f_\\text{B}(x,y)$ on any sufficiently dense non-uniform lattice.\n\nFor a concrete example consider the bandlimited function shown shown in Fig.~\\ref{Fig2DSamples}a), namely,\n\\begin{align}\\label{J1}\nf_\\text{B}(x,y)=J_1(\\pi\\,r)\/(\\pi\\,r) \n\\end{align}\nwhere $J_1$ is the first Bessel function and $r=\\sqrt{x^2+y^2}$. This function is bandlimited with $\\sqrt{k_x^2+k_y^2}<K=\\pi$ and hence critical spacing $a^*=\\pi\/K=1$. Moreover, note that this function is rotation invariant.\n\nGiven this function's bandwidth of $K=\\pi$, we can represent it via its sample values taken on a square lattice with spacing $a=1\/2\\leq a^*=1$, see Fig. \\ref{Fig2DSamples}b). We can also use a coarser square lattice with a spacing of $a=a^*=1$, see Fig. \\ref{Fig2DSamples}c). We could also use a rotated square lattice, see Fig. \\ref{Fig2DSamples}d). Sampling the function on a hexagonal lattice also works, see Fig. \\ref{Fig2DSamples}e). Finally we can use a non-uniform lattice of sample points, see Fig. \\ref{Fig2DSamples}f). From each of these discrete representations, we could recover the original bandlimited function everywhere exactly via some generalization of Eq.~\\eqref{SincRecon} in the uniform cases and Eq.~\\eqref{GenRecon} in the irregular case.\n\nThus, there is no conceptual barrier to representing a rotationally invariant bandlimited function on a square lattice. Indeed, there is no issue with representing such a function on any sufficiently dense lattice. In light of the analogy proposed in this paper, we can see this as analogous to the unsurprising fact that there is no conceptual barrier to representing rotationally invariant functions in Cartesian coordinates. There is no requirement that our representation (be it a choice of coordinates or a choice of sample points) latches onto the symmetries of what is being represented. \n\nThus we have a non-uniform sampling theory for higher dimensions. But what about a sampling theory on curved spaces? While such things are not relevant for the aims of this paper, recently notable progress has been made on developing a sampling theory for curved manifolds~\\cite{CurvedSampling,Martin2008}.\n\n\\section{Externalizing these theories - Part 2}\\label{SecExtPart2}\nHaving reviewed the sampling property of bandlimited functions, let's return to the task of embedding $\\bm{\\Phi}$ into our spacetime manifold $\\mathcal{M}$. As discussed in Sec.~\\ref{SecExtPart1}, for KG4-KG7 our translation-matching constraint forces the new field \\mbox{$\\phi:\\mathcal{M}\\to\\mathbb{R}$} to be bandlimited in the coordinate system $d_\\text{trans.}$ with bandwidth $\\omega,k_1,k_2\\in[-K,K]$ where $K=\\pi\/a$. Similarly for KG1-KG3. In either case, let's note this by giving the field a subscript $\\text{B}$ as \\mbox{$\\phi_\\text{B}:\\mathcal{M}\\to\\mathbb{R}$}. \n\nBefore continuing on, it should be noted that, the above discussed translation-matching constraint guarantees that $\\phi_\\text{B}\\circ d_\\text{trans.}$ will have the sampling property discussed in the previous section.\n\n\\subsection{A Principled Choice of Profile}\nWhat freedom remains for our choice of embedding? Focusing on KG4-KG7, in coordinate $d_\\text{trans.}$ our remaining freedom is a choice of profile $f(t,x,y)\\in F^\\pi$ appearing in Eq.~\\eqref{Embedf}. A translated and rescaled copy of this profile is associated with each basis vector $\\bm{e}_j\\otimes\\bm{e}_n\\otimes\\bm{e}_m$. As noted following Eq.~\\eqref{Eftilde}, up to a local recaling of Fourier space (which is a symmetry of the dynamics) our embedding is fixed. However, in order to completely fix this embedding, we need to pick a profile $f(t,x,y)$.\n\nThe discussion in the previous section offers us a promising candidate for this profile namely \\mbox{$f(t,x,y)=S_0(t)S_0(x)S_0(y)$}. This profile is the unique one in $F^\\pi$ with the following property\\footnote{To give the technical details, this $f$ is the unique function in $F^\\pi$ which evaluates to zero at all integer arguments except $(0,0,0)$ where it returns $1$. That is, \\mbox{$f(j,n,m)=\\delta_{j0}\\delta_{n0}\\delta_{m0}$} for $j,n,m\\in\\mathbb{Z}$. This uniqueness is a direct consequence of the Nyquist-Shannon sampling theorem.}: it makes $\\phi_\\text{B}(t,x,y)$ evaluated at $z_{j,n,m}=(j\\,a,n\\,a,m\\,a)$ take the value of $\\phi_{j,n,m}$. That is, it makes our original discrete variables $\\phi_{j,n,m}$ sample values of $\\phi_\\text{B}$ at the sample points $z_{j,n,m}$. Let us call this choice of profiles the lattice-as-sample-points constraint.\n\nTo be clear, what hinges on our choice of profile $f$ is not whether or not $\\phi_\\text{B}$ has the sampling property discussed in the previous section; This property is guaranteed independent of our choice of $f$. Rather, what is at stake is just whether our original discrete variables $\\phi_{j,n,m}$ \\mbox{\\textit{themselves}} are sample values.\n\nIt follows from the lattice-as-sample-points constraint that $E$ acts on the planewave basis as\n\\begin{align}\\label{Embed0}\nE:\\bm{\\Phi}(\\omega,k_1,k_2)\\mapsto\\phi(t,x,y;\\omega\/a,k_1\/a,k_2\/a)\n\\end{align}\nThat is, discrete planewaves are mapped onto continuous planewaves (rescaled by $a$). Moreover, this constraint forces $E$ to act on the basis $\\bm{e}_j\\otimes\\bm{e}_n\\otimes\\bm{e}_m$ as\n\\begin{align}\\label{Embed}\nE: \\bm{e}_j\\otimes\\bm{e}_n\\otimes\\bm{e}_m\\mapsto\nS_{j}(t\/a) \\, S_{n}(x\/a) \\, S_{m}(y\/a).\n\\end{align}\nIn particular, applying this to  Eq.~\\eqref{PhiVec2} gives us\n\\begin{align}\\label{PhiSincRecon1}\n&\\text{KG1-KG3:}\\\\\n\\nonumber\n&\\quad\\phi_\\text{B}(t,x)\n=\\sum_{j,n\\in\\mathbb{Z}} S_{j}(t\/a)\\,S_{n}(x\/a) \\ \\phi_{j,n}\\\\\n\\label{PhiSincRecon2}\n&\\text{KG4-KG7:}\\\\\n\\nonumber\n&\\quad\\phi_\\text{B}(t,x,y)\n=\\sum_{j,n,m\\in\\mathbb{Z}} S_{j}(t\/a) \\, S_{n}(x\/a) \\, S_{m}(x\/a) \\, \\phi_{j,n,m}.\n\\end{align}\nwhere analogous reasoning applies for KG1-KG3.\n\nThus from these translation-matching and lattice-as-sample-points constraints it follows that our new field must be the result of a certain bandlimited reconstruction. Namely, for KG4-KG7 it must come from a reconstruction using sample values $\\phi_{j,n,m}$ at sample point $z_{j,n,m}$. Similarly for KG1-KG3 it must come from using sample values $\\phi_{j,n}$ at sample points $z_{j,n}=(j\\,a,n\\,a)$.\n\n\\begin{figure*}[t]\n\\includegraphics[width=0.95\\textwidth]{PaperFigures\/RowABC.pdf}\n\\includegraphics[width=0.95\\textwidth]{PaperFigures\/RowDEF.pdf}\n\\caption{Subfigure b) shows a solution to KG3 formulated as a bandlimited function in the coordinates given by $d_\\text{trans.}$.\nThis field in these coordinates obeys Eq.~\\eqref{DKG3bandlimited}. Subfigure a) shows the values that this function takes on a square lattice in spacetime.  These discrete field values obey Eq.~\\eqref{DKG3}. From this embedding we can uniquely reconstruct the bandlimited field $\\phi_\\text{B}$. However, these sample points play no special role in the theory. We are free to redescribe the state and its dynamics by resampling it on any sufficiently dense collection of sample points. For instance, we might describe $\\phi_\\text{B}$ using the boosted sample points shown in subfigure c). Not all states will be describable on both of these lattices, but some are. If we boost both the sample values and the field as shown in subfigure d) e) and f) nothing changes. Both sets of sample values still obey Eq.~\\eqref{DKG3}. Moreover, in the bandlimited field still obeys Eq.~\\eqref{DKG3bandlimited}.}\\label{FigKleinSpaceTime}\n\\end{figure*}\n\nFig.~\\ref{FigKleinSpaceTime}a) shows for KG1-KG3 these sample points \\mbox{$z_{j,n}=(j\\,a,n\\,a)$} and sample values as they lie on the spacetime manifold $\\mathcal{M}\\cong\\mathbb{R}^2$ in the coordinates $d_\\text{trans.}$ Fig.~\\ref{FigKleinSpaceTime}b) shows for KG3 what the reconstructed bandlimited function $\\phi_\\text{B}:\\mathcal{M}\\to\\mathbb{R}$ might look like in these coordinates. One can imagine an analogous figure for KG4-KG7 with the sample points forming a cubic lattice in the coordinates $d_\\text{trans.}$.\n\nOne may worry that we are here taking a cubic 3D lattice for each of KG4-KG7 whereas for KG5 and KG7 we naturally ought to embed onto a hexagonal 3D lattice. This was after all, the lattice structure picked out by our symmetry analysis in the first interpretation. However, one need not worry for two reasons: Firstly, there is no real sense in which the sample points $z_{j,n}$ and $z_{j,n,m}$ form a square and cubic lattice respectively. These sample lattices only appear square and cubic in this particular coordinate system; in other coordinate systems they won't, see Fig.~\\ref{FigKleinSpaceTime}d). Indeed, the coordinates we are using here may lie on the spacetime manifold $\\mathcal{M}$ in a curvilinear way. As such one ought not to think of these sample points as being arranged in a square or cubic lattice on the spacetime manifold. Indeed, at this point the manifold still does not have a metric.\n\nSecondly, after we have used the sample points $z_{j,n}$ and $z_{j,n,m}$ to build $\\phi_\\text{B}$ they no longer play any special role in the theory. As discussed in the previous section, we are always free to resample a bandlimited function on any sufficiently dense sample lattice. Recall Fig.~\\ref{Fig2DSamples}. Thus even staying in this coordinate system (where our initial sample points do form a cubic 3D lattice) we are free to resample $\\phi_\\text{B}$ on a hexagonal 3D lattice. Moreover, we also might be able to resample $\\phi_\\text{B}$ on a boosted lattice as shown in Fig.~\\ref{FigKleinSpaceTime}c).\n\nThus, in a sense the lattice sites $L$ from our first interpretation still exist here they are embedded onto our new spacetime manifold, $\\ell\\in L\\mapsto z_\\ell\\in\\mathcal{M}$. However, once embedded, these lattice sites no longer play any special role in the theory; We can always resample.\n\n\\vspace{0.25cm}\n\nTo review: In Sec.~\\ref{SecExtPart1} we saw how the translation-matching constraint forced us to embed $\\bm{\\Phi}$ onto $\\mathcal{M}$ as a some bandlimited function, $\\phi_\\text{B}:\\mathcal{M}\\to\\mathbb{R}$ (i.e., bandlimited in coordinates $d_\\text{trans.}$ with $\\omega,k_1,k_2\\in[-K,K]$). In this section, we further demanded the lattice-as-sample-points constraint. This allowed us to think of our initial discrete variables $\\phi_\\ell$ as being the values that $\\phi_\\text{B}$ takes at some sample points $z_\\ell\\in\\mathcal{M}$. Taken together these two constraints completely fix how $\\phi_\\text{B}:\\mathcal{M}\\to\\mathbb{R}$ is built from $\\bm{\\Phi}$ (or equivalently $\\phi_\\ell$). In particular, in the global coordinate system given by $d_\\text{trans.}$ we must have Eq.~\\eqref{PhiSincRecon1} and Eq.~\\eqref{PhiSincRecon2}.\n\nIn the next subsection, I will rewrite the dynamics of each of KG1-KG7 in terms of this new field $\\phi_\\text{B}$ in the coordinates $d_\\text{trans.}$. Following this, in Secs.~\\ref{SecKlein3} and \\ref{SecKlein3Extra}, I will begin interpreting this theory independently of how they were constructed. In Sec.~\\ref{SecFullGenCov} I will give a coordinate-independent formulation of these theories.\n\n\\subsection{Bandlimited Dynamics}\\label{SecKlein3B}\nThe previous subsections has given us a principled way to embed $\\phi_\\ell$ onto  our new spacetime manifold $\\mathcal{M}$ as a bandlimited function. Namely, Eq.~\\eqref{PhiSincRecon1} and Eq.~\\eqref{PhiSincRecon2}. Equivalently we can think of embedding $\\bm{\\Phi}$ onto our new spacetime manifold $\\mathcal{M}$ as a bandlimited function. Namely, Eq.~\\eqref{Embed}. Either of these perspectives allow us to translate the kinematics of KG1-KG7 into this new continuous setting. This subsection will additionally translate over the dynamics into this new continuous setting.\n\nLet's begin with KG1. This translation is aided by the fact that the derivative is the generator of translations, i.e.,  $h(x+a)=\\text{exp}(a\\, \\partial_x) h(x)$. Mapping the left hand side of the dynamics of KG1 (i.e., Eq.~\\eqref{KG1Long}) onto a bandlimited function we have\n\\begin{align}\n&\\sum_{j,n\\in\\mathbb{Z}} S_{j}(t\/a)\\, S_{n}(x\/a)\\, \\left(\\text{L.H.S. of Eq.~\\eqref{KG1Long}}\\right)\\\\\n\\nonumber\n&=\\sum_{j,n\\in\\mathbb{Z}} S_{j}(t\/a)\\,S_{n}(x\/a) \\big[\\phi_{j+1,n}-2\\phi_{j,n}+\\phi_{j-1,n}\\big]\\\\\n\\nonumber\n&=\\phi_\\text{B}(t - a,x) - 2\\phi_\\text{B}(t,x) + \\phi_\\text{B}(t + a,x)\\\\\n\\nonumber\n&=[\\exp(-a\\,\\partial_t)-2+\\exp(a\\,\\partial_t)] \\phi_\\text{B}(t,x)\\\\\n\\nonumber\n&=[2\\cosh(a\\,\\partial_t)-2] \\ \\phi_\\text{B}(t,x).\n\\end{align}\nSimilarly, for the right hand side of Eq.~\\eqref{KG1Long} we have\n\\begin{align}\n&\\sum_{j,n\\in\\mathbb{Z}} S_{j}(t\/a)\\, S_{n}(x\/a)\\, \\left(\\text{R.H.S. of Eq.~\\eqref{KG1Long}}\\right)\\\\\n\\nonumber\n&=[\\mu^2+2\\cosh(a\\,\\partial_t)-2] \\ \\phi_\\text{B}(t,x).\n\\end{align}\nThus, when the sample values $\\phi_{j,n}=\\phi_\\text{B}(z_{j,n})$ obey Eq.~\\eqref{KG1Long}, the bandlimited field obeys,\n\\begin{align}\\label{DKG1bandlimited}\n\\text{KG1:}\\,[2\\cosh(a\\partial_t)\\!-\\!2] \\phi_\\text{B}\\!=\\![\\mu^2+2\\cosh(a\\partial_t)\\!-\\!2] \\phi_\\text{B}\n\\end{align}\nSimilarly for the other six theories we have:\n\\begin{align}\\label{DKG2bandlimited}\n\\text{KG2:}&\\ \\frac{1}{6}[-\\cosh(2a\\partial_t)\\!+\\!16\\cosh(a\\partial_t)\\!-\\!15]\\phi_\\text{B}\\\\\n\\nonumber\n&=\\left(\\mu^2+\\frac{1}{6} [-\\cosh(2a\\partial_x)\\!+\\!16\\cosh(a\\partial_x)\\!-\\!15]\\right)\\phi_\\text{B}\\\\\n\\label{DKG3bandlimited}\n\\text{KG3:}&\\ \\partial_t^2\\phi_\\text{B}\n=(\\mu_0^2+\\partial_x^2) \\, \\phi_\\text{B}\\\\\n\\label{DKG4bandlimited}\n\\text{KG4:}&\\  [2\\cosh(a\\partial_t)\\!-\\!2] \\phi_\\text{B}\\\\\n\\nonumber\n&=[\\mu^2+2\\cosh(a\\,\\partial_x)+2\\cosh(a\\,\\partial_y)\\!-\\!4]\\phi_\\text{B}\\\\\n\\label{DKG5bandlimited}\n\\text{KG5:}&\\ [2\\cosh(a\\,\\partial_t)\\!-\\!2] \\phi_\\text{B}=\\Big[\\mu^2+\\frac{4}{3}\\cosh(a\\,\\partial_x)\\\\\n\\nonumber\n&\\qquad\\quad+\\frac{4}{3}\\cosh(a\\,\\partial_y) +\\frac{4}{3}\\cosh(a\\,(\\partial_y-\\partial_x))\\!-\\!4\\Big]\n\\phi_\\text{B},\\\\\n\\label{DKG6bandlimited}\n\\text{KG6:}&\\  \n\\partial_t^2\\phi_\\text{B}\n=(\\mu_0^2+\\partial_x^2+\\partial_y^2) \\, \\phi_\\text{B}\\\\\n\\label{DKG7bandlimited}\n\\text{KG7:}&\\  \n\\partial_t^2\\phi_\\text{B}\n\\!=\\!\\Big(\\mu_0^2\\!+\\!\\frac{2}{3}\\big[\\partial_x^2\\!+\\!\\partial_y^2\\!+\\!(\\partial_x\\!-\\!\\partial_y)^2\\big]\\!\\Big) \\phi_\\text{B}\n\\end{align}\nwhere $\\mu_0\\coloneqq\\mu\/a$. \n\nNote that the dynamics of KG3 and KG6 are exactly the same as those of the continuum Klein Gordon equations KG00 and KG0 with a field mass \\mbox{$M=\\mu_0$}. Moreover, after a coordinate transformation, \n\\begin{align}\\label{CoorKG7KG6}\nt\\mapsto t,\\quad\nx\\mapsto x + \\frac{1}{2} y,\\quad\ny\\mapsto \\frac{\\sqrt{3}}{2}y\n\\end{align}\nthe dynamics of KG7 exactly maps onto the dynamics of KG6 (and consequently, KG0). Note that this change of coordinates is equivalent to Eq.~\\eqref{SkewKG7KG6} applied in continuum Fourier space. Moreover, note that Eq.~\\eqref{CoorKG7KG6} maps a square lattice to a hexagonal one, see Fig.~\\ref{FigSkew}. As I will discuss later, this means that KG6 and KG7 can be seen as describing the same bandlimited theory, just using different sample points.\n\n\\begin{figure}[t]\n\\includegraphics[width=0.4\\textwidth]{PaperFigures\/FigSkew.pdf}\n\\caption{As this figure shows, a certain linear transformation of coordinates (namely Eq.~\\eqref{CoorKG7KG6}) maps a square lattice to a hexagonal one. [\\textit{Reproduced with permission from} \\cite{DiscreteGenCovPart1}.]}\\label{FigSkew}\n\\end{figure}\n\nApplying this coordinate transformation to KG5's bandlimited dynamics gives us:\n\\begin{align}\\label{DKG5bandlimitedSkew}\n\\text{KG5:}\\ &[2\\cosh(a\\partial_t)\\!-\\!2] \\phi_\\text{B}=\\Big[\\mu^2-4+\\frac{4}{3}\\cosh(a\\,\\partial_x)\\\\\n\\nonumber\n&\\qquad\\qquad\\qquad+\\frac{4}{3}\\cosh(a(\\sqrt{3} \\partial_y+\\partial_x)\/2)\\\\\n\\nonumber\n&\\qquad\\qquad\\qquad+\\frac{4}{3}\\cosh(a(\\sqrt{3} \\partial_y-\\partial_x)\/2)\\Big]\n\\phi_\\text{B}.\n\\end{align}\nNote that this dynamics now manifestly has a one-sixth rotational symmetry. Sampling the above dynamics on a hexagonal lattice gives us our original dynamics for KG5. Sampling the above dynamics on a square lattice gives us the following. Given Eq.~\\eqref{ExactDerivative} sampling on a square lattice effectively means taking $a\\,\\partial_t\\to D_\\text{j}$, $a\\,\\partial_x\\to D_\\text{n}$ and $a\\,\\partial_y\\to D_\\text{m}$ and replacing $\\phi_\\text{B}$ with $\\bm{\\Phi}$. For the above dynamics of KG5 this gives:\n\\begin{align}\\label{DKG5Skew}\n\\text{KG5:}\\ \\Delta_{(1),\\text{j}}^2\\,\\bm{\\Phi}&=\\Big[\\mu^2-4+\\frac{4}{3}\\cosh(D_\\text{n})\\\\\n\\nonumber\n&+\\frac{4}{3}\\cosh((\\sqrt{3} D_\\text{m}+D_\\text{n})\/2)\\\\\n\\nonumber\n&+\\frac{4}{3}\\cosh((\\sqrt{3} D_\\text{m}-D_\\text{n})\/2)\\Big]\n\\phi_\\text{B}.\n\\end{align}\nNote that before this resampling KG5 was local in the intuitive sense nearest-neighbor couplings only, see Eq.~\\eqref{KG5Long}. After resampling it is infinite range. Thus, the intuitive notion of locality in terms of nearest neighbors is unstable under resampling. For this and other reasons (see \\cite{DiscreteGenCovPart1}) the notion of locality arising in the bandlimited formulation is preferable. Sample values can obey what appears to be non-local dynamics, because the sample values themselves are non-local objects: they ought to be associated with sinc profiles not spacetime points, see \\cite{DiscreteGenCovPart1}.\n\nWe can easily solve each of these dynamical equations. Just as in Sec. \\ref{SecSevenKG} these dynamics admit a complete basis of planewave solutions, given by Eq.~\\eqref{PlaneWaveCont}. A few key differences should be noted between these continuum planewaves and the discrete planewaves considered earlier, namely $\\bm{\\Phi}(\\omega,k)$ and $\\bm{\\Phi}(\\omega,k_1,k_2)$. Firstly, there is a difference of scale: note how our embedding Eq.~\\eqref{Embed0} rescales the planewaves by a factor of $1\/a$. Secondly, the discrete planewaves $\\bm{\\Phi}(\\omega,k)$ and $\\bm{\\Phi}(\\omega,k_1,k_2)$ repeated themselves cyclically with period $2\\pi$ outside of the regions \\mbox{$\\omega,k\\in[-\\pi,\\pi]$} and \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$}. The continuum planewaves do not do this. Now each planewave with $\\omega,k_1,k_2\\in\\mathbb{R}$ is distinct. Here, however, by its construction $\\phi_\\text{B}$ only has support over planewaves with \\mbox{$\\omega,k_1,k_2\\in[-K,K]$} with $K=\\pi\/a$.\n\nEach of these planewaves are solutions only when they obey the following dispersion relations: \n\\begin{align}\n\\text{KG1:}& \\quad \\!\\! 2-2\\,\\cos(\\omega\\,a)= \\mu^2+2-2\\,\\cos(k\\,a)\\\\\n\\text{KG2:}& \\quad \\!\\! \\frac{1}{6}\\,(\\cos(2\\,\\omega\\,a)-16\\,\\cos(\\omega\\,a)+15)\\\\\n\\nonumber\n&= \\mu^2+\\frac{1}{6}\\,(\\cos(2\\,k\\,a)-16\\,\\cos(k\\,a)+15)\\\\\n\\text{KG3:}& \\quad \\!\\! \\omega^2= \\mu_0^2+k^2\\\\\n\\text{KG4:}& \\ 2\\!-\\!2\\cos(\\omega a)\\!=\\! \\mu^2\\!+\\!4\\!-\\!2\\cos(k_1 a)\\!-\\!2\\cos(k_2 a)\\\\\n\\text{KG5:}& \\  2-2\\,\\cos(\\omega\\,a)= \\mu^2 \\\\\n\\nonumber\n&+\\frac{4}{3}\\big[3-\\cos(k_1\\,a)-\\cos(k_1\\,a)-\\cos(k_2\\,a-k_1\\,a)\\big]\\\\\n\\text{KG6:}& \\ \\omega^2 = \\mu_0^2 +k_1^2 +k_2^2\\\\\n\\text{KG7:}& \\ \\omega^2= \\mu_0^2 +\\frac{2}{3}\\big[k_1^2 +k_2^2 +(k_2-k_1)^2\\big].\n\\end{align}\nNote that the dispersion relation of KG3 and KG6 are exactly the same as those of the continuum Klein Gordon equations KG00 and KG0 with a field mass $M=\\mu_0$. Moreover, after the coordinate transformation, Eq.~\\eqref{SkewKG6KG7}, \nthe dispersion relation of KG7 exactly maps onto that of KG6 (and consequently, KG0). More will be said about this later.\n\nWe are now ready to make a third attempt at interpreting these theories. \n\n\\section{A Third Attempt at Interpreting KG1-KG7 - Part 1}\\label{SecKlein3}\nThe previous three sections have (after much effort) given us a new formulation of KG1-KG7 to interpret. In particular, this new formulation has aimed to externalize many of the internal symmetries which we discovered in Sec.~\\ref{SecKlein2}. \n\nOn this third interpretation I will be taking the formulations of KG1-KG7 in terms of $\\phi_\\text{B}$ seriously: namely Eq.~\\eqref{DKG1bandlimited}-\\eqref{DKG7bandlimited}. Taken literally as written, what are these theories about? \nIntuitively these theories are about a field $\\phi_\\text{B}$ which maps points on a manifold ($p\\in\\mathcal{M}$) into field amplitudes ($\\phi_\\text{B}(p)\\in\\mathbb{R}$). That is a field $\\phi_\\text{B}:\\mathcal{M}\\to \\mathcal{V}$ with a manifold $\\mathcal{M}$ and value space $\\mathcal{V}=\\mathbb{R}$. Thus, taking $\\phi_\\text{B}:\\mathcal{M}\\to \\mathcal{V}$ seriously as a fundamental field leads us to thinking of $\\mathcal{M}$ as the theory's underlying manifold and $\\mathcal{V}=\\mathbb{R}$ as the theory's value space. Indeed, on this interpretation\nKG1-KG7 are continuum spacetime theories of the sort we\nare used to interpreting (albeit ones which consider only bandlimited fields). \n\nNote that here we have $\\mathcal{M}\\cong\\mathbb{R}^d$ with $d=2$ for KG1-KG3 and $d=3$ for KG4-KG7. As such, in either case we have access to global coordinate systems. The fields $\\phi_\\text{B}$ considered by this interpretation are bandlimited in the following sense. There exists a fixed special diffeomorphism $d_\\text{coor.}:\\mathbb{R}^d\\to\\mathcal{M}$ which giving us a fixed special global coordinate system for $\\mathcal{M}$. In these special coordinates the field is bandlimited with $\\omega,k_1,k_2\\in[-K,K]$ where $K=\\pi\/a$. That is $\\phi_\\text{B}\\circ d_\\text{trans.}\\in F^K$. It is in these special coordinates that the fields $\\phi_\\text{B}$ obey the dynamical equations Eq.~\\eqref{DKG1bandlimited}-Eq.~\\eqref{DKG7bandlimited}.\n\nIn our construction of $\\phi_\\text{B}$, this special coordinate system is the one which is associated to the translation operations $T^\\epsilon$ by our translation-matching condition, namely $d_\\text{coor.}=d_\\text{trans.}$. Here however, we are trying to interpret KG1-KG7 as formulated above independent of how we arrived here. Thus, presently, this special coordinate system $d_\\text{coor.}$ is just something promised to us by God. Of course, we know from the lessons of general covariance to be skeptical of ``special coordinate systems''. Coordinate systems are representational artifacts, reflecting no physics. We can and should reformulate our theories in a generally covariant way so as to avoid them. A coordinate independent view of them will be given in Sec.~\\ref{SecFullGenCov}.\n\nAs I will discuss, taking $\\mathcal{M}$ to be these theories' underlying manifold has substantial consequences for these theories' locality properties and symmetries. To preview: this third interpretation either fixes all of our issues with the first and second interpretations. \n\nIn our first interpretation there was some tension between our theories' differences in locality and their differences in convergence rate in the continuum limit. The second interpretation addressed this tension in a hamfistedly way: denying that there are differences in locality in the first place. This move also had the unfortunate consequence of undercutting our ability to use the lattice sites to reason about locality. This interpretation improves on this by instead bringing the locality of these theories into harmony with their convergence rates in the continuum limit. Not much changes between the discrete heat equations considered in \\cite{DiscreteGenCovPart1} and the discrete Klein Gordon equations considered here. As such, I will leave discussion of this issue to \\cite{DiscreteGenCovPart1} and direct the interested reader there.\n\nIn our first interpretation, KG6 and KG7 were seen as distinct theories with radically different symmetries despite there being a nice one-to-one correspondence between their solutions. In our second interpretation, KG6 and KG7 were more satisfyingly seen to be completely equivalent, differing only by a change of basis in the value space. Here too KG6 and KG7 will turn out to be completely equivalent differing only by a change of coordinates. Moreover, as I will discuss, we can here see KG6 and KG7 as parts of a larger unified theory viewed in two ways with differently limited representational capacities.\n\nOur first interpretation outright denied the possibility that KG1-KG7 could have continuous symmetries (e.g., translations, rotations, Lorentz boosts). Our second interpretation then fixed this oversight by revealing KG1-KG7's hidden continuous translation and rotation symmetries and even their (limited) Lorentzian boost symmetries. However, these were unintuitively categorized as internal symmetries. This interpretation maintains all of these hidden symmetries, but more satisfyingly categorizes them as external symmetries. Moreover, seeing KG6 and KG7 as parts of a larger unified theory will guide us towards a perfectly Lorentzian lattice theory.\n\nAs I discussed in Sec.~\\ref{SecKlein2}, the improvements that the second interpretation made over the first one all centered around the following realization. The lattice structures appearing in KG1-KG7 are merely representational artifacts and as such should not be taken seriously as a substantive part of the theory. As I will discuss, this realization is deepened and clarified on the third interpretation.\n\n\\subsection{Bandlimited Symmetries}\nHow does this externalization move affect our theory's capacity for symmetry? Now that we have a continuous spacetime manifold $\\mathcal{M}$ underlying these theories, we have a greatly improved capacity for external symmetries, namely $\\text{Auto}(\\mathcal{M})=\\text{Diff}(\\mathcal{M})$. By construction all of the symmetries which we wished to externalize $G^\\text{dym}_\\text{to-be-ext}$ fit inside of $\\text{Diff}(\\mathcal{M})$. \n\nIn addition to these external symmetries, we also have some capacity for internal symmetries, namely $\\text{Auto}(\\mathcal{V})$. Note that the space of (potentially off-shell) bandlimited fields are themselves vectors $\\phi_\\text{B}(t,x,y)\\in F^K$. Namely, they are closed under addition and scalar multiplication and hence a vector space. This addition and scalar multiplication is carried out in parallel at each spacetime point. Thus, the field's value space $\\mathcal{V}=\\mathbb{R}$ is also structured like a vector space. Therefore, $\\text{Auto}(\\mathcal{V})=\\text{Aff}(\\mathbb{R})$ such that our internal symmetries are linear-affine rescalings of $\\phi_\\text{B}$. We might also find gauge symmetries by allowing these to vary smoothly across the manifold. Thus, in total the possibly symmetries for our theories under this interpretation are,\n\\begin{align}\\label{BandlimitedSymmetries}\ns:\\quad\\phi_\\text{B}\\mapsto C \\, \\phi_\\text{B}\\circ d +c\n\\end{align}\nwhere $C:\\mathcal{M}\\to\\mathbb{R}$ and $c:\\mathcal{M}\\to\\mathbb{R}$ are some real scalar functions and $d:\\mathcal{M}\\to\\mathcal{M}$ is some  diffeomorphism \\mbox{$d\\in\\text{Diff}(\\mathcal{M})$}. Note that \n\nAre these more or less possible symmetries than we had on our second interpretation? $\\text{Diff}(\\mathcal{M})$ is a tremendously large group and is in fact bigger than $\\text{Aff}(\\mathbb{R}^L)$. One might expect that we have strictly more possible symmetries here. However, looks can be deceiving. \n\nWe can translate our old potential symmetries into the new continuum setting as follows. Note that because $\\mathbb{R}^L \\cong F^K$ we have $\\text{Aff}(\\mathbb{R}^L)\\cong \\text{Aff}(F^K)$. Using $d_\\text{coor.}$ we can think of $\\text{Aff}(F^K)$ as a subgroup of $\\text{Aff}(F(\\mathcal{M}))$. Note that we can also see the above discussed transformations (i.e., \\mbox{$\\text{Aff}(\\mathbb{R})$} varying smoothly over \\mbox{$\\text{Diff}(\\mathcal{M})$}) as a subgroup of $\\text{Aff}(F(\\mathcal{M}))$. \n\nViewed this way, neither is strictly larger. Indeed, $\\text{Aff}(F^K)$ includes transformations in Fourier space which are inaccessible from $\\text{Aff}(\\mathbb{R})$ and $\\text{Diff}(\\mathcal{M})$. In particular, the local Fourier rescaling symmetry from Sec.~\\ref{SecKlein2} is no longer available to us. On the other hand, our new symmetries contain transformations which are not closed over $F^K$. However, this comparison seems unfair, what if we restrict our attention transformations closed over $F^K$? With this restriction, our new range of possible symmetries is a strict subset of $\\text{Aff}(F^K)$. \n\nThus, in moving from the second to the third interpretation, we have actually lost some capacity for symmetry. The source of this reduction in capacity is us imposing structure on the theory by our choice of manifold and value space.\n\nAs I will now discuss, besides the local Fourier rescaling symmetry, all of our previous symmetries are maintained on this third interpretation. Moreover, these remaining symmetries are reformulated and recategorized. As I will now show, on this interpretation the continuous translation, rotation and (limited) Lorentz boost symmetries identified earlier are now honest-to-goodness manifold symmetries, represented by diffeomorphisms $d\\in\\text{Diff}(\\mathcal{M})$.\n\n\\subsubsection*{Symmetries of KG1-KG7: Third Attempt}\nWhich transformations of the form Eq.~\\eqref{BandlimitedSymmetries} are symmetries for KG1-KG7? A technical investigation of the symmetries of KG1-KG7 on this interpretation is carried out in Appendix~\\ref{AppA}, but the results are the following. For KG1 and KG2 the dynamical symmetries of the form Eq.~\\eqref{BandlimitedSymmetries} are:\n\\begin{flushleft}\n\\begin{enumerate}\n    \\item[1)]  continuous translation which maps \\mbox{$\\phi_\\text{B}(t,x)\\mapsto \\phi_\\text{B}(t-t_0,x-x_0)$} for $t_0,x_0\\in\\mathbb{R}$,\n    \\item[2)] two negation symmetries which map \\mbox{$\\phi_\\text{B}(t,x)\\mapsto \\phi_\\text{B}(-t,x)$} and \\mbox{$\\phi_\\text{B}(t,x)\\mapsto \\phi_\\text{B}(t,-x)$} respectively,\n    \\item[3)] global linear rescaling which maps \\mbox{$\\phi_\\text{B}(t,x)\\mapsto c_1\\,\\phi_\\text{B}(t,x)$} for some $c_1\\in\\mathbb{R}$,\n    \\item[4)] local affine rescaling which maps \\mbox{$\\phi_\\text{B}(t,x)\\mapsto c_1\\,\\phi_\\text{B}(t,x)+c_2(t,x)$} for some $c_2(t,x)$ which is also a solution of the dynamics.\n\\end{enumerate}\n\\end{flushleft}\nThese are exactly the same symmetries that we found on the previous interpretation (sans local Fourier rescaling) just recategorized: the translation and reflection symmetries are here external symmetries. Translation here is generated by \\mbox{$\\exp(-t_0\\partial_t-x_0\\partial_x)$} for some $t_0,x_0\\in\\mathbb{R}$. Notice that translating a bandlimited function by any amount preserves its bandwidth. Thus, $\\phi_\\text{B}(t,x)$ remains in $F^K$ upon translation.\n\nThus, the first big lesson from Sec.~\\ref{SecKlein2} is repeated here: despite the fact that our discrete theories KG1 and KG2 can be represented on a lattice, they nonetheless have a continuous translation symmetry. This continuous translation symmetry was hidden on our first interpretation because we there took the lattice to be a fundamental part of the manifold. Here, we do not take the lattice structure so seriously. We have embedded it onto the manifold where it then disappears from view as just one of many possible sample lattices.\n\nAllow me to skip over KG3 temporarily. For KG4 the dynamical symmetries of the form Eq.~\\eqref{BandlimitedSymmetries} are:\n\\begin{flushleft}\\begin{enumerate}\n    \\item[1)] continuous translation which maps \\mbox{$\\phi_\\text{B}(t,x,y)\\mapsto \\phi_\\text{B}(t-t_0,x-x_0,y-y_0)$} for some $t_0,x_0,y_0\\in\\mathbb{R}$,\n    \\item[2)] three negation symmetries which map\n    \\mbox{$\\phi_\\text{B}(t,x,y)\\mapsto \\phi_\\text{B}(-t,x,y)$}, and \\mbox{$\\phi_\\text{B}(t,x,y)\\mapsto \\phi_\\text{B}(t,-x,y)$}, and \\mbox{$\\phi_\\text{B}(t,x,y)\\mapsto \\phi_\\text{B}(t,x,-y)$} respectively,\n    \\item[3)] a 4-fold symmetry which maps \\mbox{$\\phi_\\text{B}(t,x,y)\\mapsto \\phi_\\text{B}(t,y,-x)$},\n    \\item[4)] global linear rescaling which maps \\mbox{$\\phi_\\text{B}(t,x)\\mapsto c_1\\,\\phi_\\text{B}(t,x)$} for some $c_1\\in\\mathbb{R}$,\n    \\item[5)] local affine rescaling which maps \\mbox{$\\phi_\\text{B}(t,x)\\mapsto c_1\\,\\phi_\\text{B}(t,x)+c_2(t,x)$} for some $c_2(t,x)$ which is also a solution of the dynamics.\n\\end{enumerate}\\end{flushleft}\nThese are exactly the same symmetries that we found on the previous interpretation (sans local Fourier rescaling) just recategorized: the translation, reflection, and 4-fold symmetries are here external symmetries. Thus, KG4 has three continuous translation symmetries despite initially being represented on a lattice, Eq.~\\eqref{KG4Long}.\n\nLet's next consider KG5. For KG5 the dynamical symmetries of the form Eq.~\\eqref{BandlimitedSymmetries} are:\n\\begin{flushleft}\\begin{enumerate}\n    \\item[1)] continuous translation which maps \\mbox{$\\phi_\\text{B}(t,x,y)\\mapsto \\phi_\\text{B}(t-t_0,x-x_0,y-y_0)$} for some $t_0,x_0,y_0\\in\\mathbb{R}$,\n    \\item[2)] a negation symmetry which maps\n    \\mbox{$\\phi_\\text{B}(t,x,y)\\mapsto \\phi_\\text{B}(-t,x,y)$}, and an exchange symmetry which maps \\mbox{$\\phi_\\text{B}(t,x,y)\\mapsto \\phi_\\text{B}(t,y,x)$},\n    \\item[3)] a 6-fold symmetry which maps \\mbox{$\\phi_\\text{B}(t,x,y)\\mapsto \\phi_\\text{B}(t,-y,x+y)$}. (Roughly, this permutes the three terms in Eq.~\\eqref{DKG5bandlimited}),\n    \\item[4)] global linear rescaling which maps \\mbox{$\\phi_\\text{B}(t,x)\\mapsto c_1\\,\\phi_\\text{B}(t,x)$} for some $c_1\\in\\mathbb{R}$,\n    \\item[5)] local affine rescaling which maps \\mbox{$\\phi_\\text{B}(t,x)\\mapsto c_1\\,\\phi_\\text{B}(t,x)+c_2(t,x)$} for some $c_2(t,x)$ which is also a solution of the dynamics.\n\\end{enumerate}\\end{flushleft}\nThese are exactly the same symmetries that we found on the previous interpretation (sans local Fourier rescaling) just recategorized: the translation, reflection, and 6-fold symmetries are here external symmetries. Thus, KG5 has three continuous translation symmetries despite initially being represented on a lattice, Eq.~\\eqref{KG5Long}. Note that after a coordinate transformation Eq.~\\eqref{CoorKG7KG6} the above noted 6-fold symmetry is realized as one-sixth rotations, see Eq.~\\eqref{DKG5Skew}.\n\nBefore moving on to analyze the symmetries of KG6 and KG7, let's first see what this interpretation has to say about them being equivalent to one another. As discussed in Sec.~\\ref{SecKlein2} KG6 and KG7 have a solution-preserving vector space isomorphism between a substantial portion of their models, including all of their solutions. Since our second and third interpretations are related by a solution-preserving vector space isomorphism, \\mbox{$E:\\mathbb{R}^L\\to F^K$}, the same is broadly true here. However, some of the details change.\n\nAs I noted in Sec.~\\ref{SecKlein2} each of KG4-KG7 are equivalent to each other in a weak sense: approximately in the continuum limit regime, $\\vert \\omega\\vert,\\vert k_1\\vert,\\vert k_2\\vert\\ll \\pi$. Here KG4-KG7 are approximately equivalent in the $\\vert \\omega\\vert,\\vert k_1\\vert,\\vert k_2\\vert\\ll K=\\pi\/a$ regime. Note however that this is not the continuum limit regime, we are already in the continuum. Rather this is the regime which is significantly below the bandwidth $K$. While each of KG4-KG7 are equivalent in this weak sense, KG6 and KG7 are equivalent in a much stronger sense: KG6 and KG7 is in \\textit{exact} one-to-one correspondence over \\textit{the whole of} $\\sqrt{k_1^2+k_2^2} < K$ and indeed more. This includes all of their solutions but not all of $\\omega,k_1,k_2\\in[-\\pi,\\pi]$.\n\nThis exact correspondence is mediated by Eq.~\\eqref{SkewKG7KG6} and Eq.~\\eqref{SkewKG6KG7} in Fourier space or equivalently the coordinate transformations Eq.~\\eqref{CoorKG7KG6} and its inverse. On our second interpretation, we ran into technical trouble here due to the $2\\pi$ periodic identification of the discrete planewaves $\\bm{\\Phi}(\\omega,k_1,k_2)$ stemming from Euler's identity. This caused our maps between KG6 and KG7 not to be each other's inverses over the whole of $\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$. We were led to consider instead the subspaces $\\mathbb{R}^L_\\text{KG6}$ and $\\mathbb{R}^L_\\text{KG7}$ where these transformations were each other's inverse.\n\nIn the present interpretation, something similar happens but with some key differences. Unlike before, here our maps between KG6 and KG7 given wide scope are invertible and indeed each other's inverses. Wide scope here meaning seen as acting on $F(\\mathbb{R}^3)$ the set of all functions $f(t,x,y)$. However, as before, seen as acting on the relevant vector space $F^K$ we have issues. Namely, Eq.~\\eqref{SkewKG7KG6} and Eq.~\\eqref{SkewKG6KG7}, understood as acting on $F^K$ are not closed: a function supported over $\\omega,k_1,k_2\\in[-K,K]$ may have support outside of here after these transformations.\n\nIn Sec.~\\ref{SecKlein2} we fixed the non-invertible issue by focusing on the subspaces $\\mathbb{R}^L_\\text{KG6}$ and $\\mathbb{R}^L_\\text{KG7}$ where these transformations are each other's inverses. We might overcome our current not-closed-over-$F^K$ issue here in the same way defining \n\\begin{align}\nF^K_\\text{KG7}\\coloneqq&\\text{span}(\\phi(t,x,y;\\omega,k_1,k_2)\\vert\\,\\text{in }F^K\\\\\n\\nonumber\n&\\text{ before and after applying Eq.~\\eqref{SkewKG7KG6}})\\\\\nF^K_\\text{KG6}\\coloneqq&\\text{span}(\\phi(t,x,y;\\omega,k_1,k_2)\\vert\\,\\text{in }F^K\\\\\n\\nonumber\n&\\text{ before and after applying Eq.~\\eqref{SkewKG6KG7}})\\\\\n\\label{FKrotinv}\nF^K_\\text{rot.inv.}\\!\\!\\coloneqq&\\text{span}(\\phi(t,x,y;\\omega,k_1,k_2)\\vert k_1^2+k_2^2 < K^2).\n\\end{align}\nNote that under our embedding $E$ these subspaces are isomorphic to $\\mathbb{R}^L_\\text{KG6}$, $\\mathbb{R}^L_\\text{KG7}$ and $\\mathbb{R}^L_\\text{rot.inv.}$.\n\nRestricted to $F^K_\\text{KG6}$ and $F^K_\\text{KG7}$ these transformations are invertible and indeed are each other's inverses. All of KG6's solutions are in $F^K_\\text{KG6}$ (and moreover they are in $F^K_\\text{rot.inv}$ as well). Similarly, all of KG7's solutions are in $F^K_\\text{KG7}$. Thus, Eq.~\\eqref{CoorKG7KG6} maps generic solutions to KG7 onto generic solutions for KG6 in an invertible way. Therefore, Eq.~\\eqref{CoorKG7KG6} and its inverse give us not only a one-to-one correspondence between the solutions to KG6 and KG7 but a solution-preserving vector-space isomorphism between KG6 and KG7. One can gloss this situation saying: the KPMs of KG6 and KG7 are not isomorphic, but their DPMs are.\n\nThe fact that these transformations (namely, Eq.~\\eqref{CoorKG7KG6} and its inverse) are of the form Eq.~\\eqref{BandlimitedSymmetries} means that for any symmetry transformation for KG6 there is a corresponding symmetry transformation for KG7 and vice versa. In the present interpretation KG6 and KG7 are thoroughly equivalent: We have a solution-preserving vector-space isomorphism between them. Indeed, on this interpretation the only difference between KG6 and KG7 is a change of coordinates. \n\nThus our second big lesson from Sec.~\\ref{SecKlein2} is repeated here: despite KG6 and KG7 being initially presented to us with very different lattice structures (i.e., a square lattice in space versus a hexagonal lattice in space) they have nonetheless turned out to be completely equivalent to one another. This equivalence was hidden from us on our first interpretation because we there took the lattice too seriously. As I discuss in Sec.~\\ref{SecKlein2}, this reduced their continuous rotation symmetries down to quarter rotations and one-sixth rotations respectively and thereby made them inequivalent. Here, we do not take the lattice structure so seriously. We have embedded it onto the manifold where it then disappears from view as just one of many possible sample lattices.\n\nOne substantial difference from our second interpretation should be noted here. We are not forced to shrink $F^K$ down to $F^K_\\text{KG6}$ and $F^K_\\text{KG7}$ to see KG6 and KG7 as equivalent; we have another option. Namely, rather than shrinking $F^K$ we could also expand it. Consider a theory just like KG6 on this interpretation, but with a bandwidth of $2\\,K$ instead of $K$. Clearly, KG6 is a subtheory of this expanded theory. Note that the coordinate transformation which maps KG7 onto KG6, namely Eq.~\\eqref{SkewKG7KG6}, skews its support in Fourier space. However, its support remains inside of $k_1,k_2\\in[-2K,2K]$. Thus, KG7 is also a subtheory of our expanded theory. We can here see KG6 and KG7 as parts of a larger theory. As I will discuss in Sec.~\\ref{PerfectLorentz}, KG6 and KG7 are the parts of this extended theory which is visible to us when we restrict ourselves to only certain sets of representational tools. This line of thought will lead us to a perfectly Lorentzian lattice theory.\n\nFor the rest of this subsection I will only discuss the symmetries of KG6; analogous conclusions are true for KG7 after applying Eq.~\\eqref{CoorKG7KG6}. For KG6 the dynamical symmetries of the form Eq.~\\eqref{BandlimitedSymmetries} are:\n\\begin{flushleft}\\begin{enumerate}\n    \\item[1)] continuous translation which maps \\mbox{$\\phi_\\text{B}(t,x,y)\\mapsto \\phi_\\text{B}(t-t_0,x-x_0,y-y_0)$} for some $t_0,x_0,y_0\\in\\mathbb{R}$,\n    \\item[2)] three negation symmetries which map\n    \\mbox{$\\phi_\\text{B}(t,x,y)\\mapsto \\phi_\\text{B}(-t,x,y)$}, and \\mbox{$\\phi_\\text{B}(t,x,y)\\mapsto \\phi_\\text{B}(t,-x,y)$}, and \\mbox{$\\phi_\\text{B}(t,x,y)\\mapsto \\phi_\\text{B}(t,x,-y)$} respectively,\n    \\item[3)] continuous rotation which maps \\mbox{$\\phi_\\text{B}(t,x,y)$} to \\mbox{$\\phi_\\text{B}(t,x\\cos(\\theta)-y\\sin(\\theta),x\\sin{\\theta}+y\\cos(\\theta))$} for some $\\theta\\in\\mathbb{R}$. (This being a symmetry requires some qualification as I will discuss below.),\n    \\item[4)] two Lorentz boosts which map \\mbox{$\\phi_\\text{B}(t,x,y)$} to \\mbox{$\\phi_\\text{B}(t\\cosh(w)+x\\sinh(w),t\\sinh(w)+x\\cosh(w),y)$} and map \\mbox{$\\phi_\\text{B}(t,x,y)$} to \\mbox{$\\phi_\\text{B}(t\\cosh(w)+y\\sinh(w),x,t\\sinh(w)+y\\cosh(w))$} respectively for some $w\\in\\mathbb{R}$. (This being a symmetry requires some qualification as I will discuss below.),\n    \\item[5)] global linear rescaling which maps \\mbox{$\\phi_\\text{B}(t,x)\\mapsto c_1\\,\\phi_\\text{B}(t,x)$} for some $c_1\\in\\mathbb{R}$,\n    \\item[6)] local affine rescaling which maps \\mbox{$\\phi_\\text{B}(t,x)\\mapsto c_1\\,\\phi_\\text{B}(t,x)+c_2(t,x)$} for some $c_2(t,x)$ which is also a solution of the dynamics.\n\\end{enumerate}\\end{flushleft}\nThese are exactly the same symmetries that we found on the previous interpretation (sans local Fourier rescaling) just recategorized: the translation, reflection, and rotation, and Lorentz boost symmetries are here external symmetries. Thus, KG6 (and KG7) each have three continuous translation symmetries despite initially being represented on a lattice.\n\nIn addition to this, KG6 has a (qualified) continuous rotation symmetry. But in what sense is this only a qualified symmetry? Much of the discussion from the previous section following Eq.~\\eqref{RthetaDef} applies here as well with some key differences.\n\nAs I noted in Sec.~\\ref{SecKlein2}, each of KG4-KG7 are rotation invariant in a weak sense: approximately in the continuum limit regime, $\\vert \\omega\\vert,\\vert k_1\\vert,\\vert k_2\\vert\\ll \\pi$. Here KG4-KG7 are approximately rotation invariant in the $\\vert \\omega\\vert,\\vert k_1\\vert,\\vert k_2\\vert\\ll K=\\pi\/a$ regime. Note however that this is not the continuum limit regime, we are already in the continuum. Rather, this is the regime which is significantly below the bandwidth $K$. While each of KG4-KG7 are rotation invariant in this weak sense, KG6 is rotation invariant in a much stronger sense: KG6 is \\textit{exactly} rotation over \\textit{the whole of} $\\sqrt{k_1^2+k_2^2} < K$.\n\nRotation is here generated by \\mbox{$\\exp(-\\theta (x \\partial_y - y \\partial_x))$} for some $\\theta\\in\\mathbb{R}$. Acting on the continuum planewave basis this transformation merely rotates their wavenumbers in Fourier space. On our second interpretation, we ran into technical trouble here due to the $2\\pi$ periodic identification of the discrete planewaves $\\bm{\\Phi}(\\omega,k_1,k_2)$ stemming from Euler's identity. This caused $R^\\theta$ not to be invertible over the whole of $\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$. We were led to consider only the rotation invariant subspace $\\mathbb{R}^L_\\text{rot.inv.}$.\n\nIn the present interpretation, something similar happens but with some key differences. Unlike before, here given wide scope rotations are always invertible. Wide scope here meaning seen as acting on $F(\\mathbb{R}^3)$ the set of all functions $f(t,x,y)$. However, as before, seen as acting on the relevant vector space $F^K$ we have issues. Namely, understood as acting on $F^K$ rotation is not closed: some $f$ supported over $\\omega,k_1,k_2\\in[-K,K]$ may have support outside of here after being rotated.\n\nIn Sec.~\\ref{SecKlein2} we fixed the non-invertible issue by focusing on the rotation invariant subspace $\\mathbb{R}^L_\\text{rot.inv.}\\subset \\mathbb{R}^L$. We might overcome our current not-closed-over-$F^K$ issue here in the same way. While rotation is not closed over $F^K$ it is closed over $F^K_\\text{rot.inv.}$ defined in Eq.~\\eqref{FKrotinv}. Note that under our embedding $E$ this subspace is isomorphic to the vector space $\\mathbb{R}^L_\\text{rot.inv.}$ defined in Eq.~\\eqref{RLrotinv}.\n\nNote that all of KG6's solutions are within $F^K_\\text{rot.inv.}$. Within $F^K_\\text{rot.inv.}$ the above discussed rotation transformation is of the form Eq.~\\eqref{BandlimitedSymmetries} and maps solutions to KG6 onto solution to KG6 in an invertible way, and is hence a symmetry. One can gloss this situation saying: rotation does not map the KPMs of KG6 onto themselves in an invertible way, but it does for the DPMs of KG6. If we cut the KPMs of KG6 down to $F^K_\\text{rot.inv}$ this minor issue is fixed.\n\nThis adds to our first big lesson from Sec.~\\ref{SecKlein2}: despite the fact that KG6 and KG7 can be represented on a cubic 3D lattice and a hexagonal 3D lattice respectively, they nonetheless both have a continuous rotation symmetry. This, in addition to their continuous translation symmetries. These continuous translation and rotations symmetries were hidden from us on our first interpretation because we there took the lattice representations too seriously. Here, we do not take the lattice structure so seriously. We have embedded it onto the manifold where it then disappears from view as just one of many possible sample lattices.\n\nOne substantial difference from our second interpretation should be noted here. We are not forced to shrink $F^K$ down to $F^K_\\text{rot.inv.}$ to make KG6 rotation invariant; we have another option. Rather than shrinking $F^K$ down to its rotation invariant core, we could also expand it by adding in every state reachable from $F^K$ by rotation yielding $F^{\\sqrt{2}K}_\\text{rot.inv.}$. Consider a theory just like KG6 on this interpretation, but defined over $F^{\\sqrt{2}K}_\\text{rot.inv.}$ instead of $F^K$. Clearly, KG6 is a subtheory of this expanded theory. As I will discuss in Sec.~\\ref{PerfectLorentz}, KG6 is the part of this extended theory which is visible to us when we restrict ourselves to only a certain set of representational tools. This line of thought will lead us to a perfectly Lorentzian lattice theory.\n\nLet's next discuss how KG6's (limited) Lorentzian boost symmetry is realized on this interpretation. The Lorentz boosts of KG6 are here generated by \\mbox{$\\exp(-w (x \\partial_t + t \\partial_x))$} and \\mbox{$\\exp(-w (y \\partial_t + t \\partial_y))$} for some boost parameter $w\\in\\mathbb{R}$. Notice that boosting a bandlimited function also boosts its support in Fourier space. On our second interpretation, we ran into technical trouble here due to the $2\\pi$ periodic identification of the discrete planewaves $\\bm{\\Phi}(\\omega,k_1,k_2)$ stemming from Euler's identity. This caused $R^\\theta$ not to be invertible over the whole of $\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$. We were led to consider only the rotation invariant subspace $\\mathbb{R}^L_\\text{rot.inv.}$.\n\nIn the present interpretation, something similar happens but with some key differences. Unlike before, here given wide scope Lorentz boosts are always invertible. Wide scope here meaning seen as acting on $F(\\mathbb{R}^3)$ the set of all functions $f(t,x,y)$. However, as before, seen as acting on the relevant vector space $F^K$ we have issues. Namely, understood as acting on $F^K$ Lorentz boosts are not invertible nor are they even functions: some $f$ supported over $\\omega,k_1,k_2\\in[-K,K]$ may have support outside of here after being boosted. \n\nIn Sec.~\\ref{SecKlein2} we fixed the non-invertible issue by focusing on the finite-sized region around $\\omega,k_1,k_2=0$ and boost parameter $w=0$ where $\\Lambda^w_\\text{j,n}$ and $\\Lambda^w_\\text{j,m}$ are invertible. Moreover, we were even able to find a representation of the Poincar\\'e algebra. We might overcome our current not-closed-over-$F^K$ issue here in the same way. There is some finite-sized region around $\\omega,k_1,k_2=0$ and boost parameter $w=0$ where Lorentz boosts keep us inside $F^K$. Moreover, as above, we could find a representation of the Poincar\\'e algebra over $F^K$ as well as a finite portion of the Poincar\\'e group.\n\nSo far the situation here is very much like it was in Sec.~\\ref{SecKlein2}. Our first big lesson is here repeated: despite the fact that KG6 and KG7 can be represented on a cubic 3D lattice and a hexagonal 3D lattice respectively, they nonetheless both have a Lorentzian boost symmetries (in a finite but limited regime).\n\nHowever, there is one substantial difference here which ultimately points the way towards a perfectly Lorentzian lattice theory. As I will discuss in Sec.~\\ref{PerfectLorentz}, we can see KG6 as the part of some extended theory which is visible to us when we restrict ourselves to only a certain set of representational tools. This extended theory is perfectly Lorentz invariant. Viewed this way, KG6's failure to be Poincar\\'e invariant in an unqualified way ought to be seen as a problem of representational capacity and not of physics.\n\nFinally, let's consider KG3. For KG3 the dynamical symmetries of the form Eq.~\\eqref{BandlimitedSymmetries} are:\n\\begin{flushleft}\n\\begin{enumerate}\n    \\item[1)]  continuous translation which maps \\mbox{$\\phi_\\text{B}(t,x)\\mapsto \\phi_\\text{B}(t-t_0,x-x_0)$} for $t_0,x_0\\in\\mathbb{R}$,\n    \\item[2)] two negation symmetries which map \\mbox{$\\phi_\\text{B}(t,x)\\mapsto \\phi_\\text{B}(-t,x)$} and \\mbox{$\\phi_\\text{B}(t,x)\\mapsto \\phi_\\text{B}(t,-x)$} respectively,\n    \\item[3)] a Lorentz boosts which maps \\mbox{$\\phi_\\text{B}(t,x)$} to \\mbox{$\\phi_\\text{B}(t\\cosh(w)+x\\sinh(w),t\\sinh(w)+x\\cosh(w))$},\n    \\item[4)] global linear rescaling which maps \\mbox{$\\phi_\\text{B}(t,x)\\mapsto c_1\\,\\phi_\\text{B}(t,x)$} for some $c_1\\in\\mathbb{R}$,\n    \\item[5)] local affine rescaling which maps \\mbox{$\\phi_\\text{B}(t,x)\\mapsto c_1\\,\\phi_\\text{B}(t,x)+c_2(t,x)$} for some $c_2(t,x)$ which is also a solution of the dynamics.\n\\end{enumerate}\n\\end{flushleft}\nThese are exactly the same symmetries that we found on the previous interpretation (sans local Fourier rescaling) just recategorized: the translation, reflection, and (limited) Lorentz boost symmetries are here external symmetries. \n\n\\vspace{0.25cm}\n\nTo summarize: this third attempt at interpreting KG1-KG7 has fixed all of the issues with our first and second interpretations. Firstly, the tension between our theories' differences in locality and their differences in convergence rate in the continuum limit has been harmoniously resolved. This improves upon the hamfisted dissolution of tension given by our second interpretation, see \\cite{DiscreteGenCovPart1}. Secondly, like on the second interpretation, here KG6 and KG7 are seen to be completely equivalent. Here however, we can moreover see KG6 and KG7 in their entirety as parts of a larger unified theory. Finally, like on the second interpretation, we have here exposed KG1-KG7's hidden continuous translation and rotation symmetries and even limited Lorentz symmetries. Here however, these are more satisfyingly categorized as external symmetries.\n\nLike the previous interpretation, this interpretation by and large invalidates all of the first intuitions laid out in Sec.~\\ref{SecIntro}. Indeed here the lattice seems to play a merely representational role in the theory: it does not restrict our symmetries. Moreover, theories initially appearing with different lattices may nonetheless turn out to be substantially equivalent. The process for switching between lattice structures is here done by Nyquist-Shannon resampling. Indeed, the third big lesson from Sec.~\\ref{SecKlein2} is repeated here: there is no sense in which these lattice structures are essentially ``baked-into'' these theories; our bandlimited theories make no reference to any lattice structure. \n\nAs I will discuss in Sec.~\\ref{SecDisGenCov}, these three lessons lay the foundation for a strong analogy between the lattice structures which appear in our discrete spacetime theories and the coordinate systems which appear in our continuum spacetime theories. This ultimately gives rise to a discrete analog of general covariance.\n\nBefore that however, a bit more must be said about this third interpretation.\n\n\\section{A Third Attempt at Interpreting KG1-KG7 - Part 2}\\label{SecKlein3Extra}\nThe previous section presented a third interpretation of our seven discrete theories KG1-KG7. This section will flesh out this interpretation in two ways. Firstly, I will provide some more explicit demonstrations of these theories' symmetries. In particular, I will explicitly demonstrate the fact that the symmetries of these theories are independent of which sample lattice we use to represent them. Secondly, I will write our above formulations of KG1-KG7 in terms of $\\phi_\\text{B}$ in the coordinate-independent language of differential geometry. This will help clarify their symmetries and assumed background structures.\n\n\\subsection{Demonstration of Bandlimited Symmetries}\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=0.9\\textwidth]{PaperFigures\/RRow1.pdf}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{PaperFigures\/RRow2.pdf}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{PaperFigures\/RRow3.pdf}\n\\caption{The dynamics of KG4 is here shown being carried out in a variety of lattice representations. In the left most column the initial condition is shown in its bandlimited representation, given by Eq.~\\eqref{InCond}. In the rightmost column the final evolved state is shown in its bandlimited representation. Here the evolution time is $t=2$ and the field mass is $\\mu=0$. This state can be found in four different ways. Firstly by applying the bandlimited dynamics Eq.~\\eqref{DKG4bandlimited} to the bandlimited initial condition. The other three ways are shown in the three rows of this figure. The first row shows the initial condition being sampled onto a square lattice. This is then evolved forward in time via Eq.~\\eqref{KG4Long}. The bandlimited representation of the final state is then recovered through the methods discussed in Sec.~\\ref{SecSamplingTheory}. The second and third rows show the same process carried out on a hexagonal lattice and an irregular lattice. Notice that the final state has a 4-fold symmetry regardless of how the dynamics is represented. Notice that the final state is the same regardless of how the dynamics is represented. We can represent any bandlimited dynamics on any (sufficiently dense) lattice.}\\label{FigEvolutionKG4}\n\\end{figure*}\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=0.9\\textwidth]{PaperFigures\/RRow4.pdf}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{PaperFigures\/RRow5.pdf}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{PaperFigures\/RRow6.pdf}\n\\caption{The dynamics of KG5 is here shown being carried out in a variety of lattice representations. In the left most column the initial condition is shown in its bandlimited representation, given by Eq.~\\eqref{InCond}. In the rightmost column the final evolved state is shown in its bandlimited representation. Here the evolution time is $t=108$ and the field mass is $\\mu=0$. This state can be found in four different ways. Firstly by applying the bandlimited dynamics Eq.~\\eqref{DKG5bandlimitedSkew} to the bandlimited initial condition. The other three ways are shown in the three rows of this figure. The second row shows the initial condition being sampled onto a hexagonal lattice. This is then evolved forward in time via Eq.~\\eqref{KG5Long}. The bandlimited representation of the final state is then recovered through the methods discussed in Sec.~\\ref{SecSamplingTheory}. The second and third rows show the same process carried out on a square lattice and an irregular lattice. Notice that the final state has a 6-fold symmetry regardless of how the dynamics is represented. Notice that the final state is the same regardless of how the dynamics is represented. We can represent any bandlimited dynamics on any (sufficiently dense) lattice.}\\label{FigEvolutionKG5}\n\\end{figure*}\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=0.9\\textwidth]{PaperFigures\/RRow7.pdf}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{PaperFigures\/RRow8.pdf}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{PaperFigures\/RRow9.pdf}\n\\caption{The dynamics of KG6 is here shown being carried out in a variety of lattice representations. In the left most column the initial condition is shown in its bandlimited representation, given by Eq.~\\eqref{InCond}. In the rightmost column the final evolved state is shown in its bandlimited representation. Here the evolution time is $t=6$ and the field mass is $\\mu=0$. This state can be found in four different ways. Firstly by applying the bandlimited dynamics Eq.~\\eqref{DKG6bandlimited} to the bandlimited initial condition. The other three ways are shown in the three rows of this figure. The first row shows the initial condition being sampled onto a square lattice. This is then evolved forward in time via Eq.~\\eqref{DKG6}. The bandlimited representation of the final state is then recovered through the methods discussed in Sec.~\\ref{SecSamplingTheory}. The second and third rows show the same process carried out on a hexagonal lattice and an irregular lattice. Notice that the final state is rotation invariant regardless of how the dynamics is represented. Notice that the final state is the same regardless of how the dynamics is represented. We can represent any bandlimited dynamics on any (sufficiently dense) lattice.}\\label{FigEvolutionKG6}\n\\end{figure*}\n\nIn this subsection I will provide some more explicit demonstrations of these theories' symmetries. In particular, I will explicitly demonstrate the fact that the symmetries of some theory's dynamics has nothing to do with the symmetries of the lattice it is represented on. Just as we can represent any bandlimited state on any sufficiently dense lattice, so too can we represent any bandlimited dynamics on any sufficiently dense lattice. To see this consider Figs. \\ref{FigEvolutionKG4}, \\ref{FigEvolutionKG5} and \\ref{FigEvolutionKG6}. \n\nIn each of these figures we begin from some initial state with a bandlimited representation at $t=0$ of,\n\\begin{align}\\label{InCond}\n\\phi_\\text{B}(0,x,y)=\\frac{J_1(\\pi r_2)}{\\pi r_2}\n+\\frac{J_0(\\pi r_2)-J_2(\\pi r_2)}{2}\n\\end{align}\nwhere $J_n(r)$ is the $n^\\text{th}$ Bessel function and $r=\\sqrt{x^2+y^2}$ and $r_2=r\/\\sqrt{2}$. This function is shown in the first columns of Figs. \\ref{FigEvolutionKG4}, \\ref{FigEvolutionKG5} and \\ref{FigEvolutionKG6}. Note that this function is rotationally invariant.\n\nThis function is bandlimited with bandwidth of $K=\\pi\/\\sqrt{2}$. We can therefore represent this function with sampling on a square 2D lattice with $a=1<\\sqrt{2}$. We could equivalently represent this function on a hexagonal 2D lattice or even an irregular 2D lattice. For each of KG4, KG5 and KG6=KG7 Such representations are shown in the second columns of Figs. \\ref{FigEvolutionKG4}, \\ref{FigEvolutionKG5} and \\ref{FigEvolutionKG6}. \n\nFor each of these theories, we then have a choice of which representation to carry out the dynamics in. I here consider four options: as a bandlimited function, as samples on a square lattice, as samples on a hexagonal lattice or as samples on an irregular lattice. The various options for KG4, KG5 and KG6=KG7 are shown in Figs. \\ref{FigEvolutionKG4}, \\ref{FigEvolutionKG5} and \\ref{FigEvolutionKG6} respectively. \n\nLet's begin with the dynamics of KG5 represented on a hexagonal lattice. This is shown in the middle row of Fig.~\\ref{FigEvolutionKG5}. The bandlimited representation of the initial distribution is shown Fig.~\\ref{FigEvolutionKG5}b1). The initial sample points on the hexagonal lattice are shown Fig.~\\ref{FigEvolutionKG5}b2). These can be evolved forward in time using Eq.~\\eqref{KG5Long}. The resulting time-evolved sample points are shown in Fig.~\\ref{FigEvolutionKG5}b3). From these we can reconstruct a bandlimited representation for the state using the techniques discussed in Sec.~\\ref{SecSamplingTheory}. The resulting reconstruction is shown in Fig.~\\ref{FigEvolutionKG5}b4).\n\nAlternatively, we could have carried out this evolution with no lattice representation at all. That is, we could have skipped from Fig.~\\ref{FigEvolutionKG5}b1) directly to Fig.~\\ref{FigEvolutionKG5}b4). We could do this by applying the dynamics Eq.~\\eqref{DKG5bandlimitedSkew} directly to the bandlimited initial condition Eq.~\\eqref{InCond}. It is in this sense that the bandlimited and discrete representations of our dynamics are equivalent.\n\nThe first and third rows of Fig.~\\ref{FigEvolutionKG5} show the exact same evolution via KG5 represented on different lattices, namely a square lattice and an irregular lattice. In the first row the evolution is carried out by a resampled version of Eq.~\\eqref{KG5Long}, namely Eq.~\\eqref{DKG5Skew}. In the third row the evolution is carried out by whatever resampling of Eq.~\\eqref{KG5Long} corresponds to this irregular lattice. \n\nNotice that the final state has a 6-fold symmetry regardless of how the dynamics is represented. Moreover, notice that the final state is the same regardless of how the dynamics is represented. Just as we can represent any bandlimited state on any lattice, so too can we represent any bandlimited dynamics on any lattice.\n\nFig.~\\ref{FigEvolutionKG4} makes the same demonstration for KG4. Notice that the final state has a 4-fold symmetry regardless of how the dynamics is represented. Notice that the final state is the same regardless of how the dynamics is represented.\n\nLikewise, Fig.~\\ref{FigEvolutionKG6} makes the same demonstration for KG6. Notice that the final state is rotation invariant regardless of how the dynamics is represented. Notice that the final state is the same regardless of how the dynamics is represented. \n\nThese figures demonstrate clear as can be that a theory's lattice structure has nothing to do with its dynamical symmetries. We can represent any bandlimited dynamics on any lattice.\n\n\\subsection{Bandlimited General Covariance}\\label{SecFullGenCov}\nAs the discussion throughout Sec.~\\ref{SecKlein3} has shown, giving our discrete theory a bandlimited representation has had many of the same benefits one expects from a generally covariant formulation. Namely, we have exposed certain parts of our theory as merely representational artifacts and in the process we have come to a better understanding of our theory's symmetries and background structures. This is the work of the titular discrete analog of general covariance. This analogy was originally introduced in \\cite{DiscreteGenCovPart1} and has here been expanded to a Lorentzian context. The details of this analogy will be spelled out in detail in Sec.~\\ref{SecDisGenCov}.\n\nBefore that, however, I will show how to combine this discrete analog with our usual continuum notion of general covariance. Note that on this interpretation KG1-KG7 are ultimately continuum spacetime theories of the sort we are used to interpreting (albeit ones which consider only bandlimited fields). Hence we can apply to these theories the standard techniques of general covariance. (For a review see Appendix A of \\cite{DiscreteGenCovPart1}).\n\nAs I discussed at the beginning of Sec.~\\ref{SecKlein3} on this third interpretation KG1-KG7 are about a bandlimited field \\mbox{$\\phi_\\text{B}:\\mathcal{M}\\to \\mathcal{V}$} with a value space $\\mathcal{V}=\\mathbb{R}$ and a manifold $\\mathcal{M}\\cong\\mathbb{R}^d$ with $d=2$ for KG1-KG3 and $d=3$ for KG4-KG7. As such, in either case we have access to global coordinate systems for $\\mathcal{M}$. The fields $\\phi_\\text{B}$ considered by this interpretation are bandlimited in the following sense. There exists a fixed special diffeomorphism \\mbox{$d_\\text{coor.}:\\mathbb{R}^d\\to\\mathcal{M}$} which giving us a fixed special global coordinate system for $\\mathcal{M}$. In these special coordinates the field is bandlimited. That is, $\\phi_\\text{B}\\circ d_\\text{trans.}\\in F^K$. It is in these special coordinates that the fields $\\phi_\\text{B}$ obey the dynamical equations Eq.~\\eqref{DKG1bandlimited}-Eq.~\\eqref{DKG7bandlimited}.\n\nOf course, we know from the lessons of continuum general covariance that one ought to be suspicious of any ``special coordinates'' appearing in a supposedly fundamental formulation of a theory. This section will remove any reference to these (or any) coordinates from our bandlimited formulation of KG1-KG7. As is expected of such a generally covariant reformulation, this will reveal these theories' underlying geometric background structures.\n\nFor simplicity, however, I will just consider KG4 and KG6 here. To start let us first write the continuum theory KG0 in the coordinate-free language of differential geometry. After substantial work (see \\cite{Pooley2015} or Appendix A of \\cite{DiscreteGenCovPart1}) one can rewrite Eq.~\\eqref{KG0} in the coordinate-free language of differential geometry as follows:\n\\begin{align}\\label{KG0GenCov}\n\\text{KG0:}\\qquad\\text{KPMs:}\\quad&\\langle\\mathcal{M},\\eta_\\text{ab},\\varphi\\rangle\\\\\n\\nonumber\n\\text{DPMs:}\\quad&(\\eta^\\text{ab}\\nabla_\\text{a}\\nabla_\\text{b}-M^2)\\,\\varphi = 0,   \n\\end{align}\nThe geometric objects used in this formulation are as follows. $\\mathcal{M}$ is a 2+1 dimensional differentiable manifold. $\\eta_\\text{ab}$, is a fixed Lorentzian metric field with signature $(-1,1,1)$. Here $\\nabla_\\text{a}$ is the unique covariant derivative operator compatible with the metric, (i.e. with $\\nabla_\\text{c}\\,\\eta_\\text{ab}=0$). $\\varphi:\\mathcal{M}\\to\\mathbb{R}$ is a dynamical real scalar field. \n\nMathematical structures satisfying the above conditions (independent of whether it satisfies the dynamics) are the theory's kinematically possible models (KPMs). This theory's dynamically possible models (DPMs) are the subset of the KPMs which additionally satisfy the theory's dynamics.\n\nNext let's consider KG4. Rewritten in the coordinate-free language of differential geometry, KG4 becomes:\n\\begin{align}\\label{DKG4GenCov}\n\\text{KG4:}\\ \\text{KPMs:}\\ &\\langle \\mathcal{M}, \\eta_\\text{ab},T^\\text{a},X^\\text{a},Y^\\text{a},\\phi_\\text{B}\\rangle\\\\\n\\nonumber\n\\text{DPMs:}\\ &\nF(T^\\text{a}\\nabla_\\text{a})\\phi_\\text{B}\n= \\big[-\\mu^2+F(X^\\text{b}\\nabla_\\text{b})\\\\\n\\nonumber\n&\\qquad\\qquad\\qquad\\qquad \\ +F(Y^\\text{c}\\nabla_\\text{c})\\big]\\,\\phi_\\text{B}.\n\\end{align}\nwhere \n\\begin{align}\nF(z)=2\\text{cosh}(a\\,z)-2.\n\\end{align}\nAt the level of dynamics, KG4 and KG0 are radically different. Note that the dynamics of KG0 is local, only involving finite order of derivative. By contrast, the dynamics of KG4 is highly non-local.\n\nThese theories are also substantially different at the level of KPMs. The three key differences between KG4 and the continuum theory KG0 are as follows. Firstly KG4 resticts us to manifolds $\\mathcal{M}\\cong\\mathbb{R}^3$ whereas KG0 does not. Secondly, the field $\\phi_\\text{B}\\in F^K$ in KG4 is bandlimited with bandwidth $\\omega,k_1,k_2\\in[-K,K]$ whereas the field $\\psi$ in KG0 is not. I owe the reader a geometric explanation of what this means.\n\nFinally, KG4 has three extra pieces of spacetime structure which KG0 lacks. Namely, $T^\\text{a}$, $X^\\text{a}$ and $Y^\\text{a}$ are three fixed constant unit vectors which are orthogonal to each other. $T^\\text{a}$ is timelike whereas $X^\\text{a}$ and $Y^\\text{a}$ are spacelike. That is, \n\\begin{align}\n\\nabla_\\text{a}T^\\text{b}&=0 &\n\\nabla_\\text{a}Y^\\text{b}&=0 &\n\\nabla_\\text{a}Y^\\text{b}&=0\\\\\n\\nonumber\n\\eta_\\text{ab} T^\\text{a}T^\\text{b}&=-1 &\n\\eta_\\text{ab} X^\\text{a}X^\\text{b}&=1 &\n\\eta_\\text{ab} Y^\\text{a}Y^\\text{b}&=1\\\\\n\\nonumber\n\\eta_\\text{ab} T^\\text{a}X^\\text{b}&=0 &\n\\eta_\\text{ab} X^\\text{a}Y^\\text{b}&=0 &\n\\eta_\\text{ab} Y^\\text{a}T^\\text{b}&=0\n\\end{align}\nRoughly, $X^\\text{a}$ and $Y^\\text{a}$ here serve to pick out the directions for the rotational anomalies appearing in Fig.~\\ref{FigEvolutionKG4}. $T^\\text{a}$ picks out a standardized way of moving forward in time (i.e, translation generated by $T^\\text{a}\\nabla_\\text{a}$). That is, $T^\\text{a}$ provides a rest frame.\n\nIn terms of these spacetime structures, what does it here mean to say that $\\phi_\\text{B}\\in F^K$ is  ``bandlimited with bandwidth $\\omega,k_1,k_2\\in[-K,K]$''. Consider the following eigen-problem for functions $h:\\mathcal{M}\\to\\mathbb{R}$ on the manifold: \n\\begin{align}\nT^\\text{a}T^\\text{b}\\nabla_\\text{a}\\nabla_\\text{b} h = -\\omega^2 h\\\\\n\\nonumber\nX^\\text{a}X^\\text{b}\\nabla_\\text{a}\\nabla_\\text{b} h = -k_1^2 h\\\\\n\\nonumber\nY^\\text{a}Y^\\text{b}\\nabla_\\text{a}\\nabla_\\text{b} h = -k_2^2 h\n\\end{align}\nA function $f:\\mathcal{M}\\to\\mathbb{R}$ is bandlimited if and only if it is the sum of these eigensolutions over a compact range of Fourier space. The extent of this range is its bandwidth. In these terms we can state the requirement that $\\phi_\\text{B}$ is bandlimited with bandwidth $\\omega,k_1,k_2\\in[-K,K]$.\n\nLet's move on to KG6. Rewritten in the coordinate-free language of differential geometry, KG6 becomes:\n\\begin{align}\\label{DKG6GenCov}\n\\text{KG6:}\\qquad\\text{KPMs:}\\quad&\\langle \\mathcal{M}, \\eta_\\text{ab},T^\\text{a},X^\\text{a},Y^\\text{a},\\phi_\\text{B}\\rangle\\\\\n\\nonumber\n\\text{DPMs:}\\quad&(\\eta^\\text{ab}\\nabla_\\text{a}\\nabla_\\text{b}-\\mu_0^2)\\,\\phi_\\text{B} = 0\n\\end{align}\nwith spacetime structures as defined following Eq.~\\eqref{KG0GenCov}, $\\mathcal{M}\\cong\\mathbb{R}^3$ and $\\phi_\\text{B}\\in F^K$ in the same sense as KG4. \n\nAt the level of dynamics, KG6 and KG0 are nearly identical. The dynamics for KG0 has a field mass of $M$ whereas KG6 has $\\mu_0=\\mu\/a$. This can be disregarded by setting $\\mu_0=M$.\n\nHowever, at the level of KPMs, KG6 and KG0 are substantially different. The three key differences between KG6 and the continuum theory KG0 are as follows. Firstly KG6 resrticts us to manifolds $\\mathcal{M}\\cong\\mathbb{R}^3$ whereas KG0 does not. Secondly, the field $\\phi_\\text{B}$ in KG6 is bandlimited with bandwidth $\\omega,k_1,k_2\\in[-K,K]$ in the sense defined above whereas the field $\\psi$ in KG0 is not. Finally, KG6 has the same extra spacetime structures $T^a$, $X^a$ and $Y^a$ that KG4 does. Indeed, at the level of KPMs KG6 and KG4 are identical.\n\nOne might notice that these extra pieces of spacetime structure don't play any role in the dynamics of KG6. Why then are they there? $T^a$, $X^a$, and $Y^a$ are needed to spell out what it means for $\\bm{\\phi}_\\text{B}$ to be bandlimited with bandwidth $\\omega,k_1,k_2\\in[-K,K]$. This region in Fourier space is not Poincar\\'e invariant, we need some extra structure to point out in which directions the corners go.\n\nThis is a very strange sort of spacetime structure. It doesn't participate in the dynamics, all it does is allows us to articulate a certain restriction on the space of KPMs. As I will discuss in Sec.~\\ref{PerfectLorentz}, it is best to think of KG6 as being the part of an extended theory which is visible to us when we restrict our set of representational tools. On this view, while $X^a$ and $Y^a$ are real spacetime structures for KG4, for KG6 they are merely representational artifacts. \n\nBefore that, however, allow me to spell out in detail the discrete analogs of general covariance which have been developed throughout this paper.\n\n\\section{Two Discrete Analogs of General Covariance}\\label{SecDisGenCov}\nThree lessons have been repeated throughout this paper. Each of these lessons is visible in both our second and third attempts at interpreting KG1-KG7. Combined these lessons give us a rich analogy between lattice structures and coordinate systems: Lattice structure is rather less like a fixed background structure and rather more like a coordinate system, i.e., merely a representational artifact.\n\nThese three lessons run counter to the three first intuitions one is likely to have regarding lattice structure discussed in Sec.~\\ref{SecIntro}. Namely, that lattices and lattice structure: restrict our symmetries, distinguish our theories, and are fundamentally ``baked-into'' the theory. As we have seen, they do not restrict our symmetries, they do not distinguish our theories and they are merely representational not fundamental. In particular, we have learned the following three lessons.\n\nOur first lesson was that taking the lattice structure seriously as a fixed background structure or as the underlying spacetime manifold systematically under predicts the symmetries that discrete spacetime theories can and do have. Indeed, discrete theories can have significantly more symmetries than our first intuitions might allow for. As Sec.~\\ref{SecKlein2} and Sec.~\\ref{SecKlein3} have shown each of KG1-KG7 has a continuous translation symmetry despite being introduced with explicit lattice structures. Moreover, KG6 and KG7 even have a continuous rotational symmetry and a (limited) Lorentzian boost symmetry. As I will discuss in Sec.~\\ref{PerfectLorentz}, these theories can be seen as part of a larger lattice theory which is perfectly Lorentzian. Thus, the fact that a lattice structure was used in the initial statement of these theories' dynamics does not in any way restrict their symmetries. There is no conceptual barrier to having a theory with continuous symmetries represented on a discrete lattice.\n\nIn light of the proposed analogy between lattice structure and coordinate systems this first lesson is not mysterious. Coordinate systems are neither background structure nor a fundamental part of the manifold. The use of a certain coordinate system does not in any way restrict a theory's symmetries. Indeed, it is a familiar fact that there is no conceptual barrier to having a rotationally invariant theory formulated on a Cartesian coordinate system.\n\nOur second lesson was that discrete theories which are initially presented to us with very different lattice structures may nonetheless turn out to be completely equivalent theories. Indeed, as we have seen, two of our discrete theories (KG6 and KG7) have a nice one-to-one correspondence between their solutions. This despite the fact that these theories were initially presented to us with different lattice structures (i.e., a square lattice in space and a hexagonal lattice in space respectively). However, despite this nice correspondence, when in Sec.~\\ref{SecKlein1} we took these lattice structures seriously as a fixed background structure, we found that KG6 and KG7 were inequivalent; namely, they were here judged to have different symmetries. \n\nOnly in Sec.~\\ref{SecKlein2} and Sec.~\\ref{SecKlein3} when stopped taking the lattice structure so seriously did we ultimately see KG6 and KG7 as having the same symmetries. Indeed, in these later two interpretations KG6 and KG7 were seen to be completely equivalent. They are simply re-descriptions of a single theory with slightly different limitations in representational capacities. In Sec.~\\ref{SecKlein2} this re-description is a change of basis in the theory's value space, whereas in Sec.~\\ref{SecKlein3} this re-description is merely a change of coordinates. What we thought were distinct lattice theories are really just different sample points being used to describe one-and-the-same bandlimited field theory.\n\nIn light of the proposed analogy between lattice structure and coordinate systems this second lesson is not mysterious. Unsurprisingly, continuum theories presented to us in different coordinate systems may turn out to be equivalent. Moreover, we can always re-describe any continuum theory in any coordinates we wish.\n\nOur third lesson was that, in addition to being able to switch between lattice structures, we can also reformulate any discrete theory in such a way that it has no lattice structure whatsoever. I have shown two ways of doing this. In Sec.~\\ref{SecKlein2} this was done by internalizing the lattice structure into the theory's value space. In Sec.~\\ref{SecKlein3} this was done by embedding the discrete theory onto a continuous manifold using bandlimited functions. Adopting a lattice structure and switching between them was then handled using Nyquist-Shannon sampling theory discussed in Sec.~\\ref{SecSamplingTheory}.\n\nIn light of the proposed analogy between lattice structure and coordinate systems this third lesson is not mysterious. This is analogous to the familiar fact that any continuum theory can be written in a generally covariant (i.e., coordinate-free) way. Thus, the two above-discussed ways of reformulating a discrete theory to be lattice-free are each analogous to reformulating a continuum theory to be coordinate-free (i.e., a generally covariant reformulation). Thus we have not one but two discrete analogs of general covariance. See Fig.~\\ref{FigTwoAnalogies}.\n\n\\begin{figure}[t!]\n\\begin{flushleft}\n\\text{{\\bf Internal Discrete General Covariance:}}\n\\end{flushleft}\n$\\begin{array}{rcl}\n\\text{Coordinate Systems} & \\!\\!\\leftrightarrow\\!\\! & \\text{Lattice Structure}\\\\\n\\text{Changing Coordinates} & \\!\\!\\leftrightarrow\\!\\! & \\text{Changing Lattice Structures}\\\\\n\\ & \\ & \\text{by changing basis in value space}\\\\\n\\text{Gen. Cov. Formulation} & \\!\\!\\leftrightarrow\\!\\! & \\text{Basis-Free Formulation}\\\\\n\\text{(i.e., coordinate-free)} & \\ & \\text{(i.e., lattice-free)}\\\\\n\\end{array}$\\\\\n\\begin{flushleft}\n\\text{{\\bf External Discrete General Covariance:}}\n\\end{flushleft}\n$\\begin{array}{rcl}\n\\text{Coordinate Systems} & \\!\\!\\leftrightarrow\\!\\! & \\text{Lattice Structure}\\\\\n\\text{Changing Coordinates} & \\!\\!\\leftrightarrow\\!\\! & \\text{Changing Lattice Structures}\\\\\n\\ & \\ & \\text{by Nyquist-Shannon resampling}\\\\\n\\text{Gen. Cov. Formulation} & \\!\\!\\leftrightarrow\\!\\! & \\text{Bandlimited Formulation}\\\\\n\\text{(i.e., coordinate-free)} & \\ & \\text{(i.e., lattice-free)}\\\\\n\\end{array}$\n\\caption{A schematic of the two notions of discrete general covariance introduced \\cite{DiscreteGenCovPart1}. The internal strategy is applied to KG1-KG7 in Sec.~\\ref{SecKlein2} whereas the external strategy is applied to KG1-KG7 in Sec.~\\ref{SecKlein3}. These are compared in Sec.~\\ref{SecDisGenCov}. [\\textit{Reproduced with permission from} \\cite{DiscreteGenCovPart1}.]}\n\\label{FigTwoAnalogies}\n\\end{figure}\n\nBefore contrasting these two analogies, let's recap what they agree on. In either case, as one would hope, our discrete analog helps us to disentangle a discrete theory's substantive content from its merely representational artifacts. In particular, in both cases, lattice structure is revealed to be non-substantive and merely representational as is the lattice itself. Lattice structure is no more attached or baked-into to our discrete spacetime theories than coordinate systems are to our continuum theory. In either case, getting clear about this has helped us to expose our discrete theory's hidden continuous symmetries.\n\nWhat distinguishes these two notions of discrete general covariance is how they treat the lattice structure after it has been revealed as being coordinate-like and so merely representational. The approach in Sec.~\\ref{SecKlein2} was to internalize the lattice structure into the theory's value space. By contrast, the approach in Sec.~\\ref{SecKlein3} was to keep the lattice structure external, but to flesh it out into a continuous manifold such that it is no longer fundamental. Let us therefore call these two notions of discrete general covariance internal and external respectively.\n\nThese two approaches pick out very different underlying manifolds for our discretely-representable spacetime theories. As a consequence, they license very different conclusions about locality and symmetries. In particular, while for KG1-KG7 these internal and external approaches have more-or-less agreed as to what symmetries there are, they have disagreed about how they are to be classified.\n\nIn each of these differences I find reason to favor the external approach. To briefly overview my feelings: It is more natural for the continuous Poincar\\'e symmetries of KG1-KG7 to be classified as external. Moreover, keeping the lattice structure external as a part of the manifold, allows us to draw intuitions about locality from it. In particular, the external approach has a better way  of resolving some puzzles about locality and convergence in the continuum limit. However, neither of these reasons are decisive and I think either the internal or external approach is likely to be fruitful for further investigation\/use.\n\n\\section{A Perfectly Lorentzian Lattice Theory}\\label{PerfectLorentz}\nAs I will now discuss, we can leverage this analogy between lattice structures and coordinate systems to construct a perfectly Lorentz invariant lattice theory.\n\nAs discussed in Sec.~\\ref{SecFullGenCov} KG6 is not Lorentz invariant in an unqualified way due to its KPMs not being closed under boosts. Stating the condition that $\\phi_\\text{B}$ be bandlimited with $\\omega,k_1,k_2\\in[-K,K]$ requires the use of two space-like fields $X^a$ and $Y^a$ and a time-like field $T^a$ which break Lorentz invariance.\n\nAs discussed in Sec.~\\ref{SecKlein3}, we can see KG6 as a part of an extended theory. In terms of rotations, we can see KG6 as a part of an larger rotation invariant theory defined over $F^{\\sqrt{2}K}_\\text{rot.inv.}$ that is with $\\sqrt{k_1^2+k_2^2}<\\sqrt{2}K$. As I will now discuss, we can do the same thing here with Poincar\\'e invariance. Namely, we can find a Poincar\\'e invariant extension of KG6.\n\nIn particular, we can expand $F^K$ by adding to it every state reachable from $F^K$ by any Poincar\\'e transformation and then closing under addition. This defines $F^K_\\text{Poin.inv.}$ defined as follows: \\mbox{$f(t,x,y)\\in F^K_\\text{Poin.inv.}$} if and only if \n\\begin{enumerate}\n    \\item[1)] there exists some compact region $\\mathcal{K}(f)$ where the Fourier transform of $f$ is supported and \n    \\item[2)] this compact region $\\mathcal{K}(f)$ itself is contained within the portion of Fourier space satisfying \\mbox{$- d\\, K^2 \\leq w^2 - \\vert k\\vert^2 \\leq K^2$}\n\\end{enumerate}\nwith $d$ being the number of spacial dimensions and $\\vert k\\vert$ being the norm of the wavenumber, i.e., $d=2$ and \\mbox{$\\vert k\\vert=\\sqrt{k_1^2+k_2^2}$}. The first condition comes from the fact that Poincar\\'e transformations map compact regions in Fourier space to compact regions in Fourier space. The second condition comes from first closing $F^K$ under rotations in space and then Lorentz boosts then finite sums.\n\nTo get away from coordinates, let's now define $F^K_\\text{Poin.inv.}$ geometrically in terms of the metric $\\eta_\\text{ab}$. This definition comes in two parts. First let's define the space $F_\\text{B}$ which satisfies the first constraint. Consider the eigen-problem for functions $h:\\mathcal{M}\\to\\mathbb{R}$\n\\begin{align}\\label{FBdef}\n\\eta^\\text{ab} \\nabla_\\text{a}\\nabla_\\text{b} h= -\\lambda h\n\\end{align}\nA function $f:\\mathcal{M}\\to\\mathbb{R}$ is in $F_\\text{B}$ if and only if it is the sum of these eigensolutions over a compact range of $\\lambda\\in\\mathbb{R}$ of Fourier space. A function is further within $F^K_\\text{Poin.inv.}$ if and only if this compact range of $\\lambda$s is contained within \\mbox{$- d\\, K^2 \\leq \\lambda \\leq K^2$}.\n\nWe can thus define following perfectly Lorentz invariant lattice theory:\n\\begin{align}\\label{KGPoininvKGenCov}\n\\text{KG}_\\text{Poin.inv.}^K\\qquad\\text{KPMs:}\\quad&\\langle\\mathcal{M},\\eta_\\text{ab},\\phi_\\text{B}\\rangle\\\\\n\\nonumber\n\\text{DPMs:}\\quad&(\\eta^\\text{ab}\\nabla_\\text{a}\\nabla_\\text{b}-M^2)\\,\\phi_\\text{B} = 0,\n\\end{align}\nwith spacetime structures as defined following Eq.~\\eqref{KG0GenCov}, $\\mathcal{M}\\cong\\mathbb{R}^3$ and $\\phi_\\text{B}\\in F^K_\\text{Poin.inv.}$ in the sense discussed above. Note that KG6 is a subtheory of $\\text{KG}_\\text{Poin.inv.}^K$. \n\nHow does $\\text{KG}_\\text{Poin.inv.}^K$ differ from KG0? Their only difference is that we are here limited to manifolds $\\mathcal{M}\\cong\\mathbb{R}^3$ and that the field $\\phi_\\text{B}$ in $\\text{KG}_\\text{Poin.inv.}^K$ is bandlimited in $F_\\text{Poin.inv.}^K$, whereas the field $\\psi$ in KG0 is not. In fact, for either theory the dynamics guarantees that if the field amplitude starts off bandlimited it will stay bandlimited. Thus this restriction of the allowed dynamical fields is really just a restriction on the allowed initial conditions. Thus, ultimately the only difference between $\\text{KG}_\\text{Poin.inv.}^K$ and KG0 is a restriction on the initial conditions.\n\nAs innocent as this restriction on initial conditions may seem, I believe it has serious implications for the nature of the underlying manifold. As I will discuss later, $\\text{KG}_\\text{Poin.inv.}^K$ has a nice sampling property and therefore has access to discrete descriptions which make no reference to the manifold. By contrast, the manifold in KG0 seems descriptively essential. Moreover, as discussed in \\cite{DiscreteGenCovPart1}, this restriction on initial conditions has serious implications for counterfactual reasoning and locality.\n\nBut what makes $\\text{KG}_\\text{Poin.inv.}^K$ a lattice theory? Note that because every compact region is bounded in a large enough cube, every $f\\in F_\\text{B}$ is representable by a fine-enough cubic lattice. Doing so one would find its sample values on this lattice obey our original discrete equation for KG6, Eq.~\\eqref{DKG6}, with some rescaling of $\\mu$. Hence, as each state in this theory is representable on some lattice, this is a perfectly Lorentzian lattice theory. \n\nThis, should be compared with compared with the Lorentz-covariant sampling theory for fields developed in~\\cite{Pye2022}. There, only the second condition defining $\\text{KG}_\\text{Poin.inv.}^K$ is applied to the fields. Let us call this a Lorentz-covariant bandlimitation. Lorentz-covariant UV cutoffs have been considered across the physics literature recently, see~\\cite{Kempf2004a,Kempf2004b,Kempf_2010,Kempf2013,Pye2015,Kempf2018,PyeThesis,BEH_2020,Pye2022} among others.\n\nSince \\mbox{$- d\\, K^2 \\leq w^2 - \\vert k\\vert^2 \\leq K^2$} is not a compact region in Fourier space, the resulting fields are not bandlimited in the usual sense; as their support in Fourier space may be non-compact. Rather these functions are Lorentz-bandlimited. Such functions don't have the sampling property discussed in Sec.~\\ref{SecSamplingTheory}, but may have a new Lorentz sampling property. Indeed, \\cite{Pye2022} develops a new Lorentz-covariant sampling theory. Here none of that is necessary as our fields are each bandlimited in the original sense. Each field can be represented exactly on some lattice in the standard way discussed in Sec.~\\ref{SecSamplingTheory}.\n\nOne may still complain that this should not be called a lattice theory. For each state here we need a different lattice to represent it, there is no single ontologically significant lattice on which we can represent all of our states. As I will soon discuss, there is in fact such an all-representing lattice (albeit a non-fundamental one), but first let me argue against the spirit of this complaint.\n\nGiven the deflationary attitude taken in this paper towards lattice structures, the lack of an all-representing lattice wouldn't bother me. As I have argued throughout this paper, lattice structures are merely representational artifacts playing no substantive role in our theories: they do not limit our symmetries, they do not distinguish our theories, and they are not fundamentally ``baked-into'' our theories. Indeed, as I have argued (here and in \\cite{DiscreteGenCovPart1}), there is a rich analogy between the lattice structures appearing in our discrete spacetime theories and the coordinate systems appearing in our continuum spacetime theories. We wouldn't refer to our continuum spacetime theories as ``coordinate theories'', they are rather coordinate-representable theories. Analogously, I claim that properly understood, there are no such thing as lattice-fundamental theories, rather there are only lattice-representable theories. Hence, the above theory is a lattice theory in the strongest sense available.\n\nAccepting this, one might still feel there ought to be an all-representing lattice (albeit a non-fundamental one). As I will discuss, there is one but if there weren't this wouldn't be an issue. Note that we are often forced to represent our continuum theories with multiple overlapping complementary coordinate systems. That is, we often lack an all-representing coordinate system. Given the central analogy of this paper, why should the lack of an all-representing lattice bother us?\n\nReturning to answering the complaint directly however, there is such an all-representing lattice. Consider a cubic lattice with some fixed spacing. Consider another cubic lattice with twice the resolution, offset from the first lattice by some rational amount. Consider an infinite sequence of lattices each doubling in resolution with some rational offset. At some point any function $f\\in F_\\text{B}$ will be representable on one of these lattices. Consider the \\textit{deep lattice} which results from taking the infinite union of all of these lattices (but not their closure under the limit). Every function $f\\in F_\\text{B}$ is representable on this deep lattice\\footnote{Indeed any $f\\in F_\\text{B}$ can be represented on some finite length initial segment on this.}. Moreover, note that the deep lattice has a countable number of lattice sites, by construction it is a subset of the rationals $\\mathbb{Q}^3$.\n\nOne final comment on $F_\\text{Poin.inv.}^K$. Note that this space is closed under (finite) addition and scalar multiplication and is hence a vector space. As the deep lattice shows, this is a vector space with a countably infinite dimension. Thus, given its translation, rotation and Lorentz boost invariances $F_\\text{Poin.inv.}^K$ supports a countably infinite dimensional representation of the Poincar\\'e group.\n\nViewing KG6 as a part of $\\text{KG}_\\text{Poin.inv.}^{K}$ sheds some light on its failure to be Lotentz invariant in an unqualified way. To see how, it is instructive to restate this in terms of sample lattices. KG6 is a theory about all of the bandlimited functions which are representable on a fixed cubic sample lattice with spacing $a$. By contrast, $\\text{KG}_\\text{Poin.inv.}^{K}$ is a theory about\\footnote{Technically, $\\text{KG}_\\text{Poin.inv.}^{K}$ is about this collection of bandlimited functions closed under addition.} all of the bandlimited functions representable on a fixed cubic sample lattice with spacing $a$ \\textit{or any Poincar\\'e transformation thereof.}\n\nThus, KG6 is the part of $\\text{KG}_\\text{Poin.inv.}^{K}$ which is visible to us when we restrict ourselves to only a certain set of representational tools. We can thus see KG6's failure to be Lorentz invariant in an unqualified way as a problem of representational capacity and not of physics. Sure, if we limit ourselves to functions which are representable on a fixed cubic lattice, we lose rotation symmetry. But what physical reason do we have to limit our representational tools in this way?\n\nConsider an analogous situation involving coordinate systems. Almost all manifolds cannot be covered with a single global coordinate system. For instance, the 2-sphere needs at least two coordinate systems to cover it. Consider the sphere under arbitrary rotations. Note that no coordinate system is closed under these transformations. If we limit ourselves to functions which are supported entirely within the scope of a single fixed coordinate system, we lose rotation symmetry. Would it be right to say that no theory which is set on a sphere can be rotation invariant? Of course not, this just reflects the fact that each of our coordinate systems individually have a limited representational capacity. We can regain rotation invariance by using multiple representational tools, i.e., multiple coordinate systems. \n\nApplying this lesson to KG6, we ought to view it as being able to represent a Lorentz non-invariant part of a wider Lorentz invariant theory. As discussed above, making use of multiple sample lattices (i.e., all Poincar\\'e transformed cubic lattices with spacing $a$) we can describe a rotation invariant theory $\\text{KG}_\\text{Poin.inv.}^{K}$ over $F^{K}_\\text{Poin.inv.}$.\n\nThe equivalence between KG6 and KG7 can also be seen this way. Note that the coordinate transformation which maps KG7 onto KG6, namely Eq.~\\eqref{CoorKG7KG6}, maps a cubic 3D lattice to a hexagonal 3D lattice maintaining the lattice spacing $a$, see Fig.~\\ref{FigSkew}. Thus, KG7 is a theory about all bandlimited functions representable on a hexagonal 3D sample lattice with spacing $a$. The subsets of KG6 and KG7's KPMs which are equivalent to each other ($F^K_\\text{KG6}\\cong F^K_\\text{KG7}$) are exactly those parts which are representable on both a square and hexagonal lattice. All of KG6 and KG7's DPMs are representable in both ways, but not all of their KPMs. We can here see KG6 and KG7 as parts of a larger unified theory viewed in two ways with differently limited representational capacities.\n\nNote that even the extended theory $\\text{KG}_\\text{Poin.inv.}^{K}$ has an artificially limited representational capacity. What space do we find if we use all of our representational tools (i.e., every possible sample lattice)? The answer is $F_\\text{B}$ defined above.\n\nContrast $F_\\text{B}$ with $F(\\mathbb{R}^3)$ the space of all functions $f(t,x,y)$. The Gaussian distribution is in $F(\\mathbb{R}^3)$ but not $F_\\text{B}$. One might gloss this difference as follows: $F(\\mathbb{R}^3)$ contains actually infinite frequencies, whereas $F_\\text{B}$ contains arbitrarily large but always finite frequencies. That is, $F_\\text{B}$ contains only a potential infinity of frequencies. Note that while $F_\\text{B}$ is closed under finite sums it is not closed under infinite sums.\n\nUsing $F_\\text{B}$ we can thus define following perfectly Lorentz invariant lattice theory:\n\\begin{align}\\label{KGBGenCov}\n\\text{KG}_\\text{B}\\text{:}\\qquad\\text{KPMs:}\\quad&\\langle\\mathcal{M},\\eta_\\text{ab},\\phi_\\text{B}\\rangle\\\\\n\\nonumber\n\\text{DPMs:}\\quad&(\\eta^\\text{ab}\\nabla_\\text{a}\\nabla_\\text{b}-M^2)\\,\\phi_\\text{B} = 0,\n\\end{align}\nwith spacetime structures as defined following Eq.~\\eqref{KG0GenCov}, $\\mathcal{M}\\cong\\mathbb{R}^3$ and $\\phi_\\text{B}\\in F_\\text{B}$ in the sense defined following Eq.~\\eqref{FBdef}.\n\nWe can even give $\\text{KG}_\\text{B}$ a diffeomorphism invariant reformulation as:\n\\begin{align}\\label{SR2like}\\text{KG}_\\text{B}'\\text{:}\\qquad\\text{KPMs:}\\quad&\\langle\\mathcal{M},g_\\text{ab},\\phi_\\text{B}\\rangle\\\\\n\\nonumber\n\\text{DPMs:}\\quad&(g^\\text{ab}\\nabla_\\text{a}\\nabla_\\text{b}-M^2)\\,\\phi_\\text{B} = 0\\\\\n\\nonumber\n&R^\\text{a}{}_\\text{bcd}=0\n\\end{align}\nHere $\\mathcal{M}\\cong\\mathbb{R}^3$. The fixed Lorentzian metric field, $\\eta_\\text{ab}$, has been replaced with a dynamical metric field, $g_\\text{ab}$. The dynamical metric field varies from model to model. $\\nabla_\\text{a}$ is still the unique covariant derivative operator compatible with the metric, (i.e. with $\\nabla_\\text{c}\\,g_\\text{ab}=0$). The dynamical metric field obey a new dynamical equation, $R^\\text{a}{}_\\text{bcd}=0$, where $R^\\text{a}{}_\\text{bcd}$ is the Riemann tensor associated with the unique compatible derivative $\\nabla_\\text{a}$. The dynamical field $\\phi_\\text{B}$ is bandlimited in the same sense as above (i.e., $\\phi_\\text{B}\\in F_\\text{B}$) with two differences. Firstly, $g_\\text{ab}$ replaces $\\eta_\\text{ab}$ in Eq.~\\eqref{FBdef}. Secondly, this condition ought to be viewed as a compatibility constraint acting jointly on the metric $g_\\text{ab}$, the covariant derivative $\\nabla_\\text{a}$, and the field $\\phi_\\text{B}$. Comparing with \\cite{Pooley2015}, $\\text{KG}_\\text{B}$ is like SR1 and $\\text{KG}_\\text{B}'$ is like SR2. \n\nNote that the above discussion is suggestive of a possible procedure by which any spacetime theory can be given a bandlimited formulation and so gain the sampling property discussed in Sec.~\\ref{SecSamplingTheory}. \n\nPerhaps surprisingly, all of the above claims about $\\text{KG}_\\text{Poin.inv.}^K$ hold for $\\text{KG}_\\text{B}$ (and relatedly $\\text{KG}_\\text{B}'$). Its state space $F_\\text{B}$ is closed under Poincar\\'e transformation. Moreover, it is closed under generic diffeomorphisms. Every function $f\\in F_\\text{B}$ is representable on some fine-enough cubic lattice. Doing so one would find its sample values on this lattice obey Eq.~\\eqref{DKG6} with some rescaling of $\\mu$.\n\nNote that this space is closed under (finite) addition and scalar multiplication and is hence a vector space. Moreover, $F_\\text{B}$ has a deep lattice just like $F^K_\\text{Poin.inv.}$ does. Indeed, it has exactly the same deep lattice. Hence $F_\\text{B}$ is a countably infinite dimensional vector space. It supports a countably infinite dimensional representation of the Poincar\\'e group.\n\nAs I see it, the proper view of KG6 (even beginning from $\\phi_\\ell$) is the one given by Eq.~\\eqref{DKG6GenCov} understood as a part of the maximally extended theory $\\text{KG}_\\text{B}$ (or $\\text{KG}_\\text{B}'$ depending on taste). In particular, KG6 is the part of $\\text{KG}_\\text{B}$ which is visible to us if we restrict ourselves to only certain representational tools.\n\nHow does $\\text{KG}_\\text{B}$ differ from the continuum theory KG0? KG0 assents to the existence of actually infinite frequencies whereas $\\text{KG}_\\text{B}$ does not. To me $\\text{KG}_\\text{B}$ appears to be on better ground empirically speaking than KG0: I have never and will never see a literally infinite frequency photon. As small as the difference is between $\\text{KG}_\\text{B}$ and KG0, I expect there to be radical differences in views on spacetime, but this is a topic for another paper.\n\n\\section{Conclusion}\\label{SecConclusion}\nThis paper has given an exhaustive study of the seven discrete Klein Gordon equations presented in Sec.~\\ref{SecSevenKG}. These theories were initially formulated in terms of a real valued function over lattice sites, $\\phi_\\ell\\in F_L$. Next, as an infinite dimensional vector, $\\bm{\\Phi}\\in \\mathbb{R}^L$. And finally, as a bandlimited field $\\phi_\\text{B}\\in F^K$ over a continuous spacetime manifold $\\mathcal{M}$. Each of these redescriptions was carried out by a vector space isomorphism, $F_L\\cong\\mathbb{R}^L\\cong F^K$.\n\nAlong the way, we have learned three substantial lessons about the role that lattice structures play in our discrete spacetime theories. These lessons serve to undermine the three first intuitions about lattice structure laid out in Sec.~\\ref{SecIntro}. As I have shown, lattice structures don't restrict symmetries, they don't distinguish our theories, and they are not fundamentally ``baked-into'' these theories. As I have discussed, these lessons lay the foundation for a rich analogy between the lattice structures which appear in our discrete spacetime theories and the coordinate systems which appear in our continuum spacetime theories. Indeed, my analysis has shown that lattice structure is rather less like a fixed background structure or a fundamental part of some underlying manifold and rather more like a coordinate system, i.e., merely a representational artifact. This extends the results of \\cite{DiscreteGenCovPart1} to a Lorentzian setting.\n\nBased upon this analogy, we have two discrete analogs of general covariance (see Fig.~\\ref{FigTwoAnalogies}) each of which are useful for exposing hidden symmetries and background structures in our discrete spacetime theories (i.e., lattice theories). In either case, as hoped, when applied to such theories this discrete analog helps us disentangle the theory's substantive content from its representational artifacts. \n\nThese results are significant as they tell strongly against the intuitions laid out in Sec.~\\ref{SecIntro}. One might have an intuition that the world could be fundamentally set on a lattice. This lattice might be square or hexagonal and we might discover which by probing the world at the smallest possible scales looking for violations of rotational symmetry, or other lattice artifacts. Many serious efforts at quantum gravity assume that the world is set on something like a lattice at the smallest scales (although these are often substantially more complicated than the lattices assumed here). However, as this paper clearly demonstrates this just cannot be the case. \n\nThe world cannot be ``fundamentally set on a square lattice'' (or any other lattice) any more than it could be ``fundamentally set in a certain coordinate system''. Like coordinate systems, lattice structures are just not the sort of thing that can be fundamental; they are both thoroughly merely representational. Spacetime cannot be a lattice (even when it might be representable as such). Specifically, I claim that properly understood, there are no such things as lattice-fundamental theories, rather there are only lattice-representable theories.\n\nThe primary inspiration for this paper has been the work of mathematical physicist Achim Kempf~\\cite{Kempf_1997,UnsharpKempf,Kempf2000b,Kempf2003,Kempf2004a,Kempf2004b,Kempf2006,Martin2008,Kempf_2010,Kempf2013,Pye2015,Kempf2018} among others~\\cite{PyeThesis,Pye2022,BEH_2020}. For an overview of Kempf's works on this topic and his closely related thesis that ``Spacetime could be simultaneously continuous and discrete, in the same way that information can be''~\\cite{Kempf_2010}, see~\\cite{Kempf2018}. My thesis here inserts a ``representable as'' and then analyzes which parts of these representations should be thought of as substantial. Lattice structures turn out to be coordinate-like representational artifacts.\n\nFor a discussion of the impact I anticipate these results having on the philosophy of space and time generally, see the conclusion of \\cite{DiscreteGenCovPart1}. This includes some discussion of the dynamical versus geometric debate regarding how we should view spacetime structures~\\cite{EarmanJohn1989Weas,TwiceOver,BelotGordon2000GaM,BrownPooley1999,Nonentity,HuggettNick2006TRAo,StevensSyman2014Tdat,DoratoMauro2007RTbS,Norton2008,Pooley2013}. As we are now in a Lorentzian context, I will focus my attention on the aspects of this work which relate to quantum gravity and Lattice QFT.\n\nPeople often have arguments (e.g., from quantum gravity) that spacetime must be a lattice of some sort, are such arguments in disagreement with my conclusion? This would need to be checked on a case-by-case basis, but there is room for compatibility here. Upon closer investigation, such arguments may only show that spacetime must be \\textit{representable as} a lattice. In this case there is no disagreement with my conclusion. For instance, bandlimited functions can always be represented as a lattice of sample values; it's just that upon closer philosophical investigations these lattice representations cannot possibly be fundamental, as they are merely coordinate-like representational tools. However, if their arguments carry ontological force as well as representational force, then there is disagreement.\n\nWhat does my conclusion mean for those who model spacetime as a lattice (e.g., some approaches to quantum gravity)? There is no issue per se with modeling spacetime as a lattice: e.g., as I just mentioned bandlimited physics is discretely representable. My conclusion only speaks to how such models should be interpreted. Indeed, those who describe their physics on a lattice for reasons of approximation (e.g. the Lattice QFT community, people studying crystalline solid state systems) are free to ignore my conclusion. However, they ought not ignore the central analogy or the three lessons which support it. \n\nFor them my lesson is this: ``lattice artifacts'' arise under the following two conditions. They arise when 1) we represent our continuum physics on a lattice \\textit{2) and then modify the dynamics to be simple in this representation}. As this paper has shown, the blame is to be placed entirely on the second step. Square-shaped lattice artifacts do not come from using a square lattice, see Fig.~\\ref{FigEvolutionKG4}. These artifacts come from using dynamics which are relatively simple (i.e., nearest neighbor interactions) when represented on a square lattice. One can define rotation invariant dynamics on a square lattice, Fig.~\\ref{FigEvolutionKG6}, but not with nearest neighbor interactions.\n\nSo how should lattice-based approaches to quantum gravity be interpreted then? Namely, what about loop quantum gravity (LQG) and causal set theory~\\cite{Surya2019}? I am not an authority on either of these approaches, but I can offer some comments. The conclusion of this paper does not directly apply to LQG. This is because the spacetime considered in LQG consists not only of lattice sites but also links\/edges between lattice sites. In order to more directly apply, the results of this paper would need to be extended to more complicated gauge theories first.\n\nRegarding causal set theory~\\cite{Surya2019}, its proponents often note that no fixed spacetime lattice is Poincar\\'e invariant. This (apparently) spells big trouble for any lattice-based Lorentzian theories. They, however, avoid this issue by considering instead a random Poisson sprinkling of lattice points which does not pick out any preferred direction and hence does not explicitly break Poincar\\'e symmetry, at least on average. However, given the deflationary position this paper takes towards lattices, I claim there is no issue to be avoided. Like coordinate systems, lattice structures are just a representational tool for helping us express our theory in a certain way. There is no need for the symmetries of our representational tools to latch onto the symmetries of the thing being represented. Cartesian coordinates are distorted under Lorentz boosts, but we can still use them to describe our Lorentzian theories without issue. Indeed, in Sec.~\\ref{PerfectLorentz} I have presented a perfectly Lorentzian lattice theory.\n\n\\begin{acknowledgments}\nThe authors thanks James Read, Achim Kempf, Jason Pye, and Nick Menicucci and the Barrio RQI for their helpful feedback.\n\\end{acknowledgments}\n\n\\bibliographystyle{apsrev4-1}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\nLyapunov control has been proven successful in controlling highly chaotic non-linear oscillators \\cite{Ref1}\\cite{Ref2}\\cite{Ref3} . One of the fundamentals that contribute to the success or failure of any type of control strategy is the controller and system parameters. Therefore, researchers have explored different methods to find the precise parameters that would lead to achieving the best system results \\cite{Ref4}\\cite{Ref5}. \\\\\nOne of the methods that was utilized to achieve the previously mentioned goals is Genetic algorithm (GA). GAs have been successful in cases where all the system dynamics are clearly defined and known to some extent or with systems where limited system disturbances are introduced and minor parameter tuning is required \\cite{Ref6}\\cite{Ref7}. In some cases, several system assumptions are needed in order to allow the GA to run successfully.\\\\ \n\nDue to some of the limitations found in using GAs such as inability to quickly converge to the final solution or adapt to unknown system dynamics or unknown disturbances. Researchers though after different approaches that wouldn't reduce the system agility and at the same time would be able to handle unknown system characteristics. A hybrid approach of Fuzzy Control and GAs was researched \\cite{Ref8} but system linearization is a requirement in order to use the previously mentioned method .\\\\ \n\nAnother approach that is recently being researched is the use of Machine Learning to enhance the controller performance. For example through the use of Episodic learning \\cite{Ref9}\\cite{Ref10}. Most recently, there is the introduction neural lyapunov control which proposes the use of deep learning to find the control and Lyapunov functions. The approach mentioned in \\cite{Ref9}\\cite{Ref10} is suitable for find the best system parameters that would initially lead the system to stability and reduced the system error. The problem with approach in \\cite{Ref9}\\cite{Ref10} is that it assumes that the system is deterministic, time invariant, and affine in the control input. while in real life situation external perturbations might occur resulting in system failure at any moment while the system is running \\cite{Ref11}\\cite{Ref12}. The approach proposed in \\cite{Ref25} \\cite{Ref26} and \\cite{Ref27}attempts to predict the control and Lyapunov functions that would lead to system stability but under specific conditions where the system dynamics are deterministic in nature. The approach proposed in this research is novel to the best of our knowledge in that it discards the assumption of an ideal environment or fully known system dynamics and seeks continuous enhancement of the controller outcome through continuous monitoring of the system error, reference signal, system dynamics and control signal and accordingly adjust the system and controller parameters to improve the controller performance without the need to disrupt the system output.\n\n The focus of the research is to allow the Deep Learning Algorithm to learn the system from a continuously improving dataset and according to the slope of the output error the algorithm relearns the system and collects the needed information.\\\\   \n\nThe proposed method is applied to a non-linear chaotic combined system of Duffing and Van der pol oscillators\\cite{Ref24}. The aforementioned system was chosen to test the Deep learning algorithm response to unpredicted system disturbances and unknown system dynamics\\cite{Ref21}\\cite{Ref22}\\cite{Ref23}. An algorithm was developed to aid and trigger the Deep Neural Network when needed to adapt to new system dynamics and according to preset conditions. The algorithm records and feeds an updated data set to the DNN in order to relearn the system dynamics if certain conditions are detected to be true. Once the process of retraining is complete, the algorithm generates a random array of parameters and the array of parameters is fed to the DNN to predict the output error. If the DNN predicts that the error slope will be reduced from the current value improving the performance of the system, then the suggested controller parameters are allowed to be set as the new controller parameters. Otherwise the algorithm would create a new array set of randomly generated parameters and re-feed them to the DNN to provide its predictions in a continuous loop. Once the controller parameters are updated the algorithm monitors the actual error compared to the predicted error to determine the network viability and accordingly if the error difference is greater than a set value the algorithm would trigger a retraining request of the DNN. \\\\\n\\section{LYAPUNOV CONTROL ON DUFFING - VANDERPOL OSCILLATOR MODEL}\n\\label{Section II}\nThis section provides an overview of the Duffing-Van der pol system dynamics. \nDuffing-Van der pol mathematical model is  \n\\begin{equation}\n\\label{eq:1}\n\\ddot{x}+ \\delta\\dot{x}+ \\epsilon_1( x^2\\ddot{x}+x\\dot{x}^2)+\\epsilon_2x^3 = \\mu(t) + \\gamma cos(\\omega t)\t\n\\end{equation} \nIt combines both non-linear stiffness and non-linear damping [1].\nSetting\n\\begin{equation}\n\\label{eq:2}\n\\dot{x} = x_2  \n\\end{equation} \nand\n\\begin{equation}\n\\label{eq:3}\n\t x = x_1\n\\end{equation}\ntherefore\n\\begin{equation}\n\\label{eq:4}\n\t \\dot{x_1} = x_2\n\\end{equation} \nwe get the state space model\n \\begin{equation}\n\\label{eq:5}\n\t \\dot{x_2} = \\frac {1} {1+\\epsilon_1 X^2} (\\delta \\dot{x} - \\epsilon_1 x \\dot{x^2} -  \\epsilon_2 x^3 +\\mu(t) +\\gamma cos(\\omega t)) \n\\end{equation}   \n\n\\begin{equation}\n\\label{eq:6}\n\\begin{bmatrix}\n\\dot{x_1}  \\\\ \n\\dot{x_2}  \n\\end{bmatrix}\n =\n  \\begin{bmatrix}\n   x_2 \\\\\n  \\frac{1}{1+ \\epsilon_1 x^2} (\\delta x_2 - \\epsilon_1 x x_2^2 - \\epsilon_2x^3)  \n   \\end{bmatrix}\n\\end{equation}\n\\begin{equation}\n\\label{eq:7} \ny  = \\begin{bmatrix}  x_1 \\\\  x_2     \\end{bmatrix}\n\\end{equation}  \n\\section{LYAPUNOV CONTROLLER DESIGN}\n\\label{Section III}\nIn this section we give an overview of the controller design. The controller is designed with the purpose of achieving Duffing- van der pol oscillator system stability at an increased forcing frequency. Desired state q\\textsubscript{d} and actual state q.\n\\begin{equation}\n\\label{eq:8}\ne=q_d - q\n\\end{equation} \nA control LYAPUNOV candidate is then chosen to be\n\\begin{equation}\n\\label{eq:9}\nV = \\frac{1}{2} (\\gamma_1 \\dot{e} + \\gamma_2 e)\n\\end{equation} \nAs can be seen from \\cite{Ref11}, the transient behaviour of the error dynamics can be influenced by a suitable choice of the CLF and gamma parameters, respectively \\cite{Ref10}. Therefore the CLF has been chosen to be positive definite for all \\begin{equation} x,\\dot{x} \\end{equation} taking the time derivative of V\n\\begin{equation}\n\\label{eq:11}\n\\dot{V} = (\\gamma_1 \\dot{e} + \\gamma_2 e)(\\gamma_1 \\ddot{e} + \\gamma_2 \\dot{e})\n\\end{equation} \n\\begin{equation}\n\\label{eq:12}\n\\dot{V} = -kV\n\\end{equation} \nwhich is globally exponentially stable as V is globally positive definite. Hence, by substitution in Eqn.\\ref{eq:11} we find the control law\n\\begin{equation}\n\\label{eq:13}\nu = \\frac{1+\\epsilon_1 x^2}{\\gamma_2} (\\frac{k}{2} \\gamma_1 e+ \\frac {k}{2} \\gamma_2  \\dot{e} + \\gamma_1  \\dot{e} + \\gamma_2 \\ddot{x_des}) - pCos(\\omega t)+\\delta \\dot{x} + x + \\epsilon_1  \\dot{x} x^2 + \\epsilon_2 x^3\n\\end{equation} \n\\section{SIMULATION RESULTS FOR UNOPTIMIZED SYSTEM}\n\\label{Section IV}\nA Duffing- van der pol oscillator model with \\textdelta=0.5, \\textepsilon\\textsubscript{1}=1.6, \\textepsilon\\textsubscript{2}=-0.8, P=3, \\textomega=10 and the control parameters \\textgamma\\textsubscript{1} =12, \\textgamma\\textsubscript{2} =4, k=115. The parameters were manually adjusted and tuned to give low error \\ref{fig:1} and high stability \\ref{fig:2}.\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3in]{1.png}\n\\caption{Reference signal (dotted green) $5^{th}$ harmonic function with amplitudes of 0.1, 0.5 and 1 and Lyapunov controller output (red) for t=2.5s}\n\\label{fig:1}      \n\\end{figure}\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3in]{2.png}\n\\caption{Phase portrait showing a stable system with no interference under the harmonic reference signal t=2.5s}\n\\label{fig:2}      \n\\end{figure}\n\\section{LYAPUNOV CONTROLLER PARAMETER OPTIMIZATION}\n\\label{Section V}\n\\subsection{OVERVIEW}\nThe process of selecting the proper Sigmas to reduce the error of the model can be modelled as an exhaustive search problem. Where several Sigmas are tried and the one that results in a reduction in the system error is selected. Though this process results in the best Sigmas, yet it is time consuming. To overcome this problem, we developed a deep learning model that researches different system and controller parameters. A given sigma is good if it reduces the error of the system. The pipeline is presented in Fig. \\ref{fig:6}. Initially, the model is run, and the system error is monitored. If the error surpasses a curtain threshold the neural network is queried for a sigma that would reduce the system error given the current state of the system. The algorithm for the entire process is defined below.\nThe network proposed in Fig.\\ref{fig:5} is initially trained and then used to predict new Sigmas; Sigma 1 ($s_1$), and Sigma 2 ($s_2$) if the system error (sys err) exceeds a given threshold (0.8). The network is then given the current state of the system in terms of time (t), position (p), velocity (v), and the randomly generated ($s_1$) and its corresponding ($s_2$) and it predicts the system error. If the predicted error is less than the current system error, then the system is updated with the newly generated Sigmas that are speculated to reduce the error. Every iteration the newly generated Sigmas are their corresponding system parameters are stored. After each 100 iterations the system is probed for perturbations by computing the average of the system error throughout the previous 100 iterations (avg\\_sys\\_err) if this error is larger than the previous system average (prev sys avg) the network is retrained on a portion of the old data (Memo) used in previous training as well as the data from the previous 100 iterations (new data). The motive behind using the old data as well as the new data is to avoid catastrophic forgetting as mentioned in \\cite{Ref14}. \\\\\n The idea of using a memory is usually known as self-refreshing memory and was initially introduced and tested in \\cite{Ref15}\\cite{Ref16}\\cite{Ref17}. The idea here is that even when there are perturbations they might not last for so long and consequently we don't want the network to lose its ability to predict the System error if the system parameters return back to norm after the network was retrained several times.\n\\begin{algorithm}\n\\SetAlgoLined\nMemory = 10\\% of the old training data prev\\_sys\\_avg = system error average of the training data\n initialization\\;\n \\While{n $>$ 2000}{\n sys\\_err = current\\_system\\_error\\;\n\\If{sys\\_err  $>$ 0.8}\n\t           { \\While {True}\n\t                       {\n\t\t\t\t$s_1$ = randomly generated\\;\n\t                        $s_2$ = - $s_1$\/8\\;\n\t                        predicted\\_sys\\_err = predict (t, V, p, v, $s_1$, $s_2$)\\;\n\t\t\t\t\\If {predicted\\_sys\\_err  $<$  sys\\_err}\n\t                             {\n\t\t\t\t\tBreak\\;\n\t\t\t\t      \t\t}\n\t\t\t\t\t}\n\t\t\t\tupdate\\_controller ($s_1$, $s_2$)\\;\n\t                \tnew\\_data = Save (t, V, p, v, $s_1$, $s_2$, new\\_sys\\_err)\\;\n\t                \tavg\\_sys\\_err = avg\\_sys\\_err + new\\_sys\\_err\\;}\n\\If {n == 0}{\n\t              \\If { avg\\_sys\\_err \/ 100 $>$ prev\\_avg}\n\t                 { retrain\\_network(Memo + new\\_data)\\;\n\t                   Memo = Memo + 10\\% of the new\\_data\\;\n\t                   prev\\_sys\\_avg = avg\\_sys\\_err\\;\n\t                   Clear new\\_data\\;\n\t                   Clear avg\\_sys\\_err\\;}\n}\n}\n \\caption{Adaptive Algorithm Steps for Sigma prediction}\n\\end{algorithm}\n\\subsection{DATASET}\n\\label{sub1}\nThe Time, Error, velocity, position and control signal were collected every 1 millisecond and added into an array of values. The dataset was split into 60\\% training and 40\\% test. A continuous update to the dataset is done according to the calculated error slope every 10ms (100 iterations) as explained above.\n\\subsection{NETWORK ARCHITECTURE}\n\\label{sub2}\nThe network architecture is in Fig. \\ref{fig:3} The network is composed of 5 blocks (15 layers). The number of layers 15 was chosen for its efficiency and high performance. It was found that if a lower number of layers is used the DNN performance was affected and when a higher number of layers was used it had no effect on the performance but reduced the DNN efficiency.  A block is composed of 3 layers: a fully connected convolution layer followed by batch normalization layer and rectified linear unit (RELU) in Eqn. \\ref{eq:14}.\n\\begin{equation}\n\\label{eq:14} \noutput = max (0.0, input)\n\\end{equation} \nThe batch normalization layer allows us to use higher learning rates as it makes ensures that activations are not too high or low. The RELU is a nearly linear function in the sense that it is a piece wise linear function with two linear pieces. This feature increases its capability to preserve many of the properties that make linear models easy to optimize with gradient-based methods as well as the properties that make linear models generalize well. The last block has a dropout layer aside from the fully connected convolution layer and a regression layer to predict the error. Dropout \\cite{Ref18} is a regularization method that was developed to solve the problem of overfitting. The overfitting problem occurs when the neural network learns every minor detail in the training data. The effect of such a behaviour is that the network will have high accuracy on the training data but very low accuracy on the test data. This means that the network is unable to generalize on unforeseen data. The proposed solution for this problem is the dropout technique. The dropout technique ignores a number of layer outputs randomly. Such a behaviour makes the layer appear as and be treated-as a layer with a different number of nodes and connectivity to the prior layer. Consequently, each update to a layer during training is performed with a different \"perspective\" of the configured layer.\n The regression layer computes the mean squared error loss. The network is trained to learn a function F in Eqn. \\ref{eq:15}.\n\\begin{equation} \n\\label{eq:15}\nF : (t, x, v, s_1, s_2) \\rightarrow e\n\\end{equation} \n\nWhere \\textit{s\\textsubscript{1}} and \\textit{s\\textsubscript{2}} are the two Sigmas, \\textit{t} is the time and \\textit{e} is the error of the model.\n\\subsection{NETWORK TRAINING AND TESTING}\\label{sub3}\nThe network is initially trained for 5 epochs using Adam optimization algorithm. The learning rate is initially set to 0.001 and is reduced by a factor of 0.2 every 5 epochs. This allows large weight changes in the beginning of the learning process and small changes or fine-tuning towards the end of the learning process. This gives more time for fine-tuning. While the model is running if the system error passes a given threshold the Neural network is used to choose two Sigmas that would lower the error. This is done by selecting a uniformly distributed random number in the interval [-50, 50] for $s_1$. The restriction of $s_1$ and $s_2$ was added later in the research to avoid the genetic algorithm and DNN going into a continuous loop of changing the Sigmas to find the perfect candidate. It was also found that the smaller the sigma the better the system outcome but in order to maintain controller flexibility the range was set to [-50, 50] to account for unexpected changes in the system behaviour. \nThe network does not pass the proposed Sigma 1 and Sigma 2 to the model and controller unless Sigma 1 and Sigma 2 are predicted to reduce the system error. This helps in tuning the entire pipeline to take small\/large steps to change the behaviour of the duffing van der pol model given that sudden large changes in the Sigmas may result in system failure. \n The idea behind predicting the error and not the Sigmas is the fact that we can have more than one error for a given Sigma, which gives more flexibility for choosing the range of sigmas to consider.  The root of the mean square error graph for the first 5 epochs are plated in Fig. \\ref{fig:5}. The results show that the neural network presented in Fig.  \\ref{fig:6} was able to predict the error with an error of 0.03564.\n \\begin{center}\n \\begin{figure}[!t]\n\\centering\n\\includegraphics[width=0.7in]{4.png}\n\\caption{Neural network used for predicting the best Sigmas that would result in a decrease in the system error. The input to the network is the time (\\textit{t}), displacement (\\textit{x}), velocity (\\textit{v}), Sigma 1 (\\textit{$s_1$}), Sigma 2 (\\textit{$s_2$}) and the output is the expected system error (\\textit{Error}) for the system. $s_2$ = - $s_1$\/8}\n\\label{fig:3}      \n\\end{figure}\n \\end{center}\n\n\\begin{figure}[!t] \\centering\n\\includegraphics[width=3in]{5.png}\n\\caption{Neural network used for predicting the best Sigmas that would result in a decrease in the system error. The input to the network is the time (\\textit{t}), displacement (\\textit{x}), velocity (\\textit{v}), Sigma 1 (\\textit{$s_1$}), Sigma2 (\\textit{$s_2$}) and the output is the expected system error (\\textit{Error}) for the system}\n\\label{fig:5}      \n\\end{figure}\n\n\\section{RESULTS AND DISCUSSION}\n\\label{Section V}\nIn Fig. \\ref{fig:5} the system is shown to go into instability and fails 0.3 seconds from start due to the use of unfit Sigma 1 = 100 and Sigma 2 = 0.6 while in Fig. \\ref{fig:6} we show that using the proposed deep neural network to predict the appropriate Sigma1 and Sigma 2 lead to stabilizing the system and maintaining the system error under the set threshold. The results show the efficiency of the neural network in predicting the error given the sigma, as well as the efficiency of the algorithm in preventing the system from failing likewise in continuously enhancing its performance by keeping the system error as low as possible. On the other hand, maintaining a constant sigma results in large system error and the system might eventually fails.\n\\begin{figure}[!t] \\centering\n\\includegraphics[width=3in]{7.png}\n\\caption{The system error goes to infinity as shown when the algorithm is not applied}\n\\label{fig:6}      \n\\includegraphics[width=3in]{6.png}\n\\caption{The system error after using the neural network in Fig. 7 to get a good estimate of $s_1$ and $s_2$.}\n\\end{figure}\n\\subsection{Algorithm Reaction to parameter sabotage}\n\\label{sub4}\nTo elaborate on the ability of the proposed solution to tackle mid system instability or sudden changes. The system parameters are manually overwritten while the system is running to measure the NN and Algorithm effectiveness in returning the system to stability.\\\\\nAs demonstrated by Figs \\ref{fig:7} and \\ref{fig:8} the deep network was able to predict the best parameters to re-establish stability of the system and controller within 0.4 ms. \n\n\\begin{figure}[!t] \\centering\n\\includegraphics[width=3in]{8.png}\n\\caption{Error uptick at t = 0.8867}\n\\label{fig:7}\n\\end{figure}\n\\begin{figure}[!t] \\centering\n\\includegraphics[width=3in]{9.png}\n\\caption{The Algorithm reacting to the sudden change by adjusting $s_1$}\n\\label{fig:8}  \n\\end{figure}\n\\begin{figure}[!t] \\centering\n\\includegraphics[width=3in]{10.png}\n\\caption{The Algorithm reacting to the sudden change by adjusting $s_2$}\n\\label{fig:9}  \n\\end{figure}\n\n\\subsection{Improving the system performance}\n\\label{sub5}\nIn this section we show the ability of the proposed method in finding the best system parameters while the system and controllers are running.\n\\begin{figure}[!t] \\centering\n\\includegraphics[width=3in]{11.png}\n\\caption{Phase diagram of the algorithm before and after finding the optimal system parameters for the conditions which shows major improvement compared to Fig.\\ref{fig:5}}\n\\label{fig:10}  \n\\end{figure}\n\\begin{figure}[!t] \\centering\n\\includegraphics[width=3in]{12.png}\n\\caption{$s_1$ and $s_2$ are changed while the system is running to improve performance and reduce error}\n\\label{fig:11}  \n\\end{figure}\n\\begin{figure}[!t] \\centering\n\\includegraphics[width=3in]{13.png}\n\\caption{Error is shown to be reduced from the starting point by updating the system parameters and the algorithm is successful in maintain the error within bounds}\n\\label{fig:12}  \n\\end{figure}\n\\subsection{Network Retraining}\n\\label{sub5}\nDuring the re-training phase of the neural network we noticed that without the use of the memory it might take longer and, in some cases, go into an infinite state without finding the optimum solution. The speed with which the system returns back to it stable state affects the retraining phase as we only consider the change in the average of the system error. If the difference between the previous system error average and the new system error average is not large enough then the condition for retraining won't be met and the system will get stuck trying to find a pair of Sigmas that would result in error reduction. This won't be possible without memory because the moment the network is retrained the new data overwrites the information learned by the network. The use of memory allows the network to remember the stable and perturbed history of the system parameters.\n\\begin{figure}[!t] \\centering\n\\includegraphics[width=3in]{14.png}\n\\caption{Re-training to improve the deep neural network performance in order to adapt to the introduction of new system dynamics}\n\\label{fig:13}  \n\\end{figure}\n\\section{CONCLUSION AND FUTURE WORK}\n\\label{Section VII}\nLyapunov control was applied to Duffing- van der pol\noscillator model that is experiencing chaotic behaviour. The\nstudy shows the effectiveness of deep learning combination\nwith nonlinear Lyapunov control in finding the best\nparameters to maintain system stability. The study shows\nthat deep learning with the proposed algorithm enables the\nuser to effortlessly find the best parameters for the controller\nand the system initially and recalibrate the parameters if any\ndisturbances or new dynamics are introduced to the system.\nIn future work we would like to investigate the effect of\nchanging the control strategy according to the type of\ninstability detected. We speculate that depending on the type\nof perturbations and system dynamics, changing the control\nstrategy might be more effective than changing the\nparameters only. Future considerations include a network\nthat can determine which control strategy would have the\nhighest impact on returning the system to stability and cause\nerror reduction.\n\n\n \\clearpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nParticle physics candidates for dark matter abound, the most popular being the lightest neutralino of supersymmetric models such as the Minimal Supersymmetic Standard Model (MSSM) and its extensions. On the other hand, non-supersymmetric models also have their own dark matter candidate, the most popular one being the axion. Introduced by Peccei and Quinn \\cite{Peccei:1977ur} long ago to solve the strong CP problem, the axion has required an extra global $U(1)$ symmetry, attached to the fields of the Standard Model (SM).\n\nIts inclusion in a supersymmetric lagrangean has also always required the introduction of an extra global symmetry, together with a supersymmetric partner (the axino). To bring both components (the axion and the neutralino) under the spell of the gauge principle requires, quite likely, a gauging of the axionic symmetries.  \n\nThe study of axionic symmetries and of their gauging is an important aspect of string theory at lower energy, being connected with the presence, in these effective models, of several moduli fields, deprived of a potential and derived from geometrical compactifications. The attempt to match these descriptions with effective models (with or without supersymmetry and gravity) based on ordinary quantum field theory and supersymmetry (in flat space) is pursued in several works \n\\cite{Coriano:2007fw,Coriano:2007xg} \\cite{Anastasopoulos:2008jt}, where \nthe role of the anomalous interactions and of light pseudoscalars is investigated under the tenets of gauge invariance and unitarity \\cite{Armillis:2008bg}. \nIn general, these effective field theories are characterized by the presence of higher dimensional operators to correct for the exposed (axial) anomaly (in the form of Wess-Zumino terms with St\\\"uckelberg axions) \\cite{Coriano:2007fw}. Their gauge structure, in fact, requires at least one extra anomalous $U(1)$ interaction.\n  \n   The presence of gauge couplings and of a mass for this axion not related by the same suppression scale ($f_a$) - which is of the order of $10^{10}$ GeV for a traditional (ungauged) axion - makes this new axion a very attractive dark matter candidate. However, building axion models with gauged axionic symmetries is rather challenging from the field theory point of view since it requires a grasp of the unusual features of the chiral anomaly, from the organization of the effective action(s) to the presence of anomaly poles that challenge the consistency of the S-matrix  in some of their scattering amplitudes \\cite{Coriano:2008pg}.\n\\section{Supersymmetric extensions}\nOne of the realizations of field theories which contain axion-like particles in their spectrum is the Minimal Low-Scale Orientifold Model (MLSOM) \\cite{Coriano':2005js}, based on a construction involving charge assignments obtained from intersecting branes. These models introduce one St\\\"uckelberg axion for each anomalous $U(1)$ present in the gauge structure. In \\cite{Coriano':2005js} it is shown that the physical spectrum of the MLSOM contains always one physical axion, independently of the number of anomalous U(1)'s. \n\nSupersymmetric extensions of these class of models have been discussed \nrather recently  \\cite{Coriano:2008xa}, using a superpotential which is the one typical of the USSM \\cite{Cvetic:1997ky}, extending previous studies Respect to the MSSM, the USSM contains an extra singlet superfield and an extra U(1) gauge symmetry, which is anomaly free. \nIn the USSM-A model of  \\cite{Coriano:2008xa}, which is the theory proposed as a possible supersymmetric extension of the MLSOM, this symmetry has been left anomalous, and the bosonic mechanism of cancellation of these anomalies, which is \nenforced via the inclusion of Wess-Zumino (counter)terms,  has been generalized. These models contain a combined Higgs-St\\\"uckelberg mechanism for the generation of the mass of the anomalous gauge bosons. \n\nA physical axion state (the axi-Higgs) appears quite naturally, together with its supersymmetric partner, the axino. \nThis second state becomes a component of the fermionic neutralinos, after diagonalization of the mass matrix of the neutral fermion sector. \n\nCompared to other constructions, in which the axion disappeares from the physical spectrum being just a goldstone mode, in the USSM-A the presence of the extra singlet superfield allows a physical projection of the St\\\"uckelberg axion of the model on the axi-Higgs ($\\chi$). Therefore this physical state inherits direct axion-like couplings (such as $\\chi F\\tilde{F}$) to the gauge fields, becoming a gauged supersymmetric axion \\cite{Coriano:2008aw}. The particle is massless in the absence of Peccei-Quinn-breaking potentials, in which the axion appears as a phase, while the instanton vacuum can naturally lift its mass up to \n$10^{-4}$ eV as in the Peccei Quinn (invisible axion) case.\n\nIn the non-supersymmetric version of these theories, i.e. in the MLSOM, one of the special features of these theories is the presence of anomalous trilinear gauge interactions, which are totally absent in the Standard Model (SM) due to anomaly cancellation \\cite{Armillis:2007tb}. We just recall that in the SM these interactions are suppressed and appear only away from the chiral limit in $AVV$ diagrams  (for instance in $Z \\gamma \\gamma$ vertices). We show in Fig. 1 results for the production of two axions $(\\chi)$ in the MLSOM at the Tevatron (left panel) and at the LHC (right panel), as a function of the invariant mass $Q$ of the final state, obtained from a recent analysis. The gluon fusion chnnel is the most important one for their study. The processes involve the usual triangle diagrams as for the production of the Higgs sector. In our case we have considered two Higgs doublets $(h_0, H_0)$ in the CP-even sector. The s-channel exchange involves either an $h_0$, a $H_0$ or a $\\chi$. \nWe have selected a mass of the $h_0=15$ GeV and of $120$ GeV for the $H_0$, with $m_\\chi=5$ GeV.\n\n \\section{Conclusions}\n The search for light (gauged) pseudoscalars is for sure a challenge for the experimental acvitity at the LHC \\cite{Armillis:2008vp}, requiring precise evaluation of the QCD background. However, while the detection of anomalous extra Z prime which modify the neutral current sector of these theories remains difficult, the identification of a light pseudoscalar (in the mass range of 1 to 10 GeV), which is their second signature, is more favoured.  \n\\begin{figure}[t]\n\\includegraphics[width=5cm, angle=-90]{multiax_Teva_2par.ps}\n\\includegraphics[width=5cm, angle=-90]{multiax_LHC_2par.ps}\n\\caption{\\small $gg\\rightarrow$ 2-scalar processes mediated by a three-linear vertex at the Tevatron and at the LHC.}\n\\label{2scalar_prod_LHC}\n\\end{figure}\n\\newpage\n\\centerline {\\bf Acknowledgments}\nWe thank Roberta Armillis and Luigi Delle Rose for discussions.\nThis work was supported (in part) by the European Union through the Marie Curie Research and Training Network ``Universenet'' (MRTN-CT-2006-035863). \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n\\begin{quote}\n{\\em One ought to recognize that the present political chaos is connected\nwith the decay of language, and that one can probably bring about some\nimprovement by starting at the verbal end.}  --- Orwell,\n``Politics and the English language''\n\\end{quote}\n\n\\medskip\n\n\n\n\nWe have entered an era where very large amounts of politically\noriented text are now available online.\nThis\nincludes both official documents, such as the full text of \nlaws and the proceedings of legislative bodies, and unofficial\ndocuments, such as postings on weblogs (blogs) devoted to politics.\nIn some sense, \nthe availability of such data is simply a manifestation of \na general trend of ``everybody putting their records on the\nInternet''.\\footnote{It is worth pointing out that the United States'\nLibrary of Congress was an extremely early adopter of Web technology:\nthe THOMAS database (http:\/\/thomas.loc.gov) of congressional bills and related data was\nlaunched in January 1995, when Mosaic was not quite two years old and\nAltavista did not yet exist.}\nThe online accessibility of \npolitically oriented texts in particular, however,\nis a phenomenon that some have gone so far as to say will have a potentially\nsociety-changing effect.\n\nIn the \nUnited States, \nfor example,\ngovernmental bodies are providing and soliciting political\ndocuments via the Internet,\nwith lofty goals in mind: \n{\\em electronic rulemaking} (eRulemaking)\ninitiatives\ninvolving the\n``electronic collection,\ndistribution, synthesis, and analysis of public commentary in the\nregulatory rulemaking process'', may ``[alter] the citizen-government\nrelationship'' \\cite{Shulman+Schlosberg:02a}.  Additionally, much media attention\nhas been focused  recently on the potential impact that\nInternet sites \nmay have on \npolitics\\footnote{E.g., ``Internet injects sweeping change\ninto U.S. politics'', Adam Nagourney, {\\em The New York Times}, April\n2, 2006.}, or at least on political journalism\\footnote{E.g., ``The\nEnd of News?'', Michael Massing, {\\em The New York Review of Books}, \nDecember 1, 2005.}. \nRegardless of whether one views such claims as clear-sighted prophecy\nor mere hype,\nit is obviously important to help people understand and analyze\npolitically oriented text,\ngiven the importance of enabling\ninformed participation in the political process.\n\nEvaluative and persuasive documents,\nsuch as a politician's speech regarding a bill or \na blogger's commentary on a\nlegislative proposal, form a particularly interesting type of\npolitically oriented text.  \nPeople are much more likely to consult such \nevaluative statements \nthan the actual text of a bill or law\nunder discussion, \ngiven the dense nature of legislative language and the fact that (U.S.) bills\noften reach several hundred pages in length\n\\cite{Smith+Robert+VanderWielen:05a}. Moreover, \npolitical opinions are explicitly solicited in the eRulemaking scenario.\n\nIn the analysis of \nevaluative language, it is fundamentally\nnecessary to determine whether the author\/speaker supports or\ndisapproves of the topic of discussion.  \nIn this paper, we investigate\nthe following specific instantiation of this problem: \nwe seek to determine\nfrom the transcripts of U.S. Congressional floor debates whether each\n``speech'' (continuous single-speaker segment of text) represents\nsupport for or opposition to a proposed piece of legislation.  Note\nthat from an experimental point of view, this is a very convenient\nproblem to work with because we can automatically determine ground\ntruth (and thus avoid the need for manual annotation) simply by\nconsulting publicly available \nvoting records.\n\n\\paragraph{\n\\mbox{Task properties}}\n\nDetermining whether or not a speaker \nsupports\na \nproposal falls within the realm of {\\em\nsentiment analysis}, an extremely active research area devoted to the\ncomputational treatment of subjective or opinion-oriented language\n(early work includes Wiebe and Rapaport \\shortcite{Wiebe+Rapaport:88a},\nHearst \\shortcite{Hearst:92a}, Sack \\shortcite{Sack:94a}, and \\newcite{Wiebe:94a};\nsee Esuli \\shortcite{Esuli:sentbib} for an active bibliography).  In\nparticular, since we treat \neach individual speech\nwithin a \ndebate as a single ``document'', \nwe are considering \na version of\n{\\em document-level sentiment-polarity classification}, \nnamely,  automatically distinguishing between \npositive and negative\ndocuments\n\\cite{Das+Chen:01a,Pang+Lee+Vaithyanathan:02a,Turney:02a,Dave+Lawrence+Pennock:03a}.\n\n\n\n\n\n\n\nMost sentiment-polarity classifiers proposed in the recent literature  categorize\neach document independently.  A few others  incorporate\nvarious measures of inter-document \nsimilarity between the texts to be labeled\n\\cite{Agarwal+Bhattacharyya:05a,Pang+Lee:05a,Goldberg+Zhu:06a}.  \nMany interesting \nopinion-oriented documents, however, can be linked through certain relationships \nthat occur in the context of evaluative {\\em discussions}.\nFor example, we may find textual\\footnote{\nBecause we are most interested in techniques applicable across\ndomains, we\nrestrict consideration \nto \nNLP aspects of the problem, \nignoring external problem-specific information.\nFor\nexample, although most votes \nin our corpus were almost completely along party lines (and despite\nthe fact that same-party information is easily incorporated via\nthe methods we propose), \nwe did not use party-affiliation data.\nIndeed, \nin\nother settings (e.g., a movie-discussion listserv) one \nmay not be able to determine the participants' political\nleanings,  \nand such information may not lead to significantly improved results\neven if it were available.\n}\nevidence of a high likelihood of {\\em agreement} between two speakers, \n such as explicit assertions (``I second\nthat!'') or quotation of messages in emails or postings (see\n\\newcite{Mullen+Malouf:06a} but cf. \\newcite{Agrawal+al:03a}).\nAgreement evidence can be a powerful aid in our classification\ntask: for example,  we can\neasily categorize a complicated (or overly terse) document if we find\nwithin it indications of agreement with a clearly positive text.\n\n\n\n\n\n\n\n\n\n\nObviously, incorporating agreement information provides additional\nbenefit only when the \ninput\ndocuments \nare relatively difficult to classify\nindividually.  Intuition suggests that this is true of the\ndata with which we experiment, for several reasons.  \nFirst, U.S. congressional debates contain\nvery rich language and cover an extremely wide variety of topics, \nranging from flag burning to international policy to the federal budget. \nDebates are also subject to digressions, some fairly natural \nand others less so (e.g., ``Why\nare we discussing this bill when the plight of my constituents\nregarding this other issue is being ignored?'')\n\nSecond, an important characteristic of persuasive language is that \nspeakers may spend more time presenting evidence in support of their positions\n(or attacking the evidence presented by others) \n than directly stating their attitudes.  An extreme example will\nillustrate the problems involved. Consider a speech that \ndescribes the U.S. flag as \ndeeply inspirational, \nand thus\ncontains only positive language.  \nIf the bill under discussion is\na proposed flag-burning ban, then the speech is {\\em supportive}; but\nif the bill under discussion is aimed at rescinding an existing flag-burning\nban, the speech may represent {\\em opposition} to the legislation.  \nGiven the \ncurrent state of the art in sentiment analysis, it is doubtful that one \ncould determine the (probably topic-specific) relationship between \npresented evidence and speaker opinion.\n\n\n\n\n\n\n\n\n\\begin{table*}[th]\n\\begin{center}\n\\begin{tabular}{|l||r|rrl|} \\hline\n                      &total &train &test &development \\\\ \\hline\nspeech segments\\xspace              &3857  &2740     &860  &257 \\\\\n{\\debate}s\\xspace              &53    &38       &10   &5 \\\\\naverage number of speech segments\\xspace per debate\\xspace  &72.8  &72.1  &86.0  &51.4  \\\\\naverage number of speakers per debate\\xspace  &32.1  &30.9    &41.1   &22.6   \\\\ \\hline\n\\end{tabular}\n\\caption{\\label{tab:data}\nCorpus statistics. }\n\\end{center}\n\\end{table*}\n\n\n\n\\paragraph{Qualitative summary of results}\n\n\nThe above difficulties underscore the importance of enhancing standard\nclassification techniques with new information sources that \npromise to improve accuracy, \nsuch as inter-document relationships between the documents to be\nlabeled.\nIn this paper, we demonstrate that the incorporation of agreement modeling \ncan provide substantial improvements over \nthe application of\nsupport vector machines\n(SVMs) in isolation, which \nrepresents the state of the art in the individual classification \nof documents.  \nThe enhanced accuracies are obtained\nvia a fairly primitive automatically-acquired \n``agreement detector'' and a\nconceptually simple method for integrating isolated-document and\nagreement-based information.  We thus view our results as\ndemonstrating the potentially large benefits of exploiting \nsentiment-related discourse-segment relationships in\nsentiment-analysis tasks.\n\n\n\n\n\n\n\n\n\\section{Corpus}\n\\label{sec:data}\n\n\nThis section outlines the main steps of the process by which we\ncreated \nour corpus (download site:\nwww.cs.cornell.edu\/home\/llee\/data\/convote.html).\n\n\\medskip\n\nGovTrack \n(\\mbox{http:\/\/govtrack.us})\nis an independent website \nrun by Joshua Tauberer\nthat\ncollects publicly available\ndata on \nthe legislative and fund-raising activities of U.S. congresspeople.  Due to its\nextensive cross-referencing and collating of information, it \nwas nominated for a 2006 ``Webby'' award.\nA crucial characteristic of GovTrack from our point of\nview is that the information is provided in a very convenient format;\nfor instance, the floor-debate transcripts are broken into\nseparate HTML files according to the subject of the debate, so we\ncan trivially derive long sequences of speeches guaranteed to cover the\nsame topic.\n\n\n\nWe extracted from GovTrack all available transcripts of U.S.  floor\ndebates in the House of Representatives for the year 2005 (3268 pages\nof transcripts in total), together with voting records for all\nroll-call\nvotes during that year.\nWe concentrated on {\\debate}s\\xspace regarding ``controversial'' bills (ones in which the\nlosing side generated at least 20\\% of the speeches) because\nthese debates should presumably exhibit more interesting\ndiscourse structure.\n\nEach \ndebate consists of a series of {\\em speech segments\\xspace}, where each \nsegment is a\nsequence of uninterrupted utterances by a single speaker.\nSince \nspeech segments\\xspace represent\nnatural  discourse units, we \ntreat them as the basic \nunit to be classified.\nEach speech segment\\xspace was labeled by the vote (``yea''\\xspace or ``nay''\\xspace) cast for the proposed bill\nby the person who uttered the speech segment\\xspace.\n\nWe automatically discarded those speech segments\\xspace belonging to a class of \nformulaic, generally one-sentence utterances \nfocused \non the yielding of time on the house floor (for example, \n``Madam Speaker, I am pleased to yield 5 minutes to the gentleman from Massachusetts''), as such speech segments\\xspace are\nclearly off-topic.\nWe also removed speech segments\\xspace \ncontaining the term ``amendment'', since we found \nduring initial inspection that\n these speeches \ngenerally reflect a speaker's opinion on an amendment, and this\nopinion may \ndiffer from the speaker's opinion \non the underlying bill under discussion.\n\nWe randomly split the data into training, test, and development\n(parameter-tuning) sets \nrepresenting\nroughly 70\\%, 20\\%, and 10\\% of our data, respectively (see Table \\ref{tab:data}).  \nThe speech segments\\xspace remained grouped by debate\\xspace, with 38 \n{\\debate}s\\xspace\nassigned to the training set, \n10 to the test set, and 5 to the development set; \nwe \nrequire that the speech segments\\xspace from an individual debate\\xspace\nall appear in the same \nset because our goal is to\nexamine classification of speech segments\\xspace in the context of the\nsurrounding discussion.\n\\section{Method}\n\\label{sec:method}\n\nThe support\/oppose \nclassification \nproblem can be approached through \nthe use of standard classifiers such as support vector machines (SVMs), \nwhich consider each text unit in isolation.\nAs discussed in Section \\ref{sec:intro}, however, the conversational\nnature of our data implies the existence of various relationships that\ncan be exploited \nto improve cumulative classification accuracy for \nspeech segments\\xspace belonging to the same debate\\xspace.\nOur classification framework,\ndirectly inspired by \\newcite{Blum+Chawla:01a},\n integrates both perspectives, optimizing its\nlabeling of speech segments\\xspace based on \nboth individual \nspeech-segment\\xspace classification scores and preferences for groups of speech segments\\xspace to \nreceive the same label.\nIn this section, we discuss \nthe specific classification framework that we adopt\nand the set of mechanisms that we propose for  \nmodeling specific types of relationships.\n \n\n\\subsection{Classification framework}\n\\label{sec:method:graph}\n\n\nLet $s_1,s_2,\\ldots, s_{n}$ be the sequence\nof speech segments\\xspace within a given debate, and let ${\\cal Y}\\xspace$ and ${\\cal N}\\xspace$ stand\nfor the ``yea'' and ``nay'' class, respectively. Assume we have a\nnon-negative function $\\mbox{{\\it ind}}\\xspace(s,C)$ indicating the\ndegree of preference that an individual-document classifier, such as an SVM,\nhas for placing speech-segment\\xspace $s$ in class $C$.  Also, assume that \nsome pairs of speech segments\\xspace have {\\em\nweighted links} between them, where the non-negative {\\em strength} (weight)\n${\\mathit str}(\\ell)$ for a link $\\ell$ indicates the degree to which it\nis preferable that the linked speech segments\\xspace receive the same label.\nThen, any class assignment $c = c(s_1),\nc(s_2),\\ldots, c(s_{n})$\ncan be assigned a {\\em cost}\n$$\n\\sum_{s}\n\\mbox{{\\it ind}}\\xspace(s,\\overline{c}(s)) +\n\\sum_{s,s':\\: c(s) \\neq c(s')}\n\\,\n  \\sum_{\\ell\\; {\\rm between}\\,  {s, s'} }{\\mathit str}(\\ell),\n$$\nwhere $\\overline{c}(s)$ is the ``opposite'' class from ${c}(s)$.\nA {\\em minimum-cost} assignment thus represents an optimum way to classify the\nspeech segments\\xspace \nso that \neach one tends not to be put into the class that the\nindividual-document classifier disprefers, but at the same time, highly associated\nspeech segments\\xspace tend not to be put in different classes.\n\nAs has been previously observed and exploited in the NLP literature\n\\cite{Pang+Lee:04a,Agarwal+Bhattacharyya:05a,Barzilay+Lapata:05a},\nthe above optimization function, unlike many others that have been proposed\nfor \ngraph or set partitioning, can be solved {\\em exactly} in\nan \nprovably\nefficient manner via methods for finding minimum cuts in\ngraphs.  In our view, the contribution of our work is the examination\nof new types of relationships, not the method by which such\nrelationships are incorporated into the classification decision.\n\n\n\n\\subsection{Classifying speech segments\\xspace in isolation}\n\\newcommand{${\\rm  SVM}^{light}$\\xspace}{${\\rm  SVM}^{light}$\\xspace}\nIn our experiments, we \nemployed\nthe well-known \nclassifier \n${\\rm  SVM}^{light}$\\xspace\n to obtain individual-document \nclassification scores, treating ${\\cal Y}\\xspace$ as the positive \nclass and using plain unigrams as features.\\footnote{${\\rm  SVM}^{light}$\\xspace is available at\nsvmlight.joachims.org.  \nDefault parameters were\nused,\nalthough experimentation with different parameter settings is an\nimportant direction for future work\n\\cite{Daelemans+Hoste:02a,Munson+Cardie+Caruana:05a}.}\nFollowing standard practice in sentiment analysis \\cite{Pang+Lee+Vaithyanathan:02a}, the input to ${\\rm  SVM}^{light}$\\xspace consisted \nof normalized presence-of-feature (rather\n  than frequency-of-feature) vectors.  The $\\mbox{{\\it ind}}\\xspace$ value for each speech segment\\xspace\n $s$ \nwas based on\nthe signed distance $\\refconfvar(\\spvar)\\xspace$ from the vector representing\n $s$ to the trained SVM decision \nplane:\n$$\n\\mbox{{\\it ind}}\\xspace(s,{\\cal Y}\\xspace) \\stackrel{{\\rm def}}{=} \n\\begin{cases}\n1 & \\refconfvar(\\spvar)\\xspace > 2\\sigma_{s}; \\cr\n\\left(1 + \\frac{\\refconfvar(\\spvar)\\xspace}{2\\sigma_{s}}\\right)\/2 & \n\\vert \\refconfvar(\\spvar)\\xspace \\vert \\leq 2\\sigma_{s};\n\\cr\n0 & \\refconfvar(\\spvar)\\xspace < -2\\sigma_{s}\n\\end{cases}\n$$\nwhere $\\sigma_{s}$ is the standard deviation of  $d(s)$ over all\nspeech segments\\xspace $s$ in the\ndebate in question,\nand $\\mbox{{\\it ind}}\\xspace(s,{\\cal N}\\xspace) \\stackrel{{\\rm def}}{=} 1 - \\mbox{{\\it ind}}\\xspace(s,{\\cal Y}\\xspace)$.\n\n\nWe now turn to the more interesting problem of representing the\npreferences that\nspeech segments\\xspace may have for being assigned to the same class.\n\n\n\\subsection{Relationships between speech segments\\xspace}\n\\label{sec:method:relationship}\n\nA wide range of relationships between text segments \ncan be modeled as positive-strength links.\nHere we discuss two types of constraints that are considered in this work. \n\n\n\n\n\n\n\\paragraph{Same-speaker constraints:} \nIn Congressional debates and in general \nsocial-discourse contexts, \na single speaker may make \na number of comments regarding \na\ntopic.  \nIt is reasonable to expect that in many settings, the participants in\na discussion may be convinced\nto change their opinions midway through a debate.\nHence,\nin the general case we wish to be able \nto express\n``soft'' preferences for all of an author's statements\nto receive the same label, \nwhere\nthe strengths of such constraints could, \nfor instance, vary according to the time elapsed between the statements.\nWeighted links are an appropriate means to express such variation.\n\n\n\nHowever, if\nwe assume that most speakers do not change their positions in the \ncourse of a discussion, we can conclude that all \ncomments made by the same speaker\nmust\nreceive the same label.\nThis assumption\nholds by fiat for the ground-truth labels in our dataset because these\nlabels were derived from the single vote cast by the speaker on the\nbill being discussed.\\footnote{\nWe are attempting to\ndetermine whether a speech segment\\xspace represents support or not.  This differs\nfrom the problem of determining what the speaker's actual opinion is,\na problem that, as an anonymous reviewer \nput it, is complicated\nby ``grandstanding, backroom deals, or, more innocently, plain change\nof mind \n(`I voted for it before I voted against it')''.}\nWe can implement this assumption\nvia links whose weights are essentially infinite.  \nAlthough\none can also implement this assumption via concatenation of\nsame-speaker speech segments\\xspace (see\nSection \\ref{sec:eval:global}), we view the fact that our graph-based\nframework  incorporates both hard and soft constraints in a principled\nfashion as an\nadvantage of our approach.\n\n\n\\paragraph{Different-speaker agreements} In House discourse, it is common for one \nspeaker to make reference to another\nin the context of an\nagreement or disagreement over the topic of discussion.\nThe systematic identification of instances of \nagreement \ncan, as we have discussed, \nbe a powerful tool \nfor the development of intelligently\nselected weights\nfor links between speech segments\\xspace.  \n\n\n\nThe problem of agreement identification \ncan be decomposed into two\nsub-problems: identifying references and their \ntargets, and deciding whether\neach reference represents an instance of agreement.\nIn our case, the first task is straightforward\nbecause we focused solely on by-name references.\\footnote {One subtlety\n  is that for the purposes of mining agreement cues \n(but {\\em not} for\n  evaluating overall support\/oppose classification accuracy), we \ntemporarily\nre-inserted into\nour dataset previously filtered speech segments\\xspace containing the term\n``yield'', since the yielding of time on the House floor typically\nindicates agreement even though the yield statements contain little\nrelevant text on their own.} \nHence, we will now concentrate on the second, more interesting task.\n\nWe approach the problem of classifying references by \nrepresenting each reference with a word-presence vector derived from a \nwindow of text surrounding \nthe reference.\\footnote{\nWe found good \ndevelopment-set performance using\nthe 30 tokens before,\n20 tokens after,\nand the name itself.}  \nIn the training set, we classify each reference connecting two speakers \nwith a positive or negative label \ndepending on whether the two\nvoted the same way on the bill under discussion\\footnote{Since we are \nconcerned with references that potentially represent relationships \nbetween speech segments\\xspace, we ignore\nreferences for which the target of the reference did not speak in the\ndebate\\xspace in which the reference was made.}.  These labels are then used to\ntrain an SVM classifier, the output of which is subsequently used to\ncreate weights\non {\\em agreement links} in the test set as follows.\n\n Let $\\agrconfvar(\\refvar)$ denote the distance from the vector representing\n reference $r$ to the agreement-detector SVM's decision plane,\n and let\n$\\sigma_\\refvar$ be the standard deviation of  $d(r)$ over all\n references in the debate\\xspace in question.\nWe then define the strength $\\mathit{agr}\\xspace$ of the {\\em agreement link} corresponding\nto the reference\nas:\n$$\n\\mathit{agr}\\xspace(r) \\stackrel{{\\rm def}}{=} \\begin{cases}\n0           & \\agrconfvar(\\refvar) < \\theta_{\\mbox{agr}}\\xspace; \\cr\n\\alpha\\xspace \\cdot \\agrconfvar(\\refvar)\/ 4\\sigma_\\refvar & \\theta_{\\mbox{agr}}\\xspace \\le \\agrconfvar(\\refvar) \\le 4\\sigma_\\refvar; \\cr\n\\alpha\\xspace & \\agrconfvar(\\refvar) > 4\\sigma_\\refvar.\n\\end{cases}\n$$\nThe free parameter $\\alpha\\xspace$ specifies the relative importance of the\n$\\mathit{agr}\\xspace$ scores.\nThe threshold $\\theta_{\\mbox{agr}}\\xspace$ controls \nthe precision of\nthe agreement\nlinks,\nin that values of  $\\theta_{\\mbox{agr}}\\xspace$ greater than zero mean that greater\nconfidence is required before an agreement link can be added.\n\\footnote{Our implementation\nputs a link between\njust one arbitrary pair of speech segments\\xspace among all those uttered by a given\npair of\napparently agreeing speakers.  The ``infinite-weight'' same-speaker\nlinks propagate the agreement information to all other such pairs.\n} \n\n\n\n\n\n\n\n\n\n\\section{Evaluation}\n\\label{sec:eval}\n\nThis section presents experiments testing the utility  of \nusing\nspeech-segment\\xspace relationships,\nevaluating against a number of baselines.\nAll reported results use values for the\nfree parameter $\\alpha\\xspace$ \nderived via tuning on the development set. \nIn the tables, {\\bf boldface} indicates the development- and\ntest-set results for the {\\em development-set-optimal} parameter\nsettings, as one would make algorithmic choices based on\ndevelopment-set performance.\n\n\n\\subsection{Preliminaries: Reference classification}\n\\label{sec:eval:agr}\n\n\nRecall that to gather inter-speaker agreement information, the\nstrategy employed in this paper is to classify \nby-name \nreferences to other speakers as to whether they\nindicate agreement or not.\n\n\n\n\n\\begin{table}[t]\n\\begin{tabular}{|l|cc|}\n \\hline\n\\parbox[b]{4.5cm}{\\mbox{Agreement classifier} \\\\ \\hspace*{.2cm}(``{reference}$\\Rightarrow$agreement?'')}      &\n\\parbox[b]{1cm}{Devel. set}  &\\parbox[b]{1cm}{Test set} \\\\\n\\hline\\hline\n\nmajority baseline\\xspace & 81.51 & 80.26 \\\\\n\\mbox{Train: no \\amendabbrev}; \\useposref & 84.25 & 81.07 \\\\\n\\mbox{Train: with \\amendabbrev}; \\useposref & \\bestdev{86.99} & \\testbestdev{80.10} \\\\\n\\hline\n\n\\end{tabular}\n\\caption{\\label{tab:agr} \nAgreement-classifier accuracy, in percent.  ``Amdmts''=``speech segments\\xspace containing the\nword {`amendment'}''.\nRecall that boldface indicates results for\n  development-set-optimal settings.\n}\n\\end{table}\n\n\nTo train \nour agreement classifier,\nwe\nexperimented with undoing the deletion of amendment-related speech segments\\xspace\nin the training set.  Note that such speech segments\\xspace were {\\em never}\nincluded in the development or test set, since, as discussed in \nSection\n \\ref{sec:data}, their labels are probably noisy; however, including them in\nthe {\\em training} set allows the classifier to examine more instances even\nthough some of them are labeled incorrectly.  As Table\n\\ref{tab:agr} shows, using more, if noisy, data yields better\nagreement-classification results on the {development} set, and so we use\nthat policy in all subsequent experiments.\n\\footnote{\nUnfortunately, this policy leads to inferior {\\em test-set}\nagreement classification.  Section \\ref{sec:eval:devtest}\ncontains further discussion.\n}\n\n\n\nAn important observation is that precision may be more important than\naccuracy in deciding which agreement links to add: false positives\nwith respect to \nagreement\ncan cause speech segments\\xspace to be\nincorrectly assigned the same label, whereas false negatives mean only\nthat agreement-based information about other speech segments\\xspace is not employed.  As described above, we\ncan raise agreement precision by \nincreasing the threshold $\\theta_{\\mbox{agr}}\\xspace$,\nwhich specifies \nthe required confidence for the addition of an agreement link.\nIndeed, Table \\ref{tab:agr-highprec} shows that we can \nimprove \nagreement precision\nby \nsetting  $\\theta_{\\mbox{agr}}\\xspace$  to the (positive) mean agreement score $\\mu\\xspace$ assigned by the SVM\nagreement-classifier over all references in the given\n{debate\\xspace}\\footnote{We elected not to explicitly tune the value of\n$\\theta_{\\mbox{agr}}\\xspace$ in order to minimize the number of free parameters to deal\nwith.}.  However, this comes at the cost of\ngreatly reducing agreement accuracy (development: 64.38\\%; test:\n66.18\\%) due to \nlowered recall levels.  Whether or not better\nspeech-segment\\xspace classification \nis ultimately achieved \nis discussed\nin the next sections.\n\n\n\n\n\\begin{table}[t]\n\\begin{tabular}{|l|cc|}\n \\hline\n{{Agreement classifier}}      &\n\\multicolumn{2}{|c|}{Precision (in percent):} \\\\\n& \n{Devel. set}  &\n{Test set} \\\\\n\\hline\\hline\n\n$\\refthres = 0$ & 86.23 & 82.55 \\\\\n$\\refthres = \\refmean$ & \\bestdev{89.41} & \\testbestdev{88.47} \\\\\n\\hline\n\n\\end{tabular}\n\\caption{\\label{tab:agr-highprec} \nAgreement-classifier precision.\n}\n\\end{table}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{\\Localsvmsetting}\n\\label{sec:eval:local}\n\n\n\n\\paragraph{Baselines} The first two data rows of Table\n\\ref{tab:results-local} depict baseline performance results.  \nThe \n$\\#(\\mbox{``support''}) - \\#(\\mbox{``oppos''})$\\xspace}\\newcommand{\\ressvmbl}{SVM\\xspace\n baseline is meant to explore whether the\n speech-segment\\xspace classification task can be reduced to simple\nlexical checks.  Specifically, this method uses the signed difference\nbetween the number of words containing the stem ``support'' and the\nnumber of words containing the stem ``oppos'' (returning the\nmajority class if the difference is 0).  No better than 62.67\\%\ntest-set accuracy is obtained by either baseline.\n\n\n\\providecommand{\\per}{}\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{|l|cc|} \\hline\n\\parbox[b]{4.3cm}{Support\/oppose classifer \\\\ \\hspace*{.2cm}(``{speech segment\\xspace}$\\Rightarrow$yea?'')}      & \\parbox[b]{1cm}{Devel. set}  &\\parbox[b]{1cm}{Test set} \\\\ \\hline\\hline\nmajority baseline\\xspace  &54.09\\per        &58.37\\per \\\\\n$\\#(\\mbox{``support''}) - \\#(\\mbox{``oppos''})$\\xspace}\\newcommand{\\ressvmbl}{SVM\\xspace      &59.14\\per        &62.67\\per \\\\ \\hline\n\\ressvmbl [{speech segment\\xspace}]      &70.04\\per        &66.05\\per \\\\\nSVM + same-speaker links\\xspace &79.77\\per        &67.21\\per \\\\ \nSVM + same-speaker links\\xspace $\\ldots$  & & \\\\\n{~~~} \\spaceplus \\refs,  \\useposref \\xspace        &\\bestdev{89.11\\per}        &\\testbestdev{70.81\\per} \\\\ \n{~~~} \\spaceplus \\refs, \\useavgref \\xspace     &87.94\\per        &71.16\\per \\\\ \\hline\n\\end{tabular}\n\\caption{\\label{tab:results-local}\n\\Localsvmsetting accuracy, in percent.}\n\\end{center}\n\\end{table}\n\n\n\n\\providecommand{\\per}{}\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{|l|cc|} \\hline\n\\parbox[b]{4.3cm}{Support\/oppose classifer \\\\ \\hspace*{.2cm}(``{speech segment\\xspace}$\\Rightarrow$yea?'')}      & \\parbox[b]{1cm}{Devel. set}  &\\parbox[b]{1cm}{Test set} \\\\ \\hline\\hline\n\\ressvmbl [speaker]      &71.60\\per        &70.00\\per \\\\\nSVM + agreement links\\xspace $\\ldots$ & & \\\\\n{~~~} with $\\refthres = 0$\n     &\\bestdev{88.72}\\per        &\\testbestdev{71.28}\\per \\\\ \n{~~~} with $\\refthres = \\refmean$\n &84.44\\per        &76.05\\per \\\\ \n\\hline\n\\end{tabular}\n\\caption{\\label{tab:results-global}\n\\Globalclass \\spunith classification\\xspace accuracy,\nin percent.  \nHere,\nthe\ninitial SVM is run on the concatenation of all of a given speaker's\nspeech segments\\xspace,\nbut \nthe results \nare computed over speech segments\\xspace\n(not speakers),\nso that \nthey can be\ncompared to those in Table \\ref{tab:results-local}.\n}\n\\end{center}\n\\end{table}\n\n\n\\paragraph{Using relationship information}\nApplying an SVM to classify each speech segment\\xspace in isolation leads to clear\nimprovements over the two baseline methods, as demonstrated in \nTable \\ref{tab:results-local}.\nWhen we impose the constraint that all speech segments\\xspace uttered by the same speaker\nreceive the same label\nvia ``same-speaker links'', \nboth test-set\nand development-set accuracy increase even more, in the latter case\nquite substantially so.\n\n\n\nThe last two lines of Table \\ref{tab:results-local} show that the best\nresults are obtained by incorporating agreement information as well.\nThe highest test-set result, 71.16\\%, is obtained by using a high-precision\nthreshold to determine which agreement links to add.  While the\ndevelopment-set results would induce us to utilize the standard threshold\nvalue of 0, which is sub-optimal on the test set, the\n$\\refthres = 0$\\xspace \nagreement-link\npolicy still achieves noticeable improvement \nover  not using  agreement links\n(test set: 70.81\\% vs. 67.21\\%).\n\n\n\n\n\n\\subsection{\\Globalclass \\spunith classification\\xspace}\n\\label{sec:eval:global}\n\nWe \nuse speech segments\\xspace as the  unit of classification \nbecause\nthey represent natural discourse \nunits.\nAs a  consequence, \nwe \nare able to\nexploit \nrelationships \nat the speech-segment\\xspace\nlevel.  \nHowever, it is interesting to consider whether we really need to\nconsider relationships specifically between speech segments\\xspace themselves, or whether it\nsuffices to simply consider relationships between the {\\em speakers} of\nthe speech segments\\xspace.\nIn particular, as an alternative to using same-speaker links, \nwe tried a {\\em speaker-based\\xspace} approach wherein \nthe way we determine\nthe initial\nindividual-document classification score for \neach \nspeech segment\\xspace uttered by a\nperson $p$ in a given debate \nis to run\nan SVM \non the\nconcatenation of\n{\\em all} of $p$'s speech segments\\xspace\nwithin that debate.\n(We also ensure that agreement-link information is propagated from\nspeech-segment\\xspace \nto speaker pairs.)\n\n\n\n\n\nHow \ndoes the use of \nsame-speaker links compare to the concatenation of\neach speaker's speech segments\\xspace?   \nTables  \\ref{tab:results-local} and \\ref{tab:results-global} \nshow that, not surprisingly, \nthe SVM individual-document\nclassifier works better on the concatenated speech segments\\xspace than on the\nspeech segments\\xspace in isolation.\nHowever, the effect on overall\nclassification accuracy is less clear: the development set favors\nsame-speaker links over concatenation, \nwhile\nthe test set does not.\n\n\n\n\n\n\n\n\n\nBut we stress that\nthe most important observation we can make from \n Table \\ref{tab:results-global} is  that once again, the\naddition of agreement information leads to substantial improvements in\naccuracy.  \n\n\n\\subsection{``Hard'' agreement constraints}  Recall that in \nin our experiments, \nwe created \nfinite-weight\nagreement links, so that speech segments\\xspace appearing in pairs flagged by our (imperfect)\nagreement detector can potentially receive different labels.  We also\nexperimented with \n{\\em forcing} such speech segments\\xspace to receive the same label,\neither through infinite-weight agreement links or through a speech-segment\\xspace\nconcatenation strategy similar to that described in the previous\nsubsection.  Both strategies resulted in clear degradation in\nperformance on both the development and test sets, a finding that \nvalidates our encoding of agreement information as ``soft'' preferences.\n\n\n\\subsection{On the development\/test set split}\n\\label{sec:eval:devtest}\n\nWe have seen several cases in which the\nmethod that performs best on the development set does not yield the\nbest test-set performance.  However, we felt that it would be\nillegitimate to change the train\/development\/test \nsets \nin a post hoc\nfashion, that is, after seeing the experimental results.  \n\nMoreover, and crucially, it is very clear that using agreement\ninformation, encoded as preferences within our graph-based approach\nrather than as hard constraints, yields substantial improvements on\nboth the development and test set; this, we believe, is our most\nimportant finding.\n\n\n\n\\section{Related work}\n\n\n\\paragraph{Politically-oriented text}  \nSentiment analysis has specifically been proposed as a key enabling\ntechnology in eRulemaking, allowing the  automatic analysis of\nthe opinions that people submit\n\\cite{Shulman+al:05a,Cardie+al:06a,Kwon+Shulman+Hovy:06a}.\nThere has also been work focused upon\ndetermining the political leaning (e.g.,\n``liberal'' vs. ``conservative'') of a document or \nauthor, \nwhere most\npreviously-proposed methods make no direct use of relationships\nbetween the documents to be classified (the ``unlabeled'' texts)\n\\cite{Laver+Benoit+Garry:03a,Efron:04a,Mullen+Malouf:06a}.\nAn exception is \\newcite{Grefenstette+Qu+Shanahan+Evans:04a}, who experimented with\n  determining the political orientation of websites essentially by\n  classifying the concatenation of all the documents found on that site.\n\n\n\n\n\nOthers have applied \nthe NLP technologies of near-duplicate detection and\ntopic-based text categorization to politically oriented text\n\\cite{Yang+Callan:05a,Purpura+Hillard:06a}.\n\n\\paragraph{Detecting agreement} We used a simple method to learn to\nidentify \ncross-speaker\nreferences \nindicating\nagreement.  More sophisticated approaches\nhave been proposed\n\\cite{Hillard+Ostendorf+Shriberg:03a}, including an extension that, in\nan interesting reversal of our problem, makes use of\nsentiment-polarity indicators within \nspeech segments\\xspace \\cite{Galley+McKeown+Hirschberg+Shriberg:04a}.  \nAlso relevant is work on the general problems of dialog-act tagging\n\\cite{Stolcke+al:00a},  citation analysis\n\\cite{Lehnert+Cardie+Riloff:90}, and computational rhetorical analysis \\cite{Marcu:00a,Teufel+Moens:02a}.\n\nWe currently do not have an efficient means to encode {\\em\ndisagreement} information as hard constraints; we plan to\ninvestigate incorporating\nsuch information in future work.\n\n\n\\paragraph{Relationships between the unlabeled items}  \n\\newcite{Carvalho+Cohen:05a} consider sequential relations between different\ntypes of emails (e.g.,\nbetween requests and satisfactions thereof) to classify\nmessages, and thus also explicitly exploit the structure of \nconversations.\n\nPrevious\nsentiment-analysis work in different domains\nhas considered inter-document similarity\n\\cite{Agarwal+Bhattacharyya:05a,Pang+Lee:05a,Goldberg+Zhu:06a} or \nexplicit inter-document references in the form of hyperlinks  \\cite{Agrawal+al:03a}.\n\n\n\nNotable early papers on graph-based semi-supervised learning include\nBlum and Chawla \\shortcite{Blum+Chawla:01a}, Bansal, Blum, and Chawla\n\\shortcite{Bansal+Blum+Chawla:02a}, Kondor and Lafferty \\shortcite{Kondor+Lafferty:02a}, and\nJoachims \\shortcite{Joachims:03a}.  Zhu \\shortcite{Zhu:semi-survey} maintains a survey\nof this area.\n\nRecently, several alternative, often quite sophisticated approaches to {\\em collective\n  classification} have been proposed\n\\cite{Neville+Jensen:00a,Lafferty+McCallum+Pereira:01a,Getoor+al:02a,Taskar+al:02a,Taskar+Guestrin+Koller:03a,Taskar+Chatalbashev+Koller:04a,McCallum+Wellner:04a}.\nIt would be interesting to investigate the application of such methods\nto our problem.  However,  we also believe that our approach has\nimportant advantages, including  conceptual simplicity and \nthe fact that it\nis based on an  underlying optimization problem that is provably\nand \nin practice\neasy to solve.\n\n\n\n\\section{Conclusion and future work}\n\\label{sec:conc}\n\n\nIn this study, we focused on very general \ntypes\nof cross-document\nclassification preferences, \nutilizing constraints based only on speaker\nidentity and on direct textual references between statements.  \nWe showed that the integration of \neven very limited information regarding\ninter-document relationships\ncan significantly increase the accuracy\nof support\/opposition classification.\n\nThe simple constraints \nmodeled in our study, however, represent \njust a small \nportion \nof the rich network of relationships that connect\nstatements \nand speakers\nacross the political universe and in the wider realm of\nopinionated social discourse.  \nOne intriguing possibility is to take advantage of (readily identifiable)\ninformation regarding interpersonal relationships, making use of \nspeaker\/author affiliations, positions\nwithin a social hierarchy, and so on.\nOr, we could even  attempt to model relationships\nbetween\ntopics or concepts, \nin a kind of extension of collaborative filtering.\nFor example, \nperhaps we could infer that \ntwo\nspeakers sharing a common opinion on evolutionary biologist Richard\nDawkins (a.k.a. ``Darwin's rottweiler'') will be likely to agree\nin a debate centered on Intelligent Design.  While such functionality is\nwell beyond the scope of our current study, we are optimistic that we\ncan develop methods to exploit\nadditional types of relationships in future work.\n\n\n\n\n\n\n\\paragraph*{Acknowledgments}  We thank \nClaire Cardie, \nJon Kleinberg, Michael Macy, Andrew\nMyers, \nand the six anonymous EMNLP referees for valuable discussions\nand comments.  We also thank Reviewer 1 for generously providing\nadditional {\\em post hoc} feedback, and the EMNLP chairs Eric Gaussier\nand Dan Jurafsky for facilitating the process (as well as for\nallowing authors an extra proceedings page$\\ldots$).\nThis paper is based upon work supported in part by the National\nScience Foundation under grant no. IIS-0329064 and an Alfred P. Sloan\nResearch Fellowship. Any opinions,\nfindings, and conclusions or recommendations expressed are those of\nthe authors and do not necessarily reflect the views or official\npolicies, either expressed or implied, of any sponsoring institutions,\nthe U.S. government, or any other entity.\n\n\\vspace*{-.1in}\n\n{\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}