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+{"text":"\n\\section{The proposed discriminative residual coding}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[height=0.85in]{figs\/Bitrate.pdf}\n    \\caption{The proposed thresholding mechanism uses different quantization schemes depending on the target bitrate.}\n    \\label{fig:bitrate}\n     \\vspace{-4mm}\n\\end{figure}\n\n\n\nTo improve the coding gain further, we apply \\textit{discriminative} coding to the residual signals $\\bar{\\bm r}_n$, which is the information sent to the receiver in place of the full cepstrum $\\bm{c}_n$. \nDue to the overall smoothness of speech, the prediction in the cepstrum domain results in the residuals that follow a Gaussian distribution with zero mean and small variance. The larger residual values mainly occur in transient events, such as plosives.\nTo fully make use of the residual signal's statistical advantage, we apply \\textit{discriminative} coding that distinguishes the more ``code-worthy\" frames from the rest by thresholding the $L_1$ norm of the residuals. This way, frames with significant residual energy are assigned more bits, and bits assigned to the less significant frames are minimized. \n\n\n\nScanning the entire training set, we define a threshold value $\\theta$ depending on the target bitrate. The quantization process eq.\\eqref{qtz} is therefore expanded to: \n\\begin{equation} \n  \\bar{\\bm r} = \\mathcal{Q}({\\bm r}) =  \n    \\begin{cases}\n      \\mathcal{Q}_\\text{L}(\\bm{r}) & \\text{if } ||\\bm r||_1 \\geq \\theta \\\\\n      \\mathcal{Q}_\\text{S}(\\bm{r}) & \\text{otherwise},\n    \\end{cases}       \n\\end{equation}\nwith $\\mathcal{Q}_\\text{L}$ and $\\mathcal{Q}_\\text{S}$ representing quantization schemes with large and small $L_1$ norms that use large and small codebooks, respectively. Particularly, when the target bitrate is extremely low, we \\textit{discard} low $L_1$ norm frames entirely, i.e., $\\mathcal{Q}_\\text{S}(\\bm{r})=\\bm{0}$. \n\nThe thresholding mechanism is illustrated in Fig.\\ref{fig:bitrate}. The low-bitrate scheme ($\\sim$ 0.93 kbps) uses $\\mathcal{Q}_\\text{L}$ for the top 25 \\%  residual frames while discarding the rest without any coding. The intermediate bitrate ($\\sim$ 1.47 kbps) case keeps the top 7\\% for the $\\mathcal{Q}_\\text{L}$ quantization and the rest 90\\% for $\\mathcal{Q}_\\text{S}$. The $\\sim$ 2.87 kbps case uses $\\mathcal{Q}_\\text{L}$ quantization for all residual frames with no thresholding. \n\nSimilar to LPCNet's coding scheme, we separately code the first component $\\bm r_0$ of the residuals vector and the rest of dimensions $\\bm r_{1-17}$; Note here that we dropped the frame index $n$ and used subscript to indicate one of the 18 cepstrum coefficients within the frame. Also, since we noticed that $\\bm r_0$ and $\\bm r_{1-17}$ have different $L_1$ norm distributions, we define thresholds and apply discriminative coding to the scalar and vector components independently.\n\n\n\n\n\n\\begin{table}[]\n    \\centering\n    \\resizebox{\\columnwidth}{!}{%\n    \\begin{tabular}{c||c|c||c|c||c|c}\n       \n       \n        Target bitrate (kbps) & \\multicolumn{2}{c||}{$\\sim$ 0.93} & \\multicolumn{2}{c||}{$\\sim$ 1.47} & \\multicolumn{2}{c}{ $\\sim$ 2.87} \\\\\n        \\hline\n        $\\mathcal{Q}_\\text{L}$ percentage & \\multicolumn{2}{c||}{25 $\\%$} & \\multicolumn{2}{c||}{7 $\\%$} & \\multicolumn{2}{c}{ 100 $\\%$} \\\\\n        \\hline\n      \\multicolumn{7}{c}{Codebook Size (bits) : Bits-per-frame according to Huffman coding}\\\\\n        \\hline\n        Stages & 1st & 2nd & 1st & 2nd & 1st & 2nd\\\\\n        \\hline\n        $\\mathcal{Q}_\\text{L}(\\bm{r}_0)$ & 8 : 7.0 & - & 8 : 7.4 & - &8 : 7.2 & - \\\\\n        \\hline\n        $\\mathcal{Q}_\\text{S}(\\bm{r}_0)$ & - & - & 4 : 2.9 & - & - & - \\\\\n        \\hline\n        $\\mathcal{Q}_\\text{L}(\\bm{r}_{1:17})$ & 10 : 9.8 & 10 : 9.9 & 10 : 9.2 & 10 : 9.4 & 10 : 9.2 & 10 : 9.6\\\\\n        \\hline\n        $\\mathcal{Q}_\\text{S}(\\bm{r}_{1:17})$ &  - & - & 9 : 8.0 & - & - & -\\\\\n    \\end{tabular}\n    }\n    \\caption{Codebook sizes and bitrate assignments.}\n    \\label{tab:codebooks}\n    \\vspace{-3mm}\n\\end{table}\n\nTable \\ref{tab:codebooks} summarizes how we conduct discriminative and multi-stage quantization depending on the target bitrate. For scalar quantization, we use the same codebook size of $2^9=512$ in all $\\mathcal{Q}_\\text{L}$ cases, while only $16$ codewords for $\\mathcal{Q}_\\text{S}$ in the mid-bitrate case or skips coding in the low-bitrate case. All scalar quantizers use a single-stage quantization scheme. \nAs for the VQ for $\\bm{c}_{1:17}$, we employ either one or two-stage quantization for $\\mathcal{Q}_\\text{L}$ with a codebook of size 1024 in each stage; $\\mathcal{Q}_\\text{S}$ cases use a single 512-size codebook or skip coding in the low-bitrate case ($\\sim$0.93). We also estimate the bitrate considering Huffman coding by computing the frequencies $\\bm p$ of all codewords from coding randomly-selected 2-second segments per training samples and derive the average bit-per-frame by $\\sum_i^N \\bm p_ilog_2\\bm p_i$. \nApart from the bits we have stated in the \\tabref{tab:codebooks}, we also need to count in the bits for coding pitch parameters in LPCNet's original way, which takes up 0.275 kbps. We use the bitrates of Huffman coding in the rest of the paper, although it is close to the bitrates based on the codebook. In the $\\sim$0.93kbps case, for example, the target bitrate in our table is calculated by $0.25 \\times(7\\times100+(9.9+9.8 )\\times100) + 275 = 932 \\text{bps}$, given that each frame is for 10 ms (meaning 100 frames per second), and only $25\\%$ of the frames are coded in this example.\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\n\nIn this work, we proposed a lightweight, low-latency, low-bitrate speech coding framework. In line with the parametric coding paradigm, we designed a feature predictor to capture the temporal redundancy and reduce the burden of coding raw feature frames. Moreover, we applied the discriminative coding scheme to the residual signal to further improve coding gain.\nWe showed that the proposed combination of predictive coding and discriminative residual coding can be harmonized well with the original LPCNet-based codec by providing a more effective quantization scheme than the original multi-stage VQ. We open-source our codes at \\url{https:\/\/saige.sice.indiana.edu\/research-projects\/predictive-LPCNet}. \n\n\\section{Experiments}\n\\subsection{Data}\nWe use the Librispeech \\cite{PanayotovV2015Librispeech} corpus's \\texttt{train-clean-100} fold for training, and \\texttt{dev-clean} for validation, at 16kHz sampling rate. \n18 Bark-scale cepstral coefficients are produced for each 20 ms frame with an overlap of 10ms. In addition, we extract and quantize the 2-dimensional pitch parameters using LPCNet's open-sourced framework. \n\n\\subsection{Training}\nThe training process consists of three steps and is conducted sequentially: prediction model training, codebook learning, and vocoder training. Hence, the results from the preceding steps will be used in the following training. \nCompared to a potential end-to-end learning approach, our modularized learning can circumvent the issue of dealing with non-differentiable quantization. \n\nBoth the feature predictor and the vocoder will eventually operate in a synthesis mode, where the inputs to the model are the synthesized results from the previous step. Therefore, we add noise to the input during training for a more robust development, as suggested in  \\cite{ValinJ2019lpcnetcoding,JinZ2018fftnet}. \nFinally, the vocoder is finetuned with the quantized input features.\n\nCodebook training is based on the residuals ${\\bm{r}}$ produced from the encoder of the feature predictor $\\mathcal{F}_{pred}$. For both vector and scalar codebooks, we run k-means clustering and pick the learned centroids as the codewords. \nWhen generating residuals for codebook training, the encoder skips the quantization step (eq. \\eqref{qtz}) but will consider the residual thresholding. That is, the residual $\\bm r_n$ will be added back to the prediction result $\\hat{\\bm c}_n$ (as in eq. \\ref{add-back}) only if $||\\bm r||_1 \\geq \\theta$. \nWe randomly pick 2-second segments from each utterance in training set to generate the residual vector for codebook training. Codebooks are trained exclusively for each bitrate.\n\n\nThe feature predictor model we used in the experiments contains  $0.65$M parameters, and the entire codec, including the LPCNet vocoder, has $2.5$M parameters. Our codec is suitable for the real-time coding task because of the causality preserved in the frame-level prediction. The algorithmic delay of our codec is $75$ ms, to which the LPCNet vocoder contributes $60$ms-latency from its convolution operation. Another $15$ ms-delay comes from our feature predictor, which occurs while waiting for the ground-truth cepstral-frame of $10$ms with an extra $5$ms look-ahead to compute a cepstrum.\n\n\\subsection{Evaluation and baseline}\n\nWe employ two state-of-the-art low-bitrate codecs as baselines, LPCNet at 1.6kbps and Lyra V2 \\footnote{https:\/\/opensource.googleblog.com\/lyra-v2-a-better-faster-and-more-versatile-speech-codec.html} at 3.2kbps. Lyra V2 is an improved version of Lyra\\footnote{https:\/\/ai.googleblog.com\/lyra-new-very-low-bitrate-codec-for.html} \\cite{kleijn2021generative}, integratin SoundStream \\cite{Zeghidour2021soundstream} in its original architectures for a better coding gain.\n\n\nWe perform a MUSHRA test \\cite{mushra} on our codec at three different bitrates and the two baselines. Ten gender-balanced clean utterances from the LibriSpeech \\texttt{test-clean} set are used. The trials also include a hidden reference and a low-pass-filtered anchor at 3.5kHz. Ten speech experts participated in the test, and no one was excluded per the listener exclusion policy.\n\n\\section{Introduction}\n\nA speech codec, in general, comprises modules for speech compression, quantization, and reconstruction. It has been used in various communication and entertainment applications after standardization \\cite{BessetteB2002amrwb, SchroederM1985celp} or open-sourcing \\cite{ValinJM2012opus}. The common goal in speech coding is to achieve the maximum coding gain, i.e., maintaining the perceptual quality and intelligibility of the reconstructed speech signals with a minimum bitrate.\n\nThe involvement of neural networks has greatly benefited the coding trade-off, effectively eliminating the codes' redundancy while improving the reconstruction quality. More recently, the advances of generative models and their applications in speech coding led to a trend in very low-bitrate speech codecs. The first WaveNet-based speech codec \\cite{KleijnW2018wavenet} demonstrates the usage of neural synthesis in both waveform and parametric coding. The latter is more favored in subsequent studies because of its inherent advantages in dealing with very condensed speech features. These neural vocoders work on the decoder side, leveraging the powerful neural synthesis architecture. Their encoding parts are relatively simplified, relying on existing Codec 2 codes  \\cite{codec2} as in the original WaveNet-based speech codec \\cite{KleijnW2018wavenet} or the dimension-reduced frequency-domain speech representations, e.g., cepstrum features \\cite{ValinJ2019lpcnet, ValinJ2019lpcnetcoding}, and LPC analysis \\cite{KlejsaJ2019samplernn}. \n\nIn this line of work, the performance bottleneck comes from the very compact codes, leading to poorer reconstruction quality. To mitigate the issue, some efforts apply more complex models in the encoder to improve the quality of the features \\cite{kim2021neurally, yoshimura2018wavenet} or use generative models for post-processing \\cite{zhao2018convolutional, skoglund2019improving, biswas2020audio} at the end of the existing codec to facilitate signal restoration. However, the output performance is still bounded by the quality of the coding features. \n\nEnd-to-end neural codecs that train the encoder, quantizer, and decoder jointly work as an alternative to the low-bitrate generative speech vocoders \\cite{KankanahalliS2018icassp, ZhenK2019interspeech, ZhenK2022taslp, Zeghidour2021soundstream}.\nIn this way, the neural encoder participates in removing the redundancy in the source signal and produces features that are more associated with the decoder, in contrast to the traditional speech features. Various other methods have been developed to improve the quality of the features, regarding robustness \\cite{casebeer2021enhancing, Lim2020robust}, scalability \\cite{jiang2022cross} and the variability issues \\cite{kleijn2021generative}.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=.8\\columnwidth]{figs\/overview.pdf}\n    \\caption{The overview of the proposed neural speech coding system with feature predictors and LPCNet-based vocoder.}\n    \\label{fig:overview}\n    \\vspace{-6mm}\n\\end{figure}\n\nHowever, end-to-end codecs tend to suffer in very low-bitrates cases ($<$2 kbps) because that requires the coding features to be extremely small and expressive simultaneously. To deal with that, an ultra-low bitrate codec \\cite{SiahkoohiA2022interspeech} borrows the embeddings from a self-supervised training task to increase the expressiveness of the state-of-the-art codec SoundStream's features \\cite{Zeghidour2021soundstream}, and can obtain a decent speech quality with a very low bitrate, $0.6$ kbps.\nTF-Codec \\cite{jiang2022predictive} addresses the problem by reducing the temporal redundancy in the latent features with a predictive model and reports decent reconstruction quality at $1$ kbps. However, both models entail high complexity. Besides, because TF-Codec's prediction model runs on a latent space that requires a specific pair of encoder and decoder, it brings an extra cost for other existing codecs to mount its predictive module directly. \n\n\nIn this paper, we aim at a low-bitrate, low-delay, and low-complexity neural speech codec that utilizes neural feature prediction to reduce the temporal redundancy from the sequence of feature frames.\nWe introduce a gated recurrent unit (GRU)-based \\cite{ChoK2014emnlp} frame-level feature predictor that can forecast the feature vector at time frame $t$ using its preceding frames. Since the decoder also employs the exact feature predictor, it can ``generate\" most of the feature vector at no cost of bitrate, while the imperfectly generated feature vectors are compensated by the coded residual coming from the encoder side. \nAdditionally, we employ \\textit{discriminative coding} in the residual space. This idea is demonstrated in source-aware neural audio coding by distinguishing speech and noise sources in the latent feature space \\cite{YangH2021sanac}. In this paper, we use different entropy coding strategies at each frame depending on the amount of information they carry. Compared to the TF-Codec, our model explicitly codes only the prediction residuals, and the proposed predictive modules are designed to work in combination with existing low-complexity neural codecs. In particular, we are based on the efficient LPCNet-based speech coding framework \\cite{ValinJ2019lpcnetcoding}, and our analysis and model training mainly focuses on the cepstral coefficients.\n\n\n\n\n\n\\section{The proposed predictive coding}\n\\subsection{Overview}\nIn conjunction with the LPCNet's sample-level vocoding, the proposed feature prediction model performs hierarchical prediction: first in the feature space and then in the sample level. As shown in Fig \\ref{fig:overview}, the frame-level feature predictor $\\mathcal{F}_{pred}$ works on both the encoder and decoder sides. The encoder computes and quantizes the frame-level prediction residuals $\\bar{\\bm r}$ and passes them to the decoder. Then, the decoder adds the received residuals to its own feature predictions $\\hat{\\bm c}$ to obtain the recovered frame-level features. The sample-level predictive coding (i.e., the LPCNet vocoder) works only on the decoder that synthesizes waveform samples $\\hat{\\bm s}$ from the recovered features as the codec's output.\n\n\n\n\n\\subsection{The frame-level feature predicion}\n\n\\subsubsection{Feature predictor}\nWe apply a WaveRNN-based model \\cite{KalchbrennerN2018wavernn} to make a frame-level prediction on the 18-dimensional continuous cepstral coefficients. \nWaveRNN explicitly considers the output at time $n-1$ as the estimation of the $n$-th's sample. \nIn our frame-by-frame feature prediction scenario, the recurrent neural network  $\\mathcal{H}(\\cdot)$ takes in previous hidden state $\\bm h_{n-1}$ and the previous feature vector $\\bm c_{n-1}$, to predict the next frame $\\hat{\\bm c}_n $. Additionally, we condition the frame-level prediction with pitch parameters $\\bm m_n$ (period and correlation) used in LPCNet. Our model consists of two gated recurrent unit (GRU) layers \\cite{ChoK2014emnlp}, with 384 and 128 hidden units, respectively, followed by a fully connected layer. The feature predictor $\\mathcal{F}_\\text{pred}: \\bm c_{n-1} \\mapsto \\hat{\\bm c}_n$ can therefore be recursively defined as,\n\\begin{equation}\\label{eq:prediction}\n    \\bm h_n = \\mathcal{H}(\\bm c_{n-1}, \\bm h_{n-1}, \\bm m_n), \\quad\n    \\hat{\\bm c}_n = \\text{tanh}(\\bm W \\bm h_{n}), \n\\end{equation}\nwhere $n$ represents the time-domain index. We found the results are more stable by scaling input and output features to the range of $[-1,1]$. To this end, the output linear layer employs a tanh activation after a linear combination with parameter $\\bm W$. Biases are omitted for brevity.\n\nWe optimize the model by minimizing the mean squared error (MSE) between the prediction and target $\\mathcal{L} = MSE(\\bm c_n, \\hat{\\bm c}_n)$. We chose it over the maximum log-likelihood approach with explicit Gaussian modeling of the features because modeling the cepstrum coefficients with Gaussian distributions was unreliable. \n\n\n\\subsubsection{Feature residual coding}\n\n\nWe employ the predictor in both the encoder and decoder to cover the information that can be inferred from the temporal dependency. Thus, for the decoder to recover the features, it is only necessary to provide the decoder with the residuals between the prediction and ground-truth features. This kind of explicit residual coding can lead to a more efficient coding scheme, given that our predictor model makes reliable predictions, especially in the areas of smooth signals, reducing the entropy of the residual. \n\n\n\nThe primary pipeline for residual coding is then summarized recursively as follows:\n\\begin{align}\n    \\text{Encoder:} \\quad \\hat{\\bm c}_n &=\\mathcal{F}_\\text{pred}(\\bar{\\bm{c}}_{n-1}) \\label{enc_pred}\\\\\n    \\bm r_n &= \\bm c_n - \\hat{\\bm c}_n\\\\\n    \\bar{\\bm r}_n &= \\mathcal{Q}(\\bm r_n)\\quad \\text{(send it to the decoder)} \\label{qtz}\\\\\n    \\bar{\\bm c}_n &= \\hat{\\bm c}_n + \\bar{\\bm r}_n \\quad \\text{(input for the next round } n+1 \\text{)} \\label{add-back}\n\\end{align}\n\nThe encoder explicitly computes the residual $\\bm r_n$, and then the quantizer $\\mathcal{Q}(\\cdot)$ converts it into a bitstring $\\bar{\\bm{r}}_n$ as the final code.\nNote that we opt to input the \\textit{noisy feature} $\\bar{\\bm c}$ instead of the original feature $\\bm c$ into the encoder's feature predictor (eq. \\eqref{enc_pred}) in order to match the encoder's output to the decoder's circumstance. Since the decoder does not have access to the original features, it has no choice but to use the noisy ones $\\bar{\\bm c}$ as the predictor's input. Therefore, by repeating the decoder's behavior in the encoder, we aim to guarantee that the residuals provided by the encoder are the accurate compensation for the decoder's feature prediction.  \n\nThe decoder first pre-computes the prediction $\\hat{\\bm c}_n$, and then supplement it with the quantized residual $\\bar{\\bm r}_n$ received from the encoder to finalize the feature reconstruction $\\bar{\\bm c}_n$.\n\\begin{align}\n    \\text{Decoder:} \\quad \\hat{\\bm c}_n &=\\mathcal{F}_\\text{pred}(\\bar{\\bm{c}}_{n-1})\\\\\n    \\bar{\\bm c}_n &= \\hat{\\bm c}_n + \\bar{\\bm r}_n.\n\\end{align}\nWhen running $\\mathcal{F}_\\text{pred}$ in either the encoder or decoder, we starts with zero-initialized input $\\bar{\\bm c}_0 = \\bm 0$ , and iteratively update the input tensor with the model predictions . \n\n\n\n\n\n\n\n\\subsection{The sample-level vocoder: LPCNet}\n\nWe borrow LPCNet to complete time-domain synthesis on the decoder side. LPCNet takes as input pitch parameters $\\bm m_n$ and cepstrum features $\\bm c_n$. Then, it integrates LPC analysis into the neural generative model, i.e., $p_t = \\sum_{\\tau=1}^T a_\\tau \\hat{s}_{t-\\tau}$, which computes the prediction $p_t$ for the sample index $t$ by using $T$ previously estimated samples $\\hat{s}_{t-T:t-1}$. In this way, the burden of spectral envelop modeling is taken away from the neural network. The prediction coefficient $a_\\tau$ is computed only from the 18-band Bark-frequency cepstrum (the transmitted code in the original LPCNet coder \\cite{ValinJ2019lpcnetcoding}), forming a very compact bitstring. \nOn top of the DSP-based linear prediction, LPCNet also employs a WaveRNN network $\\mathcal{G}$ to estimate the prediction residuals ${e}_t$ directly in a causal manner:\n\\begin{equation}\n    \\hat{e}_t = \\mathcal{G}(p_t, \\hat{s}_{<t}, \\hat{e}_{<t}), \\quad\n    \\hat{s}_t = p_t + \\hat{e}_t, \n\\end{equation}\nwhere the quality of the estimated excitation signal $\\hat{e}_t$ significantly matters for better speech quality. The network $\\mathcal{G}$ mainly consists of two GRU layers, followed by two fully connected layers.\n\nWe employ the same LPCNet vocoder for our coding system, except that its input feature $\\bm c_n$ is replaced with our proposed feature reconstruction $\\bar{\\bm{c}}_n$, necessitating re-training the vocoder. Note that in the speech coding version \\cite{ValinJ2019lpcnetcoding}, LPCNet does take its own compressed feature representation as input, whose compression ratio is something our method competes against. \n\n\n\n\n\\section{Results}\n\n\n\\begin{figure}[t]\n   \n        \\centering\n        \\includegraphics[height=1.5in]{figs\/mushra.pdf}\n        \\vspace{-4mm}\n        \\caption{MUSHRA score. The reference at MUSHRA score 100 is not shown in the graph.}\n        \\label{fig:subjective}\n        \\vspace{-4mm}\n\\end{figure}\n\n\\begin{figure}[t]\n        \\centering\n        \\includegraphics[width=\\columnwidth]{figs\/cepstrum_analysis.pdf}\n        \\vspace{-2mm}\n        \\caption{The original cepstra, coded cepstra, the $L1$ norm curve for the vector residuals (red), and the $L_1$ norm curve for the scalar residuals (blue) of a 3s sample. The black dash lines mark the thresholds for the 1.48 kbps case. }\n        \\label{fig:cepstral}\n        \\vspace{-6mm}\n\\end{figure}\n\n\\subsection{Subjective test results}\nFig. \\ref{fig:subjective} shows the scores from the MUSHRA test. We can see that the proposed model outperforms LPCNet at a lower bitrate ($\\sim$ 1.47 vs. 1.6 kbps). At $\\sim$ 0.93kbps, our codec is slightly worse than LPCNet but its performance could be acceptable given the 40\\% of bitrate reduction. The proposed codec at 2.87 kbps is perceptually better than Lyra V2 at $3.2$kbps. The results demonstrate the proposed model's scalability and effectiveness across different bitrate ranges. \n\n\\subsection{Analysis on the discriminative residual coding}\nIn Fig.~\\ref{fig:cepstral}, we picked a random utterance sample and aligned its cepstra with the coded version of the cepstra. In between, it lays the $L_1$ norm curves of the scalar and vector residuals in blue and red. At the points where any curve is over the black dash threshold, the predictor failed to make a good prediction, thus requiring more bits to represent these residuals using a $\\mathcal{Q}_\\text{L}$ scheme. Those properly coded frames take up only $7 \\%$ of the total frames in this $\\sim 1.48$ kbps case. As for the below-threshold area, conversely, the quantization falls back to the $\\mathcal{Q}_\\text{S}$ mode while being frugal in assigning bits to these well-predictable frames.\n\nWe made three main observations from the graph. Firstly, our codec does a good job estimating and coding the original cepstra (by comparing the top and bottom cepstra), especially at the lower dimensions, although with some energy loss at the higher dimension. Even at the low-bitrate coding area, the cepstral patterns are still well captured. \nSecondly, we can observe that the curves' peaks align with the original cepstrum's transient events. To show the matches, we marked some arrows and dash lines that connect the residual $L_1$ norm curves to the cepstra as examples. The alignment of cepstral changes and the peaks indicates that, although the performance of the feature predictor degrades at the transient events, the discriminative coding can make accurate compensation at a minimal cost.\nFinally, it is also noteworthy that despite embracing similarity, the $L_1$ norm of the vector and scalar can have peaks and valleys at different places. Hence, they could compensate for the predictor model's different behavior at individual subbands, which also justifies the advantage of band-specific discriminative coding for the SQ and VQ parts. \n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\subsection{Quantum gravity}\nThe Sachdev-Ye-Kitaev (SYK) model \\cite{sachdevye,sachdev15,maldacena2016remarks,suh} is conjectured to be a toy model for quantum gravity in two dimensional spacetime.   The nature of the proposed correspondence is holographic \\cite{maldacena}:  this model of quantum mechanics with $N\\gg 1$ degrees of freedom should be equivalent to a quantum theory in two spacetime dimensions, which contains dynamical (quantum) gravity with coupling constant $G_{\\mathrm{N}} \\sim 1\/N$  \\cite{maldacena2016remarks, suh, almheiri, Jensen, Maldacena1606}.   If this correspondence is true, then at least one model of non-perturbative quantum gravity reduces to the physics of a special quantum mechanical model with a finite number of degrees of freedom.   \n\nA particularly elegant feature of these holographic theories of quantum gravity is the ``maximally rapid\" exponential growth of certain out-of-time-ordered correlation functions (OTOCs) \\cite{shenker13,stanfordbound}.   If $\\mathcal{O}_1$ and $\\mathcal{O}_2$ represent few-body operators in the holographic quantum system, \n\\begin{equation}\n\\mathrm{tr}\\left(\\sqrt{\\rho} [\\mathcal{O}_1(t),\\mathcal{O}_2)]^\\dagger \\sqrt{\\rho} [\\mathcal{O}_1(t),\\mathcal{O}_2)] \\right) \\sim \\frac{1}{N} \\mathrm{e}^{\\lambda t},  \\label{eq:introlyapunov}\n\\end{equation}\nwith $\\lambda$ the many-body Lyapunov exponent.  One implicitly assumes that $N$ is very large, so that there might be a parametrically long range of times over which (\\ref{eq:introlyapunov}) holds.  Here $\\rho$ is the density matrix associated to a mixed quantum state, usually taken to be a thermal ensemble at inverse temperature $\\beta$, where it was conjectured in \\cite{stanfordbound} that \\begin{equation}\n\\lambda \\le \\frac{2\\text{\\unslant[-.18]\\pi}}{\\beta}\n\\end{equation}\nin many quantum systems -- especially those holographically dual to quantum gravity, where this bound is saturated \\cite{shenker13}.  Indeed, the discovery of this maximal growth in the SYK model  demonstrated its apparent connections to quantum gravity.  \n\n\nIn certain circumstances, these OTOCs can be interpreted as measuring the average size of a growing operator \\cite{nahum, tibor, lucas1809}.  At leading order in $1\/N$, there are hints that the time evolution of the operator size distribution in the SYK model \\cite{stanford1802, alex1811} may reconstruct its holographically dual geometry \\cite{lenny1,lenny2}.  \n\n\n The expectation that $\\lambda < \\infty$ as $N\\rightarrow \\infty$ (at least for broad classes of models) goes by the name of the ``fast scrambling conjecture\" \\cite{sekino}.  While there are certainly counterexamples to the fast scrambling conjecture \\cite{lucas1805, lucas1903} (at least at infinite temperature, where $\\rho$ is proportional to the identity), they are arguably rather finely tuned.  For many models, especially those related via the holographic correspondence to quantum gravity, the fast scrambling conjecture is expected to hold.  However, it has never been mathematically demonstrated that the fast scrambling conjecture holds  in a genuine model of (any kind of) quantum gravity.  All that is known is that the fast scrambling conjecture holds at leading non-trivial order in a perturbative $1\/N$ expansion of OTOCs.  Unfortunately, this $1\/N$ expansion is expected to have zero radius of convergence: see e.g. \\cite{stanford1903}.\n\nThe purpose of this paper is to prove the fast scrambling conjecture in the SYK model at infinite temperature.  We do so by proving non-perturbative constraints on the dynamics of the operator size distribution.   If the holographic correspondence is true, these are non-perturbative constraints on the early time dynamics in a theory of quantum gravity.  We find that a previous perturbative calculations of operator growth at infinite temperature \\cite{stanford1802} essentially saturates our bounds, after a suitable rescaling of time.  So we conjecture that the perturbative description of operator growth in the SYK model is robust to all non-perturbative quantum effects.  This suggests that a semiclassical treatment of many-body chaos in holographic quantum gravity can be sensible, and that non-perturbative quantum gravitational effects do not destroy the emergence of (semi)classical space and time. \n\nWhile we focus on the SYK model in this paper, we expect that our bounds on operator growth can be generalized to other random quantum systems on regular factor graphs \\cite{chen1}.   We emphasize that our approach completely bypasses the ordinary diagrammatic and manifestly perturbative approach to many-body physics, where non-perturbative effects must be obtained by sophisticated and generally non-unique resummation techniques \\cite{stanford1903}.    Specifically, in our approach, non-perturbative results are rigorously obtained by classical combinatorial calculations with $\\mathrm{O}(\\log N)$ ``loops\", which can themselves be bounded in a simple way.  In contrast, in an ordinary diagrammatic expansion, non-perturbative effects often become important at $\\mathrm{O}(N)$ loops, which is usually prohibitively challenging to reach in analytic calculations.\n \n \\subsection{Comparison to the Lieb-Robinson Theorem}\n\nThe techniques we use in our proof are quite different from the conventional Lieb-Robinson bounds \\cite{liebrobinson,hastings}.  Here, one assumes a tensor product structure for the Hilbert space $\\mathcal{H}$ (though see \\cite{rezakhani}).  Define \\begin{equation}\nV := \\lbrace 1,\\ldots,N \\rbrace \\label{eq:Vdef}\n\\end{equation} \nto be the set of all quantum degrees of freedom, and then define \n\\begin{equation}\n\\mathcal{H} := \\bigotimes_{i=1}^N \\mathcal{H}_i.\n\\end{equation}\n Let $X,Y\\subset V$, and denote with $\\mathrm{dist}(X,Y)$ a ``minimal path length\" between any two points induced by the Hamiltonian which generates time evolution (we leave details to the references).  Then the Lieb-Robinson theorem states that there are constants $0 <C,\\lambda,\\mu<\\infty$ such that \\begin{equation}\n\\lVert [A_X(t),B_Y] \\rVert \\le C\\exp[\\lambda t - \\mu \\mathrm{dist}(X,Y)].\n\\end{equation}\nHowever, we emphasize that the $\\lambda$ here is not the same as $\\lambda$ in (\\ref{eq:introlyapunov}) in typical strictly local lattice models, as this bound is not qualitatively tight \\cite{chen1}.\n\nThere are two reasons why the conventional Lieb-Robinson approaches do not seem relevant to the SYK model.  Firstly, the SYK model  (defined in Section \\ref{sec:ensemble}) has no spatial locality on a lattice \\, so $\\mathrm{dist}(X,Y) = 0\\text{ or }1$.  The primary ``control parameter\" of the Lieb-Robinson bounds therefore is irrelevant.  Furthermore, it is not possible to show that $\\lambda$ remains finite as $N\\rightarrow \\infty$, using a conventional combinatorial derivation of Lieb-Robinson bounds.  Secondly,  conventional Lieb-Robinson bounds are on the operator norm (maximal singular value) of commutators, which is far stronger than the trace-like norms of (\\ref{eq:introlyapunov}). \n\nIn this paper, we develop a technique to bypass bounding conventional operator norms, and bound (\\ref{eq:introlyapunov}) directly whenever $\\rho$ is the infinite temperature density matrix.  Our approach is based on interpreting Heisenberg operator growth as a many-body continuous-time quantum walk, which naturally implements the constraints of unitary time evolution in a far stronger way.  In particular, unitarity demands that a certain ``operator size distribution\" is normalized as a conventional probability distribution.  The inability for conventional Lieb-Robinson bounds to maintain this normalization is, in large part, responsible for their inability to give sharp bounds on OTOCs such as (\\ref{eq:introlyapunov}).     \n\nPrecursors to our approach include \\cite{chen1,chen2}, which have also attempted to simplify the combinatorics of Lieb-Robinson bounds and obtain stronger results.\\footnote{These methods share some similarity to Feynman diagram resummation methods in mathematical kinetic theory \\cite{erdos}.   We do not know the extent to which the kinetic methods could be used to further improve our own techniques.  Our ``diagrammatic\" resummation is far simpler than the one used to understand kinetic theory.}  We expect that the methods developed in this paper will be broadly applicable to many problems in quantum dynamics where it is not necessary to bound the operator norm of a commutator, such as quantum state transfer \\cite{lucas2001}. \n\n\\section{Preliminaries}\n\\subsection{Majorana fermions}\n Let $q,N\\in2\\mathbb{Z}^+$ be positive even integers, and define the set $V$ as in (\\ref{eq:Vdef}).\nDefine a finite dimensional quantum mechanical Hilbert space \\begin{equation}\n\\mathcal{H} := \\left(\\mathbb{C}^2\\right)^{\\otimes \\frac{N}{2}} .\n\\end{equation}\nTo each of the $N$ elements $i\\in V$, we associate a Majorana fermion $\\psi_i$:  a Hermitian operator on $\\mathcal{H}$ obeying the anticommutation relations \\begin{equation} \n\\lbrace \\psi_i, \\psi_j\\rbrace := 2\\mathbb{I}[i=j].  \\label{eq:anticommutator}\n\\end{equation}\nHere $\\mathbb{I}[\\cdots]$ denotes the indicator function, which returns 1 if its argument is true and 0 if false.\n\nLet $\\mathcal{B} = \\mathrm{End}(\\mathcal{H})$ be the set of quantum mechanical operators.  This is a complex inner product space if we define the inner product \\begin{equation}\n(A|B) := 2^{-N\/2} \\mathrm{tr}_{\\mathcal{H}}(A^\\dagger B).\n\\end{equation}\nWe denote $\\lVert A\\rVert = \\sqrt{(A|A)}$: we emphasize that this is \\emph{not} the conventional operator norm.   \n\n\\begin{prop}\\label{prop1}\nAn orthonormal basis for $\\mathcal{B}$ is spanned by $\\lbrace |Y):Y\\subseteq V\\rbrace$, where \\begin{equation}\n|Y) := \\psi_Y := \\prod_{i\\in Y}\\psi_i.\n\\end{equation}\nWe define $\\psi_\\emptyset = 1$ (the identity operator).  Here and below the product over anticommuting fermions is ordered by the smallest $i$ first.  \n\\end{prop}\n\\begin{proof}\nWe must show that \\begin{equation}\n(Y_1|Y_2) = \\mathbb{I}[Y_1=Y_2].\n\\end{equation}\nIf $Y_1=Y_2$, this identity follows immediately from (\\ref{eq:anticommutator}).   Letting \\begin{equation}\nA\\triangle B  = A \\cup B - A\\cap B\n\\end{equation}\n denote the symmetric difference of two sets, we conclude from (\\ref{eq:anticommutator}) that \\begin{equation}\n \\psi_{Y_1}\\psi_{Y_2} = \\alpha \\psi_{Y_1\\triangle Y_2}\\text{ for }\\alpha \\in \\lbrace \\pm 1 \\rbrace.\n \\end{equation}\n   It remains to show that \\begin{equation}\n   \\mathrm{tr}_{\\mathcal{H}}(\\psi_Y) = 0\\text{, if } Y\\ne\\emptyset.  \\label{eq:trpsi0} \n   \\end{equation}\n   To obtain (\\ref{eq:trpsi0}) when $|Y|$ is an odd integer, we choose any $i\\notin Y$ (which must exist as $N$ is even).  Since \\begin{equation}\n   \\psi_Y\\psi_i  = (-1)^{|Y|} \\psi_i \\psi_Y \\;\\;\\; (i\\notin Y),  \\label{eq:psiYpsii}\n   \\end{equation}\n   we conclude that for $|Y|$ odd \\begin{equation}\n   0 = \\mathrm{tr}_{\\mathcal{H}}(\\psi_i \\lbrace \\psi_Y,\\psi_i\\rbrace) = \\mathrm{tr}_{\\mathcal{H}}(\\psi_i  \\psi_Y\\psi_i + \\psi_i^2 \\psi_Y) =\\mathrm{tr}_{\\mathcal{H}}(2 \\psi_Y),  \n   \\end{equation}\n   where in the last step we used cyclic trace properties and (\\ref{eq:anticommutator}).   If on the other hand $|Y| > 0$ is even, let $k$ be the maximal element of $Y$ and write $Y= Y_0 \\cup \\lbrace k\\rbrace$.   Then we obtain \\begin{equation}\n0=\\mathrm{tr}_{\\mathcal{H}}(\\lbrace \\psi_{Y_0}, \\psi_k \\rbrace) = 2\\mathrm{tr}_{\\mathcal{H}}(\\psi_{Y_0} \\psi_k) = 2\\mathrm{tr}_{\\mathcal{H}}(\\psi_{Y}).\n   \\end{equation}\n  Thus we obtain (\\ref{eq:trpsi0}) for any non-empty set $Y$.\n\\end{proof}\n\nLet the Hamiltonian $H$ be a Hermitian operator.  $H$ generates a one parameter family of automorphisms, called time evolution, on $\\mathcal{B}$: defining the Liouvillian \\begin{equation}\n\\mathcal{L} = \\mathrm{i}[H,\\cdot], \\label{eq:Liouvillian}\n\\end{equation}\nfor any $\\mathcal{O}\\in\\mathcal{B}$ we define \\begin{equation}\n|\\mathcal{O}(t)) = \\mathrm{e}^{\\mathcal{L}t} |\\mathcal{O}).\n\\end{equation}\nIn fact, $\\mathrm{e}^{\\mathcal{L}t}$ is unitary, since \\begin{equation}\n(\\mathcal{O}(t)|\\mathcal{O}(t)) = 2^{-N\/2} \\mathrm{tr}_{\\mathcal{H}}\\left( \\left(\\mathrm{e}^{\\mathrm{i}Ht}\\mathcal{O}\\mathrm{e}^{-\\mathrm{i}Ht}\\right)^\\dagger\\left(\\mathrm{e}^{\\mathrm{i}Ht}\\mathcal{O}\\mathrm{e}^{-\\mathrm{i}Ht}\\right)\\right) =  2^{-N\/2}\\mathrm{tr}_{\\mathcal{H}}\\left( \\mathcal{O}^\\dagger \\mathcal{O}\\right) = (\\mathcal{O}|\\mathcal{O}). \\label{eq:Lunitary}\n\\end{equation}\nMore generally, using the cyclic properties of the trace, we conclude that for any $A,B\\in\\mathcal{B}$: \\begin{equation}\n(A|\\mathcal{L}|B) = -(B|\\mathcal{L}|A). \\label{eq:Lantisymmetric}\n\\end{equation}\n\nDefine the projection matrices\\begin{equation}\n\\mathbb{Q}_s |Y) = \\mathbb{I}[|Y| = s] |Y).\n\\end{equation}\nNote that \\begin{equation}\n\\sum_{s=0}^N \\mathbb{Q}_s = 1 \\label{eq:sumPs}\n\\end{equation}\n(with 1 the identity of $\\mathrm{End}(\\mathcal{B})$).  We say that the non-null vectors of $\\mathbb{Q}_s$ correspond to \\emph{operators of size $s$}.   \nGiven $\\mathcal{O}\\in \\mathcal{B}$, we say that \\begin{equation}\nP_s(\\mathcal{O},t) = \\frac{(\\mathcal{O}(t)| \\mathbb{Q}_s |\\mathcal{O}(t))}{(\\mathcal{O}(t)|\\mathcal{O}(t))} \\label{eq:PsOt}\n\\end{equation}\nis the probability that operator $\\mathcal{O}$ is size $s$ at time $t$.   To see that this is a well-defined probability measure on $\\lbrace 0, 1,\\ldots, N\\rbrace$, observe that $\\mathbb{Q}_s$ is positive semidefinite and hence $P_s \\ge 0$; then from  (\\ref{eq:sumPs}), \\begin{equation}\n\\sum_{s=0}^N P_s(\\mathcal{O},t) = 1,\n\\end{equation} \nfor any $\\mathcal{O}\\in \\mathcal{B}$ and $t\\in\\mathbb{R}$.    For simplicity, we will drop the explicit $\\mathcal{O}$ in $P_s(t)$, as our formalism does not depend on the particular choice of operator.\n\n\n\\subsection{The SYK ensemble}\\label{sec:ensemble}\nThe SYK model corresponds to a random ensemble of Hamiltonians.  Define \\begin{equation}\nF := \\lbrace X\\subset V : |X| = q\\rbrace\n\\end{equation}\nto be the set of all subsets of $V$ with exactly $q$ elements.\nFor each $X\\in F$, let $J_X$ be an independent and identically distributed (iid) Rademacher\\footnote{In the physics literature, the random variables $J_X$ are typically taken to be Gaussian.  We expect that a very similar result to ours will hold in this case as well, but we found the combinatorial problem discussed in Section \\ref{sec:trans} to be a bit more elegant for Rademacher random variables.} random variable: \\begin{equation}\n\\mathbb{P}\\left[J_X = \\sigma \\right] = \\mathbb{P}\\left[J_X = -\\sigma \\right]  = \\frac{1}{2}, \n\\end{equation}\nwhere \\begin{equation}\n\\sigma := \\left[2q\\left(\\begin{array}{c} N-1 \\\\ q-1 \\end{array}\\right) \\right]^{-1\/2}.  \\label{eq:sigmadef}\n\\end{equation}\nThe $q$-local SYK model is the random ensemble of Hamiltonians $H$, corresponding to the random Hermitian matrix\n\\begin{equation}\nH := \\mathrm{i}^{q\/2} \\sum_{X\\in F} J_X \\prod_{i \\in X} \\psi_i := \\sum_{X\\in F} H_X.\n\\end{equation}\nThe randomness in the SYK ensemble is essential in our proof of operator growth bounds.   Averages over the ensemble of random variables $\\lbrace J_X\\rbrace $ are denoted as $\\mathbb{E}[\\cdots]$, and probability is denoted as $\\mathbb{P}[\\cdots]$, as above.\nWe define $\\mathcal{L}_X := \\mathrm{i}[H_X,\\cdot]$.\n\\begin{prop} \\label{propk}\nIf $\\mathbb{Q}_s|\\mathcal{O}_s) = |\\mathcal{O}_s)$, $X\\in F$, and $\\mathbb{Q}_{s^\\prime} \\mathcal{L}_X|\\mathcal{O}_s) \\ne 0$, then there exists $k\\in\\mathbb{Z}^+$ for which \\begin{equation}\ns^\\prime -s = q+2-4k.  \\label{eq:kdef}\n\\end{equation}  \nIn particular, \\begin{equation}\n|s^\\prime - s| \\le q-2. \\label{eq:qm2max}\n\\end{equation}\n\\end{prop}\n\\begin{proof}\nSince $\\mathcal{L}_X|Y)$ is proportional to $|[\\psi_X,\\psi_Y])$, we analyze when $[\\psi_X,\\psi_Y]\\ne 0$ is possible.   Without loss of generality we write \\begin{equation}\nZ = X\\cap Y, \\;\\;\\;\\;\\; V=X-Z, \\;\\;\\;\\;\\; W = Y-Z,\n\\end{equation}in which case it suffices to constrain\n\\begin{equation}\n\\lVert [\\psi_X,\\psi_Y]\\rVert = \\lVert [\\psi_V\\psi_Z,\\psi_W\\psi_Z]\\rVert  = \\lVert [\\psi_V\\psi_Z,\\psi_W]\\psi_Z + \\psi_W [\\psi_V,\\psi_Z]\\psi_Z\\rVert.\n\\end{equation}\nBy repeated use of (\\ref{eq:psiYpsii}), if $A\\cap B = 0$, \\begin{equation}\n\\psi_A \\psi_B = (-1)^{|A||B|}\\psi_B\\psi_A, \\label{eq:psiApsiB}\n\\end{equation}\nhence $[\\psi_A,\\psi_B]\\ne 0$ if and only if $|A|$ and $|B|$ are both odd.   Since $|V\\cap Z|$ is even, we conclude that $[\\psi_X,\\psi_Y]=0$ unless $|V|$ and $|Z|$ are both odd, in which case $|[\\psi_X,\\psi_Y])$ is proportional to $|\\psi_V\\psi_W)$.  If $X\\in F$, then $|X|=q$, and so $\\mathbb{Q}_{s^\\prime}\\mathcal{L}_X\\mathbb{Q}_s|Y)\\ne 0$ only if $|Y|=s$ and \\begin{equation}\ns^\\prime = |X|+|Y|-2|X\\cap Y| = s+q - 2|X\\cap Y|.\n\\end{equation}\nSince $|X\\cap Y|$ is odd, we obtain the desired result.\n\\end{proof}\nSince by definition $s^\\prime > s$, we conclude that \\begin{equation}\n2k-1 < \\frac{q}{2}.  \\label{eq:2kminus1}\n\\end{equation}\n\nIt will be useful to define the following partition of the set of all non-trivial operator sizes: \\begin{equation}\n\\lbrace 1,\\ldots ,N \\rbrace = \\bigcup_{l=0}^{N^\\prime} R_l \\label{eq:ldef}\n\\end{equation}\nwhere \\begin{equation}\nN^\\prime := \\left\\lceil \\frac{N-1}{q-2} \\right\\rceil \\label{eq:Nprimedef}\n\\end{equation}\nand \\begin{equation}\nR_l := \\left\\lbrace \\begin{array}{ll}  \\lbrace 1\\rbrace &\\ l=0 \\\\ \\lbrace m \\in \\mathbb{Z}:  (l-1)(q-2)+1<m\\le  l(q-2)+1 \\rbrace &\\ 0<l<N^\\prime \\\\\n \\lbrace m \\in \\mathbb{Z}:  (N^\\prime-1)(q-2)+1<m\\le  N \\rbrace &\\  l = N^\\prime  \\end{array} \\right..\n\\end{equation}\nAnalogous to before, we say that the probability that $\\mathcal{O}(t)$ is in \\emph{block} $l$ is given by \\begin{equation}\nP_l(t) :=  \\sum_{s\\in R_l} P_s(t).\n\\end{equation} \nWe define the projectors \\begin{equation}\n\\mathbb{Q}_l := \\sum_{s\\in R_l} \\mathbb{Q}_s\n\\end{equation}\nWe use the same symbols $P$ and $\\mathbb{Q}$ to denote operator probability distributions and projectors, and will use the subscript ($l$ or nearby letters vs. $s$ or nearby letters) to distinguish whether we are referring to blocks or individual sizes.\n\n\\section{Operator growth}\n\\subsection{Quantum walk of the size distribution}\nWe now show that the time evolution of the operator block distributions $P_l(t)$ can be interpreted as a continuous time quantum walk on a finite one dimensional chain.   The results of this section are very general, and apply to a broad class of quantum many-body models beyond the SYK model, for suitable redefinitions of operator size.  Define $\\varphi_l(t)$ as\\footnote{The authors of \\cite{stanford1802} called this the ``operator wave function\".} \\begin{equation}\n\\varphi_l(t) := \\sqrt{P_l(t)}.   \\label{eq:varphisqrt}\n\\end{equation}  For $\\mathcal{M} \\in \\mathrm{End}(\\mathcal{B})$, we define \\begin{equation}\n\\lVert \\mathcal{M}\\rVert := \\sup_{\\mathcal{O} \\in \\mathcal{B}} \\frac{\\lVert \\mathcal{M}|\\mathcal{O})\\rVert}{\\lVert \\mathcal{O}\\rVert} = \\sup_{\\mathcal{O},\\mathcal{O}^\\prime \\in \\mathcal{B}} \\frac{| (\\mathcal{O}^\\prime| \\mathcal{M}|\\mathcal{O}) |}{\\lVert \\mathcal{O}\\rVert \\lVert \\mathcal{O}^\\prime \\rVert} ,  \\label{eq:endBnorm}\n\\end{equation}\nin analogy with the usual operator norm on linear transformations. \n  \\begin{prop}\\label{prop2}\nLet $0\\le s<s^\\prime \\le N$, let $H$ be a Hamiltonian drawn from the SYK ensemble, and let \\begin{equation}\n\\mathcal{K}_{s^\\prime s} = \\lVert \\mathbb{Q}_{s^\\prime}\\mathcal{L}\\mathbb{Q}_s\\rVert.  \\label{eq:calKss}\n\\end{equation}\nThen there exist functions $K_{s^\\prime s}: \\mathbb{R} \\rightarrow [-\\mathcal{K}_{ss^\\prime}, \\mathcal{K}_{ss^\\prime} ]$ such that \\begin{equation}\n\\frac{\\mathrm{d}}{\\mathrm{d}t} \\sqrt{P_s(t)}= \\sum_{s^\\prime < s} K_{s s^\\prime}(t) \\sqrt{P_{s^\\prime}(t)} -  \\sum_{s>s^\\prime} K_{ s^\\prime s}(t) \\sqrt{P_{s^\\prime}(t)} \\label{eq:prop2}\n\\end{equation}\nAnalogously, there exist functions $K_l: \\mathbb{R} \\rightarrow [-\\mathcal{K}_l, \\mathcal{K}_l]$ such that \\begin{equation}\n\\frac{\\mathrm{d}\\varphi_l}{\\mathrm{d}t} = K_{l-1}(t) \\varphi_{l-1}(t) - K_l(t) \\varphi_{l+1}(t), \\label{eq:prop2dos}\n\\end{equation}\n(recall $l$ was defined in (\\ref{eq:ldef})) so long as \\begin{equation}\n\\mathcal{K}_l = \\max\\left(\\max_{s\\in R_l} \\sum_{s^\\prime \\in R_{l+1}} \\mathcal{K}_{s^\\prime s},  \\max_{s^\\prime \\in R_{l+1}} \\sum_{s\\in R_{l}} \\mathcal{K}_{s^\\prime s} \\right)  \\label{eq:Kldef}\n\\end{equation}\nand $K_{-1}(t) = K_{N^\\prime}(t)=0$.  These latter restrictions simply restrict the dynamics to operators in blocks $R_0$ to $R_{N^\\prime}$.\n\\end{prop}\n\\begin{proof}\nFor simplicity in this proof, the $t$-dependence of $\\mathcal{O}$ is implicit; without loss of generality, we may take $\\lVert \\mathcal{O}\\rVert = 1$ by (\\ref{eq:Lunitary}).  For $s\\in \\lbrace0,\\ldots,N\\rbrace$, let $|\\mathcal{A}_s)$ be a unit norm operator such that \\begin{equation}\n\\mathbb{Q}_s|\\mathcal{O}) = \\sqrt{P_s}|\\mathcal{A}_s),\n\\end{equation}\nand note that if $P_s \\ne 0$, $|\\mathcal{A}_s)$ is unique.   Now from (\\ref{eq:Liouvillian}) and (\\ref{eq:PsOt}), \\begin{align}\n\\frac{\\mathrm{d}P_s}{\\mathrm{d}t} &= (\\mathcal{O}| [\\mathbb{Q}_s,\\mathcal{L}] |\\mathcal{O}) = \\sum_{s^\\prime} \\sqrt{P_sP_{s^\\prime}} \\left[(\\mathcal{A}_s|\\mathcal{L}|\\mathcal{A}_{s^\\prime})-(\\mathcal{A}_{s^\\prime}|\\mathcal{L}|\\mathcal{A}_{s})\\right] \\notag \\\\\n&= 2\\sqrt{P_s} \\sum_{s^\\prime < s} K_{ss^\\prime}(t)\\sqrt{P_{s^\\prime}} - 2\\sqrt{P_s} \\sum_{s^\\prime > s} K_{s^\\prime s}(t)\\sqrt{P_{s^\\prime}} \\label{eq:prop2first}\n\\end{align}\nwhere in the first line, we used (\\ref{eq:sumPs}) and in the second line we used (\\ref{eq:Lantisymmetric}) and defined \\begin{equation}\nK_{ss^\\prime}(t) = (\\mathcal{A}_s|\\mathbb{Q}_s\\mathcal{L}\\mathbb{Q}_{s^\\prime}|\\mathcal{A}_{s^\\prime}). \\label{eq:proofKss}\n\\end{equation}\nSince $\\mathrm{d}\\sqrt{P_s} = \\mathrm{d}P_s \/ 2\\sqrt{P_s}$, we obtain (\\ref{eq:prop2}).  Combining (\\ref{eq:endBnorm}), (\\ref{eq:calKss}) and (\\ref{eq:proofKss}), we obtain (\\ref{eq:prop2}). \n\nThe analogue result for block probabilities is identically derived.  In addition, observe that (\\ref{eq:qm2max}) implies that \\begin{equation}\n\\mathbb{Q}_{l^\\prime} \\mathcal{L} \\mathbb{Q}_l \\ne 0 \\text{ only if } |l^\\prime - l| \\le 1.\n\\end{equation}\nHence we obtain (\\ref{eq:prop2dos}) where \\begin{equation}\nK_l(t) := (\\mathcal{O}(t) | \\mathbb{Q}_{l+1} \\mathcal{L}\\mathbb{Q}_l | \\mathcal{O}(t)) \\le \\lVert \\mathbb{Q}_{l+1}\\mathcal{L}\\mathbb{Q}_l \\rVert := \\mathcal{K}_l.\n\\end{equation}\nUsing (\\ref{eq:endBnorm}): \n\\begin{align}\n\\mathcal{K}_l &= \\sup_{\\mathcal{O},\\mathcal{O}^\\prime} \\frac{(\\mathcal{O}^\\prime|\\mathbb{Q}_{l+1}\\mathcal{L}\\mathbb{Q}_l|\\mathcal{O})}{\\sqrt{(\\mathcal{O}|\\mathcal{O})(\\mathcal{O}^\\prime|\\mathcal{O}^\\prime)}} \\le \\sup_{\\mathcal{O},\\mathcal{O}^\\prime} \\sum_{\\substack{ s\\in R_l \\\\s^\\prime \\in R_{l+1} } }\\sqrt{P_s(\\mathcal{O})P_{s^\\prime}(\\mathcal{O}^\\prime)} \\lVert \\mathbb{Q}_{s^\\prime} \\mathcal{L}\\mathbb{Q}_s\\rVert \\notag \\\\\n&\\le \\sup_{\\mathcal{O},\\mathcal{O}^\\prime} \\sum_{\\substack{ s\\in R_l \\\\s^\\prime \\in R_{l+1} } }\\frac{P_s(\\mathcal{O})+P_{s^\\prime}(\\mathcal{O}^\\prime)}{2} \\mathcal{K}_{s^\\prime s}.\n\\end{align}\nA simple identity leads to (\\ref{eq:Kldef}).\n \\end{proof}\n\n(\\ref{eq:prop2dos}) can be interpreted as follows.  $\\varphi_l(t)$ are the coefficients of the real-valued quantum wave function $|\\varphi(t)\\rangle$ of an auxiliary quantum mechanical system defined on the Hilbert space \\begin{equation}\n\\mathcal{H}_{\\mathrm{aux}} := \\mathbb{C}^{1+N^\\prime} :=  \\mathrm{span}\\lbrace |0\\rangle, |1\\rangle, \\ldots, |N^\\prime\\rangle \\rbrace;\n\\end{equation} \nthe latter basis states are defined such that \\begin{equation}\n\\varphi_l(t) := \\langle l|\\varphi(t)\\rangle.  \\label{eq:auxvarphil}\n\\end{equation}\nThe auxiliary Hamiltonian is \\begin{equation}\nH_{\\mathrm{aux}}(t) := \\sum_{l=0}^{N^\\prime - 1} \\mathrm{i} K_l(t) \\left( |l\\rangle\\langle l-1| - |l-1\\rangle\\langle l| \\right). \\label{eq:Haux}\n\\end{equation}\nThe Schr\\\"odinger equation for this auxiliary quantum system is (\\ref{eq:prop2dos}).   \n\n\\subsection{Lyapunov exponent}\nDefine the operator (block) size distribution \\begin{equation}\n\\mathbb{E}_{\\mathrm{s},t} \\left[ f(l) \\right] := \\sum_{l=0}^{N^\\prime} f(l) P_l(t).\n\\end{equation}   A formal definition of the many-body Lyapunov exponent, heuristically defined in (\\ref{eq:introlyapunov}), is given by the growth rate of the logarithm of the average operator size $\\mathbb{E}_{\\mathrm{s},t}[l]$ (recall $l$ was defined in (\\ref{eq:ldef}).  This Lyapunov growth is constrained by the following theorem, which is our first main result:\n\\begin{thm} \\label{lyapunovtheor}\nSuppose that there exist $c\\in \\mathbb{R}^+$ and $M \\in \\mathbb{Z}^+$ such that \\begin{equation}\n\\mathcal{K}_l \\le c(l+1)\\;\\;\\;\\; \\text{ if }l \\le M. \\label{eq:Kll}\n\\end{equation}\nThen for any $\\epsilon \\in \\mathbb{R}^+$, the many-body Lyapunov exponent obeys \\begin{equation}\n\\frac{\\log \\mathbb{E}_{\\mathrm{s},t}[l]}{t} := \\lambda(t) \\le 2c (1+\\epsilon)  \\label{eq:lambdadef}\n\\end{equation}\nfor times \\begin{equation}\n|t| < \\frac{1}{4c(1+\\mathrm{e})}\\left[ \\log M - 2 - \\log \\log \\frac{N^{\\prime 3}}{2\\epsilon}\\right] .  \\label{eq:scramblingtime}\n\\end{equation}\n\\end{thm}\n\\begin{proof}\nWithout loss of generality we assume $t\\ge 0$.  We begin with the following lemma.    Note that here and below, we write $\\mathbb{E}_{\\mathrm{s},t}$ as $\\mathbb{E}_{\\mathrm{s}}$ for convenience.\n \\begin{lem}\nIf (\\ref{eq:Kll}) holds with $M\\ge N^\\prime-1$, then for $n\\in \\mathbb{Z}^+$, \n\\begin{equation}\n\\frac{\\mathrm{d}}{\\mathrm{d}t} \\mathbb{E}_{\\mathrm{s}} \\left[l^n \\right] \\le 4cn (1+\\mathrm{e}) \\left(\\mathbb{E}_{\\mathrm{s}}\\left[l^n \\right] + (\\mathrm{e}n)^n\\right)  \\label{eq:ddtln}\n\\end{equation}\nIn the special case $n=1$, the following stronger inequality holds: \\begin{equation}\n\\frac{\\mathrm{d}}{\\mathrm{d}t} \\mathbb{E}_{\\mathrm{s}} \\left[l \\right] \\le c \\left(2\\mathbb{E}_{\\mathrm{s}} \\left[l \\right]  + 1\\right).  \\label{eq:ddtln1}\n\\end{equation}\\label{lmael}\n\\end{lem}\n\\begin{proof}\nWe begin by using (\\ref{eq:prop2dos}):  for any non-decreasing function $f:\\mathbb{Z} \\rightarrow \\mathbb{R}$, \\begin{align}\n\\frac{\\mathrm{d}}{\\mathrm{d}t} \\mathbb{E}_{\\mathrm{s}} \\left[f(l) \\right]  &=  \\sum_{l=0}^{N^\\prime } f(l) \\left(2 \\varphi_l \\frac{\\mathrm{d}\\varphi_l}{\\mathrm{d}t}\\right) = 2\\sum_{l=0}^{N^\\prime} f(l) \\varphi_l \\left[ K_{l-1}\\varphi_{l-1} - K_l \\varphi_l\\right] = 2 \\sum_{l=0}^{N^\\prime- 1} K_l \\varphi_l \\varphi_{l+1} [ f(l+1)-f(l)] \\notag \\\\\n&\\le 2c\\sum_{l=0}^{N^\\prime- 1}  \\varphi_l \\varphi_{l+1} (l+1) [ f(l+1)-f(l)] \\le c\\sum_{l=0}^{N^\\prime - 1} (P_l + P_{l+1}) (l+1) [f(l+1)-f(l)]. \\label{eq:fl1fl}\n\\end{align}\nIn particular, choosing $f(l)=l^n$, we may further loosen this inequality using elementary inequalities: \\begin{equation}\n\\frac{\\mathrm{d}}{\\mathrm{d}t} \\mathbb{E}_{\\mathrm{s}} \\left[f(l) \\right]  \\le 2c \\sum_{l=0}^{N^\\prime -1} P_l \\left( (l+1)^{n+1} - l^{n+1}\\right). \\label{eq:ddtln2}\n\\end{equation}\nNow observe that \\begin{equation}\n(l+1)^{n+1}-l^{n+1} = (n+1) l^n + \\sum_{k=0}^{n-1} \\left(\\begin{array}{c} n+1 \\\\ k \\end{array}\\right) l^k \\le (n+1)l^n + n(n+1) (l+1)^{n-1}.\n\\end{equation}\nNext, note the inequality \\begin{equation}\nn(l+1)^{n-1} < \\mathrm{e} l^n +  (\\mathrm{e}n)^n \\label{eq:lma5ineqinterm}\n\\end{equation}\nwhich we derive by multiplying both sides of (\\ref{eq:lma5ineqinterm}) by $l^{-n}$, assuming $l>1$ (the inequality is trivial when $l=0$): \\begin{equation}\n\\frac{n}{l} \\left(1+\\frac{1}{l}\\right)^{n-1} < \\frac{n}{l} \\mathrm{e}^{n\/l} < \\mathrm{e} +  \\left(\\frac{\\mathrm{e}n}{l}\\right)^n.\n\\end{equation}\nFor $n\\le l$, the first term on the right hand side is always at least as large as the middle term; for $n> l$, the second term on the right is larger.  Combining (\\ref{eq:ddtln2}) and (\\ref{eq:lma5ineqinterm}), we obtain (\\ref{eq:ddtln}). \n\nFor the case $n=1$, we use that $f(l+1)-f(l)=1$.  Directly plugging in to (\\ref{eq:fl1fl}) we obtain (\\ref{eq:ddtln1}).\n\\end{proof}\nThe next lemma shows that even when $K_l$ grow faster than (\\ref{eq:Kll}) at large $l$, $P_l(t)$ is very small for $l>M$ at early times. \n\\begin{lem}\\label{lmalarge}\nIf $K_l(t)$ obeys (\\ref{eq:Kll}), then \\begin{equation}\n\\frac{\\mathrm{d}}{\\mathrm{d}t} \\mathbb{P}_{\\mathrm{s}}[ l>M] \\le 2\\mathrm{e}c^2(M+1)t \\exp\\left[ -M \\mathrm{e}^{-2-4c(1+\\mathrm{e})t} \\right].  \\label{eq:PslM}\n\\end{equation}\n\\end{lem}\n\\begin{proof}\nWe begin by employing (\\ref{eq:prop2dos}): \\begin{equation}\n\\frac{\\mathrm{d}}{\\mathrm{d}t} \\mathbb{P}_{\\mathrm{s}}[ l>M] = 2\\sum_{l=M+1}^{N^\\prime} \\varphi_l (K_{l-1} \\varphi_{l-1} - K_{l+1}\\varphi_{l+1}) = 2K_M \\varphi_M \\varphi_{M+1} \\le 2c(M+1)\\varphi_{M+1}.  \\label{eq:PsgM}\n\\end{equation}\nIn the last inequality, we used (\\ref{eq:Kll}) along with $\\varphi_l(t)\\le 1$ for any $l$.  Hence, to obtain (\\ref{eq:PslM}), it suffices to bound $\\varphi_{M+1}(t)$. \n\nLet $K \\in \\mathbb{R}^{(N^\\prime+1) \\times (N^\\prime+1)}$ correspond to the transition matrix whose entries are \\begin{equation}\nK_{l^\\prime l}(t) = K_l \\mathbb{I}(l=l^\\prime - 1) - K_{l^\\prime}(t) \\mathbb{I}(l^\\prime=l - 1).\n\\end{equation}\n(indices run from $l=0$ to $l=N^\\prime$).   Hence $K$ is tridiagonal and antisymmetric.  Let us define the orthogonal matrix $U(t,t^\\prime)$  by the differential equation \\begin{equation}\n\\frac{\\mathrm{d}}{\\mathrm{d}t} U(t,t^\\prime)  = K(t) U(t,t^\\prime) , \\;\\;\\;\\;\\;\\; U(t^\\prime,t^\\prime) = 1.\n\\end{equation}\n$U(t,t^\\prime)$ generates the continuous time quantum walk with transition rates $K_l(t)$.\n\nNext, we define the quantum walk transition matrix $\\widetilde{K}(t)$ as follows: \\begin{equation}\n\\widetilde{K}_{l^\\prime l}(t) := K_l \\mathbb{I}(M>l=l^\\prime - 1) - K_{l^\\prime}(t) \\mathbb{I}(M>l^\\prime=l - 1).\n\\end{equation}\nThis matrix corresponds to excising the sites $l>M$ from the walk.  We define an analogous time evolution operator \\begin{equation}\n\\frac{\\mathrm{d}}{\\mathrm{d}t} \\widetilde{U}(t,t^\\prime)  = \\widetilde{K}(t) \\widetilde{U}(t,t^\\prime) , \\;\\;\\;\\;\\;\\; \\widetilde{U}(t^\\prime,t^\\prime) = 1.\n\\end{equation}\n\nNow we use the following integral identity:\\footnote{In the physics literature, this is called the integral form of the memory matrix formula \\cite{zwanzig, mori, forster}.} \\begin{align}\n\\varphi_{M+1}(t) &= U_{M+1,0}(t,0) = \\widetilde{U}_{M+1,0}(t,0) + \\int\\limits_0^t \\mathrm{d}t^\\prime \\sum_{l,l^\\prime}  U_{M+1,l^\\prime}(t,t^\\prime) (K_{l^\\prime l}(t) - \\widetilde{K}_{l^\\prime l}(t) )  \\widetilde{U}_{l,0}(t^\\prime,0).  \\label{eq:memorymatrix}\n\\end{align}\nDue to the fact that $\\widetilde{U}$ does not evolve into sites with $l>M$, we can immediately simplify (\\ref{eq:memorymatrix}):\n\\begin{equation}\n\\varphi_{M+1}(t) = \\int\\limits_0^t \\mathrm{d}t^\\prime \\; K_M(t^\\prime) \\; U_{M+1,M+1}(t,t^\\prime)\\widetilde{U}_{M,0}(t^\\prime,0).\n\\end{equation}\nUsing (\\ref{eq:Kll}) along with orthogonality of $U(t,t^\\prime)$ and the triangle inequality: \\begin{equation}\n\\varphi_{M+1}(t) \\le c(M+1) \\int\\limits_0^t \\mathrm{d}t^\\prime \\; \\widetilde{U}_{M,0}(t^\\prime,0).  \\label{eq:memorymatrix2}\n\\end{equation}\n\nWe now recognize that \\begin{equation}\n\\widetilde{U}_{M,0}(t^\\prime,0) =\\widetilde{\\varphi}_M(t^\\prime) \\label{eq:widetildevarphi}\n\\end{equation}\nis the solution to the blocked quantum walk generated by $\\widetilde{K}$.  This blocked quantum walk obeys Lemma~\\ref{lmael}; integrating (\\ref{eq:ddtln}), we obtain \\begin{equation}\n\\mathbb{E}_{\\widetilde{\\mathrm{s}}} \\left[ l^n \\right] \\le (\\mathrm{e}n)^n \\left( \\mathrm{e}^{4c(1+\\mathrm{e})nt}-1 \\right).\n\\end{equation}\nHere $\\mathbb{E}_{\\widetilde{\\mathrm{s}}} [\\cdots]$ denotes averages in the probability distribution of the blocked quantum walk.  Using Markov's inequality, \\begin{equation}\n\\widetilde{\\varphi}_M(t) \\le \\inf_{n\\in\\mathbb{Z}^+}  \\frac{\\mathbb{E}_{\\widetilde{\\mathrm{s}}} \\left[ l^n \\right]}{M^n} < \\inf_{n\\in\\mathbb{Z}^+} \\left(\\frac{\\mathrm{e}^{1+4c(1+\\mathrm{e})t}n}{M} \\right)^n \\le \\exp\\left[1 -M \\mathrm{e}^{-2-4c(1+\\mathrm{e})t} \\right], \\label{eq:widetildevarphiM}\n\\end{equation}\nwhere in the last step we used the following sequence of inequalities for $z\\in \\mathbb{R}^+$:  \\begin{equation}\n\\inf_{n\\in\\mathbb{Z}^+} \\left(\\frac{n}{z}\\right)^n \\le \\left(\\frac{1}{z} \\left\\lfloor \\frac{z}{\\mathrm{e}}\\right\\rfloor \\right)^{\\lfloor z\/\\mathrm{e}\\rfloor} \\le \\exp \\left[  - \\left\\lfloor \\frac{z}{\\mathrm{e}}\\right\\rfloor\\right] < \\exp \\left[ 1- \\frac{z}{\\mathrm{e}}\\right] . \n\\end{equation}\n\nCombining (\\ref{eq:PsgM}), (\\ref{eq:memorymatrix2}), (\\ref{eq:widetildevarphi}) and (\\ref{eq:widetildevarphiM}), and using the fact that our bound on $\\widetilde{\\varphi}_M(t)$ is a monotonically increasing function of time,  we obtain (\\ref{eq:PslM}).\n\\end{proof}\n\nThe last step is to combine (\\ref{eq:ddtln1}) with Lemma \\ref{lmalarge} to bound the true Lyapunov exponent.   Defining the non-decreasing functions \\begin{subequations}\\begin{align}\nf_>(l) &:= (l-M) \\mathbb{I}[l>M], \\\\\nf_<(l) &:= l - f_>(l),\n\\end{align}\\end{subequations} we write \\begin{equation}\n\\mathbb{E}_{\\mathrm{s}}[l] = \\mathbb{E}_{\\mathrm{s}}[ f_<(l) + f_>(l)]\n\\end{equation}\nand bound each piece separately.   Using the fact that (\\ref{eq:PslM}) is an increasing function of $t$: \\begin{equation}\n\\mathbb{E}_{\\mathrm{s}}[f_>(l)] \\le (N^\\prime - M) \\mathbb{P}_{\\mathrm{s}}[l>M] \\le 2\\mathrm{e}c^2(M+1)N^\\prime t^2 \\exp\\left[ -M \\mathrm{e}^{-2-4c(1+\\mathrm{e})t} \\right].\n\\end{equation}\nThen using (\\ref{eq:fl1fl}) and $f_<(l+1) \\le f_<(l) + 1$: \\begin{equation}\n\\frac{\\mathrm{d}}{\\mathrm{d}t} \\mathbb{E}_{\\mathrm{s}}[f_<(l)] \\le c \\sum_{l=0}^{N^\\prime } (2l+1) P_l(t) = c\\left(2\\mathbb{E}_{\\mathrm{s}} \\left[l \\right]  + 1\\right).\n\\end{equation}\nWe conclude that \\begin{equation}\n\\mathbb{E}_{\\mathrm{s}}[l] \\le \\frac{\\mathrm{e}^{2ct}-1}{2} + 2\\mathrm{e}c^2(M+1)N^\\prime t^2 \\exp\\left[ -M \\mathrm{e}^{-2-4c(1+\\mathrm{e})t} \\right].\n\\end{equation}\nUsing the definition of $\\lambda(t)$ in (\\ref{eq:lambdadef}) and the concavity of the logarithm, along with $\\log x < x$: \\begin{equation}\n\\lambda(t) \\le 2c \\left[ 1 + \\mathrm{e}(M+1)N^\\prime c t \\exp\\left[ -M \\mathrm{e}^{-2-4c(1+\\mathrm{e})t} \\right] \\right].\n\\end{equation}\n\nLet us define \\begin{equation}\nt = \\frac{\\log M - 2 - r}{4c(1+\\mathrm{e})}.\n\\end{equation}\nThen, using $M+1 \\le 2N^\\prime$ and $\\log M \\le N^\\prime$: \\begin{equation}\n\\frac{\\lambda}{2c} \\le 1 + \\frac{\\mathrm{e} N^{\\prime 3} }{2(1+\\mathrm{e})} \\exp\\left[- \\mathrm{e}^r \\right] < 1 + \\frac{ N^{\\prime 3} }{2} \\exp\\left[- \\mathrm{e}^r \\right].\n\\end{equation}\nDemanding that the inequality in (\\ref{eq:lambdadef}) holds and solving for $r$, we obtain (\\ref{eq:scramblingtime}).\n\\end{proof}\n\nThis theorem can be interpreted as follows.   For any $0<\\kappa<1$, define the operator scrambling time \\begin{equation}\nt_{\\mathrm{s},\\kappa} = \\inf \\left\\lbrace t\\in \\mathbb{R}^+ : \\mathbb{E}_{\\mathrm{s},t}[l] \\ge \\kappa N^\\prime  \\right\\rbrace .\n\\end{equation} \n(Recall that $N^\\prime$ is the maximal value of $l$, as defined in (\\ref{eq:Nprimedef})).  It was conjectured in \\cite{sekino} that a quantum ``scrambling time\" $t_{\\mathrm{s}} = \\mathrm{\\Omega}(\\log N)$ would necessarily grow at least logarithmically with the number of degrees of freedom in any system with few-body interactions.   For example, in the SYK model, we would demand that $q$ is finite.  In recent years, this operator scrambling time has become the preferred definition of scrambling in the physics literature, though this is likely out of convenience \\cite{lucas1805}.\nTheorem \\ref{lyapunovtheor} implies that $t_{\\mathrm{s},\\kappa} = \\mathrm{O}(\\log N)$, as summarized in the following corollary:\n\\begin{cor} \\label{corolscramble}\nIf (\\ref{eq:Kll}) holds for $M = N^\\alpha$ for $0<\\alpha<1$, then there exists an $N$-independent $b \\in \\mathbb{R}^+$ for which \\begin{equation}\nt_{\\mathrm{s},\\kappa } \\ge b \\log N \\label{eq:scrambleb}\n\\end{equation}\nfor $N>N_0$, for some finite $N_0\\in \\mathbb{Z}^+$.\n\\end{cor}\n\\begin{proof}\nThere exists an $N_*\\in\\mathbb{Z}^+$ such that \\begin{equation}\n\\frac{\\alpha \\log N_* }{8c(1+\\mathrm{e})} < \\frac{1}{4c(1+\\mathrm{e})} \\left[ \\alpha \\log N_* - 2 -\\log\\log N_*^{\\prime 3}\\right].\n\\end{equation}\nSuppose that $N>N_*$.   Using Theorem \\ref{lyapunovtheor}, we conclude that at time $t=t_{\\mathrm{s}}$, where $t_{\\mathrm{s}}$ is given by (\\ref{eq:scrambleb}) where  \\begin{equation}\nb :=\\frac{\\alpha }{8c(1+\\mathrm{e})} ,\n \\end{equation}\n \\begin{equation}\n \\mathbb{E}_{\\mathrm{s}}[l] \\le \\exp\\left[ \\frac{3\\alpha \\log N}{8(1+\\mathrm{e})}\\right] = N^{3\\alpha\/8(1+\\mathrm{e})}\n \\end{equation}\n We conclude that the corollary holds so long as $N_0$ is chosen such that $\\kappa N_0 > N_0^{3\\alpha\/8(1+\\mathrm{e})}$ and $N_0 \\ge N_*$.\n\\end{proof}\n\nWe emphasize that the results of this section are completely general, and apply to a large family of models beyond the SYK model, as soon as (\\ref{eq:Kll}) can be proved.\n\n\\section{Operator growth in the SYK ensemble}\n\n\\subsection{Bounding the transition rates}\\label{sec:trans}\nWhat remains is to show that (\\ref{eq:Kll}) holds in the SYK ensemble, with very high probability, at large $N$.   Proving this fact constitutes the second main result of this paper.   The result is summarized in the following theorem:\n\\begin{thm}\\label{theorSYK}\nLet $\\kappa \\in \\mathbb{R}^+$ and $\\theta \\in \\mathbb{R}^+$ obey \\begin{subequations}\\begin{align}\n2\\kappa \\log N + 2 &< \\sqrt{N},   \\label{eq:kappasqrtNN} \\\\\n2(q-2) &< N^{\\kappa\\theta} - 1, \\\\\n2q (1+\\sqrt{N})&<(q\\kappa \\log N - 1) \\sqrt{N} . \\label{eq:annoying} \n\\end{align}\\end{subequations}\nLet us also assume that \\begin{equation}\nq < \\frac{N}{2}.  \\label{eq:qlessN2}\n\\end{equation}Then, in the SYK model introduced in Section \\ref{sec:ensemble}, with probability at least \\begin{equation}\n\\mathbb{P}_{\\mathrm{success}} \\ge 1 - \\frac{2(q-2)}{N^{\\kappa \\theta}-1},\n\\end{equation}\n(\\ref{eq:Kll}) is obeyed with \n\\begin{subequations}\\label{eq:theorcm}\\begin{align}\n c &= \\mathrm{e}^{\\theta + 1\/\\kappa} \\left[\\sqrt{\\frac{2(q-2)}{q}} \\left(1-\\frac{2\\theta}{5\\kappa \\sqrt{N}\\log N}\\right)^{-1} + \\frac{8}{q^{q-9\/2}N^{1\/4}} \\left(\\frac{4\\theta}{5\\kappa \\log N} \\right)^{(q-3)\/2}  \\right] , \\\\\n M &= \\left\\lfloor  \\frac{\\theta}{5\\kappa} \\frac{\\sqrt{N}}{q^3\\log N} \\right\\rfloor -1 .\n\\end{align}\\end{subequations}\n\\end{thm}\n\\begin{proof}\nOur strategy will be to work primarily with $\\mathcal{K}_{s^\\prime s}$.   At the very end of the calculation, we will use (\\ref{eq:Kldef}) to bound $\\mathcal{K}_l$.   We begin with the following proposition:\n\\begin{prop}\nDefine the symmetric and positive semidefinite matrix \\begin{equation}\nM_{s^\\prime s} = \\mathbb{Q}_s \\mathcal{L}^{\\mathsf{T}} \\mathbb{Q}_{s^\\prime}\\mathcal{L}\\mathbb{Q}_s.  \\label{eq:MssDef}\n\\end{equation}\nIf the maximal eigenvalue of $M_{s^\\prime s}$ is $\\mu_{s^\\prime s}$, \\begin{equation}\n\\mathcal{K}_{s^\\prime s} = \\sqrt{\\mu_{s^\\prime s}}. \\label{eq:Kmu}\n\\end{equation} \\label{propmuss}\n\\end{prop}\n\\begin{proof}\n Let $\\mathcal{O}\\in \\mathcal{B}$ obey $\\lVert \\mathcal{O}\\rVert = 1$, and define \\begin{equation}\n|\\mathcal{O}^\\prime) = \\mathbb{Q}_{s^\\prime}\\mathcal{L}\\mathbb{Q}_s|\\mathcal{O}).\n\\end{equation} From (\\ref{eq:endBnorm}) and (\\ref{eq:calKss}), we see that $\\mathcal{K}_{s^\\prime s}$ is simply the maximal length of the vector $|\\mathcal{O}^\\prime)$.  Now observe that $M_{s^\\prime s}$ gives us a very simple way of measuring the length of $|\\mathcal{O}^\\prime)$.  Therefore,\n \\begin{equation}\n\\mathcal{K}_{s^\\prime s}^2 = \\sup_{\\mathcal{O}\\in\\mathcal{B}} (\\mathcal{O}^\\prime |  \\mathcal{O}^\\prime ) = \\sup_{\\mathcal{O}\\in\\mathcal{B}} (\\mathcal{O}|M_{s^\\prime s}|\\mathcal{O}) = \\mu_{s^\\prime s},\n\\end{equation}\nwhere for the last equality we used a variational principle which holds for a symmetric matrix.\n\\end{proof}\n\nDenote \\begin{equation}\nC_s := \\frac{N!}{s!(N-s)!},\n\\end{equation}\nand observe that $M_{s^\\prime s} \\in \\mathbb{R}^{C_s\\times C_s}$ is a positive semidefinite random matrix.   From Markov's inequality, for any $p\\in\\mathbb{Z}^+$, \\cite{furedi} \n\\begin{equation}\n\\mathbb{P}\\left[\\mu_{s^\\prime s} \\ge a \\right] \\le \\frac{\\mathbb{E}\\left[ \\mu_{s^\\prime s}^p\\right]}{a^p}\\le \\frac{\\mathbb{E}\\left[ \\mathrm{tr}(M_{s^\\prime s}^p)\\right]}{a^p}.  \\label{eq:markov4}\n\\end{equation}\nWe will choose $p=\\mathrm{O}(\\log C_s)$, so that the number of eigenvalues $C_s$ accounted for in the trace is irrelevant ($C_s^{1\/p} \\rightarrow 1$).    Importantly, at finite size $s$, $p = \\mathrm{O}(s\\log N)$.   We will see that this is sufficiently small to make bounding $\\mu_{s^\\prime s}$ analytically tractable.   \n\n \nHence, let us define \\begin{equation}\nB_{s^\\prime s}^{(p)} := \\mathbb{E}\\left[\\mathrm{tr}\\left(M^p_{s^\\prime s}\\right)\\right]. \\label{eq:BssDef}\n\\end{equation}\nWe analyze the average $\\mathbb{E}[\\cdots]$ over the random variables $J_X$ by converting it to a combinatorial problem.   To do so, let us write out \\begin{equation}\nB^{(p)}_{s^\\prime s} = \\mathbb{E}\\left[ \\sum_{X_1,\\ldots, X_p, Y_1,\\ldots,Y_p \\in F} \\sum_{Z\\subseteq V} (Z| \\prod_{i=1}^p \\mathbb{Q}_s \\mathcal{L}^{\\mathsf{T}}_{X_i}\\mathbb{Q}_{s^\\prime}\\mathcal{L}_{Y_i}\\mathbb{Q}_s |Z)   \\right] \\label{eq:explicittrace}\n\\end{equation}\nwhere the sum over $Z$ is a sum over the basis of Proposition \\ref{prop1}, without loss of generality.  We now read (\\ref{eq:explicittrace}) from right to left, starting with $\\mathbb{Q}_s|Z)$, which restricts the subset $Z\\subseteq V$ to have exactly $s$ elements: $|Z|=s$, and $Z=\\lbrace i_1,i_2,\\ldots, i_s\\rbrace$.  We draw a graph $G$ which we associate to $\\mathbb{Q}_s |Z)$: \\begin{equation} \\label{eq:minimalgraph}\n\\begin{tikzpicture}\n\\draw (-0.5,0) node[left] {$\\mathbb{Q}_s |i_1\\cdots i_s) \\sim$};\n\\draw (1.15, -0.4) -- (0.3, 0.25);\n\\draw (1.15, -0.4) -- (0.8, 0.25);\n\\draw (1.15, -0.4) -- (2, 0.25);\n\\fill[color=orange] (1,-0.5) -- ++(60:0.25) -- ++(-60:0.25) -- cycle;\n\\fill[color=blue] (0.3, 0.25) circle (3pt);\n\\fill[color=blue] (0.8, 0.25) circle (3pt);\n\\fill[color=blue] (2, 0.25) circle (3pt);\n\\draw (1.4, 0.25) node {$\\cdots$};\n\\draw (0.3, 0.35) node[above] {\\color{blue} \\footnotesize $i_1$};\n\\draw (0.8, 0.35) node[above] {\\color{blue} \\footnotesize $i_2$};\n\\draw (2, 0.35) node[above] {\\color{blue} \\footnotesize $i_s$};\n\\end{tikzpicture}\n\\end{equation}\nwhere the (blue) circles denote the fermions, and the orange triangle is a ``root\" to the graph -- it has edges drawn to the $s$ original fermions in the operator $\\mathbb{Q}_s |Z)$ (recall that $|Z)$ corresponds to a product operator).  We have written a $\\sim$ in (\\ref{eq:minimalgraph}) because we will not bother to keep track of an overall sign in the vector, although its orientation in $\\mathcal{B}$ and its dependence on any random variables $J_X$ are each important. For simplicity, let us assume that $s^\\prime=s+q-2$.  Without loss of generality, we assume that $Y_p \\cap Z = \\lbrace i_s\\rbrace$,  $Y=\\lbrace i_s,j_1,\\ldots, j_{q-1}\\rbrace$.   Then, we draw \n\\begin{equation} \\label{eq:graphtree}\n\\begin{tikzpicture}\n\\draw (-0.5, 0.3)  node[left] {$\\mathbb{Q}_{s^\\prime}\\mathcal{L}_{Y_p}\\mathbb{Q}_s|Y)\\sim$};\n\\draw (1.15, -0.4) -- (0.3, 0.25);\n\\draw (1.15, -0.4) -- (0.8, 0.25);\n\\draw (1.15, -0.4) -- (2, 0.25) -- (3, 0.25) -- (2.2, 0.85);\n\\draw (2.7, 0.85) -- (3,0.25) -- (3.7, 0.85);\n\\fill[color=orange] (1,-0.5) -- ++(60:0.25) -- ++(-60:0.25) -- cycle;\n\\fill[color=blue] (0.3, 0.25) circle (3pt);\n\\fill[color=blue] (0.8, 0.25) circle (3pt);\n\\fill[color=blue] (2, 0.25) circle (3pt);\n\\fill[color=blue] (2.2, 0.85) circle (3pt);\n\\fill[color=blue] (2.7, 0.85) circle (3pt);\n\\fill[color=blue] (3.7, 0.85) circle (3pt);\n\\fill[color=red] (2.9, 0.15) rectangle (3.1, 0.35);\n\\draw (1.4, 0.25) node {$\\cdots$};\n\\draw (3.2, 0.85) node {$\\cdots$};\n\\draw (0.3, 0.35) node[above] {\\color{blue} \\footnotesize $i_1$};\n\\draw (0.8, 0.35) node[above] {\\color{blue} \\footnotesize $i_2$};\n\\draw (2, 0.35) node[above] {\\color{blue} \\footnotesize $i_s$};\n\\draw (2.2, 0.95) node[above] {\\color{blue} \\footnotesize $j_1$};\n\\draw (2.7, 0.95) node[above] {\\color{blue} \\footnotesize $j_2$};\n\\draw (3.7, 0.95) node[above] {\\color{blue} \\footnotesize $j_{q-1}$};\n\\draw (3, 0.15) node[below] {\\color{red} \\footnotesize $Y_p$};\n\\end{tikzpicture}.\n\\end{equation}\nThe way to read this graph is as follows:  the coupling (factor, drawn as a red square) $Y_p$ connected to the fermion $i_s$, and spawned the fermions $j_1,\\ldots, j_{q-1}$.   Each fermion (circle) with an odd degree is present in the operator; those with an even degree are not present.   Because of the projectors $\\mathbb{Q}_{s^\\prime}$ and $\\mathbb{Q}_s$, we had to start with an operator of size $s$ and add exactly $q-2$ net fermions.    From (\\ref{eq:psiApsiB}), we know that  this vector is proportional to one of our simple basis vectors (a product operator), which is why we can simply draw the graph (so long as we neglect the proportionality coefficient).  The fermions do not directly connect to each other, but rather connect through the factors.   \n\nLet us continue and study the operator $\\mathbb{Q}_s \\mathcal{L}^{\\mathsf{T}}_{X_i}\\mathbb{Q}_{s^\\prime}\\mathcal{L}_{Y_i}\\mathbb{Q}_s |Z)$.  It is easiest to first illustrate the possibilities with a simple example.  Consider the theory with $s=3$, $s^\\prime=5$, $q=4$.  Let us first consider the theory where $X_p=Y_p = \\lbrace 3,4,5,6\\rbrace$ and $Z=\\lbrace 1,2,3\\rbrace$.   Then we draw \n\\begin{equation}\\label{eq:Xpexample1}\n\\begin{tikzpicture}\n\\draw (-0.5,0) node[left] {$\\mathbb{Q}_s \\mathcal{L}^{\\mathsf{T}}_{X_p}\\mathbb{Q}_{s^\\prime}\\mathcal{L}_{Y_p}\\mathbb{Q}_s |Z)\\sim$};\n\\draw (1.15, -0.4) -- (0.3, 0.25);\n\\draw (1.15, -0.4) -- (1.15, 0.25);\n\\draw (1.15, -0.4) -- (2, 0.25);\n\\fill[color=orange] (1,-0.5) -- ++(60:0.25) -- ++(-60:0.25) -- cycle;\n\\fill[color=blue] (0.3, 0.25) circle (3pt);\n\\fill[color=blue] (1.15, 0.25) circle (3pt);\n\\fill[color=blue] (2, 0.25) circle (3pt);\n\\draw (0.3, 0.35) node[above] {\\color{blue} \\footnotesize $1$};\n\\draw (1.15, 0.35) node[above] {\\color{blue} \\footnotesize $2$};\n\\draw (2, 0.35) node[above] {\\color{blue} \\footnotesize $3$};\n\\end{tikzpicture}\n\\end{equation}\nwhere the absence of the factor $Y_p$ reminds us that since $J_{Y_p}$ has appeared twice in the sequence, this sequence is non-trivial under random averaging.   We neglect to draw any fermion or factor which has degree zero, which is why the fermions 4, 5 and 6 are not shown.   \n\nHowever, suppose instead $X_p = \\lbrace 2,4,5,7\\rbrace$.  In this case, \n\\begin{equation}\\label{eq:Xpexample2}\n\\begin{tikzpicture}\n\\draw (-0.5,0.3) node[left] {$\\mathbb{Q}_s \\mathcal{L}^{\\mathsf{T}}_{X_p}\\mathbb{Q}_{s^\\prime}\\mathcal{L}_{Y_p}\\mathbb{Q}_s |Z)\\sim$};\n\\draw (1.15, -0.4) -- (0.3, 0.25);\n\\draw (1.15, -0.4) -- (1.15, 0.25) -- (1.8, 1.1) -- (1, 1.1);\n\\draw (1.15, -0.4) -- (2, 0.25) -- (3, 0.25) -- (2.3, 0.85);\n\\draw (3, 0.85) -- (3,0.25) -- (3.7, 0.85);\n\\draw (3, 0.85) -- (1.8, 1.1) -- (2.3, 0.85);\n\\fill[color=orange] (1,-0.5) -- ++(60:0.25) -- ++(-60:0.25) -- cycle;\n\\fill[color=blue] (0.3, 0.25) circle (3pt);\n\\fill[color=blue] (1.15, 0.25) circle (3pt);\n\\fill[color=blue] (2, 0.25) circle (3pt);\n\\fill[color=blue] (2.3, 0.85) circle (3pt);\n\\fill[color=blue] (3, 0.85) circle (3pt);\n\\fill[color=blue] (3.7, 0.85) circle (3pt);\n\\fill[color=blue] (1,1.1) circle (3pt);\n\\fill[color=red] (2.9, 0.15) rectangle (3.1, 0.35);\n\\fill[color=red] (1.7, 1) rectangle (1.9, 1.2);\n\\draw (0.3, 0.35) node[above] {\\color{blue} \\footnotesize $1$};\n\\draw (1.15, 0.35) node[above] {\\color{blue} \\footnotesize $2$};\n\\draw (2, 0.35) node[above] {\\color{blue} \\footnotesize $3$};\n\\draw (2.3, 0.95) node[above] {\\color{blue} \\footnotesize $4$};\n\\draw (3, 0.95) node[above] {\\color{blue} \\footnotesize $5$};\n\\draw (3.7, 0.95) node[above] {\\color{blue} \\footnotesize $6$};\n\\draw (1,1.2) node[above] {\\color{blue} \\footnotesize $7$};\n\\draw (1.8, 1.2) node[above] {\\color{red} \\footnotesize $X_p$};\n\\draw (3, 0.15) node[below] {\\color{red} \\footnotesize $Y_p$};\n\\end{tikzpicture}\n\\end{equation}\nBecause the factors $X_p \\ne Y_p$, we must draw both of them, together with an edge to all vertices\/fermions $i\\in X_p$ or $Y_p$.   We can only remove a factor when that exact factor shows up a second time.   And if a factor shows up a third time, it is redrawn in, and so on.   Note that the only factors that $X_p$ can be are those which destroy 3 fermions and create one, in this simple example. \n\nIt is straightforward to generalize these rules, which we summarize one more time.  If the next factor in the sequence is present in the existing factor graph,  that factor is deleted along with its edges to $q$ fermions.  Any fermions which subsequently have degree zero are removed.  If the factor is new, we draw that factor and $q$ edges to its fermions.  The number of odd degree fermions in each graph is fixed by the projectors to alternate between $s$ and $s^\\prime$.  \n\n\nOur next goal is to throw away detailed information about what specific factors and fermions appeared, and to only keep track of the sequence of graphs.  Let $\\mathcal{G}$ be the space of all graphs, modulo graph isomorphism.  Two graphs $G_1$ and $G_2$ are isomorphic if and only if there is a permutation on fermions $\\pi \\in \\mathrm{S}^V$ such that $\\pi \\cdot G_1 = G_2$ (the group action on fermions and factors is canonical, while the root is invariant).   We define $G_\\triangle$ to be the unique (up to isomorphism) element of $\\mathcal{G}$  with zero factors and $s$ fermions connected to the root, as in (\\ref{eq:minimalgraph}).  Let $\\mathcal{G}_-$ be the subset of $\\mathcal{G}$ consisting of $s$ odd degree fermions and no more than $p$ factors, and $\\mathcal{G}_+$ be the subset of $\\mathcal{G}$ with $s^\\prime$ odd degree fermions, subject to the constraint that any graph in $\\mathcal{G}_+$ or $\\mathcal{G}_-$ can be reached by adding and removing factors to $G_\\triangle$, according to the rules above, and with no intermediate graphs containing more than $p$ factors.  \n\nDefine $\\langle G_2| \\mathcal{N}_+|G_1 \\rangle$ to be the number of factors $X$ which can be added (or removed) to any fixed graph $G$ isomorphic to $G_1 \\in \\mathcal{G}_-$, to create any graph isomorphic to $G_2 \\in \\mathcal{G}_+$.   Similarly, we define  $\\langle G_1| \\mathcal{N}_-|G_2 \\rangle$ to be the number of factors which take a fixed graph isomorphic to $G_2\\in \\mathcal{G}_+$ to any graph isomorphic to $G_1 \\in \\mathcal{G}_-$.   We interpret $\\mathcal{N}_+ : \\mathbb{Z}^{\\mathcal{G}_+ \\times \\mathcal{G}_-} \\rightarrow \\mathbb{Z}$ and $\\mathcal{N}_- : \\mathbb{Z}^{\\mathcal{G}_- \\times \\mathcal{G}_+} \\rightarrow \\mathbb{Z}$ as integer-valued matrices, using the angle bra-ket notation to denote matrix elements.   Many of these matrix elements are zero.  $\\mathcal{N}_+$ and $\\mathcal{N}_-$ are both non-negative matrices.\n\n\\begin{prop}\n\\label{propcomb}\nIf  $G_0 = G_p = G_\\triangle$, then \\begin{equation}\nB_{s^\\prime s}^{(p)} \\le \\left(4\\sigma^2\\right)^p C_s \\sum_{ \\substack{G_1,\\ldots, G_{p-1} \\in \\mathcal{G}_- \\\\H_1,\\ldots, H_p \\in \\mathcal{G}_+ } }\\prod_{i=1}^p \\langle G_i | \\mathcal{N}_-|H_i\\rangle\\langle H_i|\\mathcal{N}_+|G_{i-1}\\rangle . \\label{eq:combineq}\n\\end{equation}\n\\end{prop}\n\n\\begin{proof} We expand out the sums in (\\ref{eq:explicittrace}) over all possible couplings.  Using the algorithm described above to associate a graph with each of the $2p+1$ operators $\\mathbb{Q}_s|Z)$, $\\mathbb{Q}_{s^\\prime}\\mathcal{L}_{Y_p}\\mathbb{Q}_s|Z)$, etc., we may convert the sequence of factors \\begin{equation}\n\\mathcal{Y}_Z := (Y_p, X_p, Y_{p-1},X_{p-1},\\ldots, Y_1,X_1)_Z \\label{eq:calYZ}\n\\end{equation}\nread right to left, along with the initial  operator $|Z)$, into a sequence of $2p+1$ graphs \\begin{equation}\n\\mathcal{Z}(\\mathcal{Y}_Z) := (G_0, H_1, G_1, H_2,\\ldots, H_p, G_p) \\label{eq:calZdef}\n\\end{equation}\nwith $G_0=G_\\triangle$.  Due to the projectors $\\mathbb{Q}_s$ and $\\mathbb{Q}_{s^\\prime}$, any sequence $\\mathcal{Y}$ which (before disorder averaging) is not zero must map to a sequence $\\mathcal{Z}(\\mathcal{Y}_Z)$ in which $G_i \\in \\mathcal{G}_-$ and $H_i \\in \\mathcal{G}_+$.  When calculating $B_{s^\\prime s}^{(p)}$,  we require that $G_p = G_\\triangle$;  otherwise, there is a coupling which appears an odd number of times in $\\mathcal{Y}$, so the disorder average of that sequence vanishes.   We define \\begin{equation}\n\\mathcal{Z}_{s^\\prime s}^{(p)} := \\lbrace (G_\\triangle, H_1, G_1, H_2,\\ldots, H_p, G_\\triangle) : H_i \\in \\mathcal{G}_+,G_i \\in \\mathcal{G}_- \\rbrace. \n\\end{equation}   Since only one factor can be added or removed in each step, it is not possible to have more than $p$ factors in any graph in $\\mathcal{Z}(\\mathcal{Y}_Z)$.   We define the equivalence relation $\\mathcal{Y}_1 \\sim \\mathcal{Y}_2$ if and only if $\\mathcal{Z}(\\mathcal{Y}_1) = \\mathcal{Z}(\\mathcal{Y}_2)$, and denote $\\mathcal{Y}_{1} \\in \\mathcal{Z}(\\mathcal{Y})$.\n\nWe then write \\begin{equation}\nB_{s^\\prime s}^{(p)} = \\sum_{Z\\subseteq V} \\sum_{\\mathcal{Z} \\in \\mathcal{Z}_{s^\\prime s}^{(p)}} \\sum_{\\mathcal{Y}_Z\\in \\mathcal{Z}}  (Z| \\prod_{i=1}^p \\mathbb{Q}_s \\mathcal{L}^{\\mathsf{T}}_{X_i}\\mathbb{Q}_{s^\\prime} \\mathcal{L}_{Y_i}\\mathbb{Q}_s |Z).\n\\end{equation}\nDue to the Rademacher distribution on the random variables $J_X$, the expectation value has become trivial, encoded in the fact that the graph sequence $\\mathcal{Z}$ ends at $G_\\triangle$.  For other distributions on $J_X$, the sum above must be weighted in a more complicated way when the sum involves $\\mathbb{E}[J_X^{2k}]$ for $k>1$.   We now apply the triangle inequality together with $(Z|Z)=1$: \\begin{equation}\nB_{s^\\prime s}^{(p)} \\le   \\sum_{Z\\subseteq V : |Z|=s} \\sum_{\\mathcal{Z} \\in \\mathcal{Z}_{s^\\prime s}^{(p)}} \\sum_{\\mathcal{Y}_Z\\in \\mathcal{Z}} \\prod_{i=1}^p \\lVert \\mathcal{L}_{X_i}\\rVert  \\lVert \\mathcal{L}_{Y_i}\\rVert =  \\sum_{\\mathcal{Z} \\in \\mathcal{Z}_{s^\\prime s}^{(p)}}   \\sum_{Z\\subseteq V : |Z|=s} \\sum_{\\mathcal{Y}_Z\\in \\mathcal{Z}} \\left(2\\sigma\\right)^{2p}.\n\\end{equation}\nIn the last step, we used (\\ref{eq:psiApsiB}) along with the fact that we may exchange the first two sums, whose summands are independent.  It now remains to evaluate each sum in turn.   By definition of $\\mathcal{N}_+$ and $\\mathcal{N}_-$: \\begin{equation}\n\\sum_{\\mathcal{Y}_Z\\in \\mathcal{Z}} \\left(2\\sigma\\right)^{2p} = \\left(4\\sigma^2\\right)^p \\prod_{i=1}^p \\langle G_i |\\mathcal{N}_-|H_i\\rangle \\langle H_i | \\mathcal{N}_+ |G_{i-1}\\rangle,\n\\end{equation}\nusing (\\ref{eq:calZdef}) and $G_0=G_\\triangle$.   By permutation symmetry, \\begin{equation}\n\\sum_{Z\\subseteq V : |Z|=s}\\left(4\\sigma^2\\right)^p \\prod_{i=1}^p \\langle G_i |\\mathcal{N}_-|H_i\\rangle \\langle H_i | \\mathcal{N}_+ |G_{i-1}\\rangle = \\left(4\\sigma^2\\right)^pC_s \\prod_{i=1}^p \\langle G_i |\\mathcal{N}_-|H_i\\rangle \\langle H_i | \\mathcal{N}_+ |G_{i-1}\\rangle\n\\end{equation}\nHence we obtain (\\ref{eq:combineq}).\n\\end{proof}\n\n\\begin{prop}\n\\label{propsym}\nLet $\\mathcal{Z} = (G_\\triangle, H_1, G_1, \\ldots, G_{p-1}, H_p, G_\\triangle) \\in \\mathcal{Z}_{s^\\prime s}^{(p)}$.  Then  \\begin{equation}\n\\prod_{i=1}^p \\langle G_i | \\mathcal{N}_-|H_i\\rangle\\langle H_i|\\mathcal{N}_+|G_{i-1}\\rangle = \\prod_{i=1}^p \\langle G_{i-1} | \\mathcal{N}_-|H_i\\rangle\\langle H_i|\\mathcal{N}_+|G_{i}\\rangle .\\label{eq:symid}\n\\end{equation}\n\\end{prop}\n\\begin{proof}\nPick any $Z\\subseteq V$ with $|Z|=s$.  Given a sequence of factors $\\mathcal{Y}_Z$, given by (\\ref{eq:calYZ}), with $\\mathcal{Y}_Z \\in \\mathcal{Z}$, define the reversed sequence \\begin{equation}\n\\mathcal{Y}_Z^{\\mathrm{r}} := (X_1,Y_1,\\ldots, X_p, Y_p)_Z.\n\\end{equation}\nwhich corresponds to factors in (\\ref{eq:explicittrace}) read left to right instead.  By construction, $\\mathcal{Y}_Z^{\\mathrm{r}} \\in \\mathcal{Z}^{\\mathrm{r}}$, defined by \\begin{equation}\n\\mathcal{Z}^{\\mathrm{r}} := (G_\\triangle, H_p, G_{p-1}, \\ldots, G_1, H_1, G_\\triangle).\n\\end{equation}\nClearly, $\\left(\\mathcal{Y}_Z^{\\mathrm{r}}\\right)^{\\mathrm{r}} = \\mathcal{Y}_Z$ and $\\left(\\mathcal{Z}^{\\mathrm{r}}\\right)^{\\mathrm{r}} = \\mathcal{Z}$.  As each sequence $\\mathcal{Y}_Z$ has a unique reverse $\\mathcal{Y}_Z^{\\mathrm{r}}$, \\begin{equation}\n\\prod_{i=1}^p \\langle G_i | \\mathcal{N}_-|H_i\\rangle\\langle H_i|\\mathcal{N}_+|G_{i-1}\\rangle = \\sum_{\\mathcal{Y}_Z \\in \\mathcal{Z}} 1 = \\sum_{\\mathcal{Y}_Z^{\\mathrm{r}} \\in \\mathcal{Z}^{\\mathrm{r}}} 1 = \\prod_{i=1}^p \\langle G_{i-1} | \\mathcal{N}_-|H_i\\rangle\\langle H_i|\\mathcal{N}_+|G_{i}\\rangle ,\n\\end{equation}\nwhich completes the proof.\n\\end{proof}\n\n\\begin{lem}\\label{lma12}\nDefine a transfer matrix $\\mathcal{M}_{s^\\prime s}^{(p)}\\in \\mathbb{R}^{\\mathcal{G}_s^{(p)} \\times \\mathcal{G}_s^{(p)}} $ component wise as \\begin{equation} \n\\langle G_1|\\mathcal{M}_{s^\\prime s}^{(p)} |G_2\\rangle = \\sum_{H\\in \\mathcal{G}_{s^\\prime}^{(p)}} \\langle G_1|\\mathcal{N}_-|H\\rangle\\langle H|\\mathcal{N}_+|G_2\\rangle.\n\\end{equation}\nThen if $\\nu_{s^\\prime s}^{(p)}$ is the maximal (left or right) eigenvalue of $\\mathcal{M}_{s^\\prime s}^{(p)}$,\n\\begin{equation}\nB_{s^\\prime s}^{(p)} \\le C_s \\left(\\nu_{s^\\prime s}^{(p)}\\right)^p.  \\label{eq:Bfinalbound}\n\\end{equation}\n\\end{lem}\n\\begin{proof}\nRewriting (\\ref{eq:combineq}) in terms of the transfer matrix: \\begin{equation}\nB_{s^\\prime s}^{(p)} \\le \\langle G_\\triangle | \\left(\\mathcal{M}_{s^\\prime s}^{(p)} \\right)^p |G_\\triangle\\rangle.\n\\end{equation}\nNow, letting $G_0=G_p=G_\\triangle$, and using the property that \\begin{equation}\n\\sum_{H_1,\\ldots, H_p \\in \\mathcal{G}_+}\\prod_{i=1}^p \\langle G_i | \\mathcal{N}_-|H_i\\rangle\\langle H_i|\\mathcal{N}_+|G_{i-1}\\rangle =\\sum_{H_1,\\ldots, H_p \\in \\mathcal{G}_+} \\prod_{i=1}^p \\langle G_{i-1} | \\mathcal{N}_-|H_i\\rangle\\langle H_i|\\mathcal{N}_+|G_{i}\\rangle \\label{eq:presymmetrizing}\n\\end{equation} which follows from Proposition \\ref{propsym},\n\\begin{align}\nB_{s^\\prime s}^{(p)} &\\le \\left(4\\sigma^2\\right)^p C_s  \\sum_{ \\substack{G_1,\\ldots, G_{p-1} \\in \\mathcal{G}_s^{(p)} \\\\H_1,\\ldots, H_p \\in \\mathcal{G}_{s^\\prime}^{(p)} } }\\prod_{i=1}^p \\langle G_i | \\mathcal{N}_-|H_i\\rangle\\langle H_i|\\mathcal{N}_+|G_{i-1}\\rangle \\notag \\\\\n&= C_s  \\sum_{   G_1,\\ldots, G_{p-1} \\in \\mathcal{G}_- }\\prod_{i=1}^p \\left[4\\sigma^2 \\sum_{H_i \\in \\mathcal{G}_+}  \\langle G_i | \\mathcal{N}_-|H_i\\rangle\\langle H_i|\\mathcal{N}_+|G_{i-1}\\rangle\\right] \\notag \\\\\n&= C_s \\sum_{   G_1,\\ldots, G_{p-1} \\in \\mathcal{G}_- }\\prod_{i=1}^p \\left[4\\sigma^2 \\sqrt{\\sum_{H_i \\in \\mathcal{G}_+}  \\langle G_i | \\mathcal{N}_-|H_i\\rangle\\langle H_i|\\mathcal{N}_+|G_{i-1}\\rangle} \\sqrt{\\sum_{H_i \\in \\mathcal{G}_+}  \\langle G_{i-1} | \\mathcal{N}_-|H_i\\rangle\\langle H_i|\\mathcal{N}_+|G_{i}\\rangle} \\right] \\notag \\\\\n&= C_s \\langle G_\\triangle | \\left(\\widetilde{\\mathcal{M}}_{s^\\prime s}^{(p)} \\right)^p |G_\\triangle\\rangle \\label{eq:symmetrizing}\n\\end{align}\nwhere we have defined the symmetrized transfer matrix \\begin{equation}\n\\langle G_1|\\widetilde{\\mathcal{M}}_{s^\\prime s}^{(p)} |G_2\\rangle := \\sqrt{\\langle G_1|\\mathcal{M}_{s^\\prime s}^{(p)} |G_2\\rangle\\langle G_2|\\mathcal{M}_{s^\\prime s}^{(p)} |G_1\\rangle}.\n\\end{equation}\nIn the third line of (\\ref{eq:symmetrizing}), we have used the distributive property along with (\\ref{eq:presymmetrizing}) and the trivial identity that $x=y$ implies $\\sqrt{xy}=x$.\n\nSince $\\mathcal{M}_{s^\\prime s}^{(p)}$ is non-negative, $\\widetilde{\\mathcal{M}}_{s^\\prime s}^{(p)}$ is a symmetric and positive semidefinite and acts on a finite dimensional vector space.  Let $\\widetilde{\\nu}_{s^\\prime s}^{(p)}$ be its maximal eigenvalue.  We conclude (for example, using elementary variational methods) that, since $\\langle G_\\triangle | G_\\triangle \\rangle = 1$, \\begin{equation}\n \\langle G_\\triangle | \\left(\\widetilde{\\mathcal{M}}_{s^\\prime s}^{(p)} \\right)^p |G_\\triangle\\rangle \\le \\left(\\widetilde{\\nu}_{s^\\prime s}^{(p)}\\right)^p. \\label{eq:nutildebound}\n\\end{equation}\n\nIt remains to relate $\\widetilde{\\nu}_{s^\\prime s}^{(p)}$ to $\\nu_{s^\\prime s}^{(p)}$.  By definition of the set $\\mathcal{G}_-$ (and the fact it has a finite number of elements), for any two graphs $G_{1,2} \\in \\mathcal{G}_-$, there exists an integer $n<\\infty$ such that \\begin{equation}\n\\langle G_1 | \\left(\\mathcal{M}_{s^\\prime s}^{(p)}\\right)^n |G_2\\rangle > 0.\n\\end{equation}\nThis identity follows from the fact that there exist a sequence of factors from $G_\\triangle$ to $G_1$ and $G_2$, as well as the reverse sequences from $G_2$ or $G_1$ to $G_\\triangle$.  Hence $\\mathcal{M}_{s^\\prime s}^{(p)}$ is an irreducible non-negative matrix, and it follows that $\\widetilde{\\mathcal{M}}_{s^\\prime s}^{(p)}$ is also irreducible.  By the Perron-Frobenius Theorem \\cite{meyer}: (\\emph{1}) $\\mathcal{M}_{s^\\prime s}^{(p)}$ has a maximal eigenvector $|\\phi \\rangle$ and $\\widetilde{\\mathcal{M}}_{s^\\prime s}^{(p)}$ has a maximal eigenvector $|\\widetilde{\\phi} \\rangle$, which each obey \\begin{equation}\n\\langle G_\\triangle|\\phi\\rangle \\ne 0 \\text{ and } \\langle G_\\triangle|\\widetilde{\\phi}\\rangle \\ne 0; \\label{eq:phiGne0}\n\\end{equation}\n(\\emph{2}) $\\nu_{s^\\prime s}^{(p)}$ and $\\widetilde{\\nu}_{s^\\prime s}^{(p)}$ are non-degenerate; (\\emph{3}) as stated in the lemma, $\\nu_{s^\\prime s}^{(p)}$ is the maximal left and maximal right eigenvalue of $\\mathcal{M}_{s^\\prime s}^{(p)}$.  As $\\mathbb{R}^{\\mathcal{G}_-}$ is a finite dimensional vector space, (\\ref{eq:phiGne0}) implies that\n\\begin{align}\n\\nu_{s^\\prime s}^{(p)} = \\lim_{n\\rightarrow \\infty} \\frac{\\log \\langle G_\\triangle | \\left(\\mathcal{M}_{s^\\prime s}^{(p)}\\right)^n |G_\\triangle\\rangle }{n}= \\lim_{n\\rightarrow \\infty} \\frac{\\log \\langle G_\\triangle | \\left(\\widetilde{\\mathcal{M}}_{s^\\prime s}^{(p)}\\right)^n |G_\\triangle\\rangle }{n} = \\widetilde{\\nu}_{s^\\prime s}^{(p)}.  \\label{eq:nunutilde}\n\\end{align}\nCombining (\\ref{eq:nutildebound}) and (\\ref{eq:nunutilde}) we obtain (\\ref{eq:Bfinalbound}).\n\\end{proof}\n\nUsing Lemma \\ref{lma12}, we now begin to bound the maximal eigenvalue of the matrix $M_{s^\\prime s}$ defined in (\\ref{eq:MssDef}).   Let \\begin{equation}\np = \\lceil \\kappa s \\log N \\rceil,  \\label{eq:pkappa}\n\\end{equation}\nwhere the parameter $\\kappa \\in \\mathbb{R}^+$ will be O(1).  We first combine (\\ref{eq:markov4}) and (\\ref{eq:BssDef}).    Using the inequality \\begin{equation}\nC_s > \\left(\\frac{\\mathrm{e}N}{s}\\right)^s,\n\\end{equation}\nwe find that since $N\\ge 4$ and $s\\ge 1$: \\begin{equation}\n\\mathbb{P}\\left[\\mu_{s^\\prime s}> \\nu_{s^\\prime s}^{(p)} \\mathrm{e}^{\\theta + 2\/\\kappa}\\right] \\le \\left(\\frac{\\mathrm{e}^{-2\/\\kappa}}{\\mathrm{e}^\\theta}  \\right)^p \\left(\\frac{\\mathrm{e}N}{s}\\right)^s = \\left(\\frac{\\mathrm{e}^{-2\/\\kappa}}{\\mathrm{e}^\\theta} \\left(\\frac{\\mathrm{e}N}{s}\\right)^{1\/\\kappa \\log N} \\right)^p  \\le \\mathrm{e}^{-p\\theta} = \\frac{1}{N^{\\kappa\\theta s}}.\n\\end{equation}\nMoreover, since there are at most $2(q-2)$ non-vanishing $\\mathcal{K}_{s^\\prime s}$ coefficients involving a fixed operator size, we conclude that \\begin{align}\n\\mathbb{P}\\left[\\mu_{s^\\prime s}> \\nu_{s^\\prime s}^{(p)} \\mathrm{e}^{\\theta + 2\/\\kappa}, \\text{ for any } s, s^\\prime \\right] &\\le \\sum_{|s^\\prime -s| \\le q-2} \\mathbb{P}\\left[\\mu_{s^\\prime s}> \\nu_{s^\\prime s}^{(p)} \\mathrm{e}^{\\theta + 2\/\\kappa}\\right] \\le    2(q-2) \\sum_{s=1}^N \\frac{1}{N^{\\kappa\\theta s}} \\notag \\\\\n&< \\frac{2(q-2)}{N^{\\kappa\\theta}-1} =1- \\mathbb{P}_{\\mathrm{success}}.\n\\end{align}\nHence, with probability $\\mathbb{P}_{\\mathrm{success}}$, we may assume that $\\mu_{s^\\prime s} \\le \\mathrm{e}^{\\theta + 2\/\\kappa} \\nu_{s^\\prime s}^{(p)}$.\n\nOf course, it remains to bound $\\nu_{s^\\prime s}^{(p)}$, which we do in the following lemma:\n\n\\begin{lem}\\label{lma13}\nIf $p$ is given by (\\ref{eq:pkappa}), $k$ is defined in (\\ref{eq:kdef}), and we assume (\\ref{eq:kappasqrtNN}), then \\begin{equation}\n\\nu_{s^\\prime s}^{(p)} < \\left( \\frac{s+q-2}{q-1} \\frac{2s}{q} \\left(\\frac{2q^2s}{N}\\right)^{2k-2}+ \\frac{2^q (q-1)! (s+q)^{q-1}}{N^{(q-2)\/2}} \\right) \\exp\\left[ \\frac{5q^2 \\kappa s \\log N}{\\sqrt{N}}\\right].  \\label{eq:lma13}\n\\end{equation}\n\n\\end{lem}\n\\begin{proof}\nLet us interpret $\\nu_{s^\\prime s}^{(p)}$ as the maximal left eigenvalue of $\\mathcal{M}_{s^\\prime s}^{(p)}$.  Defining $\\mathbb{R}^{\\mathcal{G}_-}_+$ as the set of all vectors with strictly positive entries, we begin by invoking the Collatz-Wielandt bound \\cite{meyer}: \\begin{equation}\n\\nu_{s^\\prime s}^{(p)} = \\inf_{|\\phi\\rangle \\in \\mathbb{R}^{\\mathcal{G}_-}_+} \\sup_{G\\in \\mathcal{G}_-} \\frac{\\langle \\phi| \\mathcal{M}_{s^\\prime s}^{(p)} | G\\rangle}{\\langle \\phi|G\\rangle}.  \\label{eq:collatzwieland}\n\\end{equation}\nClearly, we can bound $\\nu_{s^\\prime s}^{(p)}$ by simply guessing any $|\\phi\\rangle \\in \\mathbb{R}^{\\mathcal{G}_-}_+$.   We choose  \\begin{equation}\n\\langle \\phi |G\\rangle = N^{-|V\\cap G|\/2}\n\\end{equation}\nwhere $|V\\cap G|$ denotes the number of fermions (of non-zero degree!) in the graph $G$.   \n\nNow let us write out \\begin{equation}\n\\sup_{G\\in \\mathcal{G}_-} \\frac{\\langle \\phi| \\mathcal{M}_{s^\\prime s}^{(p)} | G\\rangle}{\\langle \\phi|G\\rangle} = \\sup_{G\\in \\mathcal{G}_-}  4\\sigma^2\\sum_{H^\\prime \\in \\mathcal{G}_+, H\\in \\mathcal{G}_-} N^{(|G\\cap V| - |H\\cap V|)\/2} \\langle H|\\mathcal{N}_-|H^\\prime\\rangle\\langle H^\\prime | \\mathcal{N}_+|G\\rangle .  \\label{eq:supGsum}\n\\end{equation}\nGiven graphs $G$, $H$ and $H^\\prime$, let us define the following four parameters: \\begin{subequations}\\begin{align}\na_+ := |(H^\\prime - H^\\prime \\cap G)\\cap V|, \\\\ \na_- := |(H - H^\\prime \\cap H)\\cap V|, \\\\ \nb_+ := |(G - H^\\prime \\cap G)\\cap V|, \\\\ \nb_- := |(H^\\prime - H^\\prime \\cap H)\\cap V|.\n\\end{align}\\end{subequations}\n$a_+$ and $a_-$ are the number of \\emph{new} fermions added to the graph in the first and second step respectively; $b_+$ and $b_-$ represent the number of fermions \\emph{removed} from the graph in the first and second step respectively.   Note the following constraints: \\begin{subequations}\\label{eq:abbounds}\\begin{align}\n0\\le a_+ \\le q+1-2k, \\\\\n0 \\le a_- \\le 2k-1, \\\\\n0 \\le b_+ \\le 2k-1, \\\\\n0 \\le b_- \\le q+1-2k.\n\\end{align}\\end{subequations} \\begin{equation}\n|G\\cap V| - |H\\cap V| = b_+ + b_- - a_+ - a_-.  \\label{eq:GVHVab}\n\\end{equation}\nLastly, note that $a_+$ and $a_-$ are non-negative if and only if a factor is added to the graph, and $b_+$ and $b_-$ are non-negative if and only if a factor is removed from the graph in that step.\n\nThere are four possible kinds of sequences of $H$ and $H^\\prime$, corresponding to whether a factor is added (A) or removed (R) from the graph in each step: RR, AA, RA, AR.  Because we keep the starting graph fixed, and sum over all possible ways to add or remove factors to the graph, we can efficiently overestimate the sum over all possible modifications to the graph with fixed $a_\\pm$ or $b_\\pm$.   Let \\begin{equation}\nv=|G\\cap V|\n\\end{equation}\nto be the number of vertices in $G$.  In the first step, the number of ways to add a factor is \\begin{equation}\nN_{\\mathrm{A}}(a_+) := \\sum_{H^\\prime \\in \\mathcal{G}_+ : |H^\\prime \\cap V| = a_+ + |G\\cap V|} \\langle H^\\prime|\\mathcal{N}_+|G\\rangle =   \\left(\\begin{array}{c} N - v \\\\ a_+ \\end{array}\\right) \\left(\\begin{array}{c} s \\\\ 2k-1 \\end{array}\\right) \\left(\\begin{array}{c} v-s \\\\ q+1-2k-a_+ \\end{array}\\right),\n\\end{equation}\nwhere the first combinatorial factor is the choice of $a_+$ distinct fermions to add to the graph, the second  is the number of $(2k-1)$-tuples of the $s$ odd degree fermions present in the graph, and the third is the number $(q+1-2k-a_+)$-tuples of even degree fermions to add an extra edge to.   If instead we remove a factor, we find \\begin{equation}\nN_{\\mathrm{R}}(b_+) \\le  \\left\\lbrace \\begin{array}{ll} \\displaystyle \\left\\lfloor \\dfrac{s}{b_+} \\right\\rfloor &\\ b_+ > 0 \\\\ p &\\ b_+ = 0 \\end{array}\\right.,\n\\end{equation}\nwhere the first line corresponds to the maximal number of factors that can have $b_+ > 0$ odd degree fermions, and the second line is a crude bound: we can remove no more factors than the maximal number $p$ allowed in any graph in $\\mathcal{G}_-$.\n\nNext we look at the AA sequences, where two factors are added sequentially.  Here we find \\begin{equation}\nN_{\\mathrm{AA}}(a_+, a_-) := N_{\\mathrm{A}}(a_+)  \\left(\\begin{array}{c} N - v - a_+ \\\\ a_- \\end{array}\\right) \\left(\\begin{array}{c} s+q+2-4k \\\\ q+1-2k \\end{array}\\right) \\left(\\begin{array}{c} v+a_+-s-q-2+4k \\\\ 2k-1-a_- \\end{array}\\right).\n\\end{equation}\nIn the RA sequences, we find \\begin{equation}\nN_{\\mathrm{RA}}(b_+, a_-) \\le N_{\\mathrm{R}}(b_+) \\times \\left(\\begin{array}{c} N - v + b_+ \\\\ a_- \\end{array}\\right) \\left(\\begin{array}{c} s+q+2-4k \\\\ q+1-2k \\end{array}\\right) \\left(\\begin{array}{c} v-b_+-s-q-2+4k \\\\ 2k-1-a_- \\end{array}\\right).\n\\end{equation}\nFor the RR sequences, we find \\begin{equation}\nN_{\\mathrm{RR}}(b_+,b_-) \\le N_{\\mathrm{R}}(b_+) \\times  \\left\\lbrace \\begin{array}{ll} \\displaystyle \\left\\lfloor \\dfrac{s+q+2-4k}{b_-} \\right\\rfloor &\\ b_- > 0 \\\\ p-1 &\\ b_- = 0 \\end{array}\\right.,\n\\end{equation}\nwhile for the AR sequences: \\begin{equation}\nN_{\\mathrm{AR}}(a_+,b_-) \\le N_{\\mathrm{A}}(a_+) \\times  \\left\\lbrace \\begin{array}{ll} \\displaystyle \\left\\lfloor \\dfrac{s+q+2-4k}{b_-} \\right\\rfloor &\\ b_- > 0 \\\\ p &\\ b_- = 0 \\end{array}\\right..\n\\end{equation}\n\nNow we must perform the sum over $a_\\pm$ and $b_\\pm$ in (\\ref{eq:supGsum}).    We start with the sum over AA sequences, where we will crudely bound the six distinct choose functions for convenience.  Using (\\ref{eq:GVHVab}), \\begin{align}\n\\sum_{a_+=0}^{q+1-2k} \\sum_{a_-=0}^{2k-1} \\frac{N_{\\mathrm{AA}}(a_+, a_-) }{N^{(a_+ + a_-)\/2} }&< \\sum_{a_+=0}^{q+1-2k} \\sum_{a_-=0}^{2k-1} \\frac{N^{a_+\/2}  s^{2k-1} v^{q+1-2k-a_+}}{a_+! (2k-1)!(q+1-2k-a_+)!} \\times \\frac{N^{a_-\/2} (s+q)^{q+1-2k} (v+q)^{2k-1-a_-}}{a_-! (q+1-2k)! (2k-1-a_-)!} \\notag \\\\\n&< \\frac{(s+q)^q}{(q+1-2k)!(2k-1)!}\\frac{\\left(\\sqrt{N}+v\\right)^{q+1-2k}}{(q+1-2k)!} \\frac{\\left(\\sqrt{N}+v+q\\right)^{2k-1}}{(2k-1)!} \\notag \\\\\n&<  \\frac{(s+q)^q \\left(\\sqrt{N}+v+q\\right)^q }{(q+1-2k)!^2(2k-1)!^2}. \\label{eq:lastsum1}\n\\end{align}\nNext, we bound the RA sequences.  For convenience, we may just use $N_{\\mathrm{R}}(b_+) \\le p$: \\begin{align}\n\\sum_{b_+=0}^{2k-1} \\sum_{a_-=0}^{2k-1} \\frac{N_{\\mathrm{RA}}(a_+, a_-) }{N^{( a_- - b_+)\/2} }&< \\sum_{b_+=0}^{2k-1} \\sum_{a_-=0}^{2k-1} p N^{b_+\/2} \\frac{N^{a_-\/2} (s+q)^{q+1-2k} v^{2k-1-a_-} }{a_-! (q+1-2k)! (2k-1-a_-)!} \\notag \\\\\n&< \\frac{p N^{(2k-1)\/2}}{1-N^{(1-2k)\/2}} \\frac{(s+q)^{q+1-2k}}{(q+1-2k)!} \\frac{\\left(\\sqrt{N} + v\\right)^{2k-1}}{(2k-1)!} \\notag \\\\\n&< \\frac{p (s+q)^{q+1-2k}}{(1-N^{-1\/2})(q+1-2k)!(2k-1)!} \\left(N + v\\sqrt{N}\\right)^{q\/2} \\label{eq:lastsum2}\n\\end{align}\nSimilarly, \\begin{align}\n\\sum_{b_+=0}^{2k-1} \\sum_{b_-=0}^{q+1-2k} N^{(b_+ + b_-)\/2} N_{\\mathrm{RR}}(b_+, b_-) < p(p-1) \\frac{N^{q\/2}}{(1-N^{-1\/2})^2} \\label{eq:lastsum3}\n\\end{align}\nLastly, and most importantly, we bound the AR sequences.  In this sum, we will split off the $b_- = q+1-2k$ contribution, and bound that more carefully: \\begin{align}\n\\sum_{b_-=0}^{q+1-2k}  \\sum_{a_+=0}^{q+1-2k} \\frac{N_{\\mathrm{AR}}(a_+, b_-)}{N^{(a_+ - b_-)\/2}} &<  \\left(p + \\sum_{b_-=1}^{q+1-2k}   N^{b_-\/2} \\left\\lfloor \\frac{s+q+2-4k}{b_-} \\right\\rfloor \\right)\\left( \\sum_{a_+=0}^{q+1-2k} \\frac{N^{a_+\/2}  s^{2k-1} v^{q+1-2k-a_+}}{a_+! (2k-1)!(q+1-2k-a_+)!} \\right) \\notag \\\\\n&< \\left(\\frac{p N^{(q-2k)\/2}}{1-N^{-1\/2}} + N^{(q+1-2k)\/2}  \\left\\lfloor \\frac{s+q+2-4k}{q+1-2k} \\right\\rfloor \\right) \\frac{s^{2k-1}}{(2k-1)!} \\frac{\\left(\\sqrt{N}+v\\right)^{q+1-2k} }{(q+1-2k)!}\\notag \\\\\n&<  \\left(\\frac{pN^{-1\/2}}{1-N^{-1\/2}} +\\left\\lfloor \\frac{s+q+2-4k}{q+1-2k} \\right\\rfloor \\right) \\frac{s^{2k-1}}{(2k-1)!} \\frac{\\left(N+v\\sqrt{N}\\right)^{q+1-2k}}{(q+1-2k)!}. \\label{eq:lastsum4}\n\\end{align}\n\nThe combinatorial bounds above, by construction, did not depend on the initial graph $G$.  Therefore, combining (\\ref{eq:lastsum1}), (\\ref{eq:lastsum2}), (\\ref{eq:lastsum3}) and (\\ref{eq:lastsum4}),  and employing (\\ref{eq:sigmadef}), (\\ref{eq:collatzwieland}) and (\\ref{eq:supGsum}), along with \\begin{equation}\nv < qp + s\n\\end{equation}\nand other simple inequalities, we obtain \n\\begin{align}\n\\nu_{s^\\prime s}^{(p)} &< \\nu_1 + \\nu_2\n\\end{align}\nwhere \\begin{subequations}\\begin{align}\n\\nu_1 &= \\left(\\frac{N+(qp+s)\\sqrt{N}}{N-q}\\right)^{q+1-2k} \\left(\\frac{q}{N-q}\\right)^{2k-2} \\frac{2s^{2k-1}}{q(2k-1)!} \\left(\\frac{pN^{-1\/2}}{1-N^{-1\/2}} + \\frac{s+q+2-4k}{q+1-2k} \\right) \\\\\n\\nu_2 &= \\frac{(q-1)!}{(N-q)^{(q-2)\/2}}\\left(\\frac{\\sqrt{N}+(p+1)q+s}{\\sqrt{N-q}}\\right)^q  \\left(\\frac{p}{1-N^{-1\/2}} + \\frac{(s+q)^{q+1-2k}}{(2k-1)!(q+1-2k)!}\\right)^2.\n\\end{align}\\end{subequations}\nNow we simplify.  Using that $N-q > \\frac{1}{2}N$ from (\\ref{eq:qlessN2}), along with $p<1+\\kappa s \\log N$,\n\\begin{align}\n\\nu_1 &< \\left(1 + 2\\frac{q + s\\sqrt{N}(1+q\\kappa \\log N) + q\\sqrt{N}}{N}\\right)^{q+1-2k} \\left(\\frac{2q}{N}\\right)^{2k-2} \\frac{2s^{2k-1}}{q(2k-1)!} \\left(\\frac{s+q-2}{q+1-2k} + \\frac{2(p+1)}{\\sqrt{N}} \\right) \\notag \\\\\n&< \\frac{s+q-2}{q+1-2k} \\frac{2s}{q} \\left(\\frac{2qs}{N}\\right)^{2k-2} \\exp\\left[ \\frac{5q^2 \\kappa s \\log N}{\\sqrt{N}}\\right].\n\\end{align}\nIn the second line, we used (\\ref{eq:annoying}), $(s+q-2)\\ge q+1-2k$, and $1+x < \\mathrm{e}^x$, to simplify further.   Next,  \\begin{align}\n\\nu_2 &< \\frac{2^{(q-2)\/2}(q-1)!}{N^{(q-2)\/2}} \\left(\\sqrt{2} \\left(1 + \\frac{s+2q+q\\kappa s \\log N}{\\sqrt{N}} \\right)\\right)^q\\left(\\frac{2\\kappa s \\log N}{\\sqrt{N}} + \\frac{(s+q)^{q+1-2k}}{(2k-1)!(q+1-2k)!} \\right) \\notag \\\\\n&< \\frac{2^q (q-1)! (s+q)^{q+1-2k}}{N^{(q-2)\/2}}\\exp\\left[ \\frac{2q^2 \\kappa s \\log N}{\\sqrt{N}}\\right] \n\\end{align}\nwhere in the second line we used the fact that $2\\kappa s\\log N + 2 < \\sqrt{N}(s+q)^{q+1-2k}$, since $q+1-2k>0$ and we assumed (\\ref{eq:kappasqrtNN}).   Making a few final simplifications, we obtain (\\ref{eq:lma13}).\n\\end{proof}\n\nThe last step to prove our theorem is to simply bound $\\mathcal{K}_l$.  Recall the relation between $l$ and $s$ defined in (\\ref{eq:ldef}).  With probability no smaller than $\\mathbb{P}_{\\mathrm{success}}$ we may invoke Lemma \\ref{lma13} as a bound on every $\\mu_{s^\\prime s}$.  Hence from Proposition~\\ref{propmuss} and (\\ref{eq:Kldef}), \\begin{align}\n\\mathcal{K}_l &\\le \\max \\left \\lbrace \\max_{s\\in R_l} \\sum_{s^\\prime \\in R_{l+1}} \\sqrt{\\mathrm{e}^{\\theta + 2\/\\kappa}\\nu_{s^\\prime s}^{(p)}},  \\max_{s^\\prime \\in R_{l+1}} \\sum_{s\\in R_{l}}  \\sqrt{\\mathrm{e}^{\\theta + 2\/\\kappa}\\nu_{s^\\prime s}^{(p)}}  \\right\\rbrace \\notag \\\\\n&< \\mathrm{e}^{\\theta\/2+1\/\\kappa} \\sum_{k=1}^{q\/2-1} \\sqrt{\\nu_{1+(q-2)l + q+2-4k, 1+(q-2)l}^{(p)}} \\notag \\\\\n&<\\mathrm{e}^{\\theta\/2+1\/\\kappa} \\left[ \\sqrt{\\frac{s+q-2}{q-1} \\frac{2s}{q}} \\dfrac{1}{\\displaystyle 1 -\\frac{2q^2s}{N}}   + q\\sqrt{\\frac{2^q (q-1)! (s+q)^{q-1}}{N^{(q-2)\/2}}} \\right] \\exp\\left[ \\frac{5q^2 \\kappa s \\log N}{2\\sqrt{N}}\\right] \\notag \\\\\n&< \\mathrm{e}^{\\theta\/2+1\/\\kappa} \\left[ \\sqrt{\\frac{2(1+(q-2)(l+1))(1+(q-2)l)}{q(q-1)}} \\dfrac{1}{\\displaystyle 1 -\\frac{2q^3(l+1)}{N}}   + \\frac{2^{q} q! (l+1)^{(q-1)\/2}}{N^{(q-2)\/4}} \\right] \\notag \\\\ \n&\\;\\;\\;\\;\\;\\; \\times \\exp\\left[ \\frac{5q^3 \\kappa (l+1) \\log N}{2\\sqrt{N}}\\right] \\notag \\\\\n&<c (l+1)\n\\end{align}\nwhere in the second line, we have used that the largest of the bounded transition rates come from the maximal value of $s$; in the third line, we used concavity of $\\sqrt{x}$; in the fourth line, we plugged in for $s=1+(q-2)l$ and used further elementary inequalities;  in the fifth line, we used $l+1\\le M$ along with various other elementary inequalities to further weaken the bound, and the values of $c$ and $M$ from (\\ref{eq:theorcm}).\n\\end{proof}\n\nCombining Corollary \\ref{corolscramble} and Theorem \\ref{theorSYK}, we immediately obtain \n\n\\begin{cor}\nIf there exist $\\gamma,\\delta  \\in \\mathbb{R}^+$ such that \\begin{equation}\nq < \\gamma N^{(1-\\delta)\/6},\n\\end{equation} \nthen as $N\\rightarrow \\infty$ in the SYK ensemble, there is a constant $b\\in \\mathbb{R}^+$ such that the operator scrambling time obeys the fast scrambling conjecture: \\begin{equation}\nt_{\\mathrm{s}} \\ge b\\log N.\n\\end{equation}\n\\end{cor}\n\nThis is the first complete proof of a logarithmic lower bound on the operator scrambling time in a Hamiltonian quantum system whose many-body Lyapunov exponent was expected to be finite.  But we do not expect that these constant prefactors in these bounds are tight.\n\nOne shortcoming of our proof, which perhaps has physical consequences, is that we were only able to demonstrate fast scrambling and the diverging growth time of operators if $q \\propto N^{(1-\\delta)\/6}$ for some $\\delta \\in \\mathbb{R}^+$.   Yet it may be the case that fast scrambling is robust so long as $q \\propto N^{(1-\\delta)\/2}$ \\cite{cotler}.   However, the SYK model is likely more physical at finite $q$ anyway (e.g. $q=4$), so this may be a relatively minor point.  The large $q$ limit is  a further simplification used in the theoretical physics literature to simplify Feynman diagrammatics.\n\nWe make one last remark.  The actual distribution of eigenvalues of $M_{s^\\prime s}$ (for any $s>1$) is not sharply peaked around the mean value which we have overestimated.  Instead, the distribution of eigenvalues is highly peculiar, with only a small fraction of eigenvalues, which we conjecture is $\\mathrm{O}(N^{1-s})$ for $s<(q-2)M$, within an O(1) factor of $\\mathcal{K}_{s^\\prime s}$.   We conjecture that the maximal eigenvector of $M_{s^\\prime s}$ is dominated by treelike factor graphs, analogous to (\\ref{eq:graphtree}), with $\\mathrm{O}(s\/q)$ leaves attached to a root which connects to a single fermion.   These are precisely the graphs associated to a growing operator which started from a single fermion.  Indeed, explicit calculations confirm that such treelike graphs have significantly larger weight in $\\mathrm{tr}(M_{s^\\prime s}^p)$ for $p=\\kappa s \\log N$.  It appears as though the fastest growing operators of average size $\\bar s$ is a single fermion operator $\\psi_1(t)$, evolved to an appropriate time $t$. It would be interesting if this set of conjectures can be proven or disproven.\n\n\\subsection{Comparison with perturbation theory}\\label{sec:pert}\nLet us now compare our bounds to prior calculations in the SYK model using perturbation theory.   First, let us discuss the Lyapunov exponent as $N\\rightarrow \\infty$.  We have found that \\begin{equation}\n\\lambda \\le 2\\sqrt{\\frac{2(q-2)}{q}},\n\\end{equation}\na slight improvement over \\cite{chen1}.   It is known analytically that \\cite{stanford1802} \\begin{equation}\n\\lambda_{\\mathrm{perturbative}} = 2 \\;\\;\\; (q=\\infty),\n\\end{equation}\nimplying that our result has over estimated the true value by a factor of $\\sqrt{2}$.  \n\n\\cite{stanford1802} also argued that the block probabilities $P_l(t)$ took the form \\begin{subequations}\\label{eq:SYKopprob}\\begin{align}\nP_0(t) &\\approx 1 - \\frac{4}{q} \\log \\cosh t + \\cdots , \\\\ \nP_l(t) &\\approx \\frac{2}{lq} (\\tanh t)^{2l} + \\cdots.\\;\\;\\;\\; (l>0)\n\\end{align}\\end{subequations}\nat leading order in a large $N$ and large $q$ expansion (with no bound on the subleading corrections, denoted above as $\\cdots$).  It is interesting to compare this with the following result: \\begin{prop}\nConsider the quantum walk of $|\\varphi(t)\\rangle$ generated by (\\ref{eq:Haux}) with \\begin{equation}\nK_l(t) = c(l+1),  \\label{eq:Klcl2}\n\\end{equation}\non the half-line where $N^\\prime = \\infty$.   Then \\begin{equation}\nP_l(t) = (\\tanh (ct))^{2l} \\mathrm{sech}^2 (ct).  \\label{eq:optqw}\n\\end{equation} \\label{propquantumwalk}\n\\end{prop}\n\\begin{proof}\nWithout loss of generality, we rescale time so that $c=1$.  Then, we repackage (\\ref{eq:prop2dos}) using generating functions: \\begin{equation}\nG(z,t) :=  \\sum_{l=0}^\\infty z^{l+1} P_l(t),\n\\end{equation}\nso that (\\ref{eq:prop2dos}) with (\\ref{eq:Klcl2}) implies that \\begin{equation}\n\\frac{\\partial G}{\\partial t} = z^2 \\frac{\\partial G}{\\partial z} - z \\frac{\\partial}{\\partial z} \\left(\\frac{G}{z}\\right) = \\left(z^2-1\\right)\\frac{\\partial G}{\\partial z} + \\frac{G}{z}.\n\\end{equation}\nThis equation is solved by the method of characteristics.  The characteristic curves $z(t)$ solve the differential equation \\begin{equation}\n\\frac{\\mathrm{d}z}{\\mathrm{d}t} = \\left(1-z^2\\right).\n\\end{equation}\nWith initial condition $z(0)=r$, we find \\begin{equation}\nt = \\frac{1}{2} \\log \\frac{(1-r)(1+z)}{(1+r)(1-z)},\n\\end{equation}\nor \\begin{equation}\nr = \\frac{z\\cosh t - \\sinh t}{\\cosh t - z\\sinh t}.\n\\end{equation}\nSolving the equation \\begin{equation}\n\\frac{\\partial G(r,t)}{\\partial t} = \\frac{G}{z}\n\\end{equation}\nwith $G(r,0)=r$ (corresponding to $P_0(0)=1$): \\begin{equation}\n\\log \\frac{G}{r} = \\int\\limits_0^t \\mathrm{d}t^\\prime \\frac{\\cosh t^\\prime  + r\\sinh t^\\prime}{\\sinh t^\\prime + r\\cosh t^\\prime} = \\log \\frac{\\sinh t + r\\cosh t }{r}.\n\\end{equation}\nThus, \\begin{equation}\nG(z,t)= \\frac{z\\mathrm{sech}t}{(1 - z\\tanh t)},\n\\end{equation}\nwhich leads to (\\ref{eq:optqw}) upon Taylor expanding and employing (\\ref{eq:varphisqrt}).\n\\end{proof}\n\nSome of the discrepancy between (\\ref{eq:optqw}) and (\\ref{eq:SYKopprob}) can be accounted for by our sloppy overestimate of $K_l(t)$ in the SYK model.  In particular, a more careful analysis demonstrates that $K_0(t) \\lesssim \\sqrt{2\/q}$ and $K_l(t) \\lesssim l$.  However, this slow first step does not change our estimate for the Lyapunov exponent.\n\nThis result is highly suggestive that the qualitative structure of the growing operator distribution in the SYK model, calculated perturbatively, is not substantially modified by non-perturbative physics.  Rather it appears quite similar to an ``optimal\" quantum walk that locally maximizes the transition rates from one operator size to the next.\\footnote{However, such an ``optimal\" quantum walk likely does not maximize the probability of large size operators, and perhaps does not even optimize the time-dependent average size.}  This may imply some universality to the patterns of operator growth in random regular $q$-local quantum systems.   If such universality exists, it may  have interesting implications for quantum gravity.\n\n\n\n\n\\section{Conclusion}\nIn this paper, we have proven the fast scrambling conjecture in the SYK model with a finite but large number $N$ of degrees of freedom.   While this result is not physically surprising, it is pleasing to have a mathematically careful derivation of this result.  We also expect that the methods developed here will lead to further advances in our technology for bounding quantum information dynamics and operator growth \\cite{chen1, chen2} beyond the Lieb-Robinson theorem \\cite{liebrobinson, hastings}.\n\nWe would like to say that  our demonstration of the robustness of operator growth  to  non-perturbative physics in at least one holographic model is a signature that the bulk geometry is semiclassical and that non-perturbative  fluctuations in quantum gravity are provably mild.  Unfortunately, this remains a conjecture, as the emergent geometry arises at finite temperature.  It would be interesting if our methods can be generalized to finite temperature states. \n\nLastly, we expect these techniques are useful for designing and constraining toy models of quantum gravity which can be experimentally studied using quantum simulation \\cite{Garttner2017, Li2017}.  At the very least, any tentative model must reproduce the exponential growth in operator size which is a  hallmark of particles falling towards black hole horizons.  Our  methods will not only bound the Lyapunov exponent of any proposed model, but also check whether the full time  evolution of  the operator size distribution could be consistent with a theory of quantum gravity.\n\n\\section*{Acknowledgments}\nThis work was supported by the University of Colorado.\n\n \n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe concept of homomorphisms of $(m, n)$-colored-mixed graph was introduced by J. Nes\\v{e}t\\v{r}il and A. Raspaud~\\cite{MNCM}\nin order to generalize homomorphisms of $k$-edge-colored graphs and oriented graphs.\n\nAn \\emph{$(m, n)$-colored-mixed graph} $G=(V, A_1, A_2,\\cdots, A_m, E_1, E_2,\\cdots, E_n)$ is a graph having $m$ colors of arcs and $n$ colors of edges.\nWe do not allow two arcs or edges to have the same endpoints.\nThe case $m=0$ and $n=1$ corresponds to simple graphs, $m=1$ and $n=0$ to oriented graphs and $m=0$ and $n=k$ to $k$-edge-colored graphs. For the case $m=0$ and $n = 2$\n($2$-edge-colored graphs) we refer to the two types of edges as \\emph{blue} and \\emph{red} edges.\n\nA \\emph{homomorphism} from an $(m, n)$-colored-mixed graph $G$ to another $(m, n)$-colored-mixed graph $H$ is a mapping $\\varphi:V(G) \\rightarrow V(H)$\nsuch that every edge (resp. arc) of $G$ is mapped to an edge (resp. arc) of $H$ of the same color (and orientation).\nIf $G$ admits a homomorphism to $H$, we say that $G$ is \\emph{$H$-colorable} since this homomorphism can be seen as a coloring of the vertices of $G$\nusing the vertices of $H$ as colors. The edges and arcs of $H$ (and their colors) give us the rules that this coloring must follow.\nGiven a class of graphs $\\mathcal{C}$, a graph is \\emph{$\\mathcal{C}$-universal} if for every graph $G \\in \\mathcal{C}$ is $H$-colorable.\nThe class $P_g^{(m, n)}$ contains every planar $(m, n)$-colored-mixed graph with girth at least $g$.\n\nIn this paper, we consider some planar $P_g^{(m, n)}$-universal graphs with $k$ vertices.\nThey are depicted in Figures~\\ref{fig:t_oriented} and~\\ref{fig:t_2edge}.\nThe known results about this topic are as follows.\n\n\\begin{theorem}\\label{thm:known}{\\ }\n\\begin{enumerate}\n\\item $K_4$ is a planar $P^{(0,1)}_3$-universal graph. This is the four color theorem.\n\\item $K_3$ is a planar $P^{(0,1)}_4$-universal graph. This is Gr\u00f6tzsch's Theorem \\cite{grotzsch}.\n\\item $\\overrightarrow{C_6^2}$ is a planar $P_{16}^{(1,0)}$-universal graph~\\cite{P10}.\n\\end{enumerate}\n\\end{theorem}\n\nOur first result shows that, in addition to the case of $(0,1)$-graphs covered by Theorems~\\ref{thm:known}.1 and~\\ref{thm:known}.2,\nour topic is actually restricted to the cases of oriented graphs (i.e., $(m,n)=(1,0)$) and 2-edge-colored graphs (i.e., $(m,n)=(0,2)$).\n\n\\begin{theorem}\\label{thm:Pmn}\nFor every $g\\ge3$, there exists no planar $P_g^{(m,n)}$-universal graph if $2m+n\\ge3$.\n\\end{theorem}\n\nAs Theorems~\\ref{thm:known}.1 and~\\ref{thm:known}.2 show for $(0,1)$-graphs, there might exist a trade-off between minimizing the girth $g$ and the number\nof vertices of the universal graph, for a fixed pair $(m,n)$.\nFor oriented graphs, Theorem~\\ref{thm:known}.3 tries to minimize the girth.\nFor oriented graphs and 2-edge-colored graphs, we choose instead to minimize the number of vertices of the universal graph.\n\n\\begin{theorem}\\label{thm:positive}{\\ }\n\\begin{enumerate}\n\\item $\\overrightarrow{T_5}$ is a planar $P_{28}^{(1,0)}$-universal graph on 5 vertices.\n\\item $T_6$ is a planar $P_{22}^{(0, 2)}$-universal graph on 6 vertices.\n\\end{enumerate}\n\\end{theorem}\n\nThe following results shows that Theorem~\\ref{thm:positive} is optimal in terms of the number of vertices of the universal graph.\n\n\\begin{theorem}\\label{thm:negative}{\\ }\n\\begin{enumerate}\n\\item For every $g\\ge3$, there exists an oriented bipartite cactus graph (i.e., $K_4^-$ minor-free graph) with girth at least $g$ and oriented chromatic number at least 5.\n\\item For every $g\\ge3$, there exists a 2-edge-colored bipartite outerplanar graph (i.e., $(K_4^-,K_{2,3})$ minor-free graph) with girth at least $g$ that does not map to a planar graph with at most 5 vertices.\n\\end{enumerate}\n\\end{theorem}\n\nMost probably, Theorem~\\ref{thm:positive} is not optimal in terms of girth. The following constructions give lower bounds on the girth.\n\n\\begin{theorem}\\label{thm:ce}{\\ }\n\\begin{enumerate}\n\\item There exists an oriented bipartite 2-outerplanar graph with girth $14$ that does not map to $\\overrightarrow{T_5}$.\n\\item There exists a 2-edge-colored planar graph with girth $11$ that does not map to $T_6$.\n\\item There exists a 2-edge-colored bipartite planar graph with girth $10$ that does not map to $T_6$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{figure}[H]\n\\begin{minipage}{0.5\\textwidth}\n\\begin{center}\n \\includegraphics{t_oriented}\n \\caption{The $P_{28}^{(1,0)}$-universal graph overrightarrow$(T_5)$.\\label{fig:t_oriented}}\n\n\\end{center}\n\\end{minipage}\\hfill\n\\begin{minipage}{0.5\\textwidth}\n\\begin{center}\n \\includegraphics{t_2edge}\n \\caption{The $P_{22}^{(0,2)}$-universal graph $T_6$.\\label{fig:t_2edge}}\n\\end{center}\n\\end{minipage}\n\\end{figure}\n\nNext, we obtain the following complexity dichotomies:\n\n\\begin{theorem}\\label{thm:NPC}{\\ }\n\\begin{enumerate}\n\\item For any fixed girth $g\\ge 3$, either every graph in $P_g^{(1,0)}$ maps to $\\overrightarrow{T_5}$ or it is NP-complete\nto decide whether a graph in $P_g^{(1,0)}$ maps to $\\overrightarrow{T_5}$.\nEither every bipartite graph in $P_g^{(1,0)}$ maps to $\\overrightarrow{T_5}$ or it is NP-complete to decide whether a bipartite graph in $P_g^{(1,0)}$ maps to $\\overrightarrow{T_5}$.\n\\item Either every graph in $P_g^{(0,2)}$ maps to $T_6$ or it is NP-complete to decide whether a graph in $P_g^{(1,0)}$ maps to $T_6$.\nEither every bipartite graph in $P_g^{(0,2)}$ maps to $T_6$ or it is NP-complete to decide whether a bipartite graph in $P_g^{(1,0)}$ maps to $T_6$.\n\\end{enumerate}\n\\end{theorem}\n\nFinally, we can use Theorem~\\ref{thm:NPC} with the non-colorable graphs in Theorem~\\ref{thm:ce}.\n\n\\begin{corollary}\\label{cor:cor}{\\ }\n\\begin{enumerate}\n\\item Deciding whether a bipartite graph in $P_{14}^{(1,0)}$ maps to $\\overrightarrow{T_5}$ is NP-complete.\n\\item Deciding whether a graph in $P_{11}^{(0,2)}$ maps to $T_6$ is NP-complete.\n\\item Deciding whether a bipartite graph in $P_{10}^{(0,2)}$ maps to $T_6$ is NP-complete.\n\\end{enumerate}\n\\end{corollary}\n\nA 2-edge-colored path or cycle is said to be \\emph{alternating} if any two adjacent edges have distinct colors.\n\n\\begin{proposition}[folklore]\\label{prop:3n-6}{\\ }\n\\begin{itemize}\n\\item Every planar simple graph on $n$ vertices has at most $3n-6$ edges.\n\\item Every planar simple graph satisfies $(\\mad(G)-2)\\cdot(g(G)-2)<4$.\n\\end{itemize}\n\\end{proposition}\n\n\\section{Proof of Theorem~\\ref{thm:positive}}\nWe use the discharging method for both results in Theorem~\\ref{thm:positive}. The following lemma will handle the discharging part.\nWe call a vertex of degree $n$ an $n$-vertex and a vertex of degree at least $n$ an $n^+$-vertex.\nIf there is a path made only of $2$-vertices linking two vertices $u$ and $v$, we say that $v$ is a weak-neighbor of $u$.\nIf $v$ is a neighbor of $u$, we also say that $v$ is a weak-neighbor of $u$. We call a (weak-)neighbor of degree $n$ an $n$-(weak-)neighbor.\n\n\\begin{lemma}\\label{lem:discharge}\nLet $k$ be a non-negative integer.\nLet $G$ be a graph with minimum degree 2 such that every 3-vertex has at most $k$ 2-weak-neighbors and every path contains at most $\\tfrac{k+1}2$ consecutive 2-vertices.\nThen $\\mad(G)\\ge2+\\tfrac2{k+2}$. In particular, $G$ cannot be a planar graph with girth at least $2k+6$.\n\\end{lemma}\n\n\\begin{proof}\nLet $G$ be as stated. Every vertex has an initial charge equal to its degree. Every $3^+$-vertex gives $\\tfrac1{k+2}$ to each of its 2-weak-neighbors.\nLet us check that the final charge $ch(v)$ of every vertex $v$ is at least $2+\\tfrac2{k+2}$.\n\\begin{itemize}\n \\item If $d(v)=2$, then $v$ receives $\\tfrac1{k+2}$ from both of its 3-weak-neighbors. Thus $ch(v)=2+\\tfrac2{k+2}$.\n \\item If $d(v)=3$, then $v$ gives $\\tfrac1{k+2}$ to each of its 2-weak-neighbors. Thus $ch(v)\\ge3-\\tfrac{k}{k+2}=2+\\tfrac2{k+2}$.\n \\item If $d(v)=d\\ge4$, then $v$ has at most $\\tfrac{k+1}2$ 2-weak-neighbors in each of the $d$ incident paths.\n Thus $ch(v)\\ge d-d\\paren{\\tfrac{k+1}2}\\paren{\\tfrac1{k+2}}=\\tfrac d2\\paren{1+\\tfrac1{k+2}}\\ge2+\\tfrac2{k+2}$.\n\\end{itemize}\nThis implies that $mad(G)\\ge2+\\frac2{k+2}$.\nFinally, if $G$ is planar, then the girth of $G$ cannot be at least $2k+6$, since otherwise $(\\mad(G)-2)\\cdot(g(G)-2)\\ge\\paren{2+\\tfrac2{k+2}-2}\\paren{2k+6-2}=\\paren{\\tfrac2{k+2}}\\paren{2k+4}=4$, which contradicts Proposition~\\ref{prop:3n-6}.\n\\end{proof}\n\n\n\\subsection{Proof of Theorem~\\ref{thm:positive}.1}\nWe prove that the oriented planar graph $\\overrightarrow{T_5}$ on 5 vertices from Figure~\\ref{fig:t_oriented} is $P_{28}^{(1,0)}$-universal by contradiction.\nAssume that $G$ is an oriented planar graphs with girth at least $28$ that does not admit a homomorphism to $\\overrightarrow{T_5}$\nand is minimal with respect to the number of vertices.\nBy minimality, $G$ cannot contain a vertex $v$ with degree at most one since a $\\overrightarrow{T_5}$-coloring of $G-v$ can be extended to $G$.\nSimilarly, $G$ does not contain the following configurations.\n\n\\begin{itemize}\n\\item A path with 6 consecutive 2-vertices.\n\\item A $3$-vertex with at least 12 2-weak-neighbors. \n\\end{itemize}\n\nSuppose that $G$ contains a path $u_0u_1u_2u_3u_4u_5u_6u_7$ such that the degree of $u_i$ is two for $1\\le i\\le6$.\nBy minimality of $G$, $G-{u_1,u_2,u_3,u_4,u_5,u_6}$ admits a $\\overrightarrow{T_5}$-coloring $\\varphi$.\nWe checked on a computer that for any $\\varphi(v_0)$ and $\\varphi(v_6)$ in $V\\paren{\\overrightarrow{T_5}}$\nand every possible orientation of the 7 arcs $u_iu_{i+1}$, we can always extend $\\varphi$ into a $\\overrightarrow{T_5}$-coloring of $G$, a contradiction.\n\nSuppose that $G$ contains a 3-vertex $v$ with at least 12 2-weak-neighbors. Let $u_1$, $u_2$, $u_3$ be the $3^+$-weak-neighbors of $v$\nand let $l_i$ be the number of common 2-weak-neighbors of $v$ and $u_i$, i.e., $2$-vertices on the path between $v$ and $l_i$.\nWithout loss of generality and by the previous discussion, we have $5\\ge l_1\\ge l_2\\ge l_3$ and $l_1+l_2+l_3\\ge12$.\nSo we have to consider the following cases:\n\\begin{itemize}\n\\item\\textbf{Case 1:} $l_1=5$, $l_2=5$, $l_3=2$.\n\\item\\textbf{Case 2:} $l_1=5$, $l_2=4$, $l_3=3$.\n\\item\\textbf{Case 3:} $l_1=4$, $l_2=4$, $l_3=4$.\n\\end{itemize}\n\nBy minimality, the graph $G'$ obtained from $G$ by removing $v$ and its 2-weak-neighbors admits a $\\overrightarrow{T_5}$-coloring $\\varphi$.\nLet us show that in all three cases, we can extend $\\varphi$ into a $\\overrightarrow{T_5}$-coloring of $G$ to get a contradiction.\n\nWith an extensive search on a computer we found that if a vertex $v$ is connected to a vertex $u$ colored in $\\varphi(u)$ by a path\nmade of $l$ 2-vertices ($0\\le l\\le5$) then $v$ can be colored in:\n\n\\begin{itemize}\n\\item at least 1 color if $l=0$,\n\\item at least 2 colors if $l=1$,\n\\item at least 2 colors if $l=2$ (the sets $\\acc{c, d, e}$ and $\\acc{b, c, d}$ are the only sets of size 3 that can be forbidden from $v$),\n\\item at least 3 colors if $l=3$, \n\\item at least 4 colors if $l=4$ and\n\\item at least 4 colors if $l=5$ (only the sets $\\acc{b}$, $\\acc{c}$, and $\\acc{e}$ can be forbidden from $v$).\n\\end{itemize}\n\nIn Case 1, $u_3$ forbids at most 3 colors from $v$ since $l_3=2$. If it forbids less than $3$ colors,\nwe will be able to find a color for $v$ since $u_1$ and $u_2$ forbid at most 1 color from $v$. The only sets of 3 colors that $u_3$ can forbid are $\\acc{b,c,d}$ and $\\acc{c, d, e}$.\nSince $u_1$ and $u_2$ can each only forbid $b$, $c$ or $e$, we can always find a color for $v$.\n\nIn Case 2, $u_1$ and $u_2$ each forbid at most one color and $u_3$ forbids at most $2$ colors so there remains at least one color for $v$.\n\nIn Case 3, $u_1$, $u_2$, and $u_3$ each forbid at most one color, so there remains at least two colors for $v$.\n\nWe can always extend $\\varphi$ into a $\\overrightarrow{T_5}$-coloring of $G$, a contradiction.\n\nSo $G$ contains at most 5 consecutive 2-vertices and every 3-vertex has at most 11 2-weak-neighbors.\nUsing Lemma~\\ref{lem:discharge} with $k=11$ contradicts the fact that the girth of $G$ is at least 28.\n\n\\subsection{Proof of Theorem~\\ref{thm:positive}.2}\nWe prove that the 2-edge-colored planar graph $T_6$ on 6 vertices from Figure~\\ref{fig:t_2edge} is $P_{22}^{(0,2)}$-universal by contradiction.\nAssume that $G$ is a 2-edge-colored planar graphs with girth at least $22$ that does not admit a homomorphism to $T_6$ and is minimal with respect to the number of vertices.\nBy minimality, $G$ cannot contain a vertex $v$ with degree at most one since a $T_6$-coloring of $G-v$ can be extended to $G$.\nSimilarly, $G$ does not contain the following configurations.\n\n\\begin{itemize}\n\\item A path with 5 consecutive 2-vertices.\n\\item A $3$-vertex with at least 9 2-weak-neighbors. \n\\end{itemize}\n\nSuppose that $G$ contains a path $u_0u_1u_2u_3u_4u_5u_6$ such that the degree of $u_i$ is two for $1\\le i\\le5$.\nBy minimality of $G$, $G-{u_1, u_2, u_3, u_4, u_5}$ admits a $T_6$-coloring $\\varphi$.\nWe checked on a computer that for any $\\varphi(v_0)$ and $\\varphi(v_6)$ in $V(T)$ and every possible colors of the 6 edges $u_iu_{i+1}$,\nwe can always extend $\\varphi$ into a $T_6$-coloring of $G$, a contradiction.\n\nSuppose that $G$ contains a 3-vertex $v$ with at least 9 2-weak-neighbors. Let $u_1$, $u_2$, $u_3$ be the $3^+$-weak-neighbors of $v$\nand let $l_i$ be the number of common 2-weak-neighbors of $v$ and $u_i$, i.e., $2$-vertices on the path between $v$ and $l_i$.\nWithout loss of generality and by the previous discussion, we have $4\\ge l_1\\ge l_2\\ge l_3$ and $l_1+l_2+l_3\\ge 9$. So we have to consider the following cases:\n\n\\begin{itemize}\n\\item\\textbf{Case 1:} $l_1=3$, $l_2=3$, $l_3=3$.\n\\item\\textbf{Case 2:} $l_1=4$, $l_2=3$, $l_3=2$.\n\\item\\textbf{Case 3:} $l_1=4$, $l_2=4$, $l_3=1$.\n\\end{itemize}\n\nBy minimality of $G$, the graph $G'$ obtained from $G$ by removing $v$ and its 2-weak-neighbors admits a $T_6$-coloring $\\varphi$.\nLet us show that in all three cases, we can extend $\\varphi$ into a $T_6$-coloring of $G$ to get a contradiction.\n\nWith an extensive search on a computer we found that if a vertex $v$ is connected to a vertex $u$ colored in $\\varphi(u)$\nby a path $P$ made of $l$ 2-vertices ($0\\le l\\le 4$) then $v$ can be colored in:\n\n\\begin{itemize}\n\\item at least 1 color if $l=0$ (the sets ${a, c, d, e, f}$ and ${b, c, d, e, f}$ of colors are the only sets of size 5 that can be forbidden from $v$\nfor some $\\varphi(u)\\in T$ and edge-colors on $P$),\n\\item at least 2 colors if $l=1$ (the sets ${a, b, c, f}$ and ${b, c, e, f}$ are the only sets of size 4 that can be forbidden from $v$),\n\\item at least 3 colors if $l=2$ (the sets ${b, c, f}$, ${c, e, f}$ and ${d, e, f}$ are the only sets of size 3 that can be forbidden from $v$),\n\\item at least 4 colors if $l=3$ (the set ${c, b}$ is the only set of size 2 that can be forbidden from $v$), and\n\\item at least 5 colors if $l=4$ (the sets ${c}$ and ${f}$ are the only sets of size 1 that can be forbidden from $v$).\n\\end{itemize}\n\nSuppose that we are in Case 1. Vertices $u_1$, $u_2$, and $u_3$ each forbid at most 2 colors from $v$ since $l_1=l_2=l_3=3$.\nSuppose that $u_1$ forbids 2 colors. It has to forbid colors $c$ and $f$ (since it is the only pair of colors that can be forbidden by a path made of 3 2-vertices).\nIf $u_2$ or $u_3$ also forbids 2 colors, they will forbid the exact same pair of colors. We can therefore assume that they each forbid 1 color from $v$.\nThere are 6 available colors in $T_6$, so we can always find a color for $v$ and extend $\\varphi$ to a $T_6$-coloring of $G$, a contradiction.\nWe proceed similarly for the other two cases.\n\nSo $G$ contains at most 4 consecutive 2-vertices and every 3-vertex has at most 8 2-weak-neighbors.\nThen Lemma~\\ref{lem:discharge} with $k=8$ contradicts the fact that the girth of $G$ is at least 22.\n\n\\section{Proof of Theorem~\\ref{thm:negative}.1}\nWe construct an oriented bipartite cactus graph with girth at least $g$ and oriented chromatic number at least 5. Let $g'$ be such that $g'\\ge g$ and $g'\\equiv4\\pmod{6}$.\nConsider a circuit $v_1,\\cdots,v_{g'}$. Clearly, the oriented chromatic number of this circuit is 4 and the only tournament on 4 vertices it can map to\nis the tournament $\\overrightarrow{T_4}$ induced by the vertices $a$, $b$, $c$, and $d$ in $\\overrightarrow{T_5}$.\nNow we consider the cycle $C=w_1,\\cdots,w_{g'}$ containing the arcs $w_{2i-1}w_{2i}$ with $1\\le i\\le g'\/2$, $w_{2i+1}w_{2i}$ with $1\\le i\\le g'\/2-1$, and $w_{g'}w_1$.\n\nSuppose for contradiction that $C$ admits a homomorphism $\\varphi$ such that $\\varphi(w_1)=d$.\nThis implies that $\\varphi(w_2)=a$, $\\varphi(w_3)=d$, $\\varphi(w_4)=a$, and so on until $\\varphi(w_{g'})=a$.\nSince $\\varphi(w_{g'})=a$ and $\\varphi(w_1)=d$, $w_{g'}w_1$ should map to $ad$, which is not an arc of $\\overrightarrow{T_4}$, a contradiction.\n\nOur cactus graph is then obtain from the circuit $v_1,\\cdots,v_{g'}$ and $g'$ copies of $C$ by identifying every vertex $v_i$ with the vertex $w_1$ of a copy of $C$.\nThis cactus graph does not map to $\\overrightarrow{T_4}$ since one of the $v_i$ would have to map to $d$ and then the copy of $C$ attached to $v_i$ would not be $\\overrightarrow{T_4}$-colorable.\n\n\\section{Proof of Theorem~\\ref{thm:negative}.2}\nWe construct a 2-edge-colored bipartite outerplanar graph with girth at least $g$ that does not map to a 2-edge-colored planar graph with at most 5 vertices.\nLet $g'$ be such that $g'\\ge g$ and $g'\\equiv2\\pmod{4}$. Consider an alternating cycle $C=v_0,\\cdots,v_{g'-1}$.\nFor every $0\\le i\\le g'-3$, we add $g'-2$ 2-vertices $w_{i,1},\\cdots,w_{i,g'-2}$ that form the path $P_i=v_iw_{i,1}\\cdots w_{i,g'-2}v_{i+1}$\nsuch that the edges of $P_i$ get the color distinct from the color of the edge $v_iv_{i+1}$. Let $G$ be the obtained graph.\nThe 2-edge-colored chromatic number of $C$ is 5.\nSo without loss of generality, we assume for contradiction that $G$ admits a homomorphism $\\varphi$ to a 2-edge-colored planar graph $H$ on 5 vertices.\nLet us define $\\mathcal{E}=\\bigcup_{i\\texttt{ even}}\\varphi(v_i)$ and $\\mathcal{O}=\\bigcup_{i\\texttt{ odd}}\\varphi(v_i)$.\nSince $C$ is alternating, $\\varphi(v_i)\\ne\\varphi(v_{i+2})$ (indices are modulo $g'$). Since $g'\\equiv2\\pmod{4}$, there is an odd number of $v_i$ with an even (resp. odd) index.\nThus, $\\abs{\\mathcal{E}}\\ge3$ and $\\abs{\\mathcal{O}}\\ge3$. Therefore we must have $\\mathcal{E}\\cap\\mathcal{O}\\ne\\emptyset$.\n\nNotice that every two vertices $v_i$ and $v_j$ in $G$ are joined by a blue path and a red path such that the lengths of these paths have the same parity as $i-j$.\nThus, the blue (resp. red) edges of $H$ must induce a connected spanning subgraph of $H$. Since $|V(H)|=5$, $H$ contains at least 4 blue (resp. red) edges.\nSince red and blue edges play symmetric roles in $G$ and since $|E(H)|\\le9$ by Proposition~\\ref{prop:3n-6}, we assume without loss of generality that $H$ contains exactly 4 blue edges.\nMoreover, these 4 blue edges induce a tree. In particular, the blue edges induce a bipartite graph which partitions $V(H)$ into 2 parts.\nThus, every $v_i$ with even index is mapped into one part of $V(H)$ and every $v_i$ with odd index is mapped into the other part of $V(H)$.\nSo $\\mathcal{E}\\cap\\mathcal{O}=\\emptyset$, which is a contradiction.\n\n\\section{Proof of Theorem~\\ref{thm:Pmn}}\nLet $T$ be a $P_g^{(m, n)}$-universal planar graph for some $g$ that is minimal with respect to the subgraph order.\n\nBy minimality of $T$, there exists a graph $G \\in P_g^{(m, n)}$ such that every color in $T$ has to be used at least once to color $G$.\nWithout loss of generality, $G$ is connected, since otherwise we can replace $G$ by the connected graph obtained from $G$\nby choosing a vertex in each component of $G$ and identifying them. We create a graph $G'$ from $G$ as follows:\n\nFor each edge or arc $uv$ we create $4m+n$ paths starting at $u$ and ending at $v$ made of vertices of degree 2:\n\n\\begin{itemize}\n\\item For each type of edge, we create a path made of $g-1$ edges of this type.\n\n\\item For each type of arc, we create two paths made of $g-1$ arcs of this type such that the paths alternate between forward and backward arcs.\nWe make the paths such that $u$ is the tail of the first arc of one path and the head of the first arc of the other path.\n\n\\item Similarly, for each type of arc we create two paths made of $g$ arcs of this type such that the paths alternate between forward and backward arcs.\nWe make the paths such that $u$ is the tail of the first arc of one path and the head of the first arc of the other path.\n\\end{itemize}\n\nNotice that $G'$ is in $P_g^{(m, n)}$ and thus admits a homomorphism $\\varphi$ to $T$.\nSince $G$ is connected and every color in $T$ has to be used at least once to color $G$, we can find for each pair of vertices and $(c_1, c_2)$ in $T$\nand each type of edge a path $(v_1, v_2,\\cdots, v_l)$ in $G'$ made only of edges of this type such that $\\varphi(v_1)=c_1$ and $\\varphi(v_l)=c_2$. \\newline\n\nThis implies that for every pair of vertices $(c_1, c_2)$ in $T$ and each type of edge, there exists a walk from $c_1$ to $c_2$ made of edges of this type.\nTherefore, for $1\\le j\\le n$, the subgraph induced by $E_j(T)$ is connected and contains all the vertices of $T$.\nSo $E_j(T)$ contains a spanning tree of $T$. Thus $T$ contains at least $|V(T)|-1$ edges of each type.\\newline\n\nSimilarly, we can find for each pair of vertices $(c_1, c_2)$ in $T$ and each type of arc a path of even length $(v_1, v_2,\\cdots, v_{2l-1})$ in $G'$ made only of arcs of this type,\nstarting with a forward arc and alternating between forward and backward arcs such that $\\varphi(v_1)=c_i$ and $\\varphi(v_l)=c_2$.\nWe can also find a path of the same kind with odd length.\\newline\n\nThis implies that for every pair of vertices $(c_1, c_2)$ in $T$ and each type of arc there exist a walk of odd length and a walk of even length\nfrom $c_1$ to $c_2$ made of arcs of this type, starting with a forward arc and alternating between forward and backward arcs.\nLet $p$ be the maximum of the length of all these paths. Given one of these walks of length $l$, we can also find a walk of length $l+2$\nthat satisfies the same constraints by going through the last arc of the walk twice more.\nTherefore, for every $l\\ge p$, every pair of vertices $(c_1, c_2)$ in $T$, and every type of arc,\nit is possible to find a homomorphism from the path $P$ of length $l$ made of arcs of this type, starting with a forward arc and alternating\nbetween forward and backward arcs to $T$ such that the first vertex is colored in $c_1$ and the last vertex is colored in $c_2$.\\newline\n\nWe now show that this implies that $|A_j(T)|\\ge2|V(T)|-1$ for $1\\le j\\le m$.\nLet $P$ be a path $(v_1, v_2,\\cdots, v_p, v_{p+1})$ of length $p$ starting with a forward arc and alternating between forward and backward arcs of the same type.\nWe color $v_1$ in some vertex $c$ of $T$. Let $C_i$ be the set of colors in which vertex $v_i$ could be colored.\nWe know that $C_1=c$ and $C_2$ is the set of direct successors of $c$. Set $C_3$ is the set of direct predecessors of vertices in $C_2$ so $C_1\\subseteq C_3$ and,\nmore generally, $C_i \\subseteq C_i+2$. Let $uv$ be an arc in $T$. If $u\\in C_i$ with $i$ odd, then $v\\in C_{i+1}$.\nIf $v\\in C_i$ with $i$ even then $u\\in C_{i+1}$. We can see that $uv$ is capable of adding at most one vertex to a $C_i$ (and every $C_j$ with $j\\equiv i\\mod 2$ and $i\\le j$).\nWe know that $C_{p+1}=V(T)$ hence $T$ contains at least $2|V(T)|-1$ arcs of each type.\\newline\n\nTherefore, the underlying graph of $T$ contains at least $m\\paren{2|V(T)|-1}+n\\paren{|V(T)|-1}=\\paren{2m+n}|V(T)|-m-n$ edges, which contradicts Proposition~\\ref{prop:3n-6} for $2m+n\\ge3$.\n\n\\section{Proof of Theorem~\\ref{thm:ce}.1}\nWe construct an oriented bipartite 2-outerplanar graph with girth $14$ that does not map to $\\overrightarrow{T_5}$.\n\nThe oriented graph $X$ is a cycle on 14 vertices $v_0,\\cdots,v_{13}$ such that the tail of every arc is the vertex with even index, except for the arc $\\overrightarrow{v_{13}v_0}$.\nSuppose for contradiction that $X$ has a $\\overrightarrow{T_5}$-coloring $h$ such that no vertex with even index maps to $b$.\nThe directed path $v_{12}v_{13}v_0$ implies that $h(v_{12})\\ne h(v_0)$.\nIf $h(v_0)=a$, then $h(v_1)\\in\\acc{b,c}$ and $h(v_2)=a$ since $h(v_2)\\ne b$. By contagion, $h(v_0)=h(v_2)=\\cdots=h(v_{12})=a$, which is a contradiction. Thus $h(v_0)\\ne a$.\nIf $h(v_0)=c$, then $h(v_1)=d$ and $h(v_2)=c$ since $h(v_2)\\ne b$. By contagion, $h(v_0)=h(v_2)=\\cdots=h(v_{12})=c$, which is a contradiction. Thus $h(v_0)\\ne c$.\nSo $h(v_0)\\not\\in\\acc{a,b,c}$, that is, $h(v_0)\\in\\acc{d,e}$. Similarly, $h(v_{12})\\in\\acc{d,e}$.\nNotice that $\\overrightarrow{T_5}$ does not contain a directed path $xyz$ such that $x$ and $z$ belong to $\\acc{d,e}$.\nSo the path $v_{12}v_{13}v_0$ cannot be mapped to $\\overrightarrow{T_5}$.\nThus $X$ does not have a $\\overrightarrow{T_5}$-coloring $h$ such that no vertex with even index maps to $b$.\n\nConsider now the path $P$ on 7 vertices $p_0,\\cdots,p_6$ with the arcs $\\overrightarrow{p_1p_0}$, $\\overrightarrow{p_1p_2}$, $\\overrightarrow{p_3p_2}$, $\\overrightarrow{p_4p_3}$, $\\overrightarrow{p_5p_4}$, $\\overrightarrow{p_5p_6}$. It is easy to check that there exists no $\\overrightarrow{T_5}$-coloring $h$ of $P$ such that $h(p_0)=h(p_6)=b$.\n\nWe construct the graph $Y$ as follows: we take 8 copies of $X$ called $X_{\\texttt{main}}$, $X_0$, $X_2$, $X_4$, $\\cdots$, $X_{12}$.\nFor every couple $(i,j)\\in\\acc{0,2,4,6,8,10,12}^2$, we take a copy $P_{i,j}$ of $P$, we identify the vertex $p_0$ of $P_{i,j}$\nwith the vertex $v_i$ of $X_{\\texttt{main}}$ and we identify the vertex $p_6$ of $P_{i,j}$ with the vertex $v_j$ of $H_i$.\n\nSo $Y$ is our oriented bipartite 2-outerplanar graph with girth $14$. Suppose for contradiction that $Y$ has a $\\overrightarrow{T_5}$-coloring $h$.\nBy previous discussion, there exists $i\\in\\acc{0,2,4,6,8,10,12}$ such that the vertex $v_i$ of $X_{\\texttt{main}}$ maps to $b$.\nAlso, there exists $j\\in\\acc{0,2,4,6,8,10,12}$ such that the vertex $v_j$ of $X_i$ maps to $b$.\nSo the corresponding path $P_{i,j}$ is such that $h(p_0)=h(p_6)=b$, a contradiction. Thus $Y$ does not map to $\\overrightarrow{T_5}$.\n\n\\section{Proof of Theorem~\\ref{thm:ce}.2}\nWe construct a 2-edge-colored 2-outerplanar graph with girth $11$ that does not map to $T_6$.\nWe take 12 copies $X_0,\\cdots,X_{11}$ of a cycle of length $11$ such that every edge is red.\nLet $v_{i,j}$ denote the $j^{\\text{\\tiny th}}$ vertex of $X_i$.\nFor every $0\\le i\\le 10$ and $0\\le j\\le 10$, we add a path consisting of 5 blue edges between $v_{i,11}$ and $v_{j,i}$.\n\nNotice that in any $T_6$-coloring of a red odd cycle, one vertex must map to $c$.\nSo we suppose without loss of generality that $v_{0,11}$ maps to $c$.\nWe also suppose without loss of generality that $v_{0,0}$ maps to $c$.\nThe blue path between $v_{0,11}$ and $v_{0,0}$ should map to a blue walk of length 5 from $c$ to $c$ in $T_6$.\nSince $T_6$ contains no such walk, our graph does not map to $T_6$.\n\n\\section{Proof of Theorem~\\ref{thm:ce}.3}\nWe construct a 2-edge-colored bipartite 2-outerplanar graph with girth $10$ that does not map to $T_6$.\nBy Theorem~\\ref{thm:negative}.2, there exists a bipartite outerplanar graph $M$ with girth at least $10$\nsuch that for every $T_6$-coloring $h$ of $M$, there exists a vertex $v$ in $M$ such that $h(v)=c$.\n\nLet $X$ be the graph obtained as follows. Take a main copy $Y$ of $M$.\nFor every vertex $v$ of $Y$, take a copy $Y_v$ of $M$. Since $Y_v$ is bipartite, let $A$ and $B$ the two independent sets of $Y_v$.\nFor every vertex $w$ of $A$, we add a path consisting of 5 blue edges between $v$ and $w$.\nFor every vertex $w$ of $B$, we add a path consisting of 4 edges colored (blue, blue, red, blue) between $v$ and $w$.\n\nNotice that $X$ is indeed a bipartite 2-outerplanar graph with girth $10$.\nWe have seen in the previous proof that $T_6$ contains no blue walk of length 5 from $c$ to $c$.\nWe also check that $T_6$ contains no walk of length 4 colored (blue, blue, red, blue) from $c$ to $c$.\nBy the property of $M$, for every $T_6$-coloring $h$ of $X$, there exist a vertex $v$ in $Y$ and a vertex $w$ in $Y_v$ such that $h(v)=h(w)=c$.\nThen $h$ cannot be extended to the path of length 4 or 5 between $v$ and $w$.\nSo $X$ does not map to $T_6$.\n\n\n\\section{Proof of Theorem~\\ref{thm:NPC}.1}\nLet $g$ be the largest integer such that there exists a graph in $P_g^{(1,0)}$ that does not map to $\\overrightarrow{T_5}$.\nLet $G\\in P_g^{(1,0)}$ be a graph that does not map to $\\overrightarrow{T_5}$ and such that the underlying graph of $G$ is minimal with respect to the homomorphism order.\n\nLet $G'$ be obtained from $G$ by removing an arbitrary arc $v_0v_3$ and adding two vertices $v_1$ and $v_2$ and the arcs $v_0v_1$, $v_2v_1$, $v_2v_3$.\nBy minimality, $G'$ admits a homomorphism $\\varphi$ to $\\overrightarrow{T_5}$. Suppose for contradiction that $\\varphi(v_2)=c$. This implies that $\\varphi(v_1)=\\varphi(v_3)=d$.\nThus $\\varphi$ provides a $\\overrightarrow{T_5}$-coloring of $G$, a contradiction. So $\\varphi(v_2)\\ne c$ and, similarly, $\\varphi(v_2)\\ne e$.\n\nGiven a set $S$ of vertices of $\\overrightarrow{T_5}$, we say that we force $S$ if we specify a graph $H$ and a vertex $v\\in V(H)$ such that \nfor every vertex $x\\in V\\paren{\\overrightarrow{T_5}}$, we have $x\\in S$ if and only if there exists a $\\overrightarrow{T_5}$-coloring $\\varphi$ of $H$ such that $\\varphi(v)=x$.\nThus, with the graph $G'$ and the vertex $v_2$, we force a non-empty set $\\mathcal{S}\\subset V\\paren{\\overrightarrow{T_5}}\\setminus\\acc{c,e}=\\acc{a,b,d}$.\n\nWe use a series of constructions in order to eventually force the set $\\acc{a,b,c,d}$ starting from $\\mathcal{S}$.\nRecall that $\\acc{a,b,c,d}$ induces the tournament $\\overrightarrow{T_4}$.\nWe thus reduce $\\overrightarrow{T_5}$-coloring to $\\overrightarrow{T_4}$-coloring, which is NP-complete for subcubic bipartite planar graphs with any given girth~\\cite{GO15}.\n\nThese constructions are summarized in the tree depicted in Figure~\\ref{fig:oriented}. The vertices of this forest contain the non-empty subsets of $\\acc{a,b,d}$ and a few other sets.\nIn this tree, an arc from $S_1$ to $S_2$ means that if we can force $S_1$, then we can force $S_2$. Every arc has a label indicating the construction that is performed.\nIn every case, we suppose that $S_1$ is forced on the vertex $v$ of a graph $H_1$ and we construct a graph $H_2$ that forces $S_2$ on the vertex $w$.\n\n\\begin{figure}[htpb]\n\\begin{center}\n \\includegraphics[scale=0.8]{oriented}\n \\caption{Forcing the set $\\acc{a,b,c,d}$.\\label{fig:oriented}}\n\\end{center}\n\\end{figure}\n\n\\begin{itemize}\n\\item Arcs labelled \"out\": The set $S_2$ is the out-neighborhood of $S_1$ in $\\overrightarrow{T_5}$. We construct $H_2$ from $H_1$ by adding a vertex $w$ and the arc $vw$.\nThus, $S_2$ is indeed forced on the vertex $w$ of $H_2$.\n\\item Arcs labelled \"in\": The set $S_2$ is the in-neighborhood of $S_1$ in $\\overrightarrow{T_5}$. We construct $H_2$ from $H_1$ by adding a vertex $w$ and the arc $wv$.\nThus, $S_2$ is indeed forced on the vertex $w$ of $H_2$.\n\\item Arc labelled \"Z\": Let $g'$ be the smallest integer such that $g'\\ge g$ and $g'\\equiv4\\pmod{6}$. We consider a circuit $v_1,\\cdots,v_{g'}$.\nFor $2\\le i\\le g'$, we take a copy of $H_1$ and we identify its vertex $v$ with $v_i$. We thus obtain the graph $H_2$ and we set $w=v_2$. Let $\\varphi$ be any $T_6$-coloring of $H_2$.\nBy construction, $\\acc{\\varphi(v_2),\\cdots,\\varphi(v_{g'})}\\subset S_1=\\acc{a,b,d}$.\nA circuit of length $\\not\\equiv0\\pmod{3}$ cannot map to the 3-circuit induced by $\\acc{a,b,d}$, so $\\varphi(v_1)\\in\\acc{c,e}$.\nIf $\\varphi(v_1)=c$ then $\\varphi(v_2)=d$ and if $\\varphi(v_1)=e$ then $\\varphi(v_2)=a$. Thus $S_2=\\acc{ad}$.\n\\end{itemize}\n\n\\section{Proof of Theorem~\\ref{thm:NPC}.2}\nLet $g$ be the largest integer such that there exists a graph in $P_g^{(0,2)}$ that does not map to $T_6$.\nLet $G\\in P_g^{(0,2)}$ be a graph that does not map to $T_6$ and such that the underlying graph of $G$ is minimal with respect to the homomorphism order.\n\nLet $G'$ be obtained from $G$ by subdividing an arbitrary edge $v_0v_3$ twice to create the path $v_0v_1v_2v_3$\nsuch that the edges $v_0v_1$ and $v_1v_2$ are red and the edge $v_2v_3$ gets the color of the original edge $v_0v_3$.\nBy minimality, $G'$ admits a homomorphism $\\varphi$ to $T_6$.\nSuppose for contradiction that $\\varphi(v_1)=f$. This implies that $\\varphi(v_0)=\\varphi(v_2)=b$. Thus $\\varphi$ provides a $T_6$-coloring of $G$, a contradiction.\n\nGiven a set $S$ of vertices of $T_6$, we say that we force $S$ if we specify a graph $H$ and a vertex $v\\in V(H)$ such that \nfor every vertex $x\\in V(T_6)$, we have $x\\in S$ if and only if there exists $T_6$-coloring $\\varphi$ of $H$ such that $\\varphi(v)=x$.\nThus, with the graph $G'$ and the vertex $v_1$, we force a non-empty set $\\mathcal{S}\\subset V(T_6)\\setminus\\acc{f}=\\acc{a,b,c,d,e}$.\n\nRecall that the core of a graph is the smallest subgraph which is also a homomorphic image.\nWe say that a subset $S$ of $V(T_6)$ is \\emph{good} if the core of the subgraph induced by $S$ is isomorphic\nto the graph $T_4$ which is a a clique on 4 vertices such that both the red and the blue edges induce a path of length $3$.\nWe use a series of constructions in order to eventually force a good set starting from $\\mathcal{S}$.\nWe thus reduce $T_6$-coloring to $T_4$-coloring, which is NP-complete for subcubic bipartite planar graphs with any given girth~\\cite{MO17}.\n\nThese constructions are summarized in the forest depicted in Figure~\\ref{fig:2edge}.\nThe vertices of this forest are the non-empty subsets of $\\acc{a,b,c,d,e}$ together with a few auxiliary sets of vertices containing $f$.\nIn this forest, an arc from $S_1$ to $S_2$ means that if we can force $S_1$, then we can force $S_2$. Every set with no outgoing arc is good.\nWe detail below the construction that is performed for each arc. In every case, we suppose that $S_1$ is forced on the vertex $v$ of a graph $H_1$\nand we construct a graph $H_2$ that forces $S_2$ on the vertex $w$.\n\n\\begin{figure}\n\\begin{center}\n \\includegraphics[scale=0.8]{2edge}\n \\caption{Forcing a good set.\\label{fig:2edge}}\n\\end{center}\n\\end{figure}\n\n\\begin{itemize}\n\\item Blue arcs: The set $S_2$ is the blue neighborhood of $S_1$ in $T_6$. We construct $H_2$ from $H_1$ by adding a vertex $w$ adjacent to $v$ such that $vw$ is blue.\nThus, $S_2$ is indeed forced on the vertex $w$ of $H_2$.\n\\item Red arcs: The set $S_2$ is the red neighborhood of $S_1$ in $T_6$. The construction is as above except that the edge $vw$ is red.\n\\item Dashed blue arcs: The set $S_2$ is the set of vertices incident to a blue edge contained in the subgraph induced by $S_1$ in $T_6$. We construct $H_2$ from two copies of\n$H_1$ by adding a blue edge between the vertex $v$ of one copy and the vertex $v$ of the other copy. Then $w$ is one of the vertices $v$.\n\\item Dashed red arcs: The set $S_2$ is the set of vertices incident to a red edge contained in the subgraph induced by $S_1$ in $T_6$.\nThe construction is as above except that the added edge is red.\n\\item Arc labelled \"X\": Let $g'=2\\ceil{g\/2}$. We consider an even cycle $v_1,\\cdots,v_{g'}$ such that $v_1v_{g'}$ is red and the other edges are blue.\nFor every vertex $v_i$, we take a copy of $H_1$ and we identify its vertex $v$ with $v_i$. We thus obtain the graph $H_2$ and we set $w=v_1$.\nLet $\\varphi$ be any $T_6$-coloring of $H_2$. In any $T_6$-coloring of $H_2$, the cycle $v_1,\\cdots,v_{g'}$ maps to a 4-cycle with exactly one red edge contained\nin the subgraph of $T_6$ induced by $S_1=\\acc{a,b,c,d,e}$. These 4-cycles are $aedb$ with red edge $ae$ and $cdba$ with red edge $cd$.\nSince $w$ is incident to the red edge in the cycle $v_1,\\cdots,v_{g'}$, $w$ can be mapped to $a$, $e$, $c$, or $d$ but not to $b$. Thus $S_2=\\acc{a,c,d,e}$.\n\\item Arc labelled \"Y\": We consider an alternating cycle $v_0,\\cdots,v_{8g-1}$. \nFor every vertex $v_i$, we take a copy of $H_1$ and we identify its vertex $v$ with $v_i$.\nWe obtain the graph $H_2$ by adding the vertex $x$ adjacent to $v_0$ and $v_{4g+2}$ such that $xv_0$ and $xv_{4g+2}$ are blue. We set $w=v_0$.\nIn any $T_6$-coloring $\\varphi$ of $H_2$, the cycle $v_1,\\cdots,v_{g'}$ maps to the alternating $4$-cycle $acde$ contained in $S_1=\\acc{a,c,d,e}$ such that $\\varphi(v_i)=\\varphi(v_{i+4\\pmod{8g}})$.\nSo, a priori, either $\\acc{\\varphi(v_0),\\varphi(v_{4g+2})}=\\acc{a,d}$ or $\\acc{\\varphi(v_0),\\varphi(v_{4g+2})}=\\acc{c,e}$.\nIn the former case, we can extend $\\varphi$ to $H_2$ by setting $\\varphi(x)=b$. In the latter case, we cannot color $x$ since $c$ and $e$ have no common blue neighbor in $T_6$.\nThus, $\\acc{\\varphi(v_0),\\varphi(v_{4g+2})}=\\acc{a,d}$ and $S_2=\\acc{a,d}$.\n\\end{itemize}\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Comparing results of \\nameA and \\nameB}\nIn this section, we provide results from our analysis of measurements generated\nby the \\nameA and \\nameB platforms. In particular, we use measurements from each\nplatform to (1) understand which countries are the least free -- \\ie have the\nhighest amounts of censorship and (2) to demonstrate challanges in finding\nground truth when conducting censorship measurements. We use these results\nto demonstrate how the \\nameA and \\nameB platforms can provide complementary\ninsights into censor behavior.\n\n\\subsection{Identifying the least free countries}\nUsing the tests described in Section \\ref{sec:platforms}, we now report the\ncountries found to be the least free based on measurements obtained from the\n\\nameA and \\nameB platforms.\n\n\\begin{figure}[ht]\n\\centering\n\\begin{subfigure}[h]{0.45\\textwidth}\n\\includegraphics[width=1\\textwidth]\n{plots\/iclab_overall_censorship_stats\/out_top_6_url_method_combs_no_null-crop.pdf}\n\\end{subfigure}\n\n\\begin{subfigure}[h]{0.45\\textwidth}\n\\includegraphics[width=1\\textwidth]\n{plots\/ooni_overall_censorship_stats\/top_6_censorship_with_errors-crop.pdf}\n\\end{subfigure}\n\n\\caption{The six most censored countries according to measurements from \\nameA\n(top) and \\nameB (bottom).}\n\\label{fig:iclab-ooni-least-free}\n\\end{figure}\n\nFigure \\ref{fig:iclab-ooni-least-free} illustrates the fraction of URLs that were\ncensored in each of the six least free countries -- Iran (IR), Saudi Arabia\n(SA), India (IN), Cyprus (CY), China (CN), and Russia (RU) -- based on tests\nconducted by the \\nameA platform; and Iran (IR), Saudi Arabia\n(SA), India (IN), Greece (GR), Qatar (QA), and Turkey (TR) according to \\nameB's\nmeasurements. We note that the remaining countries only displayed marginal\namounts of censorship -- \\ie under 5\\% of all tested URLs were censored.\nWe find that both platforms find the most censorship in Iran, Saudi Arabia,\nand India.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.5\\textwidth]\n{plots\/ooni-iran\/ooni-iran}\n\\caption{The number of domains seen blocked in Iran spiked in \\nameB measurement\ndata in fall of 2015, several months before the election.}\n\\label{fig:iranblocking}\n\\end{figure}\n\nIn the specific case of Iran, both the \\nameA and \\nameB platforms show\ncomparably high levels of censorship. We see that both platforms are able to\ndetect the large fraction of blockpages served by Iranian censors and that in\naddition to identifying the Iran blockpage, the TTL and RST anomaly detectors\nin the \\nameA platform are also triggered. We attribute the extremely high\nlevels of blocking observed to the fact that the measurements from both\nplatforms were carried out around the same time period as the Iranian\nparliamentary elections. We investigate further using past data from the \\nameB\nplatform and confirm that in October 2015, four months before the elections, a\nsharp rise in censorship was observed. This is illustrated in Figure\n\\ref{fig:iranblocking}. The URLs tested by the \\nameB platform during this time\nincluded content relating to political news and speech, social media,\ncensorship circumvention tools, and pornography.\n\nAnalyzing the results for Saudi Arabia and India we find that measurements\nperformed on the \\nameA platform see significantly less information controls\nthan on the \\nameB platform. In particular, while measurements from the \\nameA\nplatform detected a number of blockpages and RSTs in each of these countries, we\nfind that it did not encounter the large number of DNS anomalies and incomplete\nTCP connections that are observed by the \\nameB platform. We attribute this to\nthe fact that measurements from the \\nameB platform are usually obtained from\nresidential networks where covert censorship is observed (\\ie censorship without\nexplicitly serving a blockpage).\n\nFinally, the results from both platforms also provide an insight into the\ncensorship infrastructure in place in each country. The presence of a single\ndominant method of censorship in Iran and Saudi Arabia are indicative of the\npresence of a central censorship apparatus, while the case of India -- where\nmultiple equally dominant methods are observed -- is indicative of censorship\nbeing implemented by local ISPs rather than the central government.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{ |l|c|l|c| }\n \\hline\n URL & \\# of VPs &  URL & \\# of VPs \\\\\n \\hline\n battle.net    &        1459 & uol.com.br    &         842 \\\\\n 163.com       &        1417 & alibaba.com   &         748 \\\\\n baidu.com     &        1350 & yahoo.com     &         700 \\\\\n hao123.com    &        1333 & directrev.com &         564 \\\\\n youth.cn      &         918 & roblox.com    &         415 \\\\\n \\hline\n\\end{tabular}\n\\caption{Websites with the highest number of TCP \\texttt{RST} packets in \\nameA.}\n\\label{tab:rsts}\\end{center}\n\\end{table}\n\n\\subsection{The elusive ground truth}\nGround truth plays a crucial role in  analyzing censorship measurement data, and\nthere are several challenges associated with gathering ground-truth censorship data\nat scale. Comparing measurement data collected in the field against a baseline \ncollected in well-provisioned network settings (\\ie in the lab) helps delineate \ncensorship from server-side blocking caused by VPN blocking or automated measurements \nnot looking like real user traffic.\nTable \\ref{tab:rsts} shows websites with the highest number of TCP \\texttt{RST} \npackets in their streams across \\nameA's vantage points, pointing to possible \nserver-side blocking.\n\nAmong the list of websites with many observed \\texttt{RST}s are several\nwebsites hosted in China (\\eg \\texttt{163.com} and \\texttt{baidu.com}) that\nexhibit anomalous TCP behaviors when queried by \\nameA -- \\ie the IPID values \nfrom \\texttt{SYN} and \\texttt{SYN-ACK} packets are different from the rest of \nthe packets receieved, and sequence numbers overlapping between packets.\n\nThis anomaly is hard to distinguish from anomalous traits that are caused by the\nGreat Firewall of China. Similarly, gaming websites \\texttt{roblox.com} and \\texttt{battle.net}\naggressively block VPN users, while \\texttt{yahoo.com} and \\texttt{directrev.com} (an ad\nmarketplace website) do the same to a lesser degree. Other websites\ncan also respond unexpectedly (\\eg due to server misconfiguration) and\ntrigger false alarms. As an example, the Iranian retail\nwebsite \\texttt{digikala.com} shows sequence number anomalies as tested by 587 of\n\\nameA's vantage points, but is not censored in any of them.\n\n\\nameB faces a similar problem in determining ground truth. Many of the `control'\nmeasurements used by the local client to determine what sites should look like\nare conducted through Tor. In practice, many websites either fully deny, or\ndisplay substantially different content to visitors through the Tor network,\nmaking it difficult for the probe to determine if the local result is correct\nor not.\n\nAn additional challenge arises due to websites that are suddenly unavailable for\nnon-censorship reasons -- \\eg a dead website with a registered domain and\nunavailable webserver. For these cases, the \\nameA platform verifies if the\nwebpage could be loaded from any one of its other vantage points. If the page\nwas unable to be loaded successfully, it is discarded from the test outputs.\nFigure \\ref{fig:iclab-dead-sites} illustrates the URLs that were censored in the\n20 least free countries. We observe several vertical bands in this figure. These\nare indicative of dead websites, ones which could not be loaded from any vantage\npoint.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.5\\textwidth]\n{plots\/url_scatter\/t-top_20_url_vs_country-crop.pdf}\n\\caption{URLs censored in the 20 least free countries according to \\nameA.\nColors are indicative of the type of blocking observed (see Figure\n\\ref{fig:iclab-ooni-least-free}). The black vertical lines show websites\nthat are either no longer available or have blocked access to all of\nthe \\nameA vantage points.}\n\\label{fig:iclab-dead-sites}\n\\end{figure}\n\n\\section{Conclusions}\n\nIn this paper we presented the fundamental design decisions faced by\ndevelopers of large-scale longitudinal censorship measurement platforms --\nwhere to obtain vantage points, how to use these vantage points to collect\nmeasurements, what measurements to collect, and how to analyze them. We then\ndescribed the decisions that were made in development of the \\nameA and \\nameB\nplatform and their influence on the measurements obtained by these platforms.\n\nIn particular, we find that the \\nameA platform is able to provide a more\nreliable and global picture of censorship by harnessing dedicated global VPN\ninfrastructures in addition to on-the-ground volunteers. However, we also find\nthat this dependence on VPNs can result in measurements being carried out\non vantage points further away from residential networks which impacts the\nconclusions drawn from the platform. For example, the \\nameA platform sees\nsignificantly less censorship than the \\nameB platform in India and Saudi\nArabia. To this extent, it is important to work with representatives in affected\ncountries, and the responsive nature of \\nameB has been successful in gaining\nsupport to measure several important political events.\nThe challenge of obtaining representative, global, reliable, and response\nmeasurements remains a goal we continue to aspire to.\n\nIn addition, we showed how the results obtained from each of these platforms can\nbe used to provide a deeper insight into understanding regional censorship at a\nglobal scale. By analyzing the types of censorship observed in several countries\nwe were able to identify characteristics of the implemented censorship apparatus\n-- \\ie results obtained by both platforms suggest the presence of a\ndecentralized censorship infrastructure in India and a mostly centralized\ninfrastructure in Iran and Saudi Arabia.\n\nFinally, our current investigation also uncovers open challenges that remain in\nbeing able to distinguish censorship from anomalies that arise from phenomena\nsuch as misconfigured webservers, network outages, end-point discrimination,\nand unresponsive websites. Our platforms plan on addressing these limitations\nin future work.\n\n\n\\section{Design Decisions} \\label{sec:challenges}\n\nWe now discuss the choices of\n(1) where\nmeasurements are run,\n(2) how measurements are run,\nand\n(3) how data is intepreted .\nThese three design decisions are central to\nthe design of  a global censorship measurement platform.\n\n\\subsection{Where measurements are run} \nThere are two  options when considering\nvantage points: crowd-sourced and dedicated infrastructure vantage points.\nPlatforms using a crowd-sourced approach rely on volunteers running measurement\nsoftware. They have the ability to turn citizens in any location into vantage\npoints. Dedicated infrastructure, on the other hand are  \ndistributed and operated exclusively for the platform. Both approaches have their own\nbenefits and drawbacks.\n\n\\myparab{Cost and availability.} A hurdle in setting up a\ndedicated infrastructure is distributing infrastructure globally.\nHowever, once this infrastructure is in place, it \n has the capability of performing on-demand measurements; limited only\nby the reliability and uptime of the infrastructure. Crowd-sourced platforms, \non the other hand,\nincur no setup cost\nbut are dependent on the availability of volunteers to execute\nmeasurements. As a consequence,\ncrowd-sourced\nplatforms are unable to provide a reliable flow of measurements from a region.\n\n\\myparab{Representativeness and diversity of measurements.}\nCrowd-sourced platforms\nhave \nthe potential to obtain a view of the Internet from a wide variety of networks\n(\\eg residential, academic, and corporate).\nIn contrast, dedicated infrastructure faces an uphill battle of distributing  devices \n or may leverage existing infrastructure (\\eg academic networks, or dedicated\nhosting networks). \nVantage point location can \n impact conclusions drawn from their measurements. As an example,\nmeasurements conducted from the UK academic network (JANET) do not \nobserve the \n``Great Firewall of Cameron'' \\cite{GFC}, since they are placed\noutside of its purview. Crowd-sourced platforms can also leverage \npublic interest and news coverage \nto introduce additional vantage points.\n\n\\myparab{Safety and risk.} Information controls\nmeasurements using humans in the field poses a significant and hard to quantify risk. In\nmany regions (\\eg Syria) this risk has been determined to be too high for \nvolunteers. Such risks impede crowd-sourced measurements, while infrastructure \n(such as VPN or hosting networks) allow for\nmeasurements while posing little to no risk to users.\n\n\n\\subsection{Measurement autonomy} \nA censorship measurement platform\ncan either \nuse a \ncentral server to schedule experiments, or leave these tasks to each vantage point. This\ndichotomy has an impact on several capabilities of the platform.\n\n\\myparab{Time and context sensitive measurements.} The time and political\ncontext of  measurements are important for understanding evolving and\nabrupt policy changes. As an example, during the rise of ISIS in 2014, the\nIndian government blocked (and subsequently unblocked) access to 32 websites\nincluding GitHub, Vimeo, and PasteBin for propagating ``Anti India content''\n\\cite{ZDNet-IndiaDoT}. Centrally controlled measurement platforms have the\nadvantage of being able to evolve existing and schedule new measurements \n in response to changing political and social situations. Locally controlled\nplatforms, however, do not have this capability. Instead they are dependent on\nthe update schedule of the local vantage point.\n\n\n\\myparab{Infrastructure requirements.} The ability to remotely schedule \nexperiments and aggregate data centrally allows for the use of\ncomputationally constrained infrastructure, not needing technically savvy local\nmaintenance efforts. This comes with the cost of bandwidth requirements\nassociated with shipping  unprocessed  data to a central\nserver. Locally controlled platforms require local management of the platform\ninfrastructure to ensure up-to-date experiments, higher computational\ncapabilities for processing gathered data, and lower bandwidth for communicating\nprocessed results of measurements.\n\n\\subsection{Gathering and interpreting data}\n A censorship measurement\nplatform must specify data collected and how it will identify censorship \nin this\ndata.\n\n\\myparab{Type and quantity of data gathered.} A platform may record packet\ncaptures of entire tests or selectively gather data such as packet headers and\nresponses. While complete packet captures are ideal\nfor deep aposteriori analysis and to identify censorship not\nvisible at the application layer. However, they require root privileges, high storage \nand bandwidth \nrequirements, and may accidentally collect data of other system users.\n\n\\myparab{Identifying censorship events.} Another challenge that arises\nduring the processing of gathered data is defining when ``censorship'' has occurred. \nThis task is complicated by \nstrange protocol implementations (\\eg  load\nbalancers that cause gaps in TCP sequence numbers), server side blocking~\\cite{torndss16}, and \nregular network failures.\n\n\\section{Introduction} \\label{sec:introduction}\n\nThe last five years have cemented the Internet as critical infrastructure for\ncommunication. In particular, it has demonstrated high\nutility for citizens and political activists to obtain accurate information,\norganize political movements, and express dissent. This fact has not gone\nunnoticed, with governments clamping down on this medium \\via censorship and\ninformation controls.\nConsequently, there has been a surge of interest in measuring various aspects\nof online information controls. More specifically, data obtained\nfrom such measurements has been used by (1) political activists to understand\nthe motivation for and the impact of such government policies \nand\n(2) researchers to build safer and more secure censorship circumvention tools\nby understanding the techniques used to implement these policies \\cite{tor-pt}.\n\nWhile there have been numerous efforts to characterize\nonline information controls \\cite{Chaabance-IMC14, censmon, Roberts2011a,\nWright2011a, Aryan2013, Aceto2015a}, the data gathered or\nused by these measurements have limited scope due to the specificity of\nlocations and time-periods considered. In order to gain a nuanced understanding\nof the evolution of Internet censorship, in terms of policy and techniques, a\nmeasurement platform needs to be able to gather longitudinal data\nfrom a diverse set of regions while performing accurate analysis using robust \nand well specified techniques. We present and compare two such platforms --\n\\nameA and \\nameB -- that represent different points in the censorship\nmeasurement design space.\n\n\nIn this paper, we first identify three primary design decisions made in \nthe development of censorship measurement platforms. Then, we describe how \n\\nameA and \\nameB address these decisions, \nwhile\nconsidering the impact of these decisions on the measurement results produced by the systems. \n Finally, we show how \\nameA and \\nameB, when used together, provide a unique \n  insight into the current state of information\ncontrols around the globe.\n\n\n\\section{The \\nameA and \\nameB Platforms}\\label{sec:platforms}\n\nIn this section we describe the design decisions made during the development of\nthe \\nameA and \\nameB platforms.\n\n\\subsection{Vantage points}\nThe most fundamental difference between the \\nameA and \\nameB platforms is the \napproach each system takes to recruit vantage points. \n\\nameA relies on a dedicated infrastructure to perform measurements.\nThis allows measurements that require permissions that may not be compatible with \nsoftware to be run on end-user systems. As a consequence, the system has thus far \nfocused on deployment on VPN vantage points and a limited deployment of \n Raspberry Pi's installed with \\nameA software.\nIn contrast, \\nameB takes a lighter-weight software-based approach. \nand assumes some amount of technical savvy on the \npart of volunteers.\nThis leads to differences in the\navailability and representativeness of measurements from each platform.\n In Figure \\ref{fig:choropleth}, we see that as\na result of the decision to use VPN end points, \\nameA is\nable to provide vantage points for measurements in significantly more countries\nthan \\nameB (151 for \\nameA and 46 for \\nameB in the last 100\ndays\\footnote{Since its release in 2012, \\nameB has received nearly 10M\nmeasurements from volunteers in 95 countries.}). However, we found that that \\nameB's crowd-sourced model is\nable to provide more AS-level diversity -- \\ie \\nameB provides vantage points\nfrom an average of 3.15 different networks (ASes) in each country, compared to\n\\nameA's average of 1.46 networks per country. \n\n\n\n\\begin{figure}[ht]\n\\centering\n\\begin{subfigure}[h]{0.5\\textwidth}\n\\includegraphics[trim=0cm 0.125cm 0cm 0cm, clip=true,width=\\textwidth]\n{figures\/combined-choro-sol.pdf}\n\\caption{Global availability of measurements.}\n\\label{fig:choropleth}\n\\end{subfigure}\n\\begin{subfigure}[h]{0.5\\textwidth}\n\\includegraphics[trim=0cm 0.125cm 0cm 0cm, clip=true,width=\\textwidth]\n{figures\/country-availability.pdf}\n\\caption{Country-level temporal availability. Green and red indicate \nthe availability of measurement data from \\nameA and \\nameB, respectively. %\nBlack indicates the availability of measurements from\nboth platforms from the same region on the given day.}\n\\label{fig:country-availability-heat}\n\\end{subfigure}\n\\caption{{Availability of measurements from the \\nameA and \\nameB platforms in\nthe last 100 days.}}\n\\label{fig:vantage-points}\n\\end{figure}\n\n\n\n\\subsection{System architecture}\nIn terms of system architectures, \\nameA uses a central controller to schedule\nexperiments while \\nameB processes data both on the vantage point and inside of\nits data processing pipeline.\nThis introduces several key differences in how each platform\nhandles its vantage points.\n\n\\myparab{Measurement scheduling.}\n\\nameA takes a centralized approach to scheduling experiments, leveraging a single \nserver that is able to schedule experiments on all deployed nodes (\\eg VPNs, Raspberry Pis) or \na subset thereof (e.g, a given country). This facilitates the execution of ongoing or one-off measurements.\nIn contrast, \\nameB takes a decentralized approach. Recommended measurements\nare hard-coded into the \\nameB platform source-code and require vantage points\n(technically savvy volunteers) to regularly download updates in order to execute\nnew measurements. Repetition of measurements is dependent on individual\nvolunteer availability. Volunteers\nalso have the option to add their own tests and modify inputs to existing tests\n(\\eg they may change the set of URLs being used by a test).\n\n\\myparab{Performing measurements.}\n\\nameA and \\nameB also differ in their approach to performing measurements. \n\\nameA takes a ``simple node'' approach, with nodes largely being responsible for collecting \ndata and transmitting it back to the central server for later analysis. This lowers the \ncomputational requirement of the vantage points but increased demands on bandwidth. \nIn contrast, \\nameB performs measurements and analysis on the device and ships \nprocessed data back to a central server. Importantly, \\nameB allows volunteers to\nopt-out of submitting measurement reports to the \\nameB publishing server, while \\nameA \ntakes an informed approach with vantage points opting into participate in the system.\n\nFigure \\ref{fig:country-availability-heat} shows the impacts of these decisions on the platforms.\n\\nameB has a core set of vantage points that continuously measure and a few opportunistic \nmeasurements. \\nameA on the other hand exhibits large coordinated testing as a result \nof its VPN vantage points. \n\n\\subsection{Tests and analysis}\n\nBoth platforms perform a battery of tests to identify censors that may be\nblocking or manipulating content. The \\nameA test infrastructure is extensible,\nallowing new tests to be scheduled on vantage points without the need for\nupdating their software. In addition to custom tests, the \\nameA platform\nperiodically schedules a baseline test on each vantage point. This baseline\nexperiment tests connectivity to a set of URLs that are composed of the Alexa\nTop 500 websites and a country-specific list of potentially blocked URLs\n(obtained from the CitizenLab). In contrast, tests on the \\nameB platform are\nnot scheduled remotely and new tests need to be obtained by software updates.\nExisting tests, however, do not require software updates to evolve the list of\ndomains that they test connectivity to. The default experiments included in the\n\\nameB platform test connectivity to the global and country-specific lists of\npotentially blocked URLs (also obtained from the CitizenLab).\n\nIn terms of analysis, the \\nameA platform does not perform analysis on the\nvantage points, rather it leaves all post-processing to the centralized servers.\nThis allows \\nameA to perform retroactive analysis on existing results. The\n\\nameB platform, on the other hand, performs data analysis on the vantage\npoints. This allows independent and private deployments by in-country watchdog\ngroups. \n\nWe now briefly describe the tests conducted to identify censorship by each\nplatform. \n\n\\myparab{DNS anomaly detection.}\nFor each URL to be tested on a given vantage point, the \\nameA and \\nameB vantage\npoints perform DNS name resolution queries for the domain name associated with\nthat URL using both the default DNS resolver configured on the machine as well\nas Google's DNS at \\texttt{8.8.8.8}. The \\nameA platform concludes that an\nanomaly (\\eg DNS injection, tampering, \\etc) has occurred if a second DNS\nresponse is received within 2 seconds of the first. The \\nameB platform on\nthe other hand, makes several requests at once and does not wait between\nrequests. Requests are also made to control resolver that binds to a non\nstandard DNS port. The client is able to report failures to resolve directly,\nand resolutions are included in the generated report to allow further analysis\nby the central analysis infrastructure.\n\n\\myparab{HTTP tampering, proxy, and blockpage detection.}\nFor each URL to be tested on a given vantage point, the \\nameA and \\nameB\nvantage points issue HTTP GET requests and record received responses, with\n\\nameA to follow redirects. The responses received from these tests are\nprocessed to identify blockpages and evidence of HTTP tampering. The \\nameA\nplatform uses regular expression pattern matching to identify known blockpages\nand responses obtained by the same test executed from a censor-free vantage\npoint in the US to identify instances of content manipulation. The \\nameB\nplatform uses meta-data (\\eg status codes, response sizes, \\etc) obtained from a\nTor control channel to identify HTTP tampering. Additionally, the \\nameB\nplatform is also able to detect the presence of HTTP proxies. It does this by\ngenerating malformed HTTP requests that cause proxies on the vantage point\nnetwork to reveal their presence (\\eg by modifying the malformed headers). The\ndata processing pipeline is then capable of identifying specific types of proxy\nsoftware based on known fingerprints.\n\n\\myparab{TLS man-in-the-middle detection.}\nThe \\nameA platform also performs tests on HTTPS compatible URLs. For each such\nURL, a TLS handshake is performed and all server certificates that are received\nare checked for validity. If they are found to have been expired or signed by an\nuntrusted certificate authority, a TLS anomaly is reported.\n\n\\myparab{Sequence number, TTL, and RST anomaly detection.}\nFor each of the above tests, the \\nameA platform analyzes raw data (packet\ncaptures) of TCP streams to identify inconsistent sequence number and TTL values\nin packet headers. Additionally, the presence of pre-mature RST packets is also\nrecorded. If any of these are identified, the \\nameA platform reports anomalies\nthat may be the result of a censor injecting packets into a TCP stream. \n\n\\myparab{TCP connectivity test.}\nThe \\nameB platform attempts to establish TCP connections to a specified set of\nhosts to validate that the handshake can be completed and detect instances of\nIP level blocking. In this test, the vantage point attempts to establish a\nconnection to the end host directly and also via a Tor control channel. If the\ncontrol channel succeeds, while the direct connection fails, an anomaly is\nreported.\n\n\\myparab{Circumvention protocol tests.}\nFinally, the \\nameB platform also includes a set of tests designed to detect the\navailability of several circumvention system by (1) mimicking the protocols\ninvolved and (2) by launching bundled instances of the actual tools and checking\nwhether they are able to successfully complete connections. Currently the test\nconsiders the connectivity of all Tor pluggable transports (scramblesuit, meek,\nfteproxy, obfsproxy versions 2 to 4), Psiphon, Lantern, and the OpenVPN protocol.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\nScalar mesons are still a puzzle in light meson spectroscopy: they have complex structure,\nand there are too many states to be accommodated within the quark model without difficulty~\\cite{polosa}. \nIn particular, the structure of the $I=1\/2$  $K \\pi$ $\\mathcal{S}$-wave is a longstanding problem. In recent years many experiments have performed \naccurate studies of the decays of heavy-flavored hadrons producing a $K \\pi$ system in the final state.\nThese studies include searches for \\ensuremath{C\\!P}\\xspace violation~\\cite{cp}, and searches for, and observation of, new exotic resonances~\\cite{zs} and\ncharmed mesons~\\cite{bs}.\nHowever, the still poorly known structure of \nthe  $I=1\/2$  $K \\pi$ $\\mathcal{S}$-wave is a source of large systematic uncertainties.\nThe best source of information on the scalar structure of the $K \\pi$ system comes from the LASS experiment, which studied the reaction $\\mbox{${K^{-}}$} p \\mbox{$\\rightarrow$} \\mbox{${K^{-}}$} \\mbox{${\\pi^{+}}$} n$~\\cite{lass_kpi}.\nPartial wave analysis of the $K \\pi$ system reveals a large contribution from the $I=1\/2$  $K \\pi$ $\\mathcal{S}$-wave amplitude over the mass\nrange studied.\nIn the description of the $I=1\/2$  scalar amplitude up to a $K \\pi$ mass of about 1.5 \\mbox{${\\mathrm{GeV}}\/c^2$}\\ the $K^*_0(1430)$ resonant amplitude is added coherently to an effective-range\ndescription in such a way that the net amplitude\nactually decreases rapidly at the resonance mass. The $I=1\/2$  $\\mathcal{S}$-wave amplitude representation is given explicitly in Ref.~\\cite{babar_z}.\nIn the LASS analysis, in the region above 1.82 \\mbox{${\\mathrm{GeV}}\/c^2$}, the $\\mathcal{S}$-wave suffers from a two-fold ambiguity, but in both solutions it is understood in terms of the presence of a $K^*_0(1950)$ resonance. It should be noted that the extraction of the $I=1\/2$ $\\mathcal{S}$-wave amplitude is complicated by the presence of an $I=3\/2$ contribution. \n\nFurther information on the $K \\pi$ system has been extracted from Dalitz plot analysis of the decay $D^+ \\mbox{$\\rightarrow$} \\mbox{${K^{-}}$} \\mbox{${\\pi^{+}}$} \\mbox{${\\pi^{+}}$}$ where, in order to fit \nthe data, the presence of an additional resonance, the $\\kappa(800)$, was claimed~\\cite{aitala}. Using the same data, a Model Independent Partial Wave Analysis (MIPWA)\nof the $K \\pi$ system was developed for the first time~\\cite{aitala1}.\nThis method allows the amplitude and phase of the $K \\pi$ $\\mathcal{S}$-wave to be extracted as\n functions of mass (see also Refs.~\\cite{cleo} and ~\\cite{focus}). However in these analyses the phase space is limited to mass values less than 1.6 \\mbox{${\\mathrm{GeV}}\/c^2$}\\ due to the kinematical limit imposed by the $D^+$ mass.\nA similar method has been used to extract the $\\mbox{${\\pi^{+}}$} \\mbox{${\\pi^{-}}$}$ $\\mathcal{S}$-wave amplitude in a Dalitz plot analysis of $D^+_s \\mbox{$\\rightarrow$} \\mbox{${\\pi^{+}}$} \\mbox{${\\pi^{-}}$} \\mbox{${\\pi^{+}}$}$~\\cite{marco}.\n\nIn the present analysis, we consider three-body \\ensuremath{\\eta_c}\\xspace decays to $K \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace \\pi$ and obtain new information on the $K \\pi$ $I=1\/2$  $\\mathcal{S}$-wave amplitude extending up to a mass of 2.5 \\mbox{${\\mathrm{GeV}}\/c^2$}. We emphasize that, due to isospin conservation in the \\ensuremath{\\eta_c}\\xspace hadronic decay to $(K \\pi) \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$,\nthe $(K \\pi)$ amplitude must have $I=1\/2$ , and there is no $I=3\/2$ contribution.\nThe {\\em B}{\\footnotesize\\em A}{\\em B}{\\footnotesize\\em AR}\\ experiment first performed a Dalitz plot analysis of $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$} \\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}$ and $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$} \\mbox{${K^{-}}$} \\eta$ using an isobar model~\\cite{etakk}. The analysis reported the first observation of $K^*_0(1430) \\mbox{$\\rightarrow$} K \\eta$, and observed that \\ensuremath{\\eta_c}\\xspace decays to three pseudoscalars are dominated by intermediate\nscalar mesons. A previous search for charmonium resonances decaying to \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace in two-photon interactions is reported in Ref.~\\cite{kkpipipi0}. \nWe continue these studies of \\ensuremath{\\eta_c}\\xspace decays and extract the $K \\pi$ $\\mathcal{S}$-wave amplitude by performing a MIPWA of both \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace and \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace final states. \n\nWe describe herein studies of the $K \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace \\pi$ system produced in two-photon interactions.\nTwo-photon events in which at least one of the interacting photons is not quasi-real are\nstrongly suppressed by the selection \ncriteria described below. This implies that the allowed $J^{PC}$ values of\nany produced resonances are $0^{\\pm+}$, $2^{\\pm+}$, $3^{++}$, $4^{\\pm+}$...~\\cite{Yang}.       \nAngular momentum conservation, parity conservation, and charge conjugation\ninvariance imply that these quantum numbers also apply to\nthe final state except that the $K \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace \\pi$ state cannot be in a $J^P=0^+$ state.\n\nThis article is organized as follows. In Sec.\\ II, a brief description of the\n{\\em B}{\\footnotesize\\em A}{\\em B}{\\footnotesize\\em AR}\\ detector is given. Section III is devoted to the event reconstruction and data selection of the \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace system. In Sec.\\ IV, we describe studies of efficiency and resolution,\nwhile in Sec.\\ V we describe the MIPWA. In Secs. VI and VII we perform Dalitz plot analyses of \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace and \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace decays. Section VIII is devoted to discussion of the measured $K \\pi$ $\\mathcal{S}$-wave amplitude, and finally results are summarized in Sec.~IX.\n\n\\section{The {\\em B}{\\footnotesize\\em A}{\\em B}{\\footnotesize\\em AR}\\ detector and dataset}\n\nThe results presented here are based on data collected\nwith the {\\em B}{\\footnotesize\\em A}{\\em B}{\\footnotesize\\em AR}\\ detector\nat the PEP-II asymmetric-energy $e^+e^-$ collider\nlocated at SLAC, and correspond \nto an integrated luminosity of 519~\\mbox{${\\mathrm{fb}^{-1}}$}~\\cite{luminosity} recorded at\ncenter-of-mass energies at and near the $\\Upsilon (nS)$ ($n=2,3,4$)\nresonances. \nThe {\\em B}{\\footnotesize\\em A}{\\em B}{\\footnotesize\\em AR}\\ detector is described in detail elsewhere~\\cite{BABARNIM}.\nCharged particles are detected, and their\nmomenta are measured, by means of a five-layer, double-sided microstrip detector,\nand a 40-layer drift chamber, both operating  in the 1.5~T magnetic \nfield of a superconducting\nsolenoid. \nPhotons are measured and electrons are identified in a CsI(Tl) crystal\nelectromagnetic calorimeter. Charged-particle\nidentification is provided by the measurement of specific energy loss in\nthe tracking devices, and by an internally reflecting, ring-imaging\nCherenkov detector. Muons and \\KL\\ mesons are detected in the\ninstrumented flux return  of the magnet.\nMonte Carlo (MC) simulated events~\\cite{geant}, with reconstructed sample sizes \nmore than 10  times larger than the corresponding data samples, are\nused to evaluate the signal efficiency and to determine background features. \nTwo-photon events are simulated  using the GamGam MC\ngenerator~\\cite{BabarZ}.\n\n\\section{ {\\boldmath Reconstruction and selection of $\\protect \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$} events}\n\nTo study the reaction\n\\begin{equation}\n\\gamma \\gamma \\mbox{$\\rightarrow$} \\KS \\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}\n\\end{equation}\nwe select events in which the $e^+$ and $e^-$  beam particles are scattered  \nat small angles, and hence are undetected in the final state. \nWe consider only events for which the number of well-measured charged-particle tracks with\ntransverse momentum greater than 0.1~\\mbox{${\\mathrm{GeV}}\/c$}\\ is exactly equal to four, and for which there are no more than five photon candidates\nwith reconstructed energy in the electromagnetic calorimeter greater than 100 MeV.\nWe obtain $K^0_S \\mbox{$\\rightarrow$} \\mbox{${\\pi^{+}}$} \\mbox{${\\pi^{-}}$}$ candidates by means of a vertex fit of pairs of oppositely charged tracks which\nrequires a $\\chi^2$ fit probability greater than 0.001. Each \\KS candidate is then combined with \ntwo oppositely charged tracks, and fitted to a common vertex, with the requirements that the fitted vertex be within the\n$e^+ e^-$ interaction region and have a $\\chi^2$ fit probability greater than 0.001.\nWe select kaons and pions by applying high-efficiency particle identification criteria.\nWe do not apply any particle identification requirements\nto the pions from the \\KS decay.\nWe accept only $K_S^0$ candidates with decay lengths from the main vertex of the event greater than 0.2 cm, and\nrequire $\\cos \\theta_{\\KS}>0.98$, where $\\theta_{\\KS}$ is defined as the angle between the \\KS momentum direction and the\nline joining the primary and the \\KS vertex.\nA fit to the $\\mbox{${\\pi^{+}}$} \\mbox{${\\pi^{-}}$}$ mass spectrum using a linear function for the background and a Gaussian\nfunction with mean $m$ and width $\\sigma$ gives $m=497.24$ \\mbox{${\\mathrm{MeV}}\/c^2$}\\ and $\\sigma=2.9$ \\mbox{${\\mathrm{MeV}}\/c^2$}. We select the $\\KS$ signal region to be within\n$\\pm 2 \\sigma$ of $m$ and reconstruct the \\KS 4-vector by adding the three-momenta of the pions and computing the energy using the \\KS PDG mass value~\\cite{pdg}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=9cm]{fig1.eps}\n\\caption{Distributions of \\mbox{$p_T$}\\xspace\\ for $\\gamma \\gamma \\mbox{$\\rightarrow$} \\KS \\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}$.  The data are shown as (black) points with error bars,\nand the signal MC simulation as a (red) histogram; the vertical dashed line indicates the selection applied to select two-photon events.}\n\\label{fig:fig1}\n\\end{center}\n\\end{figure}\nBackground arises mainly from random combinations of particles from\n\\ensuremath{e^+e^-}\\xspace\\ annihilation, from other two-photon processes, and from events with initial-state photon radiation (ISR). The ISR \nbackground is dominated by $J^{PC}=1^{--}$ resonance production~\\cite{isr}.\nWe discriminate against \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace events produced via ISR by requiring $\\ensuremath{{\\rm \\,mm}}\\xspace\\equiv(p_{\\ensuremath{e^+e^-}\\xspace}-p_{\\mathrm{rec}})^2>10$~GeV$^2$\/$c^4$, where\n$p_{\\ensuremath{e^+e^-}\\xspace}$ is the four-momentum of the initial state and $p_{\\mathrm{rec}}$ is the four-momentum of the \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace system. \n\nThe \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace mass spectrum shows a prominent \\ensuremath{\\eta_c}\\xspace signal.\nWe define \\mbox{$p_T$}\\xspace\\ as the magnitude of the vector sum of the transverse momenta, in the \\ensuremath{e^+e^-}\\xspace\\ rest frame, of the final-state particles with respect to the beam axis.\nSince well-reconstructed two-photon events are expected to have low values of \\mbox{$p_T$}\\xspace, we optimize the selection as a function of this variable.\nWe produce $\\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ mass spectra with different \\mbox{$p_T$}\\xspace selections and fit the mass spectra to extract the number of \\ensuremath{\\eta_c}\\xspace signal events ($N_s$) and the number\nof background events below the \\ensuremath{\\eta_c}\\xspace signal ($N_b$). We then compute the purity, defined as $P = N_s\/(N_s + N_b)$, and the significance $S = N_s\/\\sqrt{N_s + N_b}$. To obtain the best significance with the highest purity, \nwe optimize the selection by requiring  the maximum value of the product of purity and significance, $P \\cdot S$, and find that this corresponds to the requirement $\\mbox{$p_T$}\\xspace<0.08~\\mbox{${\\mathrm{GeV}}\/c$}$.\n\nFigure~\\ref{fig:fig1} shows the  measured \\mbox{$p_T$}\\xspace\\ distribution in comparison to the corresponding  \\mbox{$p_T$}\\xspace\\ distribution obtained from simulation of the signal process.\nA peak at low \\mbox{$p_T$}\\xspace\\ is observed indicating\nthe presence of the two-photon process. The shape of the peak agrees well with that seen in the MC simulation. \nFigure~\\ref{fig:fig2} shows the $\\KS \\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}$ mass spectrum in the \\ensuremath{\\eta_c}\\xspace mass region. A clear \\ensuremath{\\eta_c}\\xspace signal over a background of about 35\\% can be seen, together with a residual \\ensuremath{{J\\mskip -3mu\/\\mskip -2mu\\psi\\mskip 2mu}}\\xspace signal. Information on the fitting procedure is given at the end of Sec. IV.\nWe define the \\ensuremath{\\eta_c}\\xspace signal region as the range 2.922-3.039~\\mbox{${\\mathrm{GeV}}\/c^2$}\\ ($m(\\ensuremath{\\eta_c}\\xspace) \\pm 1.5 \\ \\Gamma$), which contains 12849 events with a purity of\n$(64.3 \\pm 0.4)$\\% . \nSideband regions are defined by the ranges 2.785-2.844~\\mbox{${\\mathrm{GeV}}\/c^2$} \\ and 3.117-3.175~\\mbox{${\\mathrm{GeV}}\/c^2$}\\ (within 3.5-5 $\\Gamma$), respectively as indicated (shaded) in\nFig.~\\ref{fig:fig2}.\n\nDetails on data selection, event reconstruction, resolution, and efficiency measurement for the \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace decay can be found in Ref.~\\cite{etakk}.\nThe \\ensuremath{\\eta_c}\\xspace signal region for this decay mode contains 6710 events with a purity of $(55.2 \\pm 0.6)$\\%. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=9cm]{fig2.eps}\n\\caption{The $\\KS \\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}$ mass spectrum in the \\ensuremath{\\eta_c}\\xspace mass region after requiring $\\mbox{$p_T$}\\xspace<0.08~\\mbox{${\\mathrm{GeV}}\/c$}$. The solid curve shows the total fitted function,\nand the dashed curve shows the fitted background contribution. The shaded areas show signal and sideband regions.}\n\\label{fig:fig2}\n\\end{center}\n\\end{figure}\n\n\\section{Efficiency and resolution}\n\nTo compute the efficiency, MC signal events are generated using a detailed detector simulation~\\cite{geant} in which the \\ensuremath{\\eta_c}\\xspace decays uniformly in phase space.\nThese simulated events are reconstructed and analyzed in the same manner as data. The efficiency is computed as the ratio of \nreconstructed to generated events. \nDue to the presence of long tails in the Breit-Wigner (BW) representation of the resonance, we apply \nselection criteria to restrict the generated events to the \\ensuremath{\\eta_c}\\xspace mass region. \nWe express the efficiency as a function of the invariant mass, $m(\\mbox{${K^{+}}$} \\mbox{${\\pi^{-}}$})$~\\cite{conj}, and $\\cos \\theta$, where $\\theta$ is the angle, in the $\\mbox{${K^{+}}$} \\mbox{${\\pi^{-}}$}$ \nrest frame, between the directions of the \\mbox{${K^{+}}$}\\ and the boost from the $\\KS \\mbox{${K^{+}}$} \\mbox{${\\pi^{-}}$}$ rest frame.\n\nTo smooth statistical fluctuations, this efficiency map is parametrized as follows.\nFirst we fit the efficiency as a function of  $\\cos \\theta$ in separate intervals of $m(\\mbox{${K^{+}}$} \\mbox{${\\pi^{-}}$})$, using \nLegendre polynomials up to $L=12$:\n\\begin{equation}\n\\epsilon(\\cos\\theta) = \\sum_{L=0}^{12} a_L(m) Y^0_L(\\cos\\theta),\n\\end{equation}\nwhere $m$ denotes the $\\mbox{${K^{+}}$} \\mbox{${\\pi^{-}}$}$ invariant mass.\nFor each value of $L$, we fit the mass dependent coefficients $a_L(m)$ with a seventh-order polynomial in $m$.\nFigure~\\ref{fig:fig3} shows the resulting fitted efficiency map $\\epsilon(m,\\cos \\theta)$.\nWe obtain $\\chi^2\/N_{\\rm cells}=217\/300$ for this fit, where $N_{\\rm cells}$ is the number of cells in the efficiency map.\nWe observe a significant decrease in\nefficiency in regions of $\\cos\\theta \\sim \\pm 1$ due to the impossibility of reconstructing $\\ensuremath{K^\\pm}\\xspace$ mesons with laboratory momentum\nless than about 200~\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c}}\\xspace, and \\mbox{${\\pi^{\\pm}}$} and $\\KS(\\mbox{$\\rightarrow$} \\mbox{${\\pi^{+}}$} \\mbox{${\\pi^{-}}$})$ mesons with laboratory momentum less than about 100~\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c}}\\xspace (see Fig. 9 of Ref.~\\cite{babar_z}). These effects result from energy loss in the beampipe and inner-detector material.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=9cm]{fig3.eps}\n\\caption{Fitted detection efficiency in the $\\cos \\theta \\  vs. \\ m(\\mbox{${K^{+}}$} \\mbox{${\\pi^{-}}$})$ plane. Each interval shows the average value of the fit for that region.}\n\\label{fig:fig3}\n\\end{center}\n\\end{figure}\n\nThe mass resolution, $\\Delta m$, is measured as the difference between the generated and reconstructed \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace invariant-mass values.\nThe distribution has a root-mean-squared value of 10 \\mbox{${\\mathrm{MeV}}\/c^2$}, and is parameterized by the sum of a Crystal Ball~\\cite{cb} and a Gaussian function. \nWe perform a binned fit to the \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace mass spectrum in data using the following model. The background is described by a second-order polynomial, and the \\ensuremath{\\eta_c}\\xspace resonance is represented by a nonrelativistic BW function convolved with the resolution function. \nIn addition, we allow for the presence of a residual \\ensuremath{{J\\mskip -3mu\/\\mskip -2mu\\psi\\mskip 2mu}}\\xspace contribution modeled with a Gaussian function. Its parameter values are fixed to those obtained from a fit to the \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace mass spectrum for the ISR data sample obtained by requiring\n$\\lvert \\ensuremath{{\\rm \\,mm}}\\xspace \\rvert<1 \\ {\\rm GeV}^2\/c^4$.\nThe fitted \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace mass spectrum is shown in Fig.~\\ref{fig:fig2}. We obtain the following \\ensuremath{\\eta_c}\\xspace parameters:\n\\begin{equation}\n  \\begin{split}\n    m=2980.8 \\pm 0.4 \\ \\mbox{${\\mathrm{MeV}}\/c^2$}, \\ \\Gamma=33 \\pm 1 \\ \\mbox{${\\mathrm{MeV}}$},\\\\\n    \\ N_{\\ensuremath{\\eta_c}\\xspace}=9808 \\pm 164,\n    \\end{split}\n\\end{equation}\nwhere uncertainties are statistical only. Our measured mass value is 2.8 \\mbox{${\\mathrm{MeV}}\/c^2$}\\ lower than the world average~\\cite{pdg}.\nThis may be due to interference between the \\ensuremath{\\eta_c}\\xspace amplitude and that describing the background in the signal region~\\cite{bes_int}.\n\n\\section{Model Independent Partial Wave Analysis}\n\nWe perform independent MIPWA of the \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace and \\ensuremath{\\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace Dalitz plots in the \\ensuremath{\\eta_c}\\xspace mass region using  unbinned maximum likelihood fits.\nThe likelihood function is written as\n\\begin{eqnarray}\n\\mathcal{L} = \\nonumber\\\\     \n \\prod_{n=1}^N&\\bigg[&f_{\\rm sig}(m_n) \\epsilon(x'_n,y'_n)\\frac{\\sum_{i,j} c_i c_j^* A_i(x_n,y_n) A_j^*(x_n,y_n)}{\\sum_{i,j} c_i c_j^* I_{A_i A_j^*}} \\nonumber\\\\\n& &+(1-f_{\\rm sig}(m_n))\\frac{\\sum_{i} k_iB_i(x_n,y_n,m_n)}{\\sum_{i} k_iI_{B_i}}\\bigg]\n\\end{eqnarray}\n\\noindent where\n\\begin{itemize}\n\\item $N$ is the number of events in the signal region;\n\\item for the $n$-th event, $m_n$ is the \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace or the \\ensuremath{\\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace invariant mass;\n\\item for the $n$-th event, $x_n=m^2(K^+ \\mbox{${\\pi^{-}}$})$, $y_n=m^2(\\KS \\mbox{${\\pi^{-}}$})$ for \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace; $x_n=m^2(K^+ \\mbox{${\\pi^{0}}$})$, $y_n=m^2(K^- \\mbox{${\\pi^{0}}$})$ for $\\ensuremath{\\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace$; \n\\item $f_{\\rm sig}$ is the mass-dependent fraction of signal obtained from the fit to the \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace or \\ensuremath{\\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace mass spectrum;\n\\item for the $n$-th event, $\\epsilon(x'_n,y'_n)$ is the efficiency parametrized as a function of $x'_n=m(\\mbox{${K^{+}}$} \\mbox{${\\pi^{-}}$})$ for \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace and $x'_n=m(\\mbox{${K^{+}}$} \\mbox{${K^{-}}$})$ for \\ensuremath{\\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace, and $y'_n=\\cos \\theta$ (see Sec. IV);\n\\item for the $n$-th event, the $A_i(x_n,y_n)$ describe the complex signal-amplitude contributions;\n\\item $c_i$ is the complex amplitude for the $i$-th signal component; the $c_i$ parameters are allowed to vary during the fit process;\n\\item for the $n$-th event, the $B_i(x_n,y_n)$ describe the background probability-density functions assuming that interference between signal and background amplitudes can be ignored;\n\\item $k_i$ is the magnitude of the $i$-th background component; the $k_i$ parameters are obtained by fitting the sideband regions;\n\\item $I_{A_i A_j^*}=\\int A_i (x,y)A_j^*(x,y) \\epsilon(x', y')\\ {\\rm d}x{\\rm d}y$ and \n$I_{B_i}~=~\\int B_i(x,y) {\\rm d}x{\\rm d}y$ are normalization\n  integrals. Numerical integration is performed on phase space generated events with \\ensuremath{\\eta_c}\\xspace signal and background generated according to the experimental distributions. In case of MIPWA or when resonances have free parameters, integrals are re-computed at each minimization step.\n  Background integrals and fits dealing with amplitudes having fixed resonance parameters are computed only once. \n\\end{itemize}\nAmplitudes are described along the lines described in Ref.~\\cite{kopp}.\nFor an \\ensuremath{\\eta_c}\\xspace meson decaying into three pseudoscalar mesons via an intermediate\n  resonance $r$ of spin $J$ (i.e. $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} C r$, $r \\mbox{$\\rightarrow$} A B$), each amplitude\n  $A_i(x,y)$ is represented by the product of a complex Breit-Wigner (BW)\n  function and a real angular distribution function represented by\n  the spherical harmonic function $\\sqrt{2 \\pi} Y_J^0({\\rm cos} \\theta)$; $\\theta$\n  is the angle between the direction of $A$, in the rest frame of $r$,\n  and the direction of $C$ in the same frame. This form of the angular\n  dependence results from angular momentum conservation in the rest\n  frame of the \\ensuremath{\\eta_c}\\xspace, which leads to the production of $r$ with helicity 0.\n\n  It follows that\n\\begin{equation}\n    A_i(x,y) = BW(M_{AB}) \\sqrt{2 \\pi}  Y_J^0({\\rm cos} \\theta).\n\\label{eq:spin}\n\\end{equation}\n\n  The function $BW(M_{AB})$ is a relativistic BW function of the form\n  \\begin{equation}\n         BW(M_{AB}) = \\frac{F_{\\eta_c} F}{M_r^2 - M_{AB}^2 - i M_r \\Gamma_{\\rm tot}(M_{AB})}\n\\end{equation}\n   where $M_r$ is the mass of the resonance $r$, and $\\Gamma_{\\rm tot}(M_{AB})$ is\n   its mass-dependent total width. In general, this mass dependence\n   cannot be specified, and a constant value should be used. However,\n   for a resonance such as the $K^*_0(1430)$, which is approximately elastic,\n   we can use the partial width $\\Gamma_{AB}$, and specify the mass-dependence\n   as:\n\\begin{equation}\n\\Gamma_{AB} = \\Gamma_r \\left(\\frac{p_{AB}}{p_r}\\right)^{2J+1} \\left(\\frac{M_r}{M_{AB}}\\right)F^2\n\\end{equation}\nwhere\n\\begin{equation}\np_{AB} = \\frac{\\sqrt{\\left(M_{AB}^2-M_A^2-M_B^2\\right)^2-4M_A^2M_B^2}}{2M_{AB}}.\n\\end{equation}\nand $p_r$ is the value of $p_{AB}$ when $M_{AB}=M_r$. \n\nThe form factors $F_{\\ensuremath{\\eta_c}\\xspace}$ and  $F$ attempt to model the underlying quark structure of the parent particle and the intermediate\nresonances. We set $F_{\\ensuremath{\\eta_c}\\xspace}$ to a constant value, while for $F$  we use Blatt-Weisskopf penetration factors~\\cite{blatt} (Table~\\ref{tab:tab_blatt}), that depend on a single parameter $R$ representing the meson ``radius'', for which we assume $R=1.5 \\ \\mbox{${\\mathrm{GeV}}$}^{-1}$.\nThe $a_0(980)$ resonance is parameterized as a coupled-channel Breit-Wigner function whose parameters are taken from Ref.~\\cite{cbar}.\n\n\\begin{table}\n  \\caption{Summary of the Blatt-Weisskopf penetration form factors.}\n  \\label{tab:tab_blatt}\n\\begin{center}\n  \\begin{tabular}{cc}\n    \\hline \\\\ [-2.3ex]\nSpin & $F$ \\\\\n\\hline \\\\ [-2.3ex]\n0 & $1$ \\\\\n&\\\\\n1 & {\\Large $\\frac{\\sqrt{1+(R_r p_r)^2}}{\\sqrt{1+(R_r p_{AB})^2}}$} \\\\\n&\\\\\n2 & {\\Large $\\frac{\\sqrt{9+3(R_r p_r)^2+(R_r p_r)^4}}{\\sqrt{9+3(R_r p_{AB})^2+(R_r p_{AB})^4}}$} \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nTo measure the $I=1\/2$  $K \\pi$ $\\mathcal{S}$-wave we make use of a MIPWA technique first described in Ref.~\\cite{aitala1}.\nThe $K \\pi$ $\\mathcal{S}$-wave, being the largest contribution, is taken as the reference amplitude. \nWe divide the $K \\pi$ mass spectrum into 30 equally-spaced mass intervals 60 \\mbox{${\\mathrm{MeV}}$}\\ wide, and \nfor each interval we add to the fit two new free parameters,\nthe amplitude and the phase of the $K \\pi$ $\\mathcal{S}$-wave in that interval. \nWe fix the amplitude to 1.0 and its phase to $\\pi\/2$ at an arbitrary point in the mass spectrum, for which we choose interval 14, corresponding to a mass of 1.45 \\mbox{${\\mathrm{GeV}}\/c^2$}. The number of associated free parameters is therefore 58.\n\nDue to isospin conservation in the hadronic $\\eta_c$  and $K^*$ decays, the $(K \\pi) \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$ amplitudes are combined with positive signs, and so \ntherefore are symmetrized with respect to the two $K^* \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$ modes. In particular we write the $K \\pi$ $\\mathcal{S}$-wave amplitudes as\n\n\\begin{equation}\n  A_{\\mathcal{S}\\myhyphen\\rm{wave}} = \\frac{1}{\\sqrt{2}}(a_j^{K^+ \\pi^-}e^{i\\phi_j^{K^+ \\pi^-}} + a_j^{\\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace^0 \\pi^-}e^{i\\phi_j^{\\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace^0 \\pi^-}}),\n  \\label{eq:amp}\n\\end{equation}\n\n\\noindent where $a^{K^+ \\pi^-}(m)=a^{\\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace^0 \\pi^-}(m)$ and $\\phi^{K^+ \\pi^-}(m) = \\phi^{\\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace^0 \\pi^-}(m)$, for $\\eta_c \\mbox{$\\rightarrow$} \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace^0 \\mbox{${K^{+}}$} \\mbox{${\\pi^{-}}$}$~\\cite{conj} and\n\n\\begin{equation}\n  A_{\\mathcal{S}\\myhyphen\\rm{wave}} = \\frac{1}{\\sqrt{2}}(a_j^{K^+ \\pi^0}e^{i\\phi_j^{K^+\\pi^0}} + a_j^{K^- \\pi^0}e^{i\\phi_j^{K^- \\pi^0}}),\n\\end{equation}\nwhere $a^{K^+ \\pi^0}(m)=a^{K^- \\pi^0}(m)$ and $\\phi^{K^+ \\pi^0}(m) = \\phi^{K^- \\pi^0}(m)$, for \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace.\nFor both decay modes the bachelor kaon is in an orbital $\\mathcal{S}$-wave with respect to the relevant $K \\pi$ system, and so does not affect\nthese amplitudes. The second amplitude in Eq.(\\ref{eq:amp}) is reduced because the $\\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace^0$ is observed as a $\\KS$, but the same reduction\nfactor applies to the first amplitude through the bachelor $\\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace^0$, so that the equality of the three-body amplitudes is preserved.\n\nOther resonance contributions are described as above. The $K^*_2(1430) \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$ contribution is symmetrized in the same way as the $\\mathcal{S}$-wave amplitude.\n\nWe perform MC simulations to test the ability of the method to find the correct solution.\nWe generate \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace event samples which yield reconstructed samples having the same size as the data sample, according to arbitrary mixtures of resonances, and extract the $K \\pi$ $\\mathcal{S}$-wave using the MIPWA method. We find that the fit is able to extract correctly\nthe mass dependence of the amplitude and phase.\n\nWe also test the possibility of multiple solutions by starting the fit from random values or\nconstant parameter values very far from the solution found by the fit.\nWe find  only one solution in both final states and conclude that the fit converges to give the correct $\\mathcal{S}$-wave behaviour for\ndifferent starting values of the parameters.\n\nThe efficiency-corrected fractional contribution $f_i$ due to resonant or non-resonant contribution $i$ is defined as follows:\n\\begin{equation}\nf_i = \\frac {|c_i|^2 \\int |A_i(x_n,y_n)|^2 {\\rm d}x {\\rm d}y}\n{\\int |\\sum_j c_j A_j(x,y)|^2 {\\rm d}x {\\rm d}y}.\n\\end{equation}\nThe $f_i$ do not necessarily sum to 100\\% because of interference effects. The uncertainty for each $f_i$ is evaluated by propagating the full covariance matrix obtained from the fit.\n\nWe test the quality of the fit by examining a large sample of MC events at the generator level weighted \nby the likelihood fitting function and by the efficiency.\nThese events are used to\ncompare the fit result to the Dalitz plot and its projections with proper normalization.\nIn these MC simulations we smooth the fitted $K \\pi$ $\\mathcal{S}$-wave amplitude and phase by means of a cubic spline.\nWe make use of these weighted events to compute a \\mbox{2-D} $\\chi^2$ over the Dalitz plot. For this purpose, we divide the Dalitz plot into a grid of $25 \\times 25$\ncells and consider only those containing at least five events. We compute \n$\\chi^2 = \\sum_{i=1}^{N_{\\rm cells}} (N^i_{\\rm obs}-N^i_{\\rm exp})^2\/N^i_{\\rm exp}$, where $N^i_{\\rm obs}$ and $N^i_{\\rm exp}$ are event yields from data and simulation, respectively.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=12cm]{fig4.eps}\n\\caption{Dalitz plot for $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ events in the signal region.}\n\\label{fig:fig4}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=18cm]{fig5.eps}\n\\caption{The $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ Dalitz plot projections on (a) $m^2(\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$})$, (b) $m^2(\\KS \\mbox{${\\pi^{\\pm}}$})$, and (c) $m^2(\\KS \\ensuremath{K^\\pm}\\xspace)$. The superimposed curves result from the MIPWA described in the text. The shaded regions show the\nbackground estimates obtained  by interpolating the results of the Dalitz plot analyses of the sideband regions.}\n\\label{fig:fig5}\n\\end{center}\n\\end{figure*}\n\n\\section{Dalitz plot analysis of {\\boldmath$\\protect \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$} }\n\nFigure~\\ref{fig:fig4} shows the Dalitz plot for the candidates in the \\ensuremath{\\eta_c}\\xspace signal region, and Fig.~\\ref{fig:fig5} shows the corresponding Dalitz plot projections. Since the width of the \\ensuremath{\\eta_c}\\xspace meson is $32.3 \\pm 1.0$ MeV, no mass constraint can be applied.\n\nThe Dalitz plot is dominated by the presence of horizontal and vertical uniform bands at the position of the \\ensuremath{K^*_0(1430)}\\xspace resonance. We also observe\nfurther bands along the diagonal. Isospin conservation in \\ensuremath{\\eta_c}\\xspace decay requires that the $(K \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace)$ system have I=1, so that these structures\nmay indicate the presence of $a_0$ or $a_2$ resonances. Further narrow bands are observed at the position\nof the $K^*(892)$ resonance, mostly in the $\\KS \\mbox{${\\pi^{-}}$}$ projection; these components are consistent with originating from background, as will be shown.\n\nThe presence of background in the \\ensuremath{\\eta_c}\\xspace signal region requires precise study of its structure. This can be achieved by\nmeans of the data in the \\ensuremath{\\eta_c}\\xspace sideband regions, for which the Dalitz plots are shown in Fig.~\\ref{fig:fig6}.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=16cm]{fig6.eps}\n\\caption{Dalitz plots for the $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ sideband regions: (a) lower, (b) upper.}\n\\label{fig:fig6}\n\\end{center}\n\\end{figure*}\n\nIn both regions we observe almost uniformly populated resonant structures mostly in the $\\KS \\mbox{${\\pi^{-}}$}$ mass, especially in the regions corresponding to the $K^*(892)$`<\nand $K^*_2(1430)$ resonances. The resonant structures in $\\mbox{${K^{+}}$} \\mbox{${\\pi^{-}}$}$ mass are weaker. The three-body decay of a pseudoscalar meson into a spin-one or spin-two resonance yields a non-uniform distribution (see Eq.~\\ref{eq:spin}) in the relevant resonance band on the Dalitz plot. The presence of uniformly populated bands in the $K^*(892)$ and $K^*_2(1430)$ mass regions, indicates that these\nstructures are associated with background. Also, the asymmetry between the two $K^*$ modes in background \nmay be explained as being due to interference between the $I = 0$ and $I = 1$ isospin configurations for the $K^*(\\mbox{$\\rightarrow$} K \\pi) \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$ final\nstate produced in two-photon fusion.  \n\nWe fit the \\ensuremath{\\eta_c}\\xspace sidebands using an incoherent sum of amplitudes, which includes contributions from the $a_0(980)$, $a_0(1450)$, $a_2(1320)$, $K^*(892)$, $K^*_0(1430)$, $K^*_2(1430)$, $K^*(1680)$, and $K^*_0(1950)$ resonances. To better constrain the sum of the fractions to one, we make use of the channel likelihood method~\\cite{chafit} and include resonances\nuntil no structure is left in the background and an accurate description of the Dalitz plots is obtained.\n\nTo estimate the background composition in the \\ensuremath{\\eta_c}\\xspace signal region we perform a linear mass dependent interpolation of the\nfractions of the different contributions, obtained from the fits to the sidebands, and normalized using the results from the fit to the \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace mass spectrum. The estimated background contributions are indicated by the shaded regions in Fig.~\\ref{fig:fig5}.\n\n\\subsection{MIPWA of {\\boldmath$\\protect \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$}}\n\nWe perform the MIPWA including the resonances listed in Table~\\ref{tab:tab1}. In this table, and in the remainder of the paper,\nwe use the notation $(K \\pi) \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$ or $K^* \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$ to represent the corresponding symmetrized amplitude.\nAfter the solution is found we test for\nother contributions, including spin-one resonances, but these are found to be consistent with zero, and so are not included.\nThis supports the observation that\nthe observed \\ensuremath{K^*(892)}\\xspace structures originate entirely from background.\nWe find a dominance of the $K \\pi$ $\\mathcal{S}$-wave amplitude, with small contributions from $a_0 \\pi$ amplitudes and a\nsignificant $K^*_2(1430) \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$ contribution.\n\n\\begin{table*}\n\\caption{Results from the $\\eta_c \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ and $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$} \\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}$ MIPWA. Phases are determined relative to the $(K\\pi \\ \\mathcal{S}$-wave) $\\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$ amplitude which is fixed to $\\pi\/2$ at 1.45 \\mbox{${\\mathrm{GeV}}\/c^2$}.}\n\\label{tab:tab1}\n\\begin{center}\n  \\begin{tabular}{|l | r@{}c@{}r | r@{}c@{}r | r@{}c@{}r | r@{}c@{}r|}\n    \\hline \\\\ [-2.3ex]\n      &   \\multicolumn{6}{c|} {$\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$} &   \\multicolumn{6}{c|}{ \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace  }  \\cr\n   \\hline \\\\ [-2.3ex]\n  Amplitude & \\multicolumn{3}{c|} {Fraction (\\%)}  & \\multicolumn{3}{c|}{Phase (rad)} & \\multicolumn{3}{c|}{Fraction (\\%)} & \\multicolumn{3}{c|}{Phase (rad)}\\cr\n   \\hline \\\\ [-2.3ex]\n$(K\\pi \\ \\mathcal{S}$-wave) $\\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$ & 107.3 $\\pm$  & \\, 2.6 $\\pm$  & \\, 17.9 & &  fixed  &  & 125.5 $\\pm$  & \\, 2.4 $\\pm$  & \\, 4.2 & & fixed  &  \\cr\n$a_0(980) \\pi$ & 0.8 $\\pm$  & \\, 0.5 $\\pm$  & \\, 0.8 & 1.08 $\\pm$  & \\, 0.18  $\\pm$  & \\, 0.18 & 0.0 $\\pm$  & \\, 0.1 $\\pm$  & \\, 1.7 &  & - & \\cr\n$a_0(1450) \\pi$ & 0.7 $\\pm$  & \\, 0.2 $\\pm$  & \\, 1.4 & 2.63 $\\pm$  & \\, 0.13 $\\pm$  & \\, 0.17 & 1.2 $\\pm$  & \\, 0.4 $\\pm$  & \\, 0.7 & 2.90 $\\pm$  & \\, 0.12 $\\pm$  & \\, 0.25\\cr\n$a_0(1950) \\pi$ & 3.1 $\\pm$  & \\, 0.4 $\\pm$  & \\, 1.2 & $-$1.04 $\\pm$  & \\, 0.08 $\\pm$  & \\, 0.77& 4.4 $\\pm$  & \\, 0.8 $\\pm$  & \\, 0.8& $-$1.45 $\\pm$  & \\, 0.08 $\\pm$  & \\, 0.27\\cr\n$a_2(1320) \\pi$& 0.2 $\\pm$  & \\, 0.1 $\\pm$  & \\, 0.1 & 1.85 $\\pm$  & \\, 0.20 $\\pm$  & \\, 0.20 & 0.6 $\\pm$  & \\, 0.2 $\\pm$  & \\, 0.3& 1.75 $\\pm$  & \\, 0.23 $\\pm$  & \\, 0.42\\cr\n$K^*_2(1430) \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$ & 4.7 $\\pm$  & \\, 0.9 $\\pm$  & \\, 1.4 & 4.92 $\\pm$  & \\, 0.05 $\\pm$  & \\, 0.10 & 3.0 $\\pm$  & \\, 0.8 $\\pm$  & \\, 4.4 & 5.07 $\\pm$  & \\, 0.09 $\\pm$  & \\, 0.30\\cr\n \\hline \\\\ [-2.3ex]\nTotal & 116.8 $\\pm$  & \\, 2.8  $\\pm$ & \\, 18.1 &   & &    & 134.8 $\\pm$  & \\, 2.7 $\\pm$ & \\, 6.4 & & &   \\cr\n$-$ $2\\log {\\cal L}$ & \\multicolumn{3}{c|} {$-$4314.2} & & & & \\multicolumn{3}{c|}{$-$2339} & & & \\cr\n$\\chi^2\/N_{\\rm cells}$ & \\multicolumn{3}{c|} {301\/254=1.17} & & & &  \\multicolumn{3}{c|}{283.2\/233=1.22} & & & \\cr\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\nThe table lists also a significant contribution from the $a_0(1950) \\pi$ amplitude, where $a_0(1950)^+ \\mbox{$\\rightarrow$} \\KS \\mbox{${K^{+}}$}$ is a new\nresonance. \nWe also test the spin-2 hypothesis for this contribution by replacing the amplitude for $a_0 \\mbox{$\\rightarrow$} K^0_S K^+$ with an $a_2 \\mbox{$\\rightarrow$} K^0_S K^+$ amplitude with parameter values left free in the fit.\nIn this case no physical solution is found inside the allowed ranges of the parameters, and the additional contribution is found consistent with zero. This new state has isospin one, and the spin-0 assignment is preferred over that of spin-2.\n\nA fit without this state gives a poor description of the high mass $\\KS \\mbox{${K^{+}}$}$ projection, as can be seen in Fig.~\\ref{fig:fig7}(a). We obtain $-2\\log {\\cal L} = -$4252.9 and\n$\\chi^2\/N_{\\rm cells}=1.33$ for this fit.\nWe then include in the MIPWA a new scalar resonance decaying to $\\KS \\mbox{${K^{+}}$}$ with free parameters. \nWe obtain \n$\\Delta (\\log {\\cal L})=61$ and  $\\Delta \\chi^2=38$ for an increase of four new parameters.\nWe estimate the significance for the $a_0(1950)$ resonance using the fitted \nfraction divided by its statistical and systematic errors added in quadrature, and obtain $2.5\\sigma$.\nSince interference effects may also contribute to the significance, this procedure gives a conservative estimate.\nThe systematic uncertainties associated with the $a_0(1950)$ state are described below.\nThe fitted parameter values for this state are given in Table~\\ref{tab:tab2}.\nWe note that we obtain $\\chi^2\/N_{\\rm cells}=1.17$ for this final fit, indicating good description of the data.\nThe fit projections on the three squared masses from the MIPWA are shown in Fig.~\\ref{fig:fig5}, and they indicate that the description of the data is quite good.\n\n\\begin{table}\n\\caption{Fitted $a_0(1950)$ parameter values for the two \\ensuremath{\\eta_c}\\xspace decay modes.}\n\\label{tab:tab2}\n\\begin{center}\n\\begin{tabular}{|l|c|r@{}c@{}r|}\n  \\hline \\\\ [-2.3ex]\nFinal state & Mass (\\mbox{${\\mathrm{MeV}}\/c^2$}) & \\multicolumn{3}{c|}{Width (\\mbox{${\\mathrm{MeV}}$})}\\cr\n\\hline \\\\ [-2.3ex]\n$\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ & 1949 $\\pm$ 32 $\\pm$ 76 & 265 $\\pm$ & \\, 36 $\\pm$ & \\,110\\cr\n\\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace & 1927 $\\pm$ 15 $\\pm$ 23 & 274 $\\pm$ & \\, 28 $\\pm$ & \\, 30 \\cr\n\\hline \\\\ [-2.3ex]\nWeighted mean & 1931 $\\pm$ 14 $\\pm$ 22 & 271 $\\pm$ & \\, 22 $\\pm$ & \\, 29\\cr\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=16cm]{fig7.eps}\n\\caption{The mass projections (a) $\\KS \\ensuremath{K^\\pm}\\xspace$ from $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ and (b) $\\mbox{${K^{+}}$} \\mbox{${K^{-}}$}$ from $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace$. The histograms show the MIPWA fit projections with (solid, black) and without (dashed, red) the presence of the $a_0(1950)^+ \\mbox{$\\rightarrow$} \\KS \\ensuremath{K^\\pm}\\xspace$ resonance.  The shaded regions show the background estimates obtained by interpolating the results of the Dalitz plot analyses of the sideband regions.}\n\\label{fig:fig7}\n\\end{center}\n\\end{figure*}\n\nWe compute the uncorrected Legendre polynomial moments $\\langle Y^0_L \\rangle$ in each $\\mbox{${K^{+}}$} \\mbox{${\\pi^{-}}$}$, $\\KS \\mbox{${\\pi^{-}}$}$ and $\\KS \\mbox{${K^{+}}$}$  mass interval by weighting each event by the relevant $Y^0_L(\\cos \\theta)$ function.\nThese distributions are shown in Fig.~\\ref{fig:fig8} as functions of $K \\pi$ mass after combining $\\mbox{${K^{+}}$} \\mbox{${\\pi^{-}}$}$ and  $\\KS \\mbox{${\\pi^{-}}$}$, and in Fig.~\\ref{fig:fig9} as functions of $\\KS \\mbox{${K^{+}}$}$ mass. We also compute the expected Legendre polynomial moments from the weighted MC events and compare with the experimental distributions. We observe good agreement for all the distributions, which indicates that the fit is able to reproduce the local structures apparent in the Dalitz plot.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=16cm]{fig8.eps}\n\\caption{Legendre polynomial moments for $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ as functions of $K \\pi$ mass, and combined for $\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}$ and  $\\KS \\mbox{${\\pi^{\\mp}}$}$; the superimposed curves result from the Dalitz plot fit described in the text.}\n\\label{fig:fig8}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=16cm]{fig9.eps}\n\\caption{Legendre polynomial moments for $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ as a function of $\\KS \\ensuremath{K^\\pm}\\xspace$ mass, the superimposed curves result from the Dalitz plot fit described in the text.}\n\\label{fig:fig9}\n\\end{center}\n\\end{figure*}\n\nWe compute the following systematic uncertainties on the $I=1\/2$  $K \\pi$ $\\mathcal{S}$-wave amplitude and phase. The different contributions are added in quadrature.\n\\begin{itemize}\n\\item{} Starting from the solution found by the fit, we generate MC simulated events which are fitted using a MIPWA. In this way we estimate the bias introduced by the fitting method.\n\\item{} The fit is performed by interpolating the  $K \\pi$ $\\mathcal{S}$-wave amplitude and phase using a cubic spline.\n\\item{} We remove low-significance contributions, such as those from the $a_0(980)$ and $a_2(1320)$ resonances.\n\\item{} We vary the signal purity up and down according to its statistical uncertainty.\n\\item{} The effect of the efficiency variation as a function of $K \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace \\pi$ mass is evaluated by computing separate efficiencies\n  in the regions below and above the $\\eta_c$ mass.\n\\end{itemize}\n\nThese additional fits also allow the computation of systematic uncertainties on the amplitude fraction and phase values, as well as on the parameter values for the $a_0(1950)$ resonance; these are summarized in Table~\\ref{tab:a0_sys}. In the evaluation of overall systematic uncertainties, all effects are assumed to be uncorrelated, and are added in quadrature.\n\nThe measured amplitude and phase values of the $I=1\/2$  $K \\pi$ $\\mathcal{S}$-wave as functions of mass obtained from the MIPWA of $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ are shown in Table~\\ref{tab:tab6}. Interval 14 of the $K \\pi$ mass contains the fixed amplitude and phase values.\n\n\\begin{table*}\n  \\caption{Systematic uncertainties on the $a_0(1950)$ parameter values from the two \\ensuremath{\\eta_c}\\xspace decay modes.}\n  \\label{tab:a0_sys}\n       \\begin{center}\n         \\begin{tabular}{|l|r|r|r|r|r|r|}\n           \\hline \\\\ [-2.3ex]\n        &  \\multicolumn{3}{c|}{$\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$} & \\multicolumn{3}{c|}{$\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$} \\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}$} \\cr\n      \\hline \\\\ [-2.3ex]   \n    Effect & Mass  & Width  & Fraction (\\%) & Mass  & Width & Fraction (\\%)\\cr\n      &  (\\mbox{${\\mathrm{MeV}}\/c^2$}) &  (MeV) &  & (\\mbox{${\\mathrm{MeV}}\/c^2$}) &  (MeV) & \\cr   \n        \\hline \\\\ [-2.3ex]\nFit bias & 11 & 22 & 0.5 & 1 & 10 & 0.5 \\cr\nCubic spline & 24 & 79 & 0.6 & 14 & 9 & 0.2\\cr\nMarginal components  & 70 & 72 & 0.0 & 2 & 8 & 0.3 \\cr\n\\ensuremath{\\eta_c}\\xspace purity  & 3 & 16  & 1.0 & 18 & 26 & 0.4 \\cr\nEfficiency & 11 & 8  & 0.2 & 1  & 15  & 0.2 \\cr\n \\hline \\\\ [-2.3ex]\nTotal & 76 & 110 & 1.3 & 23 & 30 & 0.8\\cr\n\\hline\n  \\end{tabular}\n \\end{center} \n\\end{table*}\n\n\\begin{table*}\n  \\caption{Measured amplitude and phase values for the $I=1\/2$  $K \\pi$ $\\mathcal{S}$-wave as functions of mass obtained from the MIPWA of $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ and $\\eta_c \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$} \\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}$. The first error is statistical, the second systematic. The amplitudes and phases in the mass interval 14\n  are fixed to constant values.}\n\\label{tab:tab6}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n \\hline \\\\ [-2.3ex]\n  \\multicolumn{2}{|c}{} & \\multicolumn{2}{|c}{$\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$} & \\multicolumn{2}{|c|}{$\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$} \\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}$} \\cr\n \\hline \\\\ [-2.3ex]\nN & $K \\pi$ mass & Amplitude & Phase (rad) & Amplitude & Phase (rad)\\cr\n \\hline \\\\ [-2.3ex]\n   1 & 0.67 & $ \\ 0.119 \\pm  0.100 \\ \\pm 0.215$ & $ \\ \\ 0.259 \\pm  0.577 \\ \\pm 1.290$ & $ \\ 0.154 \\pm  0.350 \\ \\pm 0.337$ & $ \\ \\ 3.786 \\pm  1.199 \\ \\pm 0.857$ \\cr\n   2 & 0.73 & $ \\ 0.103 \\pm  0.043 \\ \\pm 0.113$ & $  -0.969 \\pm  0.757 \\ \\pm 1.600$ & $ \\ 0.198 \\pm  0.124 \\ \\pm 0.216$ & $ \\ \\ 3.944 \\pm  0.321 \\ \\pm 0.448$ \\cr\n   3 & 0.79 & $ \\ 0.158 \\pm  0.086 \\ \\pm 0.180$ & $ \\ \\  0.363 \\pm  0.381 \\ \\pm 1.500$ & $ \\ 0.161 \\pm  0.116 \\ \\pm 0.098$ & $ \\ \\ 1.634 \\pm  0.584 \\ \\pm 0.448$ \\cr\n   4 & 0.85 & $ \\ 0.232 \\pm  0.128 \\ \\pm 0.214$ & $ \\ \\  0.448 \\pm  0.266 \\ \\pm 1.500$ & $ \\ 0.125 \\pm  0.118 \\ \\pm 0.031$ & $ \\ \\ 3.094 \\pm  0.725 \\ \\pm 0.448$ \\cr\n   5 & 0.91 & $ \\ 0.468 \\pm  0.075 \\ \\pm 0.194$ & $ \\ \\  0.091 \\pm  0.191 \\ \\pm 0.237$ & $ \\ 0.307 \\pm  0.213 \\ \\pm 0.162$ & $ \\ \\ 0.735 \\pm  0.326 \\ \\pm 0.255$ \\cr\n   6 & 0.97 & $ \\ 0.371 \\pm  0.083 \\ \\pm 0.129$ & $ \\ \\  0.276 \\pm  0.156 \\ \\pm 0.190$ & $ \\ 0.528 \\pm  0.121 \\ \\pm 0.055$ & $  -0.083 \\pm  0.178 \\ \\pm 0.303$ \\cr\n   7 & 1.03 & $ \\ 0.329 \\pm  0.071 \\ \\pm 0.102$ & $ \\ \\  0.345 \\pm  0.164 \\ \\pm 0.273$ & $ \\ 0.215 \\pm  0.191 \\ \\pm 0.053$ & $ \\ \\ 0.541 \\pm  0.320 \\ \\pm 0.638$ \\cr\n   8 & 1.09 & $ \\ 0.343 \\pm  0.062 \\ \\pm 0.062$ & $ \\ \\  0.449 \\pm  0.196 \\ \\pm 0.213$ & $ \\ 0.390 \\pm  0.146 \\ \\pm 0.046$ & $ \\ \\ 0.254 \\pm  0.167 \\ \\pm 0.144$ \\cr\n   9 & 1.15 & $ \\ 0.330 \\pm  0.070 \\ \\pm 0.081$ & $ \\ \\  0.687 \\pm  0.167 \\ \\pm 0.221$ & $ \\ 0.490 \\pm  0.135 \\ \\pm 0.089$ & $ \\ \\ 0.618 \\pm  0.155 \\ \\pm 0.099$ \\cr\n  10 & 1.21 & $ \\ 0.450 \\pm  0.059 \\ \\pm 0.042$ & $ \\ \\  0.696 \\pm  0.156 \\ \\pm 0.226$ & $ \\ 0.422 \\pm  0.092 \\ \\pm 0.102$ & $ \\ \\ 0.723 \\pm  0.242 \\ \\pm 0.267$ \\cr\n  11 & 1.27 & $ \\ 0.578 \\pm  0.048 \\ \\pm 0.112$ & $ \\ \\  0.785 \\pm  0.208 \\ \\pm 0.358$ & $ \\ 0.581 \\pm  0.113 \\ \\pm 0.084$ & $ \\ \\ 0.605 \\pm  0.186 \\ \\pm 0.166$ \\cr\n  12 & 1.33 & $ \\ 0.627 \\pm  0.047 \\ \\pm 0.053$ & $ \\ \\  0.986 \\pm  0.153 \\ \\pm 0.166$ & $ \\ 0.643 \\pm  0.106 \\ \\pm 0.039$ & $ \\ \\ 1.330 \\pm  0.264 \\ \\pm 0.130$ \\cr\n  13 & 1.39 & $ \\ 0.826 \\pm  0.047 \\ \\pm 0.105$ & $ \\ \\  1.334 \\pm  0.155 \\ \\pm 0.288$ & $ \\ 0.920 \\pm  0.153 \\ \\pm 0.056$ & $ \\ \\ 1.528 \\pm  0.161 \\ \\pm 0.160$ \\cr\n  \\textcolor{red}{14} & \\textcolor{red}{1.45} & \\textcolor{red}{$ \\ 1.000 $} & \\textcolor{red}{$ \\ 1.570  $} & \\textcolor{red}{$ \\ 1.000   $} & \\textcolor{red}{$ \\ 1.570  $} \\cr\n  15 & 1.51 & $ \\ 0.736 \\pm  0.031 \\ \\pm 0.059$ & $ \\ \\ 1.918 \\pm  0.153 \\ \\pm 0.132$ & $ \\ 0.750 \\pm  0.118 \\ \\pm 0.076$ & $ \\ \\ 1.844 \\pm  0.149 \\ \\pm 0.048$ \\cr\n  16 & 1.57 & $ \\ 0.451 \\pm  0.025 \\ \\pm 0.053$ & $ \\ \\ 2.098 \\pm  0.202 \\ \\pm 0.277$ & $ \\ 0.585 \\pm  0.099 \\ \\pm 0.047$ & $ \\ \\ 2.128 \\pm  0.182 \\ \\pm 0.110$ \\cr\n  17 & 1.63 & $ \\ 0.289 \\pm  0.029 \\ \\pm 0.065$ & $ \\ \\ 2.539 \\pm  0.292 \\ \\pm 0.180$ & $ \\ 0.366 \\pm  0.079 \\ \\pm 0.052$ & $ \\ \\ 2.389 \\pm  0.230 \\ \\pm 0.213$ \\cr\n  18 & 1.69 & $ \\ 0.159 \\pm  0.036 \\ \\pm 0.089$ & $ \\ \\ 1.566 \\pm  0.308 \\ \\pm 0.619$ & $ \\ 0.312 \\pm  0.074 \\ \\pm 0.043$ & $ \\ \\ 1.962 \\pm  0.195 \\ \\pm 0.150$ \\cr\n  19 & 1.75 & $ \\ 0.240 \\pm  0.034 \\ \\pm 0.067$ & $ \\ \\ 1.962 \\pm  0.331 \\ \\pm 0.655$ & $ \\ 0.427 \\pm  0.093 \\ \\pm 0.063$ & $ \\ \\ 1.939 \\pm  0.150 \\ \\pm 0.182$ \\cr\n  20 & 1.81 & $ \\ 0.381 \\pm  0.031 \\ \\pm 0.059$ & $ \\ \\ 2.170 \\pm  0.297 \\ \\pm 0.251$ & $ \\ 0.511 \\pm  0.094 \\ \\pm 0.063$ & $ \\ \\ 2.426 \\pm  0.156 \\ \\pm 0.277$ \\cr\n  21 & 1.87 & $ \\ 0.457 \\pm  0.035 \\ \\pm 0.085$ & $ \\ \\ 2.258 \\pm  0.251 \\ \\pm 0.284$ & $ \\ 0.588 \\pm  0.098 \\ \\pm 0.080$ & $ \\ \\ 2.242 \\pm  0.084 \\ \\pm 0.210$ \\cr\n  22 & 1.93 & $ \\ 0.565 \\pm  0.042 \\ \\pm 0.067$ & $ \\ \\ 2.386 \\pm  0.255 \\ \\pm 0.207$ & $ \\ 0.729 \\pm  0.114 \\ \\pm 0.095$ & $ \\ \\ 2.427 \\pm  0.098 \\ \\pm 0.254$ \\cr\n  23 & 1.99 & $ \\ 0.640 \\pm  0.044 \\ \\pm 0.055$ & $ \\ \\ 2.361 \\pm  0.228 \\ \\pm 0.092$ & $ \\ 0.777 \\pm  0.119 \\ \\pm 0.075$ & $ \\ \\ 2.306 \\pm  0.102 \\ \\pm 0.325$ \\cr\n  24 & 2.05 & $ \\ 0.593 \\pm  0.046 \\ \\pm 0.065$ & $ \\ \\ 2.329 \\pm  0.235 \\ \\pm 0.268$ & $ \\ 0.775 \\pm  0.134 \\ \\pm 0.075$ & $ \\ \\ 2.347 \\pm  0.107 \\ \\pm 0.299$ \\cr\n  25 & 2.11 & $ \\ 0.614 \\pm  0.057 \\ \\pm 0.083$ & $ \\ \\ 2.421 \\pm  0.230 \\ \\pm 0.169$ & $ \\ 0.830 \\pm  0.134 \\ \\pm 0.078$ & $ \\ \\ 2.374 \\pm  0.105 \\ \\pm 0.199$ \\cr\n  26 & 2.17 & $ \\ 0.677 \\pm  0.067 \\ \\pm 0.117$ & $ \\ \\ 2.563 \\pm  0.218 \\ \\pm 0.137$ & $ \\ 0.825 \\pm  0.140 \\ \\pm 0.070$ & $ \\ \\ 2.401 \\pm  0.127 \\ \\pm 0.189$ \\cr\n  27 & 2.23 & $ \\ 0.788 \\pm  0.085 \\ \\pm 0.104$ & $ \\ \\ 2.539 \\pm  0.228 \\ \\pm 0.241$ & $ \\ 0.860 \\pm  0.158 \\ \\pm 0.123$ & $ \\ \\ 2.296 \\pm  0.131 \\ \\pm 0.297$ \\cr\n  28 & 2.29 & $ \\ 0.753 \\pm  0.097 \\ \\pm 0.125$ & $ \\ \\ 2.550 \\pm  0.234 \\ \\pm 0.168$ & $ \\ 0.891 \\pm  0.167 \\ \\pm 0.133$ & $ \\ \\ 2.320 \\pm  0.131 \\ \\pm 0.273$ \\cr\n  29 & 2.35 & $ \\ 0.646 \\pm  0.096 \\ \\pm 0.118$ & $ \\ \\ 2.315 \\pm  0.241 \\ \\pm 0.321$ & $ \\ 0.994 \\pm  0.202 \\ \\pm 0.076$ & $ \\ \\ 2.297 \\pm  0.153 \\ \\pm 0.197$ \\cr\n  30 & 2.41 & $ \\ 0.789 \\pm  0.184 \\ \\pm 0.187$ & $ \\ \\ 2.364 \\pm  0.336 \\ \\pm 0.199$ & $ \\ 0.892 \\pm  0.322 \\ \\pm 0.098$ & $ \\ \\ 2.143 \\pm  0.292 \\ \\pm 0.393$ \\cr\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n \n\\subsection{Dalitz plot analysis of {\\boldmath$\\protect \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$} using an isobar model}\n\nWe perform a Dalitz plot analysis of \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace using a standard isobar model, where all resonances\nare modeled as BW functions multiplied by the corresponding angular functions. In this case the $K \\pi$ $\\mathcal{S}$-wave is represented by a superposition of interfering $K^*_0(1430)$, \\ensuremath{K^*_0(1950)}\\xspace, non-resonant (NR), and possibly $\\kappa(800)$ contributions.\nThe NR contribution is parametrized as an amplitude that is constant in magnitude and phase\nover the Dalitz plot.\nIn this fit the $K^*_0(1430)$ parameters\nare taken from Ref.~\\cite{etakk}, while all other parameters are fixed to PDG values. We also add the $a_0(1950)$\nresonance with parameters obtained from the MIPWA analysis.\n\n\n\nFor the description of the \\ensuremath{\\eta_c}\\xspace signal, amplitudes are added one by one to ascertain the associated increase of the likelihood value and decrease of the \\mbox{2-D} $\\chi^2$. \nTable~\\ref{tab:tab4} summarizes the fit results for the amplitude fractions and phases.\nThe high value of $\\chi^2\/N_{\\rm cells}=1.82$ (to be compared with $\\chi^2\/N_{\\rm cells}=1.17$) indicates a poorer description of the data than that obtained with the MIPWA method. Including the $\\kappa(800)$ resonance does not improve the fit quality. If included, it gives a fit fraction of $(0.8 \\pm 0.5)$\\%.\n\nThe Dalitz plot analysis shows a dominance of scalar meson amplitudes, with small contributions from spin-two resonances. The $K^*(892)$ contribution is consistent with originating entirely from background. Other spin-1 $K^*$ resonances have been included in the fit, \nbut their contributions have been found to be\nconsistent with zero. We note the presence of a sizeable non-resonant contribution. However, in this case the sum of the fractions is significantly lower than 100\\%, indicating important interference effects.\nFitting the data without the NR contribution gives a much poorer description, with $-2\\log {\\cal L}=-$4115 and $\\chi^2\/N_{\\rm cells}=2.32$.\n\nWe conclude that the $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ Dalitz plot is not well-described by an isobar model in which the $K \\pi$ $\\mathcal{S}$-wave is modeled as a superposition of Breit-Wigner functions. A more complex approach is needed, and the MIPWA is able to describe\nthis amplitude without the need for a specific model.\n\n\\begin{table}\n\\caption{Results from the \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace Dalitz plot analysis using an isobar model. The listed uncertainties are statistical only.}\n\\label{tab:tab4}\n\\begin{center}\n  \\begin{tabular}{|l|r@{}c|r@{}c|}\n     \\hline \\\\ [-2.3ex]\n Amplitude & \\multicolumn{2}{c|}{Fraction \\%} & \\multicolumn{2}{c|}{Phase (rad)}\\cr\n  \\hline \\\\ [-2.3ex]\n$K^*_0(1430) \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$ & 40.8 $\\pm$ & \\, 2.2 &  0. & \\cr\n$K^*_0(1950) \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$ & 14.8 $\\pm$ & \\, 1.7 & $-$1.00 $\\pm$ & \\, 0.07 \\cr\nNR & 18.0 $\\pm$ & \\, 2.5 & 1.94 $\\pm$ & \\, 0.09 \\cr\n$a_0(980) \\pi$ & 10.5 $\\pm$ & \\, 1.2 & 0.94 $\\pm$ & \\, 0.12 \\cr\n$a_0(1450) \\pi$ & 1.7 $\\pm$ & \\, 0.5 & 2.94 $\\pm$ & \\, 0.13 \\cr\n$a_0(1950) \\pi$ & 0.7 $\\pm$ & \\, 0.2 & $-$1.76 $\\pm$ & \\, 0.24 \\cr\n$a_2(1320) \\pi$ & 0.2 $\\pm$ & \\, 0.2 & $-$0.53 $\\pm$ & \\, 0.42 \\cr\n$K^*_2(1430) \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$ & 2.3 $\\pm$ & \\, 0.7 & $-$1.55 $\\pm$ & \\, 0.11 \\cr\n \\hline \\\\ [-2.3ex]\nTotal &  88.8 $\\pm$ & \\, 4.3 & & \\cr\n$-2\\log {\\cal L}$ &  $-$4290.7 & \\,  & &  \\cr\n$\\chi^2\/N_{\\rm cells}$ & 467\/256=1.82 & \\,   & & \\cr\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\section{Dalitz plot analysis of {\\boldmath$\\protect \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace$}\\ }\n\nThe \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace Dalitz plot~\\cite{etakk} is very similar to that for $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ decays. It is dominated by uniformly populated  bands at\nthe $K^*_0(1430)$ resonance position in $\\mbox{${K^{+}}$} \\mbox{${\\pi^{0}}$}$ and $\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}$ mass squared. It also shows a broad diagonal structure indicating\nthe presence of $a_0$ or $a_2$ resonance contributions. The Dalitz plot projections are shown in Fig.~\\ref{fig:fig10}.\n\nThe \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace Dalitz plot analysis using the isobar model has been performed already in Ref.~\\cite{etakk} . It was found that\nthe model does not give a perfect description of the data. In this section we  obtain a new measurement\nof the $K \\pi$ $\\mathcal{S}$-wave by making use of the MIPWA method. In this way we also perform a cross-check of the results\nobtained from the \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace analysis, since analyses of the two $\\ensuremath{\\eta_c}\\xspace$ decay modes should give consistent results, given\nthe absence of I=3\/2 $K \\pi$ amplitude contributions.\n\n\\subsection{MIPWA of {\\boldmath$\\protect \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace$}}\n\nWe perform a MIPWA of \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace decays using the same model and the same mass grid as for \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace.\nAs for the previous case we obtain a better description of the data if we include an additional $a_0(1950)$ resonance, whose parameter values are listed in Table~\\ref{tab:tab2}. We observe good agreement between the parameter values obtained from the two \\ensuremath{\\eta_c}\\xspace decay modes.\nThe table also lists parameter values obtained as the weighted mean of the two measurements.\nTable~\\ref{tab:tab1} gives the fitted fractions from the MIPWA fit.\n\nWe obtain a good description of the data, as evidenced by the value $\\chi^2\/N_{\\rm cells}=1.22$, and observe\nthe $a_0(1950)$ state with a significance of $4.2\\sigma$.\nThe fit projections on the $\\mbox{${K^{+}}$} \\mbox{${\\pi^{0}}$}$, $\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}$, and $\\mbox{${K^{+}}$} \\mbox{${K^{-}}$}$ squared mass distributions are shown in Fig.~\\ref{fig:fig10}. As previously, there is\na dominance of the ($K \\pi$ $\\mathcal{S}$-wave) $\\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$ amplitude, with a significant $K^*_2(1430) \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$ amplitude, and small contributions from $a_0 \\pi$ amplitudes. We observe good agreement between fractions and relative phases of the\namplitudes between the \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace and \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace decay modes.\nSystematic uncertainties are evaluated as discussed in Sec. VI.A.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=18cm]{fig10.eps}\n\\caption{The $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$} \\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}$ Dalitz plot projections, (a) $m^2(\\mbox{${K^{+}}$} \\mbox{${\\pi^{0}}$})$, (b) $m^2(\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$})$, and (c) $m^2(\\mbox{${K^{+}}$} \\mbox{${K^{-}}$})$. The superimposed curves result from the MIPWA described in the text. The shaded regions show the\nbackground estimates obtained  by interpolating the results of the Dalitz plot analyses of the sideband regions.}\n\\label{fig:fig10}\n\\end{center}\n\\end{figure*}\n\nWe compute the uncorrected Legendre polynomial moments $\\langle Y^0_L \\rangle$ in each $\\mbox{${K^{+}}$} \\mbox{${\\pi^{0}}$}$, $\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}$ and $\\mbox{${K^{+}}$} \\mbox{${K^{-}}$}$  mass interval by weighting each event by the relevant $Y^0_L(\\cos \\theta)$ function.\nThese distributions are shown in Fig.~\\ref{fig:fig11} as functions of  $K \\pi$ mass, combined for $\\mbox{${K^{+}}$} \\mbox{${\\pi^{0}}$}$ and  $\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}$, and in Fig.~\\ref{fig:fig12} as functions of $\\mbox{${K^{+}}$} \\mbox{${K^{-}}$}$ mass. We also compute the expected Legendre polynomial moments from the weighted MC events and compare with the experimental distributions. We observe good agreement for all the distributions, which indicates that also in this case the fit is able to reproduce the local structures apparent in the Dalitz plot.\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=16cm]{fig11.eps}\n\\caption{Legendre polynomial moments for $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$} \\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}$ as functions of $K \\pi$ mass, combined for $\\mbox{${K^{+}}$} \\mbox{${\\pi^{0}}$}$ and  $\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}$. The superimposed curves result from the Dalitz plot fit described in the text.}\n\\label{fig:fig11}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=16cm]{fig12.eps}\n\\caption{Legendre polynomial moments for $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$} \\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}$ as a function of $\\mbox{${K^{+}}$} \\mbox{${K^{-}}$}$ mass. The superimposed curves result from the Dalitz plot fit described in the text.}\n\\label{fig:fig12}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=16cm]{fig13.eps}\n\\caption{The $I=1\/2$  $K \\pi$ $\\mathcal{S}$-wave amplitude (a) and phase (b) from $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ (solid (black) points) and \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace\n  (open (red) points); only statistical uncertainties are shown. The dotted lines indicate the $K \\eta$ and $K \\eta'$ thresholds.}\n\\label{fig:fig13}\n\\end{center}\n\\end{figure*}\n\n\\section{The $I=1\/2$  {\\boldmath$\\protect  K \\pi \\ \\mathcal{S}$}-wave amplitude and phase}\n\nFigure~\\ref{fig:fig13} displays the measured $I=1\/2$  $K \\pi$ $\\mathcal{S}$-wave amplitude and phase from both $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ and \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace.\nWe observe good agreement between the amplitude and phase values obtained from the two measurements.\n\nThe main features of the amplitude (Fig.~\\ref{fig:fig13}(a)) can be explained by the presence of a clear peak related to the $K^*_0(1430)$ resonance\nwhich shows a rapid drop around 1.7 \\mbox{${\\mathrm{GeV}}\/c^2$}, where a broad structure is present which can be related to the $K^*_0(1950)$ resonance.\nThere is some indication of feedthrough from the $K^*(892)$ background.\nThe phase motion (Fig.~\\ref{fig:fig13}(b)) shows the expected behavior for the resonance phase, which varies by about $\\pi$ in the $K^*_0(1430)$ resonance region. The phase shows a drop around 1.7 \\mbox{${\\mathrm{GeV}}\/c^2$}\\ related to interference with the $K^*_0(1950)$ resonance.\n\nWe compare the present measurement of the $K \\pi$ $\\mathcal{S}$-wave amplitude from  $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ with measurements\nfrom LASS~\\cite{lass_kpi} in Fig.~\\ref{fig:fig14}(a)(c) and E791~\\cite{aitala1} in Fig.~\\ref{fig:fig14}(b)(d).\nWe plot only the first part of the LASS measurement since it suffers from\na two-fold ambiguity above the mass of 1.82 \\mbox{${\\mathrm{GeV}}\/c^2$}.\nThe Dalitz plot fits extract invariant amplitudes. Consequently, in Fig.~\\ref{fig:fig14}(a), the LASS $I=1\/2$  $K \\pi$ scattering amplitude\nvalues have been multiplied by the factor $m(K \\pi)\/q$ to convert to invariant amplitude, and normalized so as to equal\nthe scattering amplitude at 1.5 \\mbox{${\\mathrm{GeV}}\/c^2$}\\ in order to facilitate comparison to the \\ensuremath{\\eta_c}\\xspace results. Here $q$ is the momentum of either meson in the $K \\pi$ rest frame. For better comparison, the LASS absolute phase measurements have been displaced by $-0.6$ rad before plotting them in Fig.~\\ref{fig:fig14}(c).\nIn Fig.~\\ref{fig:fig14}(b) the E791 amplitude has been obtained by multiplying the amplitude $c$ in Table III of Ref.~\\cite{aitala1} by the Form Factor $F_D^0$, for which the mass-dependence is motivated by theoretical speculation. This yields amplitude values corresponding to the E791 Form Factor having value 1, as for the \\ensuremath{\\eta_c}\\xspace analyses. In Fig.~\\ref{fig:fig14}(d), the E791 phase measurements have been displaced by $+0.9$ rad, again in order to facilitate comparison to the \\ensuremath{\\eta_c}\\xspace measurements.\n\nWhile we observe similar phase behavior\namong the three measurements up to about 1.5 \\mbox{${\\mathrm{GeV}}\/c^2$}, we observe striking differences in the mass dependence of the amplitudes.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=18cm]{fig14.eps}\n\\caption{The $I=1\/2$  $K \\pi$ $\\mathcal{S}$-wave amplitude measurements from  $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ compared to the (a) LASS and (b) E791 results: the corresponding $I=1\/2$  $K \\pi$ $\\mathcal{S}$-wave phase measurements compared to the\n (c) LASS and (d) E791 measurements. \n  Black dots indicate the results from the present analysis; square (red) points indicate the LASS or E791 results. The LASS data are plotted in the region having only one solution.}\n\\label{fig:fig14}\n\\end{center}\n\\end{figure*}\n\n\\section{Summary}\n\nWe perform Dalitz plot analyses, using an isobar model and a MIPWA method, of data on the decays $\\eta_c \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ and $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$} \\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}$, where the \\ensuremath{\\eta_c}\\xspace mesons are produced in two-photon interactions in the {\\em B}{\\footnotesize\\em A}{\\em B}{\\footnotesize\\em AR}\\ experiment at SLAC. \nWe find that, in comparison with the isobar models examined here, an improved description of the data is obtained by using a MIPWA method.\n\nWe extract the $I=1\/2$  $K \\pi$ $\\mathcal{S}$-wave amplitude and phase and find good agreement between the measurements\nfor the two $\\eta_c$ decay modes.\nThe $K \\pi$ $\\mathcal{S}$-wave is dominated by the presence of the $K^*_0(1430)$ resonance which is observed as a clear peak\nwith the corresponding increase in phase of about $\\pi$ expected for a resonance. A broad structure in the 1.95 \\mbox{${\\mathrm{GeV}}\/c^2$}\\ mass region indicates\nthe presence of the $K^*_0(1950)$ resonance.\n\nA comparison between the present measurement and previous experiments indicates a similar trend for the phase up to a mass of\n1.5 \\mbox{${\\mathrm{GeV}}\/c^2$}. The amplitudes, on the other hand, show very marked differences.\n\nTo fit the data we need to introduce a new $a_0(1950)$ resonance in both $\\eta_c \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ and $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$} \\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}$ decay modes, and their associated parameter values are in good agreement. The weighted averages for\nthe parameter values are:\n\n\\begin{equation}\n  \\begin{split}\n    m(a_0(1950))=1931 \\pm 14 \\pm 22 \\ {\\rm MeV}\/c^2, \\\\\n    \\Gamma(a_0(1950))= 271 \\pm 22 \\pm 29 \\ {\\rm MeV}\n    \\end{split}\n\\end{equation}\n\n\\noindent with significances of 2.5$\\sigma$ and 4.2$\\sigma$ respectively, including systematic uncertainties.\nThese results are, however, systematically limited, and more detailed studies of the $I=1$ $K \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$ $\\mathcal{S}$-wave will be\nrequired in order to improve the precision of these values.\n\n\n\\section{Acknowledgements}\nWe are grateful for the \nextraordinary contributions of our PEP-II\\ colleagues in\nachieving the excellent luminosity and machine conditions\nthat have made this work possible.\nThe success of this project also relies critically on the \nexpertise and dedication of the computing organizations that \nsupport {\\em B}{\\footnotesize\\em A}{\\em B}{\\footnotesize\\em AR}.\nThe collaborating institutions wish to thank \nSLAC for its support and the kind hospitality extended to them. \nThis work is supported by the\nUS Department of Energy\nand National Science Foundation, the\nNatural Sciences and Engineering Research Council (Canada),\nthe Commissariat \\`a l'Energie Atomique and\nInstitut National de Physique Nucl\\'eaire et de Physique des Particules\n(France), the\nBundesministerium f\\\"ur Bildung und Forschung and\nDeutsche Forschungsgemeinschaft\n(Germany), the\nIstituto Nazionale di Fisica Nucleare (Italy),\nthe Foundation for Fundamental Research on Matter (The Netherlands),\nthe Research Council of Norway, the\nMinistry of Education and Science of the Russian Federation,\nMinisterio de Economia y Competitividad (Spain), and the\nScience and Technology Facilities Council (United Kingdom).\nIndividuals have received support from \nthe Marie-Curie IEF program (European Union), the A. P. Sloan Foundation (USA) \nand the Binational Science Foundation (USA-Israel).\nThe work of A. Palano and M. R. Pennington was supported (in part) by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under contract DE-AC05-06OR23177.\n\n\\renewcommand{\\baselinestretch}{1}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}