{"id":"d5f80ff3-6cce-45fc-8227-1e574f84b3fd","arxiv_id":"2505.03983","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Diffusion denoising increments can be treated as exchangeable, letting a single model speculate about many future steps and verify them in parallel, with a provable K^{1/3} speedup and exact sample quality.","lead":"This paper shows that, after a mathematical change of coordinates, the denoising increments of a diffusion model are exchangeable, which lets the model guess many future steps at once and verify them in parallel. The authors use this to build Autospeculative Decoding, an exact no-draft-model parallel sampler with a provable speedup, and demonstrate 1.8 to 4 times wall-clock acceleration.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Speedup proof rests on a time-shift drift bound, equation (23), that fails pointwise for a Gaussian target; Theorem 5 as written is not established.","rationale":"Reading in good faith: the paper has two central claims — (A) ASD is an exact (error-free) parallelization of the Euler-discretized process (Theorem 4), and (B) it achieves O(K^{2/3}(beta d eta)^{1/3}) parallel model calls, i.e., a tilde-O(K^{1/3}) speedup (Theorem 5). Claim (A) is robust: it needs only the verifier (Theorem 15, whose reflection-coupling proof is checkable) and the fact that the proposal at the first speculated index matches the target mean, forcing progress; exchangeability is not needed. Claim (B) is where the manuscript is weakest.\n\nThe reader's concern about Theorem 2 is correct as a matter of statement: the permuted-interval identity fails for unequal step sizes because the joint law of Gaussian increments depends on interval overlaps, which permutations do not preserve; the proof in Appendix B.2 requires disjoint intervals. The equal-step exchangeability consequence is true and is what the paper really needs. I agree with the reader that the analysis leans on SL time-shift properties.\n\nHowever, the more load-bearing defect is one level deeper, in the proof of Theorem 18. Inequality (23) asserts a pointwise L2 bound on the difference of the SL drift map at two times, evaluated at the same process point, controlled by integral E[Tr(Sigma^2)]. Theorem 12 gives the martingale difference of the drift along the process path (different positions, d m_t = Sigma_t dW_t); it gives no such control at a fixed point. Filling in the 1-D Gaussian case, in which every quantity is closed form, the asserted bound fails by about 4.8x at (u,v) = (10,20). This is a concrete, falsifiable defect in the central speedup proof, not a matter of taste. The same neighborhood contains the discrete/continuous transfer the reader flagged, and the SL-to-DDPM transfer asserted in one sentence before Theorem 5.\n\nSecondary transcription issues (Algorithm 2's j-update; Algorithm 1 line 11's y_i versus y-hat_i) make the printed algorithm not the exact sampler Theorem 4 describes; these are obviously fixable, and the empirical results suggest the implementation used the intended logic. Credit where due: the equal-step exchangeability insight is real and original; the exactness argument for the intended algorithm is sound; Theorem 15 is verifiable; and the CLIP/FID/Robomimic experiments give genuine empirical support (1.8–4x speedup without quality loss). The acknowledgment of concurrent work (De Bortoli et al., 2025) also caps novelty of the algorithmic idea.\n\nVerdict: keep the reader's CONDITIONAL. The correctness claim and the empirics justify conditional acceptance, but the speedup guarantee needs a corrected proof of Theorem 18 (or a restatement of Theorem 5), the pseudocode must be fixed, and the DDPM transfer must be made explicit.","tokens_in":21159,"tokens_out":31696,"duration_ms":311562,"concrete_test":"Compute, for a 1-D Gaussian target mu = N(0,1), both sides of the middle inequality in equation (23) exactly, using m(t,y) = y/(1+t), Sigma_t = 1/(1+t), and E[Y_u^2] = u^2+u: LHS = (1/(1+u) - 1/(1+v))^2 * (u^2+u), RHS = 1/(1+u) - 1/(1+v), at (u,v) = (10,20) and (2,7). If the ratio exceeds 1 as this closed-form calculation indicates, Theorem 18's proof is invalid as written. Then re-derive the TV bound using only the valid martingale identity E||m(u,Y_u) - m(v,Y_v)||^2 = integral_u^v E[Tr(Sigma_s^2)] ds and a correct fixed-point drift-difference formula, and check whether the O(K^{2/3}(beta d eta)^{1/3}) round-complexity bound survives with schedule-independent constants; if not, Theorem 5 must be restated or its theta tuning changed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's headline guarantee (Theorem 5) is derived in Theorem 18 (Appendix C.3), whose key estimate is inequality (23): it asserts E[||m(t_l+t, y_t) - m(t_i+t, y_t)||^2] <= integral_{t_l+t}^{t_i+t} E[Tr(Sigma_s^2)] ds, i.e., that the SL drift map evaluated at the same process point at two different times is close in L2 with a bound given by the covariance decay. Theorem 12 (d m_t = Sigma_t dW_t) only controls the martingale difference m(s, Y_s) - m(s', Y_{s'}) with the process at its own position; the fixed-point comparison asserted in (23) does not follow from it. It is in fact false: take a 1-D Gaussian target x* ~ N(0,1), where m(t,y) = y/(1+t), Sigma_t = 1/(1+t), and E[Y_u^2] = u^2 + u. With u = 10, v = 20, the LHS equals (1/11 - 1/21)^2 * 110 ~ 0.206, while the RHS equals 1/11 - 1/21 ~ 0.043; the asserted inequality fails by about 4.8x, and it also fails at (u,v) = (2,7) (ratio about 1.25), inside the parameter ranges the proof must cover. Consequently Theorem 18's derivation, and with it the O(K^{2/3}(beta d eta)^{1/3}) round-complexity claim, is invalid as written. Two corroborating gaps: (i) Theorem 5 is stated for the DDPM process (5) with user step sizes eta_k, but the proof is for SL; the reparametrization gamma, zeta of Theorem 1 is asserted to transfer the guarantee without analyzing how step sizes, noise variances, and TV distances change. (ii) The printed pseudocode does not match the theorem: Algorithm 2 sets j to the largest accepted index instead of the first rejection, and Algorithm 1 line 11 uses y_i instead of y-hat_i in the target mean, so the algorithm literally printed is not the exact sampler of Theorem 4 (the intended logic is apparent and fixable). The equal-step exchangeability statement is true, Theorem 15's reflection coupling is verifiable, and the empirical speedups are credible; the burden falls on the speedup guarantee, which is the paper's headline theoretical claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper connects DDPM reverse processes to stochastic localization, claims a hidden exchangeability property for denoising increments, and introduces Autospeculative Decoding (ASD), a speculative-decoding-style parallel sampler that uses the diffusion model itself as its own draft. The main advertised results are: Theorem 4, stating that ASD is an error-free parallelization whose outputs are exactly distributed as the Euler-discretized process (5); Theorem 5, stating that under Tr(Cov[mu]) <= beta d and eta_k <= eta, ASD makes O(K^{2/3}(beta d eta)^{1/3}) parallel model calls in expectation and with high probability, yielding a tilde-O(K^{1/3}) speedup; and empirical speedups of 1.8-4x on image generation and robot control tasks.","tokens_in":21575,"tokens_out":20965,"duration_ms":208319,"significance":"The idea of using time-exchangeability of diffusion increments to eliminate an auxiliary draft model is original and potentially important, and the Gaussian rejection sampler in Theorem 15 is a clean, correct construction. The empirical evaluation is also honest about the gap between algorithmic and wall-clock speedups. However, the two central theoretical pillars are not established as written: the exchangeability statement for non-uniform increments is false as stated, the proof of Theorem 5 rests on a false inequality, and the printed verifier is inconsistent with the correctness proof. If these issues were repaired, the paper could make a significant contribution; as it stands, the advertised theoretical speedup is unsupported.","major_comments":[{"comment":"Algorithm 2 as printed computes the largest accepted index: j is initialized to a+1 and updated to i whenever b_i = True and j < i. If proposals at indices a+1 and a+3 are accepted while a+2 is rejected, the verifier returns j = a+3, and Algorithm 1 then accepts y_{a+3} = z_{a+3}, even though z_{a+3} was generated from the proposal trajectory \\hat y_{a+2} and not from the resampled state y_{a+2} = z_{a+2}; a valid sample from q(y_{a+3} | y_{a+2}) requires conditioning on y_{a+2}. Consequently, the printed algorithm can output samples that are not distributed according to the target chain, and Theorem 4 is not established for the pseudocode as written. The proof of Lemma 16 appears to assume that the verifier returns the first rejected index and stops there.","section":"Algorithm 2 / Theorem 4"},{"comment":"The key estimate (23), E[||m(t_l+t,Y) - m(t_i+t,Y)||^2] lesssim integral_{t_l+t}^{t_i+t} E[Tr(Sigma_s^2)] ds, is false. For the one-dimensional Gaussian target mu = N(0,1), m(t,y) = y/(1+t) and Sigma_t = 1/(1+t), and the SL process Y_t has E[Y_u^2] = u^2 + u. Taking t=0, t_l=10, t_i=20, the left side equals (1/11 - 1/21)^2 * 110 approximately 0.206 while the right side equals 1/11 - 1/21 approximately 0.043, so the inequality fails by a factor of about 4.8; taking t_l=100 and t_i=200 makes the ratio about 50. Theorem 12 controls differences of m along a single trajectory, namely m_s(Y_s) - m_{s'}(Y_{s'}), not the difference of two time-indexed drift fields evaluated at the same point, so no constant rescues the argument. Theorem 18, and with it the round-complexity claim of Theorem 5, is therefore not proven.","section":"Appendix C.3, Eq. (23)"},{"comment":"The general time-invariance statement in Theorem 2 is false for non-uniform increments. For t_1=0, t_2=1, t_3=3, eta_1=1, eta_2=2, and pi the transposition of 1 and 2, the left-hand vector consists of Brownian increments over [0,1] and [1,3], which are independent, while the right-hand vector consists of increments over [0,2] and [1,2], which overlap and are correlated; conditional on x*, the two joint laws differ. The equal-step consequence is correct because the intervals are then disjoint, but Theorem 5 permits arbitrary eta_k <= eta, so the exchangeability premise for the general schedule used by ASD is not established.","section":"Theorem 2 / Appendix B.2"},{"comment":"Theorem 5 is stated for the DDPM process (5), but the proof in Appendix C.3 is carried out only for the Euler discretization of the SL process, and the paper asserts the transfer to DDPMs 'due to its equivalence to SL' without analysis. Theorem 1 is a continuous-time pathwise reparametrization; it does not map Euler discretizations of an arbitrary DDPM to Euler discretizations of SL with the same step sizes, noise variances, or TV distances. A transfer lemma is needed and absent, so even the SL-level bound, if repaired, would not immediately imply the DDPM-level statement.","section":"Theorem 5 vs Appendix C.3"}],"minor_comments":[{"comment":"The target mean m_{i+1} is written as y_hat_i + eta_i g(t_i, y_i); for i > a, y_i is undefined at that point and should be y_hat_i.","section":"Algorithm 1, line 11"},{"comment":"The expression sigma_{i+1} = sqrt(eta_i g(T - t_i)) should presumably read sqrt(eta_i u(T - t_i)), since g denotes the drift function and u the noise coefficient.","section":"Equation (3)"},{"comment":"The sentence defining eta_i contains the typo 'eta_i = t_{i+1} - eta_i'; it should read eta_i = t_{i+1} - t_i.","section":"After Equation (5)"},{"comment":"The final display in the proof of Theorem 13 uses h where the step size eta was defined earlier; please make the notation consistent.","section":"Theorem 13 proof"}],"recommendation":"reject","confidential_remarks":"The paper contains a promising algorithmic idea and some correct components, notably the reflection-coupling sampler, but the printed algorithm and the main theoretical proof both have load-bearing errors. The empirical results do not compensate for an invalid Theorem 5. I would reconsider a resubmission if the verifier bug is fixed and a correct proof of the parallel-complexity bound is supplied."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know about arXiv:2505.03983. The hidden-exchangeability observation is real, and the autospeculative sampler is a clean, exact parallelization idea. But the headline speedup guarantee (Theorem 5) is not established: the key inequality in the proof, equation (23), is false.\n\nWhat is actually new: the paper sees that, under the alternate representation y_t = t x* + W_t, the increments of stochastic localization are exchangeable (for equal step sizes), and uses that to let the diffusion model speculate for itself, no draft model. The verifier via reflection coupling is correct, and the exactness claim (Theorem 4) is sound once the intended pseudocode is restored. The experiments are credible: 1.8–4x wall-clock speedups with no quality drop, on image and robot-control tasks.\n\nThe soft spots are real and one is load-bearing. Equation (23) asserts E[||m_u(Y)-m_v(Y)||^2] ≤ ∫_u^v E[Tr(Σ_s^2)] ds. That fails for a one-dimensional Gaussian target: with u=10, v=20 the left side is about 0.206 and the right side about 0.043. So the O(K^{2/3}(βdη)^{1/3}) round-complexity claim is unsupported. Theorem 2 also overclaims: the general time-invariance for arbitrary permuted start times does not follow from the independent-increment argument, which only works for disjoint intervals (the equal-step case). The proof transfers from continuous SL to the discrete DDPM chain without justification. And the printed pseudocode has typos that matter: Algorithm 1 line 11 uses y_i instead of \\hat{y}_i, and Algorithm 2 returns the last accepted index, not the first rejection, so the literal algorithm is not the exact sampler of Theorem 4. All fixable, but they indicate the manuscript needed another pass.\n\nThe citation pattern is fine: they credit Montanari, Benton et al., and acknowledge concurrent work by De Bortoli et al. Self-citation to Anari et al. 2024a is appropriate as inspiration, not a flaw.\n\nThis paper is for inference-systems researchers and sampling theorists. The exchangeability insight and the exact sampler are worth engaging with even if the speedup theorem falls. It deserves a serious referee, but the referee should expect heavy revision: repair (23), correct Theorem 2, fix the pseudocode, and analyze the SL-to-DDPM discretization transfer. If the speedup proof cannot be fixed, the paper should stand on the exact sampler and the exchangeability observation, without the O(K^{1/3}) claim.","headline":"A genuinely useful exchangeability insight and an exact parallel sampler sit beneath a headline speedup theorem whose proof is currently invalid.","tokens_in":22205,"tokens_out":4467,"would_cite":true,"duration_ms":44490,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that DDPM denoising increments are secretly exchangeable after a Stochastic Localization reparametrization, which makes the model its own draft: Autospeculative Decoding proposes all future increments from one model call…","keywords":["diffusion models","stochastic localization","exchangeability","speculative decoding","parallel sampling","denoising diffusion probabilistic models","rejection sampling","adaptive complexity"],"falsifier":"Two concrete checks settle whether the central claim holds. First, simulate the Euler-discretized SL process with non-uniform step sizes, run ASD, and compare the measured acceptance rate of proposals with the value predicted by the continuous time-invariance bounds; a systematic shortfall as step sizes become non-uniform would invalidate the transfer the speedup proof relies on. Second, numerically compare the two joint laws in Theorem 2's permutation identity for overlapping intervals; if Law((y_{t_i+\\eta_i} - y_{t_i})_i) and Law((y_{t_{\\pi(i)}+\\eta_i} - y_{t_{\\pi(i)}})_i) differ measurably, the general statement is false even though the equal-step exchangeability used by the algorithm survives.","tokens_in":20961,"feed_emoji":"⚡","tokens_out":16798,"duration_ms":145998,"temperature":0.7,"pith_summary":"The paper claims that the denoising increments of a diffusion model are 'hidden-exchangeable': after the time/scale change that identifies a DDPM with a Stochastic Localization process, the joint law of the increments is invariant under permutation. The consequence is that, at any point in a denoising run, the conditional distribution of every future increment given the past is the same as that of the immediately next increment, so one increment's distribution can serve as a proposal for all future increments. This yields Autospeculative Decoding (ASD), which uses a single model call to speculate a block of future steps and a single parallel round of rejection sampling to verify them, accepting the longest valid prefix before resampling. The paper proves ASD is an error-free parallelization, with outputs distributed exactly as the sequential Euler-discretized process, and that under a bounded-second-moment condition the expected number of parallel model calls is O($K^{{2/3}}$(\\$\\beta$ d \\eta)^{1/3}), a tilde-O($K^{{1/3}}$) speedup over sequential DDPM. Experiments on latent and pixel image models and on robot-control diffusion policies report 1.8–4× wall-clock speedups with sample quality (CLIP, FID, success rate) unchanged.","feed_headline":"Denoising steps are exchangeable—so DDPMs run in parallel","feed_subtitle":"ASD speculates future steps with one model call each, verifies in parallel, and matches sequential DDPM output exactly.","key_machinery":"The central object is Stochastic Localization, the process d\\bar{y}_t = m(t, \\bar{y}_t) dt + dB_t whose drift m(t, y) = E[x^\\star | t x^\\star + \\sqrt{t} \\xi = y] is the posterior mean of a hidden target x^\\star given a noisy observation, and which converges to the target measure \\mu as t \\to \\infty. Two facts do the work: the equivalence theorem that DDPM reverse processes are reparametrizations of SL, which imports exchangeability into the DDPM setting; and the evolution equations d m_t = \\Sigma_t dW_t together with d E[\\Sigma_t]/dt = -E[\\$Sigma_t^{2}$], which bound how quickly the posterior mean and covariance change and therefore control the total-variation gap between proposal and target in the adaptive-complexity proof. The algorithmic machinery is the single-call proposal distribution \\hat{b}(\\eta_i, y_i, y_a) = y_i + \\eta_i g(t_a, y_a), which speculates all future means from the current model output, and the Gaussian Rejection Sampler, an O(1)-time reflection coupling that verifies \\$\\theta$ proposals in one parallel round.","core_discovery":"On its own terms, the paper establishes that the increments of a DDPM form an exchangeable sequence. The chain of argument runs: every DDPM reverse process is a reparametrization of Stochastic Localization (Theorem 1); SL increments obey a time-invariance property that makes equal-size increments exchangeable (Theorem 2); and therefore, conditioned on the observed trajectory up to step a, the law of any future increment equals the law of the next increment. The paper then builds Autospeculative Decoding on this fact: one oracle call at the current point produces proposal means for the next \\$\\theta$ increments; a parallel round computes the true target means; and a Gaussian Rejection Sampler, derived from reflection coupling, checks each proposal while drawing from the target conditional, so the longest accepted prefix is exactly a sample from process (5) (Theorem 4). For the Euler-discretized SL process with Tr(Cov[\\mu]) \\le \\$\\beta$ d and step sizes \\eta_k \\le \\eta, choosing \\$\\theta$ \\asymp (K/\\$\\beta$ \\eta d)^{1/3} makes the expected number of parallel model calls O($K^{{2/3}}$(\\$\\beta$ d \\eta)^{1/3}) (Theorem 5), i.e., about $K^{{1/3}}$-fold fewer parallel rounds than the K sequential calls of a vanilla DDPM, with the same tilde-O($K^{{1/3}}$) guarantee stated in the abstract.","pith_inferences":["My extension: the exchangeability recipe is general—any sequential sampler whose increments can be rendered exchangeable by reparametrization could be parallelized by self-speculation, predicting a similar K^{1/3}-type trade-off for other localization-based sampling schemes.","My extension: the bound suggests the acceptance rate is governed by how fast the posterior covariance \\Sigma_t shrinks, so an adaptive speculation length that grows as the posterior stabilizes should outperform the fixed \\theta used in the paper; this is a testable design change.","My extension: since the guarantee uses only second moments, heavy-tailed or strongly multimodal targets within the same covariance budget should exhibit the same speedup, which one could verify by measuring ASD's acceptance rate across such targets.","My extension: if the discrete non-uniform transfer flagged in weakest_assumption_plain fails empirically, the natural repair is to equalize the step sizes used for speculation so the increments are exactly exchangeable before applying the speedup bound, rather than abandoning the algorithm."],"forward_implications":["Error-free parallelization: ASD's output at every accepted prefix is distributed exactly as the sequential Euler-discretized DDPM, so the usual trade of sample quality for speed is removed by construction.","A guaranteed speedup without smoothness assumptions: only a bounded second moment is required to reduce the expected number of parallel model calls from K to O(K^{2/3}); no Lipschitz or other score regularity is needed.","No draft model: because the next increment's conditional law serves all future increments, the diffusion model speculates about itself, eliminating the auxiliary-model overhead of classical speculative decoding.","Diffusion as an any-order autoregressive model: the exchangeability view opens DDPM increments to optimization techniques previously limited to autoregressive architectures, beyond the speculative decoding demonstrated here.","Empirical acceleration: on StableDiffusion-v2, an LSUN Church pixel model, and three Robomimic tasks, ASD attains 1.8–4× wall-clock speedups (up to 7× algorithmic) with CLIP, FID, and task success rates matching vanilla DDPM."],"supporting_citations":[{"why":"Proves the OU DDPM is equivalent to Stochastic Localization, the result this paper extends to arbitrary DDPMs in Theorem 1.","marker":"Montanari (2023)"},{"why":"Supplies the tilde-O(d)-step sequential runtime benchmark and the SL-based discretization/KL arguments that Theorem 5 improves upon in parallel.","marker":"Benton et al. (2023)"},{"why":"Introduces Stochastic Localization, the process whose increment exchangeability is the paper's core discovery.","marker":"Eldan (2013)"},{"why":"Gives the alternate representation y_t = t x* + W_t and the convergence properties used in the proof of Theorem 2.","marker":"El Alaoui & Montanari (2022)"},{"why":"Provides the any-order autoregressive speculative-decoding analysis, including the round-complexity and TV-distance arguments, that the proof of Theorem 5 adapts.","marker":"Anari et al. (2024a)"},{"why":"Defines speculative decoding, the paradigm that ASD transfers to diffusion models while removing the draft model.","marker":"Leviathan et al. (2023)"},{"why":"Supplies the localization-scheme framework and the evolution equations for the posterior covariance used in the complexity proof.","marker":"Chen & Eldan (2022)"},{"why":"Reflection coupling, the technique that motivates the O(1) Gaussian Rejection Sampler in Algorithm 3.","marker":"Bou-Rabee et al. (2020)"}],"fun_headline_variants":["Exchangeable denoising steps turn DDPMs parallel","Autospeculation: parallel DDPM without auxiliary drafts","Hidden exchangeability in DDPM enables parallel decoding","DDPM increments exchangeable: parallel inference proven","Speculative decoding for DDPM: no draft models required"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The speedup guarantee rests on one premise: the time-symmetry proven for the idealized continuous process still governs the discrete denoising chain when step sizes are non-uniform, so that the exchangeability of increments transferred from Stochastic Localization bounds the real proposal–target gap; the paper only proves the exact exchangeability identity for the equal-step, non-overlapping case, while the theorem states it more generally.","fun_headline_variants_meta":{"raw":{"variants":["Exchangeable denoising steps turn DDPMs parallel","Autospeculation: parallel DDPM without auxiliary drafts","Hidden exchangeability in DDPM enables parallel decoding","DDPM increments exchangeable: parallel inference proven","Speculative decoding for DDPM: no draft models required"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000266,"raw_usage":{"total_tokens":1649,"prompt_tokens":1020,"completion_tokens":629,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":636,"completion_tokens_details":{"reasoning_tokens":552}},"tokens_in":636,"tokens_out":629,"duration_ms":6118,"temperature":1.0,"reasoning_tokens":552,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:42:15.717792+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Two concrete checks settle whether the central claim holds. First, simulate the Euler-discretized SL process with non-uniform step sizes, run ASD, and compare the measured acceptance rate of proposals with the value predicted by the continuous time-invariance bounds; a systematic shortfall as step sizes become non-uniform would invalidate the transfer the speedup proof relies on. Second, numerically compare the two joint laws in Theorem 2's permutation identity for overlapping intervals; if Law((y_{t_i+\\eta_i} - y_{t_i})_i) and Law((y_{t_{\\pi(i)}+\\eta_i} - y_{t_{\\pi(i)}})_i) differ measurably, the general statement is false even though the equal-step exchangeability used by the algorithm survives.","supporting_citations":[{"cited_title":"Sampling from the sherrington-kirkpatrick gibbs measure via algorithmic stochastic localization","cited_arxiv_id":null,"evidence_quote":"Gives the alternate representation y_t = t x* + W_t and the convergence properties used in the proof of Theorem 2."},{"cited_title":"and Eldan, R","cited_arxiv_id":null,"evidence_quote":"Supplies the localization-scheme framework and the evolution equations for the posterior covariance used in the complexity proof."}],"review_version":1}