{"id":"7ff481e1-2494-496f-9bf6-52081f75886e","arxiv_id":"2608.02373","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A correlated Ising prior lowers the reconstruction threshold when it is paramagnetic, but when it is glassy it pushes the Bayes-optimal posterior into a static replica-symmetry-breaking phase under Nishimori conditions.","lead":"Using cavity and message-passing methods, this paper maps when noisy edge observations allow recovery of a spin signal drawn from a correlated Ising prior on random regular graphs. The notable claim is that when the prior itself is glassy, the Bayes-optimal posterior can enter a static replica-symmetry-breaking phase even on the Nishimori line, so correlated signals can make inference hard rather than easy.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Posterior s-RSB transition rests on the x=1 ansatz, which the paper itself says is not the correct description of the condensed phase; without the x<1 solution, β_rsb and the s-RSB branch are not yet established.","rationale":"The reader's weakest_assumption identifies the same load-bearing concern: the posterior s-RSB transition is located using the x=1 simplified 1RSB equations, and the paper explicitly concedes that the correct thermodynamic description inside a static-RSB phase requires x<1. This is the single most load-bearing issue because the headline claim is not merely the existence of some glassy slowdown but a static RSB phase on the Nishimori line. The x=1 negative-complexity criterion is a plausible indicator, but the free-energy crossing between the RS solution and the x=1 branch is used as if it were the physical transition. That step is not justified by the paper's own formalism. The finite-size BP evidence does not rescue the analytic claim because sampling from an s-RSB prior via BP-guided decimation is acknowledged to be unreliable in exactly the regime studied. At the same time, the claim is not contradicted: at β=0 the posterior equals the prior, which is known to be in a static RSB phase, so some finite-β s-RSB interval is plausible by continuity. The correct response is therefore to keep the CONDITIONAL verdict and demand the x<1 computation as the decisive check. No ad hominem or theatrical framing is needed; the paper itself flags the limitation, and the reader already captured it accurately.","tokens_in":30897,"tokens_out":6520,"duration_ms":67162,"concrete_test":"Solve the full 1RSB cavity equations (C18) at x<1 for κ=-1.2 (and one κ closer to κ_rsb, e.g., -0.97), using population dynamics over distributions. For β in [0,0.5], find x*(β) from the zero-complexity condition and compute the physical free energy Φ(x*) and q0, q1. If the x<1 branch dominates RS only for β<β_rsb≈0.398 and q0<q1 with x*<1, the claim is confirmed; if the crossing shifts or the x<1 branch is absent/not dominant, the x=1 detection is an artifact.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central red-region claim is derived entirely from the x=1 simplification of the 1RSB cavity equations (Appendix C2c, Eq. C27). The paper states in Section IIIB3 that a correct static-RSB description requires x<1 and that x=1 does not give correct thermodynamic observables. Nevertheless, β_rsb(κ) is read off from the free-energy crossing between the RS solution and this x=1 branch, whose complexity is negative. Negative complexity at x=1 may signal condensation, but it does not by itself determine whether the physical x*<1 solution is thermodynamically dominant, nor that its free energy crosses the RS branch at the reported β_rsb. Without solving the x<1 equations, the existence of a posterior static-RSB phase and the location of the transition are unsupported. Finite-size BP (Fig. 5) is only suggestive, and the paper concedes BP-guided decimation (Appendix A2) is not guaranteed accurate for sampling from the s-RSB prior, so the numerics cannot settle the point either.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the planted spin glass (censored block model) on random d-regular graphs when the planted signal is drawn from an Ising prior with coupling κ, rather than an i.i.d. prior. The posterior is an Ising model with edge couplings κ + β J_ij; the Bayes-optimal setting places the model on the Nishimori line. Using distributional cavity equations, the authors derive a phase diagram in (κ, β). For a paramagnetic prior they obtain a stability threshold β_c(κ) (Eq. 11) that decreases with |κ|; for a ferromagnetic prior they show β_p = κ separates the regime where the posterior improves over the prior estimator; for an s-RSB prior (κ < κ_rsb) they report a transition at β_rsb(κ) from a replica-symmetric easy phase to a static RSB phase, detected from negative complexity in the x=1 1RSB equations (Eq. C27). Finite-size BP simulations support the RS-regime predictions and suggest degraded performance in the low-β s-RSB regime.","tokens_in":31236,"tokens_out":7028,"duration_ms":58017,"significance":"If the posterior s-RSB transition were established, it would be a notable counterexample to the usual expectation that Bayes-optimal inference on the Nishimori line is replica symmetric, and it would provide a minimal analytically tractable model of correlated priors. The paper's RS results for paramagnetic and ferromagnetic priors are clean: Eq. (11) is a concrete closed-form stability condition, and the β_p = κ argument is elegant and non-circular. The manuscript also provides code and notebooks. However, the headline claim is currently supported only by the x=1 simplification, which the authors themselves state is not the correct thermodynamic description; the numerical evidence is also weakened by sampling caveats. The significance is high if the x<1 program can be completed or the claims are appropriately downgraded.","major_comments":[{"comment":"The posterior s-RSB transition is the central claim, but it is derived entirely from the x=1 1RSB equations. The manuscript explicitly states in Secs. IIIA and IIIB3 that a correct static-RSB description requires x<1 and that x=1 gives incorrect thermodynamic observables inside the condensed phase. Negative complexity at x=1 may indicate condensation, but it does not by itself establish that the physical x*<1 branch exists, that its free energy crosses the RS branch at the reported β_rsb(κ), or that the transition is a true thermodynamic transition. Without solving the x<1 equations, or at least giving an independent controlled argument for the onset, the headline claim is not established.","section":"IIIB3 / Eq. (C27)"},{"comment":"The free-energy comparison used to locate β_rsb compares the RS free entropy with the x=1 replicated potential of a branch with negative complexity. When Σ<0 the x=1 branch is not a physical state of the Gibbs measure; the quantity Φ_RSB(x=1) includes a negative complexity contribution and is not the physical free entropy of a condensed phase. A crossing between these two objects is therefore not automatically a thermodynamic transition. The authors should compare the RS free entropy with the physical free entropy of the x*<1 solution.","section":"Eq. (C34) / Fig. 4"},{"comment":"The finite-size BP results in the s-RSB prior regime use BP-guided decimation to sample planted configurations from the prior. The authors concede that this method gives accurate marginals only in the RS phase and is not guaranteed accurate in the RSB regime. The energy consistency check is necessary but weak. Thus the numerical simulations are suggestive but cannot independently confirm the existence or location of the posterior s-RSB phase.","section":"Fig. 5 / Appendix A2"},{"comment":"The prior Parisi parameter x0 is estimated by locating the zero of Σ0(x0), where the complexity values are of order 10^-6 near κ_rsb. The authors note this is numerically unreliable close to κ_rsb and consequently omit cavity predictions from Fig. 5. This means the quantitative value of β_rsb(κ) is uncontrolled precisely in the regime used to illustrate the transition.","section":"Appendix C1c"}],"minor_comments":[{"comment":"The abstract and introduction state the detection of a static RSB transition as a definite result, while Sec. IIIB3 says the nature of the phase 'remains unresolved' and the x=1 solution 'does not give correct thermodynamic observables.' Please align the claims with the caveats throughout.","section":"Abstract / Sec. I vs. IIIB3"},{"comment":"The sum over J is not defined; please specify that J ∈ {±1} and define the bracket notation ⟨·⟩ if used.","section":"Eq. (11)"},{"comment":"The caption says 'κ ∈ [κ_rsb, κ_c]' with κ_rsb negative; the sign convention is confusing. Please state explicitly that κ_rsb = -atanh(1/√(d-1)).","section":"Fig. 1 caption"},{"comment":"The opening sentence 'Obtaining the 1-RSB cavity equation for the prior probability is analogous...' appears to be a copy-paste from C2a and should read 'posterior probability'.","section":"Appendix C2b"},{"comment":"The definition of ̅Q_σ and the removal of the reweighting factor are quite terse; a few explanatory sentences would improve readability.","section":"Eq. (C25)"},{"comment":"The GitHub repository [34] is cited but the paper does not state which figures/notebooks correspond to which numerics; please add reproducibility notes.","section":"Code availability"}],"recommendation":"major_revision","confidential_remarks":"The paper is honest about its limitations, and the RS parts are solid. My main concern is that the abstract's central claim goes beyond what the x=1 calculation can support. If the authors can solve the x<1 equations or explicitly reframe the s-RSB transition as a conjecture with supporting numerical evidence, the paper would be suitable. The GitHub availability and the clean β_c and β_p derivations are strengths."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nYou should know two things about this paper: it has a genuinely new model, and its headline claim is real but comes with an explicit caveat that the authors do not hide. The model is the planted spin glass (censored block model) on random regular graphs, with the signal drawn from an Ising prior at coupling κ. The phase diagram is new: for a paramagnetic prior, correlations lower the reconstruction threshold; for a ferromagnetic prior, the posterior never beats the prior estimator below β=κ; for a glassy (antiferromagnetic) prior, they find a static RSB phase in the posterior on the Nishimori line at low β.\n\nThe paramagnetic and ferromagnetic parts are the cleanest. The stability formula (11) is a one-line calculation and is backed by finite-size BP. The argument that β_p=κ is exact is nice: for β≤κ all effective couplings are non-negative, so the posterior is an Ising ferromagnet and the prior estimator is already optimal. These parts are solid.\n\nThe soft spot is the sRSB region. The transition β_rsb is located from the free-energy crossing between the RS solution and the x=1 branch of the 1RSB equations, and that branch has negative complexity. The authors state explicitly that the correct thermodynamic description inside the condensed phase requires x<1 and that x=1 does not give correct observables. So the precise location of the transition and the physics inside the glassy phase are not settled. The stress-test note is right that this is the load-bearing part of the claim, but it goes too far if it implies the existence of the phase is unsupported: negative complexity at x=1 is a standard signature of condensation, and the x=1 equations are a legitimate limit. The qualitative conclusion is plausible, even if the quantitative boundary may shift when the x<1 solution is found.\n\nThe finite-size BP results in the RSB prior region are only suggestive. The planted configurations are sampled via BP-guided decimation, which the authors concede is not guaranteed accurate in an RSB prior. They do check the energy against cavity predictions, which helps, but it does not settle the phase.\n\nThis paper deserves peer review. The referee should push on the x=1 criterion and on whether the free-energy crossing is the right way to locate the static transition, but the model and the clean parts of the analysis are worth publishing. I'd bring it to a reading group for anyone working on structured priors or the Nishimori line, and I'd cite it if I were working on message passing in these models.\n\nMy recommendation: send it to a serious referee, and don't desk-reject it. The main claim is not proven in full, but it is clearly stated, carefully hedged, and backed by a tractable model that others can build on.","headline":"A solid cavity-method paper with a provocative claim—static RSB on the Nishimori line for a correlated prior—that is honestly caveated but not fully nailed down because the key phase is only analyzed at x=1.","tokens_in":31672,"tokens_out":5701,"would_cite":true,"duration_ms":51638,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["82B44","82B26","62F15"],"pacs":[],"model":"deepseek-v4-flash","headline":"A correlated, non-separable prior can push a Bayes-optimal posterior into a static replica-symmetry-breaking phase on the Nishimori line, with threshold β_rsb(κ) separating the glassy low-β phase from an easy replica-symmetric phase.","keywords":["planted spin glass","censored block model","Nishimori line","replica symmetry breaking","structured prior","cavity method","belief propagation","random regular graphs"],"falsifier":"Solve the full 1RSB cavity equations at the correct Parisi parameter x<1 inside the β<β_rsb region: if the dominant solution is not a condensed phase with zero complexity and q_0<q_1, or if the free-energy crossing shifts or disappears, the reported transition is an artifact of the x=1 simplification. Alternatively, on finite instances under Nishimori conditions, measure the posterior overlap distribution P(q) for β<β_rsb: static RSB predicts a non-trivial, multi-peaked P(q), whereas replica symmetry predicts a concentrated one.","tokens_in":30772,"feed_emoji":"🧲","tokens_out":5544,"duration_ms":48840,"temperature":0.7,"pith_summary":"The paper asks whether correlations in the signal's prior can break the usual replica-symmetric behaviour of Bayes-optimal inference. Using a minimal planted spin glass on random regular graphs whose planted configuration is sampled from an Ising model with coupling κ, it maps three regimes. In the paramagnetic regime, correlations lower the reconstruction threshold β_c(κ). In the ferromagnetic regime, the prior alone gives partial recovery and observations only help above β_p=κ. In the static RSB regime of the prior, the paper finds a posterior transition at β_rsb(κ) from an easy replica-symmetric phase at large β to a static RSB phase at small β, on the Nishimori line — a glassy phase in Bayes-optimal inference driven by the non-separable correlated prior.","feed_headline":"Bayes-optimal inference turns glassy when the signal prior is glassy","feed_subtitle":"In a planted spin glass, a correlated Ising prior puts the posterior into a static RSB phase when observations are weak.","key_machinery":"The central object is the planted spin glass (the Censored Block Model) on random d-regular graphs, with signal drawn from the Ising prior P_κ(s) ∝ ∏_{(ij)∈E} e^{κ s_i s_j} and noisy edge observations J_ij with likelihood e^{β J_ij s_i s_j}/(2 cosh β). The analysis runs through distributional cavity equations: replica-symmetric equations for the paramagnetic and ferromagnetic prior regimes, and, for the glassy prior regime, the simplified 1RSB equations at Parisi parameter x=1, where condensation is signalled by negative complexity Σ and q_0=0<q_1. The threshold β_rsb(κ) is located by the free-energy crossing between this static-RSB branch and the RS branch.","core_discovery":"When the signal is sampled from an Ising prior that is itself in a static RSB phase (κ<κ_rsb), the posterior on the Nishimori line is not always replica symmetric: for β below a threshold β_rsb(κ) the thermodynamically dominant solution is a static RSB (condensed) phase, detected through a negative complexity and q_0=0<q_1 at x=1, while for β above the threshold the RS solution dominates and inference is easy. This provides an example where a non-separable, correlated prior leads to static RSB in a Bayes-optimal inference problem, overturning the standard expectation that Bayes-optimality implies replica symmetry.","pith_inferences":["If the x=1 negative-complexity criterion is confirmed by a full x<1 solution, the mechanism suggests that any non-separable prior with a glassy component can seed a glassy posterior even when the channel is Bayes-optimal; one could look for the same phenomenon in stochastic-block-model variants with correlated priors.","The finite-size finding that a mismatched unstructured prior converges but reconstructs worse suggests a practical consequence: using an overly simple prior can make algorithms look well-behaved while silently degrading recovery; a testable extension is to compare matched versus mismatched priors across the β<β_rsb region.","One sharp quantitative consequence worth testing is that β_c(κ) vanishes as κ approaches κ_rsb from the paramagnetic side, meaning arbitrarily weak observations suffice near the prior condensation transition; this could be probed with spectral or message-passing algorithms on large instances.","The unusual scenario where the dominant x=1 solution has zero overlap with the signal while subdominant informative states exist suggests that recovery inside the glassy phase may hinge on metastable states; this is an editorial inference, not a claim the paper establishes."],"forward_implications":["In the paramagnetic prior regime, the reconstruction threshold β_c(κ) is lower than the unstructured β_c(0), reaching β_c=0 at κ=κ_rsb, so correlations in the signal make inference information-theoretically easier.","In the ferromagnetic prior regime, the prior-based trivial estimator already achieves non-zero overlap; observations improve over it only for β>β_p(κ)=κ.","When the prior is static RSB, the posterior is replica symmetric and easy for β>β_rsb(κ), but becomes a static RSB (condensed) phase for β<β_rsb(κ) even under Nishimori conditions.","The glassy posterior phase is associated with algorithmic hardness: Belief Propagation convergence deteriorates at small β, and the posterior-based estimator can do worse than the prior-based one.","The Nishimori identities still hold on the thermodynamically dominant branch, so the static RSB phase is not a violation of Bayes optimality but a breakdown of the usual replica-symmetric picture for correlated priors."],"fun_headline_variants":["Bayes-optimal inference can be glassy when the prior is","Glassy priors induce static RSB in Bayes-optimal inference","Correlated signal priors trigger static RSB in inference","When the prior breaks symmetry, Bayes-optimal inference follows"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The paper's headline transition is detected with a simplified version of the replica calculation that the authors themselves say does not give correct thermodynamic observables inside the glassy phase; if that simplified negative-complexity criterion does not mark the true onset of the static RSB phase, the central claim shifts or disappears.","fun_headline_variants_meta":{"raw":{"variants":["Bayes-optimal inference can be glassy when the prior is","Glassy priors induce static RSB in Bayes-optimal inference","Correlated signal priors trigger static RSB in inference","When the prior breaks symmetry, Bayes-optimal inference follows"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000348,"raw_usage":{"total_tokens":1734,"prompt_tokens":732,"completion_tokens":1002,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":476,"completion_tokens_details":{"reasoning_tokens":931}},"tokens_in":476,"tokens_out":1002,"duration_ms":13645,"temperature":1.0,"reasoning_tokens":931,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T08:27:10.383509+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Solve the full 1RSB cavity equations at the correct Parisi parameter x<1 inside the β<β_rsb region: if the dominant solution is not a condensed phase with zero complexity and q_0<q_1, or if the free-energy crossing shifts or disappears, the reported transition is an artifact of the x=1 simplification. Alternatively, on finite instances under Nishimori conditions, measure the posterior overlap distribution P(q) for β<β_rsb: static RSB predicts a non-trivial, multi-peaked P(q), whereas replica symmetry predicts a concentrated one.","supporting_citations":[],"review_version":1}