{"id":"1bd3debc-44b2-4dde-9451-cdf4f2612af5","arxiv_id":"2312.02089","paper_version":3,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Under local spectral assumptions from Gur et al., the sequential sweep on high-dimensional expanders has spectral gap near 1 and satisfies entropy contraction, generalizing rapid mixing results from Ramanujan complexes.","lead":"The paper proves that sequential sweep walks on n-partite high-dimensional expanders achieve spectral gaps arbitrarily close to 1 under strong local spectral assumptions, while n steps of the standard down-up walk remain bounded by a constant. Readers outside the subfield may care because this refines mixing time analysis for sampling algorithms on complex combinatorial structures.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption treats the truth of the local spectral premise for n-partite complexes as load-bearing. That premise is not asserted by the paper; the paper asserts only the implication from the premise. Hence the load-bearing question for the stated claim is whether the implication is correctly proved, not whether the premise is true. The two concerns are distinct, so the reader's identification does not match the load-bearing point for the paper's central (conditional) statement.","tokens_in":1809,"tokens_out":333,"duration_ms":42169,"concrete_test":"Locate the section deriving the spectral-gap bound for the sequential sweep and verify that every eigenvalue estimate reduces exactly to an invocation of the Gur-Lifschitz-Liu local spectral gap (or the entropy-contraction variant) with no additional global assumptions; if the reduction is direct and the cited theorem applies verbatim, the implication holds.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is conditional: under a sufficiently strong local spectral assumption (Gur-Lifschitz-Liu STOC 2022), the sequential sweep operator on an n-partite complex has spectral gap arbitrarily close to 1. The abstract and description present this as a direct consequence of the cited assumption plus the definition of the sweep as a fixed-order composition of coordinate updates. No internal gap, unstated hypothesis, or misapplication of the local assumption is visible in the stated argument. The paper likewise conditions its entropy-contraction and mixing results on local entropy contraction, again without asserting the premise holds universally.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The paper studies an alternative to the down-up walk on n-partite simplicial complexes called the sequential sweep (systematic scan Glauber dynamics), in which coordinates are updated in a fixed deterministic order rather than randomly. It claims that, while the n-step down-up walk remains bounded by a constant spectral gap, the sequential sweep achieves spectral gap arbitrarily close to 1 under a sufficiently strong local spectral assumption (Gur-Lifschitz-Liu, STOC 2022). The manuscript also derives entropy contraction for these walks under local entropy contraction assumptions and generalizes the rapid-mixing result of Lubetzky-Lubotzky-Parzanchevski (JEMS) from Ramanujan complexes to suitable high-dimensional expanders.","tokens_in":1941,"tokens_out":469,"duration_ms":13479,"significance":"If the local spectral and entropy assumptions hold for the n-partite complexes under consideration, the result supplies a concrete mechanism for obtaining near-optimal mixing via systematic scan on high-dimensional expanders, extending prior work on Ramanujan complexes. The conditional framing and explicit dependence on the Gur et al. assumption are stated clearly, which strengthens the contribution by avoiding over-claim. No machine-checked proofs or parameter-free derivations are present, but the generalization to HDX and the entropy-contraction extension are substantive if the premises apply.","major_comments":[],"minor_comments":[{"comment":"Abstract, line 8: the phrase 'arbitrarily close to 1' should be accompanied by a brief parenthetical indicating the dependence on the strength of the local spectral assumption (e.g., 'as the local gap parameter tends to 1').","section":null},{"comment":"The manuscript should add a short paragraph in the introduction clarifying whether the Gur-Lifschitz-Liu local spectral assumption is known to hold for any explicit infinite family of n-partite complexes beyond Ramanujan complexes, or whether this remains open.","section":null},{"comment":"Notation: the distinction between the single-step sequential sweep operator and its n-fold composition should be denoted consistently (e.g., P_seq vs P_seq^n) throughout the proofs to avoid reader confusion when comparing to the n-step down-up operator.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. The review correctly identifies the core contributions and the conditional nature of the results under the Gur-Lifschitz-Liu assumptions.","responses":[],"tokens_in":1368,"tokens_out":62,"duration_ms":8145,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"This paper shows that sequential sweeps achieve spectral gap close to 1 on n-partite complexes when the strong local spectral assumption from Gur-Lifschitz-Liu holds, beating the O(1/n) cap on standard down-up walks. It also gets entropy contraction under the related local entropy condition. The main new piece is carrying the Lubetzky-Lubotzky-Parzanchevski rapid-mixing result over from Ramanujan complexes to arbitrary high-dimensional expanders that meet the local assumption. The write-up does a clean job contrasting the fixed-order sweep with random coordinate selection and explaining how the deterministic order sidesteps coboundary obstructions. The argument stays consistent with the cited prior work and does not introduce circular steps or self-referential fitting. The central limitation is that everything is conditional on the local assumption being strong enough; the paper does not claim the assumption holds for most complexes, so applicability stays narrow if the premise is rare or hard to check. No load-bearing gaps appear in the abstract or stress-test description. Readers working on high-dimensional expanders, mixing times, or sampling algorithms will get value from the concrete improvement under the stated conditions. I would bring this to a reading group as maybe, since the conditional framing makes it worth discussing but not essential for everyone. I would not cite it in my own work in the next year unless I am directly in this subfield. It deserves peer review because the extension is substantive and the claims rest on external verifiable assumptions rather than internal contradictions.","headline":"Extends rapid mixing for sequential sweeps to general high-dim expanders under Gur et al local assumption, generalizing the Ramanujan case while staying conditional on that premise.","tokens_in":2437,"tokens_out":378,"would_cite":false,"duration_ms":15707,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[{"relation":"unclear","rs_module":"IndisputableMonolith/Foundation/RealityFromDistinction.lean","rs_theorem":"reality_from_one_distinction","paper_passage":"under a strong enough local spectral assumption (in the sense of Gur, Lifschitz, Liu, STOC 2022) we can show that the spectral gap of this walk can be arbitrarily close to 1"},{"relation":"unclear","rs_module":"IndisputableMonolith/Cost/FunctionalEquation.lean","rs_theorem":"washburn_uniqueness_aczel","paper_passage":"σ₂(P_seq)² ≤ 1 − ∏_{j=2}^n (1 − (ε_{[j−1]→j})²)"}],"headline":"Spectral analysis of sequential sweeps on partite complexes; no overlap with RS forcing from distinction to J-cost/φ/8-tick","alignment":"orthogonal","rationale":"Paper studies spectral gaps of P_seq = Q_1⋯Q_n (products of coordinate-update projections) on n-partite simplicial complexes under local spectral expansion (γ_i) or ε-product assumptions, deriving bounds via subspace angles cos(U_I,U_J) ≤ ε_{I→J} and products ∏(1−ε²). Central objects (colored walks C_{I→J}^α, entropy contraction η_{I→J}, SSW77 angle theorem) are standard in high-dimensional expanders and Markov chains. RS framework (reality_from_one_distinction, J(x)=½(x+x⁻¹)−1 uniqueness, φ-ladder constants, 8-tick periodicity, AlexanderDuality for D=3) contains none of these structures or assumptions; paper never invokes recognition cost, golden-ratio identities, or parameter-free derivation of constants. Domain mismatch (cs.DM vs. logic-to-physics forcing) confirms orthogonality.","tokens_in":69309,"confidence":"high","tokens_out":432,"duration_ms":8381,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Under strong local spectral assumptions, the sequential sweep on n-partite complexes achieves spectral gap arbitrarily close to 1.","keywords":["sequential sweep","high dimensional expanders","spectral gap","Glauber dynamics","simplicial complexes","local spectral expansion","entropy contraction"],"falsifier":"An explicit n-partite simplicial complex that satisfies the strong local spectral assumption yet has sequential sweep spectral gap bounded strictly below 1.","tokens_in":2707,"feed_emoji":"","tokens_out":631,"duration_ms":17045,"temperature":0.7,"pith_summary":"Standard down-up walks on n-partite simplicial complexes are limited to spectral gaps of order 1/n by obstructions such as coboundaries. The paper studies the sequential sweep, which updates coordinates in a fixed deterministic order instead of selecting them randomly. When the complex satisfies a sufficiently strong local spectral assumption, one full sweep attains a spectral gap that can be made arbitrarily close to 1. The same local condition also implies an entropy contraction inequality for the walk. The argument extends known rapid mixing of sequential sweeps on Ramanujan complexes to the broader setting of high-dimensional expanders.","feed_headline":"Sequential sweep reaches spectral gap near 1 on expanders","feed_subtitle":"Under strong local spectral conditions one ordered pass mixes faster than n random updates on partite complexes.","key_machinery":"The sequential sweep (systematic scan Glauber dynamics), which traverses the n coordinates in a fixed deterministic order and applies the local update to each coordinate in turn.","core_discovery":"Under the strong local spectral assumption of Gur, Lifschitz, Liu, the spectral gap of the sequential sweep walk can be made arbitrarily close to 1, while the spectral gap of the n-th power of the down-up walk remains bounded by a constant. Under local entropy contraction assumptions, the sequential sweep satisfies an entropy contraction inequality. This generalizes the rapid mixing result for sequential sweeps on Ramanujan complexes to suitable high dimensional expanders.","pith_inferences":["Ordered update schemes may be preferable to random coordinate selection for sampling on certain expanders.","The local-to-global spectral lifting technique could apply to other deterministic scan orders in Markov chain analysis.","Constructions of high dimensional expanders can be checked directly for the local spectral strength required to reach gap near 1."],"forward_implications":["The mixing time of the sequential sweep becomes independent of n under the local spectral assumption.","High dimensional expanders admit rapid mixing via systematic scan Glauber dynamics.","The walk satisfies an entropy contraction inequality under the related local entropy contraction condition.","Rapid mixing results extend from Ramanujan complexes to all high dimensional expanders meeting the local spectral condition."],"fun_headline_variants":["Sequential sweep nears spectral gap 1 on expanders","Sequential sweep gap approaches 1 under local spectral conditions","High dimensional expanders yield sequential sweep gap near 1","Sequential sweeps achieve near-1 gap on partite complexes","Local assumptions let sequential sweep gap near 1 while n-powers stay bounded"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The n-partite simplicial complex satisfies the strong local spectral assumption defined in Gur, Lifschitz, Liu (STOC 2022).","fun_headline_variants_meta":{"raw":{"variants":["Sequential sweep nears spectral gap 1 on expanders","Sequential sweep gap approaches 1 under local spectral conditions","High dimensional expanders yield sequential sweep gap near 1","Sequential sweeps achieve near-1 gap on partite complexes","Local assumptions let sequential sweep gap near 1 while n-powers stay bounded"]},"model":"grok-4.3","cost_usd":0.005966,"raw_usage":{"total_tokens":2860,"prompt_tokens":732,"num_sources_used":0,"completion_tokens":73,"cost_in_usd_ticks":59662000,"prompt_tokens_details":{"text_tokens":732,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2055,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":732,"tokens_out":73,"duration_ms":12188,"temperature":1.0,"reasoning_tokens":2055,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-24T04:54:11.546251+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit n-partite simplicial complex that satisfies the strong local spectral assumption yet has sequential sweep spectral gap bounded strictly below 1.","supporting_citations":[],"review_version":1}