{"id":"c835495c-5e29-4f4f-942a-7ed6d2c164b6","arxiv_id":"2603.05490","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Chromatic threshold δ_χ(L) of a homogeneous linear equation over F_p is zero iff L has a zero-sum subcollection of at least three coefficients; every infinite discrete abelian group has a topologically but not measurably recurrent set.","lead":"The paper defines a chromatic threshold for linear equations over finite fields and proves it vanishes exactly when the equation has a zero-sum subcollection of three or more coefficients. It also builds, on every infinite discrete abelian group, a set that is topologically recurrent but not measurably recurrent.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"Abstract-only review: the load-bearing quantitative chromatic lower bound for Hamming-ball Cayley graphs on Z_p^n cannot be checked, so the classification remains unverified.","rationale":"The Reader correctly identified that the quantitative lower-bound ingredient is the single point on which the classification hinges, and correctly left the paper UNVERDICTED with LOW confidence because only the abstract is present. No additional soft spot can be diagnosed without the proofs; manufacturing one would violate the good-faith rule. The concrete test simply operationalises the missing verification. Hence the verdict stays UNVERDICTED and agreement with the Reader is full.","tokens_in":2176,"tokens_out":468,"duration_ms":5288,"concrete_test":"Obtain the full manuscript and re-derive (or machine-check the topological steps of) the chromatic lower bound for the Hamming-ball generators on Z_p^n. Verify that the bound remains unbounded as density \to0 whenever L has no zero-sum subcollection of size ≥3; if the bound collapses for some positive-density regime, the “only if” direction fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is the exact classification δ_χ(L)=0 iff L has a zero-sum subcollection of ≥3 coefficients. The “only if” direction (and the resolution of Griesmer’s question) rests on a quantitative chromatic lower bound for Cayley graphs Cay(Z_p^n, Hamming ball around the all-ones vector). That bound is obtained from a newly introduced Kneser-type graph that embeds into Z_p^n together with an equivariant Borsuk–Ulam argument. Because only the abstract is available, neither the embedding, the equivariance, the resulting numerical lower bound, nor its density-independence can be inspected. Without those details it is impossible to confirm that the lower bound is strong enough to force unbounded chromatic number for every positive density when no zero-sum subcollection of size ≥3 exists. This is precisely the ingredient the Reader flagged as weakest; the absence of the full text leaves it uncheckable and therefore load-bearing for the whole classification.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript studies a chromatic analogue of Roth-type problems for homogeneous linear equations L over F_p. For L-solution-free sets A ⊆ F_p it considers the chromatic number of the Cayley graph Cay(F_p,A) and introduces the chromatic threshold δ_χ(L), the infimal density guaranteeing that this chromatic number remains bounded. The main claim is an exact classification: δ_χ(L)=0 if and only if L contains a zero-sum subcollection of at least three coefficients. The “only if” direction is obtained from a quantitative chromatic lower bound for Cayley graphs on Z_p^n generated by Hamming balls around the all-ones vector, via a newly introduced Kneser-type graph that embeds into Z_p^n together with an equivariant Borsuk–Ulam argument. As consequences the authors resolve a question of Griesmer and relate the classification to measurable, topological and Bohr recurrence, showing in particular that every infinite discrete abelian group admits a set that is topologically recurrent but not measurably recurrent.","tokens_in":2402,"tokens_out":1014,"duration_ms":13424,"significance":"If the classification and the supporting lower bound hold, the paper supplies a clean, parameter-free dichotomy for when L-solution-free sets of arbitrarily small positive density can force unbounded chromatic number of the associated Cayley graphs. The resolution of Griesmer’s question and the extension of the Kříž–Ruzsa examples to every infinite discrete abelian group would be substantial contributions at the interface of combinatorial number theory, topological combinatorics and recurrence theory. The introduction of a new Kneser-type graph with a natural embedding into Z_p^n is potentially of independent interest. These strengths, however, rest entirely on technical ingredients that cannot be inspected from the abstract alone.","major_comments":[{"comment":"The central classification δ_χ(L)=0 ⇔ L has a zero-sum subcollection of size ≥3 is load-bearing for the whole paper. The “only if” direction (and the resolution of Griesmer’s question) depends on a quantitative chromatic lower bound for Cayley graphs Cay(Z_p^n, Hamming ball about the all-ones vector). That bound is obtained from a newly introduced Kneser-type graph plus an equivariant Borsuk–Ulam argument. With only the abstract available, neither the embedding, the equivariance, the resulting numerical lower bound, nor its claimed density-independence can be verified. Until the full argument is supplied and checked, the classification remains unconfirmed.","section":"Abstract (key ingredient paragraph)"},{"comment":"The abstract asserts that the new lower bound is strong enough to force unbounded chromatic number for every positive density whenever L has no zero-sum subcollection of size ≥3. This density-independence claim is precisely the ingredient needed to conclude δ_χ(L)>0 in that case. Without the quantitative estimate (or even a sketch of how the Borsuk–Ulam degree or index produces a bound independent of density), it is impossible to assess whether the lower bound actually reaches the threshold required by the classification.","section":"Abstract (classification statement and key ingredient)"},{"comment":"The recurrence-theoretic consequences (topological but not measurable recurrence on every infinite discrete abelian group) are presented as flowing from the same classification. If the chromatic lower bound is weaker than claimed, the extension beyond the classical Kříž–Ruzsa examples may fail or require additional hypotheses. The logical dependence should be made explicit once the full text is available.","section":"Abstract (final paragraph)"}],"minor_comments":[{"comment":"The abstract is clear and well-structured, but the notation for the chromatic threshold δ_χ(L) and for the equation L itself should be fixed consistently once the full manuscript is supplied (script L versus calligraphic L, etc.).","section":"Abstract"},{"comment":"A precise statement of Griesmer’s question, even in one sentence, would help the reader locate the contribution; the abstract only says the question is resolved.","section":"Abstract"}],"recommendation":"uncertain","confidential_remarks":"This is an abstract-only review. The claimed results look interesting and the outline is plausible, but the load-bearing quantitative lower bound cannot be checked. I recommend obtaining the full manuscript before any editorial decision; with only the abstract I cannot responsibly choose accept, revise, or reject. If the full text confirms the Kneser-type embedding and the density-independent chromatic lower bound, the paper would likely merit serious consideration; if those ingredients are incomplete, the classification collapses."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing to know: this abstract claims an exact classification of when the chromatic threshold δ_χ(L) vanishes for a homogeneous linear equation L over F_p—namely, iff L has a zero-sum subcollection of at least three coefficients—and packages it with a new Kneser-type construction plus a uniform topological-but-not-measurable recurrence result for every infinite discrete abelian group.\n\nWhat looks new and useful is the framing itself. Chromatic thresholds for graphs are classical; transplanting the idea to Cayley graphs of L-solution-free sets and getting a clean combinatorial criterion is a natural and apparently original move. The claimed resolution of Griesmer’s question and the uniform extension of the Kříž–Ruzsa phenomenon are both worth having if the proofs hold. The abstract is precise about the statement and about the technical ingredients (Hamming-ball generators on Z_p^n, a new Kneser-type graph with an embedding, equivariant Borsuk–Ulam).\n\nThe soft spot is exactly the one the stress-test flags, and it is load-bearing: the “only if” direction rests on a quantitative chromatic lower bound that we cannot inspect. Without the embedding, the equivariance, or the density-independence of the bound, we cannot confirm that unbounded chromatic number really holds for every positive density when no zero-sum triple of coefficients exists. Everything else—circularity, free parameters, invented entities—looks clean; this is pure classification work. The limitation is simply that we have only the abstract.\n\nThis is for people who work on additive combinatorics, chromatic thresholds, or recurrence hierarchies. A serious referee should see the full manuscript; the claim is sharp enough and the surrounding literature is mature enough that desk rejection would be the wrong call. I would bring it to reading group once the proofs are public, and I would cite the classification if it survives. Send it to peer review.","headline":"Clean classification of when the chromatic threshold vanishes for linear equations, but only the abstract is available so the load-bearing lower bound cannot be checked.","tokens_in":3043,"tokens_out":490,"would_cite":false,"duration_ms":3977,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C15","05C25","11B30","37B20"],"pacs":[],"model":"grok-4.5","headline":"A linear equation has chromatic threshold zero exactly when it contains a zero-sum subcollection of three or more coefficients.","keywords":["chromatic threshold","linear equations","Cayley graphs","finite fields","solution-free sets","recurrence","Kneser graphs","Borsuk-Ulam"],"falsifier":"Produce a homogeneous linear equation with no zero-sum subcollection of three or more coefficients for which every solution-free set above some fixed positive density has bounded-chromatic Cayley graph, or produce an equation that does possess such a subcollection yet still admits arbitrarily sparse free sets whose Cayley graphs have unbounded chromatic number.","tokens_in":3047,"feed_emoji":"🎨","tokens_out":851,"duration_ms":17916,"temperature":0.7,"pith_summary":"This paper studies a chromatic analogue of classical Roth-type problems for a homogeneous linear equation L over a prime field. It measures L-solution-free sets A through the chromatic number of the Cayley graph Cay(F_p, A) and defines the chromatic threshold δ_χ(L) as the least density that forces every such free set to produce a Cayley graph of bounded chromatic number. The authors prove that this threshold is exactly zero if and only if L contains a subcollection of at least three coefficients that sum to zero. The proof rests on a quantitative lower bound for the chromatic numbers of certain Cayley graphs on vector spaces over F_p, obtained from a new Kneser-type graph that embeds into Z_p^n together with an equivariant Borsuk–Ulam argument. The classification resolves a question of Griesmer and yields, in every infinite discrete abelian group, a set that is topologically recurrent but not measurably recurrent.","feed_headline":"Chromatic threshold is zero iff equation has zero-sum triple","feed_subtitle":"Complete classification for Cayley graphs of free sets over prime fields, with new recurrence examples.","key_machinery":"A new Kneser-type graph that admits a natural embedding into Z_p^n, combined with an equivariant Borsuk–Ulam-type argument, producing a quantitative chromatic lower bound for Cayley graphs generated by Hamming balls around the all-ones vector.","core_discovery":"The chromatic threshold δ_χ(L) of a homogeneous linear equation L over F_p equals zero if and only if L contains a zero-sum subcollection of at least three coefficients. Equivalently, among L-solution-free sets A the Cayley graph Cay(F_p, A) has unbounded chromatic number at every positive density precisely when no such subcollection exists.","pith_inferences":["The same zero-sum criterion may control chromatic thresholds for systems of several linear equations rather than a single equation.","The Kneser-type construction may supply chromatic lower bounds for other families of Cayley graphs generated by metric balls.","Analogous thresholds over the integers could be approachable by lifting the finite-field constructions via Fourier analysis or transfer principles."],"forward_implications":["If L admits a zero-sum subcollection of size at least three, every positive-density L-free set yields a Cayley graph of bounded chromatic number.","If L admits no such subcollection, there exist L-free sets of arbitrarily small positive density whose Cayley graphs have unbounded chromatic number.","Every infinite discrete abelian group contains a set that is topologically recurrent but not measurably recurrent.","Griesmer’s question on the existence of certain recurrent sets is settled by the classification."],"fun_headline_variants":["Chromatic threshold vanishes iff equation has zero-sum triple","Zero χ-threshold exactly when L admits zero-sum subcollection ≥3","δ_χ(L)=0 iff linear equation contains a zero-sum triple of coeffs","Unbounded chromatic number at all densities without zero-sum triples","Equations with zero-sum triples force zero chromatic threshold"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The quantitative chromatic lower bound for Cayley graphs generated by Hamming balls around the all-ones vector is strong enough to force unbounded chromatic number at every positive density whenever the equation has no zero-sum subcollection of size three or larger.","fun_headline_variants_meta":{"raw":{"variants":["Chromatic threshold vanishes iff equation has zero-sum triple","Zero χ-threshold exactly when L admits zero-sum subcollection ≥3","δ_χ(L)=0 iff linear equation contains a zero-sum triple of coeffs","Unbounded chromatic number at all densities without zero-sum triples","Equations with zero-sum triples force zero chromatic threshold"]},"model":"grok-4.5","effort":"low","cost_usd":0.006854,"raw_usage":{"total_tokens":1791,"prompt_tokens":877,"num_sources_used":0,"completion_tokens":93,"cost_in_usd_ticks":68540000,"prompt_tokens_details":{"text_tokens":877,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":821,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":877,"tokens_out":93,"duration_ms":6795,"temperature":1.0,"reasoning_tokens":821,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-15T14:28:26.750294+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Produce a homogeneous linear equation with no zero-sum subcollection of three or more coefficients for which every solution-free set above some fixed positive density has bounded-chromatic Cayley graph, or produce an equation that does possess such a subcollection yet still admits arbitrarily sparse free sets whose Cayley graphs have unbounded chromatic number.","supporting_citations":[],"review_version":1}