{"id":"9456113d-99da-455c-8752-c8550d4c3686","arxiv_id":"2606.02180","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Cokernels of random p-adic matrices with inhomogeneously balanced columns converge in distribution to the Haar-random case under the condition that the sum of exp(-ε α_n(i) n) tends to zero.","lead":"The paper proves that cokernels of random n by (n+u) matrices over the p-adic integers with inhomogeneously balanced columns converge in distribution to the same limit as fully Haar-random matrices, provided a sum of exponentials involving the balance parameters goes to zero. A smart generalist might read it to see how controlled randomness in algebraic objects over p-adics still yields universal behavior, relevant to arithmetic statistics.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly isolates the modeling choice and the sufficiency of the given sum condition as the load-bearing premise. Because the provided claim contains no evident circularity, parameter dependence, or mismatch with the stated conclusion, the reader's UNVERDICTED assessment (driven by missing full text) does not require adjustment on the basis of any detectable flaw in the argument structure.","tokens_in":1759,"tokens_out":253,"duration_ms":14677,"concrete_test":"Check whether the paper's proof of the main theorem invokes the sum condition only through a union bound or first-moment estimate on the probability that any column fails to be sufficiently balanced; if so, recompute the bound with the precise definition of α_n(i)-balanced from the paper to confirm the o(1) decay.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract states a direct extension of the Nguyen-Wood universality result to the inhomogeneous α_n(i)-balanced model, with the exponential-sum condition serving as the explicit hypothesis that controls the deviation from homogeneity. No internal inconsistency, hidden assumption in the modeling, or unsupported transfer step is visible from the claim as formulated.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper proves that for a random n × (n+u) matrix A(n) over Z_p with i-th column α_n(i)-balanced, the condition ∑_{i=1}^{n+u} exp(−ε α_n(i) n) → 0 as n → ∞ for every ε > 0 implies that the cokernels of A(n) converge in distribution to the same limiting law as the cokernels of Haar-random n × (n+u) matrices over Z_p. This is presented as a direct extension of the Nguyen-Wood universality theorem to the inhomogeneous balanced-column model.","tokens_in":1824,"tokens_out":283,"duration_ms":27161,"significance":"If the result holds, it broadens the scope of cokernel universality results in p-adic random matrix theory by accommodating varying column balances under an explicit, checkable exponential-sum hypothesis. The extension preserves the limiting law without introducing new parameters or altering the target distribution, which is a strength for applications in arithmetic statistics.","major_comments":[],"minor_comments":[],"recommendation":"uncertain","confidential_remarks":"The provided abstract states the claim clearly, but the absence of the full proof, explicit definitions of α_n(i)-balanced columns, and verification of the transfer argument from the homogeneous case prevents assessment of soundness; the exponential-sum condition is asserted as sufficient without visible discussion of its tightness."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful summary of the main theorem and for recognizing the significance of extending the Nguyen–Wood universality result to the inhomogeneous balanced-column model under the stated exponential-sum hypothesis. No major comments were raised in the report, so we have no specific points to address point-by-point at this time.","responses":[],"tokens_in":1216,"tokens_out":72,"duration_ms":9886,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main new piece is the transfer of the cokernel universality limit from Haar-random matrices to the setting where each column i has its own alpha_n(i)-balance level, as long as sum exp(-eps alpha_n(i) n) goes to zero for every eps > 0. That condition is the device that keeps the inhomogeneity from affecting the limiting distribution.\n\nThe paper states the hypothesis cleanly and frames the result as a direct extension rather than a wholesale reworking. It re-uses the alpha-balanced notion from the literature without introducing new parameters or circular definitions. The claim itself is precise and matches the abstract exactly.\n\nThe soft spots are limited. The condition is shown to be sufficient but its necessity is not addressed, so it is not clear how sharp the threshold is. The full estimates that turn the sum condition into control on the cokernel probabilities are not visible in the provided text, which leaves the derivation steps unexamined. No internal contradictions appear in the statement, and the modeling choice is consistent with prior work.\n\nThis is for people already working on random p-adic matrices and cokernel statistics in arithmetic geometry. Readers who know the Nguyen-Wood theorem will see the incremental value right away.\n\nIt deserves a serious referee because the extension is targeted, the hypothesis is checkable, and the result sits in an active subfield.","headline":"This extends Nguyen-Wood to the inhomogeneous balanced-columns case under an explicit exponential-sum decay condition on the alphas.","tokens_in":2298,"tokens_out":346,"would_cite":false,"duration_ms":20896,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"A vanishing exponential sum on column balance parameters forces cokernels of random p-adic matrices to match the Haar-random limiting law.","keywords":["cokernels","random matrices","p-adic integers","universality","balanced columns","Haar measure","convergence in distribution","finite abelian p-groups"],"falsifier":"Construct a sequence α_n(i) for which the exponential sum vanishes yet, for some fixed finite abelian p-group G and some n large, the probability that the cokernel equals G differs from the corresponding Haar probability by more than a fixed positive constant.","tokens_in":2640,"feed_emoji":"","tokens_out":729,"duration_ms":22948,"temperature":0.7,"pith_summary":"The paper shows that if each column of a random n by (n+u) matrix over Z_p is α_n(i)-balanced and the sum over i of exp(-ε α_n(i) n) tends to zero for every ε>0, then the distribution of the cokernel converges to the same limit obtained from Haar-random matrices. This condition allows the balance levels to vary across columns while still guaranteeing the universality result. The extension matters because it identifies a quantitative threshold on inhomogeneity that preserves the limiting statistics without requiring every column to have the same balance parameter. The proof transfers the known convergence for the uniform case by controlling the deviation introduced by the varying α_n(i).","feed_headline":"Sum condition makes cokernels of p-adic matrices match Haar limit","feed_subtitle":"Vanishing of an exponential sum over column balance parameters transfers the known limiting law to the inhomogeneous case.","key_machinery":"The α_n(i)-balanced condition on individual columns together with the exponential-sum vanishing condition that averages out the inhomogeneity.","core_discovery":"Let u be a fixed nonnegative integer. Let A(n) be a random n by (n+u) matrix over Z_p whose i-th column is α_n(i)-balanced. If the sum from i=1 to n+u of exp(-ε α_n(i) n) tends to zero as n tends to infinity for every ε>0, then the cokernels of A(n) converge in distribution to the same limiting law as the cokernels of Haar-random n by (n+u) matrices over Z_p.","pith_inferences":["The same sum condition may be sufficient to prove universality for Smith normal forms or for cokernels over other local rings.","One could numerically sample matrices with chosen α_n sequences satisfying the sum condition and compare empirical cokernel frequencies against the known Haar probabilities for small p.","The technique might extend to models of random p-adic modules with column-dependent generation probabilities that still satisfy an averaged balance requirement."],"forward_implications":["The limiting cokernel distribution depends only on the vanishing of the sum and is otherwise insensitive to the particular choice of the sequence α_n.","The same convergence holds for any fixed excess number of columns u.","Universality statements that previously required uniform balance now apply to a strictly larger family of column distributions.","The result supplies an explicit quantitative criterion that can be checked for concrete random-matrix ensembles arising in p-adic arithmetic."],"fun_headline_variants":["Sum decay condition yields Haar cokernels for inhomogeneous p-adics","Column balance sum vanishing transfers universality to p-adic matrices","Inhomogeneous p-adics match Haar cokernel distribution via sum condition","Exponential sum condition unifies cokernels of p-adic matrices to Haar","Vanishing balance sum transfers p-adic cokernels to Haar universality limit"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The matrix entries are drawn so that each column exactly satisfies its prescribed α_n(i)-balanced property, and this modeling choice plus the sum condition suffices to equate the distributions.","fun_headline_variants_meta":{"raw":{"variants":["Sum decay condition yields Haar cokernels for inhomogeneous p-adics","Column balance sum vanishing transfers universality to p-adic matrices","Inhomogeneous p-adics match Haar cokernel distribution via sum condition","Exponential sum condition unifies cokernels of p-adic matrices to Haar","Vanishing balance sum transfers p-adic cokernels to Haar universality limit"]},"model":"grok-4.3","cost_usd":0.006126,"raw_usage":{"total_tokens":2889,"prompt_tokens":662,"num_sources_used":0,"completion_tokens":85,"cost_in_usd_ticks":61262000,"prompt_tokens_details":{"text_tokens":662,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2142,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":662,"tokens_out":85,"duration_ms":15672,"temperature":1.0,"reasoning_tokens":2142,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T12:47:32.833654+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Construct a sequence α_n(i) for which the exponential sum vanishes yet, for some fixed finite abelian p-group G and some n large, the probability that the cokernel equals G differs from the corresponding Haar probability by more than a fixed positive constant.","supporting_citations":[],"review_version":1}