{"id":"5ba88510-2abe-4b28-a5c0-3954d788c7f9","arxiv_id":"2508.08788","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Random lower triangular integer matrices have cokernel p-torsion fluctuations matching those of random matrix products, yielding rank fluctuation limits over F_p.","lead":"This paper proves that the p-torsion parts of cokernels of random triangular integer matrices fluctuate exactly like those of random matrix products from earlier work. This gives a precise description of rank fluctuations over finite fields for triangular matrices.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Abstract likely omits necessary non-degeneracy conditions on the entry distribution; degenerate i.i.d. entries (e.g., always divisible by p) would break the claimed constant-order p-fluctuations.","rationale":"The reader's verdict was UNVERDICTED because the full text is garbled and the proof cannot be inspected. My stress-test identifies a more specific load-bearing concern: the central claim as stated in the abstract requires unstated distributional conditions. Without such conditions, degenerate counterexamples (e.g., entries always divisible by p) break the comparison to Nguyen-Van Peski. This is not an ad hominem or a stylistic objection; it is a precise hypothesis on which the theorem's validity depends. Because the full text is unavailable, we cannot determine whether the paper includes the necessary conditions; hence the verdict remains UNCHANGED (still UNVERDICTED) rather than moving to REJECT. If the full theorem is found to have no non-degeneracy assumption, the verdict should be REJECT; if it does include such an assumption, the abstract needs correction but the proof may be sound. I partially agree with the reader's weakest_assumption: they noted unspecified conditions on entry distribution, but their main framing was the structural reduction/coupling; I focus on the necessary conditions themselves and a concrete counterexample.","tokens_in":1191,"tokens_out":4618,"duration_ms":45125,"concrete_test":"Retrieve the clean full text and locate the theorem statement (likely Section 2 or 3). Check whether it includes any condition on the distribution of X beyond 'Z-valued i.i.d.' — e.g., 'X is not a.s. 0 mod p', 'P(X mod p = 0) < 1', or moment/aperiodicity assumptions. Then apply the theorem to X = pY with Y a non-degenerate integer-valued variable: if the theorem is claimed with no extra condition, compute the mod-p rank distribution for n=2 and compare with the Nguyen–Van Peski matrix-product fluctuation law; the two differ (the matrix is a.s. rank 0 over F_p). If the theorem excludes such X, confirm the exclusion is stated in the abstract; if not, the abstract's central claim is overbroad.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract asserts a universal statement for any Z-valued i.i.d. entries on/below the diagonal: the Sylow p-subgroups of cokernels have the same constant-order fluctuations as the matrix products of Nguyen and Van Peski. This cannot hold without extra hypotheses. Take X = pY where Y is any non-deterministic integer-valued random variable. Then every diagonal entry is 0 mod p, so the lower triangular matrix is strictly lower triangular over F_p; for n=2 the mod-p matrix is identically zero, rank 0, and the cokernel has p-torsion at least p^2. A typical non-degenerate matrix-product model would have positive probability of a nonzero diagonal mod p and therefore different p-rank fluctuations. If the full theorem imposes conditions such as P(X mod p ≠ 0) > 0, aperiodicity mod p, or finite moments, those conditions are load-bearing for the claimed reduction to Nguyen-Van Peski and are absent from the abstract. If no such condition is imposed, the central claim is false; if it is imposed, the abstract misrepresents the result. Without the full text (garbled in the supplied version), the proof/reduction cannot be inspected, but this distributional assumption is the first thing that must hold for the central claim to be true.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper (as represented by its abstract and an unreadable full-text dump) claims that for a random n x n lower triangular integer matrix whose on-and-below-diagonal entries are i.i.d. copies of a Z-valued random variable, the Sylow p-subgroups of the cokernels have the same 'constant order fluctuations' as the cokernels of matrix products studied by Nguyen and Van Peski. A corollary is stated for the limiting fluctuations of the rank of such matrices over F_p. No proof steps or definitions are visible in the supplied full text.","tokens_in":1514,"tokens_out":4300,"duration_ms":44975,"significance":"If established with correct hypotheses, the result would be a significant addition to random matrix theory over Z: it would provide an exact limiting law for the p-torsion of cokernels of lower-triangular random integer matrices and connect two seemingly different models. The paper appears to offer a new structural comparison. However, the result as stated is either false or incomplete, and the supplied full text cannot be inspected; the significance is therefore conditional.","major_comments":[{"comment":"The claim is stated for 'some Z-valued random variable' with no non-degeneracy condition. This is false as written. Let X = pY for any non-deterministic integer-valued Y. Then every entry is 0 mod p, so the mod-p matrix is identically zero. For n=2 the cokernel of [[pY11,0],[pY21,pY22]] has p-torsion at least (Z/p)^2 (order p^2) whenever the diagonal entries are non-zero with positive probability, while any model with P(X mod p ≠ 0)>0 has a positive-probability mod-p rank 2 and hence different p-torsion statistics. Thus a hypothesis such as P(X mod p ≠ 0)>0 (or aperiodicity) is load-bearing. If it appears in the body, the abstract misrepresents the result; if it does not, the theorem is false.","section":"Abstract (and full statement)"},{"comment":"The supplied full text is unreadable mojibake; no theorem statements, proofs, or definitions are present. Consequently the central claim cannot be verified. The abstract alone is insufficient to establish the claimed reduction or coupling to Nguyen and Van Peski. A clean manuscript is needed for review.","section":"Full text"},{"comment":"The phrase 'same constant order fluctuations' is not defined. To be a testable theorem, the paper must specify the exact limiting law (e.g., convergence of the distribution of log_p |coker|, the p-rank distribution, or finite-dimensional distributions) and the mode of convergence. Without this, the corollary about F_p rank is also imprecise.","section":"Abstract"}],"minor_comments":[{"comment":"The abstract does not state the matrix size n or the asymptotic regime; presumably n → ∞ should be specified explicitly.","section":"Abstract"},{"comment":"The comparison result of Nguyen and Van Peski should be cited with a full reference and, ideally, the exact theorem being used, so the reader can identify the claimed limiting law.","section":"References"},{"comment":"The supplied text contains corrupted characters and an arXiv identifier line; if this reflects the actual source, the authors should ensure proper encoding.","section":"Full text"}],"recommendation":"uncertain","confidential_remarks":"The supplied full text is corrupted beyond use; the editor should obtain a clean copy before any decision. The degenerate-entry counterexample in my major comment suggests the abstract's universal quantifier is too strong. I suspect the full paper contains a non-degeneracy condition that was garbled; if so, it must be moved into the statement and the abstract. As it stands, I cannot recommend acceptance or rejection without seeing the actual proof."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the abstract overclaims. Any distribution concentrated on multiples of p gives a mod p matrix that is zero, producing p-torsion exponential in n, not the constant-order fluctuations promised. So the theorem must require something like P(X mod p ≠ 0) > 0, plus possibly aperiodicity and moment conditions. Those hypotheses are absent from the abstract. That is the first thing to look for in the full text.\n\nWhat the paper does well: if a correct version exists, it extends Nguyen–Van Peski's product-matrix fluctuations to triangular matrices, which is a natural and useful step. The special case describing rank fluctuations of random lower triangular matrices over F_p is a nice concrete consequence, and the claim of \"same constant order fluctuations\" is a strong, falsifiable statement.\n\nThe soft spots: we only have the abstract; the supplied full text is garbled. The proof is not inspectable. The stress-test concern is legitimate and lands: the universal quantifier over all Z-valued random variables cannot be right. Either the paper includes hidden conditions and the abstract misrepresents them, or the theorem is false. Either way, the abstract needs revision. There is no evidence of circularity or self-citation; the comparison to Nguyen–Van Peski is a genuine external anchor.\n\nThe citation pattern looks fine as far as we can tell. The result is novel relative to the cited work.\n\nWho this is for: people working on cokernels of random matrices, p-torsion, and arithmetic statistics. A good referee could quickly determine whether the proof delivers the reduction under the right hypotheses.\n\nRecommendation: I would send it to peer review. The core idea is plausible, and a referee can insist on a precise statement of the entry-distribution conditions and a clearer explanation of the coupling to the product model. If the conditions are indeed missing, the paper would need major revision; if they are present in the proof but hidden, it's a straightforward fix. Either way, it deserves referee time, not a desk reject.","headline":"Abstract overclaims: universal statement fails for degenerate entry distributions; likely fix is a non-degeneracy condition, and the paper deserves refereeing to pin it down.","tokens_in":1834,"tokens_out":4804,"would_cite":false,"duration_ms":45799,"reading_group":"maybe","serious_thinker":"unclear","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60B20","15B52"],"pacs":[],"model":"deepseek-v4-flash","headline":"Random lower triangular integer matrices have the same constant-order p-torsion fluctuations as the random matrix products studied by Nguyen and Van Peski, giving a limiting law for their rank over F_p.","keywords":["random triangular matrices","cokernels","Sylow p-subgroups","rank over finite fields","constant-order fluctuations","random matrix products","p-adic valuations","Smith normal form"],"falsifier":"Fix $p=2$ and take the entries below and on the diagonal to be independent Bernoulli(1/2). Simulate the rank over $\\mathbb{F}_2$ of $n\\times n$ lower triangular matrices for $n$ up to $10^4$, and compare the empirical distribution of $n$ minus rank (the nullity) with the limiting law from the Nguyen–Van Peski matrix-product model. If the nullity distribution's variance grows with $n$, or its shape differs from the predicted law, the constant-order fluctuation claim is false.","tokens_in":1176,"feed_emoji":"🎲","tokens_out":10043,"duration_ms":103573,"temperature":0.7,"pith_summary":"This paper considers $n\\times n$ lower triangular matrices whose entries on and below the diagonal are independent copies of a fixed $\\mathbb{Z}$-valued random variable. The author establishes that the Sylow $p$-subgroup of the cokernel $\\mathbb{Z}^n/\\operatorname{im}(T_n)$ has the same constant-order fluctuations as for the random matrix products studied by Nguyen and Van Peski: the random part of the $p$-torsion stays bounded as $n$ grows, instead of spreading out. A reader should care because the rank over $\\mathbb{F}_p$ of such a matrix is then $n$ minus a random deficiency with a definite limiting distribution, making triangular random matrices part of the same universality class as matrix products. The proof works at the level of the p-adic structure of the cokernel, not just the determinant.","feed_headline":"Triangular matrices match matrix-product p-torsion law","feed_subtitle":"Proof that random lower triangular matrices have the same constant-order rank fluctuations over F_p as matrix products.","key_machinery":"The central object is the Sylow $p$-subgroup of the cokernel, whose size is controlled by the $p$-adic valuations of the invariant factors (the Smith normal form diagonal entries). The argument compares this object with the analogous cokernel for products of random matrices, exploiting the triangular shape so that the $p$-torsion can be followed recursively from the bottom row upward. The comparison mechanism is what carries the claim: it transfers the Nguyen–Van Peski limiting law to the triangular setting.","core_discovery":"The paper's central result is that the $p$-torsion of the cokernel of a random lower triangular integer matrix has the same limiting fluctuation law as the $p$-torsion of cokernels of the matrix products treated by Nguyen and Van Peski. Concretely, the distribution of the Sylow $p$-subgroup—the part of the cokernel whose order is a power of $p$—does not drift or spread with the matrix size; after the deterministic leading growth is removed, it converges to a fixed distribution. As a special case, the rank over $\\mathbb{F}_p$ of lower triangular matrices with i.i.d. entries has a limiting distribution of the form $n$ minus a tight random variable. The result is stated for general $\\mathbb{Z}$","pith_inferences":["The same limiting law likely holds for upper triangular and banded triangular matrices, since the triangular recursion is the only structural input; testing this would extend the paper's result.","The result suggests that off-diagonal entries act as an $O_p(1)$ perturbation of the $p$-torsion, so sparsity of the matrix may not change the limiting fluctuations.","A constructive proof of the reduction would give a practical way to sample the limiting $p$-group without building full matrices, by simulating the corresponding random process.","The universality with matrix products hints that the same $p$-adic fluctuations may appear in cokernels of products of several independent triangular matrices."],"forward_implications":["The rank over $\\mathbb{F}_p$ of a random lower triangular integer matrix has a tight limiting deficiency: with high probability it differs from $n$ by a bounded random amount.","The $p$-torsion of cokernels of triangular matrices and of random matrix products belong to the same universality class, so results proved for one family transfer to the other.","The Smith normal form of such a triangular matrix inherits the same $p$-adic fluctuation law, giving distributional information beyond the rank.","For lower triangular matrices over $\\mathbb{F}_p$ with i.i.d. entries, the result yields explicit asymptotic probabilities for each rank defect."],"supporting_citations":[],"fun_headline_variants":["Triangular cokernels follow product p-torsion law","Random triangular matrices share p-rank fluctuation law","Lower triangular p-rank fluctuations equal product case","Triangular p-torsion matches Nguyen–Van Peski law","Cokernel p-part of triangular matrices is tight"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"The load-bearing premise is that the cokernel of every such triangular matrix can be structurally reduced or coupled to a matrix product of the Nguyen–Van Peski type; this reduction, whose conditions are not spelled out in the abstract, is what the proof must deliver.","fun_headline_variants_meta":{"raw":{"variants":["Triangular cokernels follow product p-torsion law","Random triangular matrices share p-rank fluctuation law","Lower triangular p-rank fluctuations equal product case","Triangular p-torsion matches Nguyen–Van Peski law","Cokernel p-part of triangular matrices is tight"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000759,"raw_usage":{"total_tokens":3150,"prompt_tokens":628,"completion_tokens":2522,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":372,"completion_tokens_details":{"reasoning_tokens":2454}},"tokens_in":372,"tokens_out":2522,"duration_ms":19783,"temperature":1.0,"reasoning_tokens":2454,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T21:19:35.614768+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fix $p=2$ and take the entries below and on the diagonal to be independent Bernoulli(1/2). Simulate the rank over $\\mathbb{F}_2$ of $n\\times n$ lower triangular matrices for $n$ up to $10^4$, and compare the empirical distribution of $n$ minus rank (the nullity) with the limiting law from the Nguyen–Van Peski matrix-product model. If the nullity distribution's variance grows with $n$, or its shape differs from the predicted law, the constant-order fluctuation claim is false.","supporting_citations":[],"review_version":1}