{"id":"336747dd-920d-4012-ab96-bd928a8afd59","arxiv_id":"2607.08607","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Large families of measures on partitions share the same moments; the p=2 sandpile distributions of Mészáros and of random bipartite graphs both sit inside one such family.","lead":"The paper builds large families of probability distributions on finite abelian p-groups that share identical moments, including the special p=2 cases that arise for sandpile groups of random regular graphs and (conjecturally) random bipartite graphs. These families show that moments alone need not determine a unique distribution once they grow faster than the classical Cohen–Lenstra bound.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The paper’s strongest claim is the construction of multi-parameter families of measures sharing identical moments, together with the observation that the Mészáros even-d/p=2 distribution and the conjectural bipartite p=2 distribution both sit inside the r=1 subfamily. All supporting arguments (Lemmas 4.1–4.9, Theorems 4.6 and 4.10, the cube structure of A_r) are purely combinatorial and are proved in full. The numerical experiments of Section 2 address only the open Conjecture 1.3; they are presented as supporting evidence, not as a premise of any theorem. Consequently the reader’s identified weakest assumption does not touch the load-bearing core, and no further correctness risk is visible. The verdict ACCEPT with high confidence remains appropriate.","tokens_in":31066,"tokens_out":435,"duration_ms":4914,"concrete_test":"Independently recompute the r=1 moment identity of Theorem 1.10 for the empty partition and for μ=(1) by direct summation of the series defining P^{Sym,(k)}_p and P^{M,(k)}_{p,α} (using the closed form of K(∞,b) and Corollary 4.5); agreement to machine precision confirms the normalization and the shared-moment claim for the two sandpile special cases.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theorems (1.8–1.10) rest on explicit evaluations of moments via Hall–Littlewood specializations and the transition kernels K(a,b). The inductive argument of Theorem 4.10 that odd-cardinality sign flips leave moments unchanged, the polytope description of A_r, and the normalization that produces genuine probability measures are all self-contained and free of hidden analytic assumptions. The reader’s weakest point concerns only the open Conjecture 1.3 and its n=100 numerics; that material is cleanly separated from the proved statements and does not underwrite the strongest claim.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper constructs large families of measures on partitions that share identical μ-moments, using Hall–Littlewood specializations and transition kernels K(a,b). For a strict partition κ of length r it produces a 2^{r-1}-parameter family ˜P^{Sym,κ}_{p,α} (α∈A_r) whose moments equal those of ˜P^{Sym,κ}_p (Theorems 1.8–1.9); the r=1 subfamily recovers both Mészáros’ even-d, p=2 distribution and the conjectural p=2 bipartite distribution, both having moments p^{n(μ)+l(μ)} (Theorem 1.10). The same machinery yields closed-form moments for Cohen–Lenstra, Malle-type and Garton-type measures (Theorems 1.11–1.13). Conjecture 1.3 on Sylow p-subgroups of random bipartite graphs is stated and supported by n=100 numerics.","tokens_in":31266,"tokens_out":788,"duration_ms":7145,"significance":"The work supplies the first systematic source of non-uniqueness for the method of moments when moments exceed the Wood bound, and cleanly embeds two previously isolated special cases (Mészáros even-d p=2 and the bipartite p=2 conjecture) into a single geometric family. The proofs are self-contained, relying only on the skew Cauchy identity and classical q-hypergeometric summations; the polytope A_r is shown to be a cube, guaranteeing non-negativity of the affine combinations. The moment formulae for the Cohen–Lenstra, Malle and Garton families are new for general real parameters and finite d. The numerical experiments and accompanying code repository give concrete, falsifiable predictions for Conjecture 1.3.","major_comments":[],"minor_comments":[{"comment":"Section 2: the n=100, 500-sample experiments for Conjecture 1.3 lack error bars or concentration bounds; a short remark that the numerics are only heuristic would clarify the separation between proved theorems and open conjecture.","section":"Section 2"},{"comment":"After (1.7): the definition of the polytope A_r is clear, but a one-sentence reminder that the inequalities are exactly those needed for non-negativity of every coordinate would help the reader.","section":"§1.5"},{"comment":"Lemma 4.1 and Corollary 4.2: the appeal to dominated convergence for the a\to∞ limit is correct but could be flagged more explicitly for readers less familiar with q-series.","section":"§4.1"},{"comment":"References: the forthcoming preprint [20] of the third author is cited for the odd-p case of Conjecture 1.3; if a public arXiv link becomes available before publication it should be added.","section":"References"}],"recommendation":"accept","confidential_remarks":"The manuscript is a clean, technically solid contribution that fits well in a combinatorics or algebraic combinatorics journal. The open conjecture and its numerics are carefully isolated from the proved statements and do not affect the recommendation."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The real news is Theorems 1.8–1.10: for any strict partition κ of length r they build a genuine 2^{r-1}-parameter family of measures (parametrized by the cube A_r) that all share the same μ-moments, and both the Mészáros even-d p=2 law and the conjectured bipartite p=2 law sit inside the r=1 subfamily. That is new and useful; previously those two special cases looked isolated.\n\nThey do the work carefully. Moments are reduced to products of the transition kernels K(a,b), evaluated with classical q-hypergeometric identities and the skew Cauchy identity for Hall–Littlewood functions. The induction that odd-cardinality sign flips leave moments unchanged is clean, the non-negativity argument via the geometry of A_r is solid, and the same machinery gives independent proofs of the Cohen–Lenstra, Malle and Garton moment formulas. Code for the n=100 experiments is public. No free parameters, no circularity.\n\nThe soft spot is exactly where the reader said: Conjecture 1.3 (the actual bipartite limit) is still open, and the numerical support is 500 samples at n=100 with no error bars or concentration. That material is cleanly separated from the proved theorems and does not underwrite the strongest claim, so it is a minor caveat rather than a load-bearing flaw. The forthcoming Singhal paper for odd p is noted but not needed here.\n\nThis is for people who work on random abelian groups, sandpile groups, or the method of moments. Anyone who has ever wondered how non-uniqueness of moments can actually arise in a concrete model will get value from it. The math is self-contained and the citation pattern is normal. I would send it to a serious referee without hesitation; the proved core is already strong enough.","headline":"Solid construction of multi-parameter moment-sharing families that cleanly unifies the Mészáros and bipartite p=2 sandpile cases; the bipartite conjecture is open but cleanly separated.","tokens_in":31842,"tokens_out":478,"would_cite":true,"duration_ms":5530,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C80","05E05","15B52","20K01"],"pacs":[],"model":"grok-4.5","headline":"Large families of partition measures share the same moments, including two distinct sandpile-group laws for p=2.","keywords":["sandpile groups","random bipartite graphs","method of moments","Hall–Littlewood functions","Cohen–Lenstra heuristics","finite abelian p-groups","partition measures"],"falsifier":"Compute the empirical distribution of the 2-Sylow of the sandpile group for random bipartite graphs with part sizes n and ⌈αn⌉, α>1/2, for n several thousand, and check whether the observed frequencies for partitions of length 1–4 converge to 2^{ℓ(λ)−1}P^{Sym}_{∞,2}(λ).","tokens_in":32002,"feed_emoji":"📊","tokens_out":738,"duration_ms":6914,"temperature":0.7,"pith_summary":"The paper builds explicit families of probability measures on partitions (equivalently, on finite abelian p-groups) that all possess identical moments, even though the measures themselves are different. These families arise naturally when one studies the Sylow p-subgroups of sandpile groups of random graphs. In particular, the special distribution that appears for even-regular graphs at p=2 and the conjectured distribution for random bipartite graphs at p=2 are distinct, yet they belong to the same one-parameter family and therefore share every moment. The construction is obtained by weighting the classical symmetric-matrix cokernel measure by monomials in the conjugate partition and then taking affine combinations of sign-twisted versions; the resulting moments are evaluated with Hall–Littlewood polynomials and basic hypergeometric identities. The work therefore shows that the method of moments, while powerful when moments grow slowly, can leave large continuous moduli of distributions once the growth rate exceeds the Wood uniqueness threshold.","feed_headline":"Same moments, different sandpile laws for p=2","feed_subtitle":"A 2^{r−1}-parameter family unifies two distinct random-graph distributions that moments alone cannot separate","key_machinery":"The measures ˜P^{Sym,κ}_{p,α} obtained by multiplying the infinite symmetric cokernel measure by p^{∑κ_i λ′_i} and then taking affine combinations of the odd-sign twists indexed by subsets of [r]; their moments are computed by expressing the surjection counts via Hall–Littlewood skew polynomials and summing the resulting products of transition kernels K(a,b) with q-hypergeometric identities.","core_discovery":"For every strict partition κ of length r there exists a 2^{r−1}-parameter family of measures on partitions, all sharing the same collection of μ-moments as the weighted measure ˜P^{Sym,κ}_p. The two special p=2 sandpile distributions—one for even-regular graphs and one conjectured for bipartite graphs—both sit inside the r=1 subfamily and therefore have identical moments p^{n(μ)+ℓ(μ)}.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Same moments, distinct p=2 sandpile laws for bipartite graphs","Bipartite sandpile p-groups share moments with even-regular case","Families of measures on partitions with identical sandpile moments","Random bipartite graphs yield p=2 sandpile laws matching moments","Distinct sandpile distributions for p=2 fit in same-moment families"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The numerical evidence offered for the bipartite-sandpile conjecture consists of only 500 samples of graphs on 100 vertices and supplies no concentration bounds that would justify the infinite-n limit.","fun_headline_variants_meta":{"raw":{"variants":["Same moments, distinct p=2 sandpile laws for bipartite graphs","Bipartite sandpile p-groups share moments with even-regular case","Families of measures on partitions with identical sandpile moments","Random bipartite graphs yield p=2 sandpile laws matching moments","Distinct sandpile distributions for p=2 fit in same-moment families"]},"model":"grok-4.5","effort":"low","cost_usd":0.004708,"raw_usage":{"total_tokens":1342,"prompt_tokens":788,"num_sources_used":0,"completion_tokens":94,"cost_in_usd_ticks":47080000,"prompt_tokens_details":{"text_tokens":788,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":460,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":788,"tokens_out":94,"duration_ms":4633,"temperature":1.0,"reasoning_tokens":460,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-10T05:04:52.295069+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Compute the empirical distribution of the 2-Sylow of the sandpile group for random bipartite graphs with part sizes n and ⌈αn⌉, α>1/2, for n several thousand, and check whether the observed frequencies for partitions of length 1–4 converge to 2^{ℓ(λ)−1}P^{Sym}_{∞,2}(λ).","supporting_citations":[],"review_version":2}