{"id":"54571fd9-28f9-420e-81c8-5aa99fb0f962","arxiv_id":"2606.17822","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves that shifted Jack Littlewood-Richardson coefficients g^λ_μν(α) for triples differing by a single box move are congruent modulo the α-hook length of the pivot box.","lead":"This paper proves a conjecture that shifted Jack Littlewood-Richardson coefficients for partitions differing by one box move are congruent modulo the alpha-hook length of the pivot box. A smart generalist might read it to see how algebraic relations in symmetric functions can be established through modular arithmetic on polynomials.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly isolates the background definition needed for the objects to exist in a ring where the congruence is meaningful. No internal gap, unverified assumption, or inconsistency in the Jack proof is apparent; the open Macdonald part is explicitly separated and does not affect the proved claim.","tokens_in":1652,"tokens_out":274,"duration_ms":21697,"concrete_test":"Extract the precise statement of the main theorem (including the ring in which the congruence holds) and compare it directly to the conjecture formulated in the author's prior work; confirm that the proof invokes only established properties of the Jack case and does not rely on the two open shift-map properties.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is a proof of the stated congruence for the shifted Jack LR coefficients g^λ_μν(α) when partitions differ by a single box move. The background fact that these are Laurent polynomials in α is taken from prior work (Alexandersson-Féray) and is required for the modulo statement to be well-defined in the appropriate ring; the manuscript notes that the Macdonald extension remains open due to two unproven properties of Lassalle's shift map, but this does not touch the Jack case that forms the main theorem.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proves the author's earlier conjecture that the shifted Jack Littlewood-Richardson coefficients g^λ_μν(α) for two triples of partitions differing by a single box move are congruent modulo the α-hook length of the pivot box. The coefficients are Laurent polynomials in the Jack parameter α (as established by Alexandersson-Féray). The note also records that the analogous statement for shifted Macdonald functions remains open, pending two properties of Lassalle's shift map.","tokens_in":1728,"tokens_out":307,"duration_ms":22264,"significance":"If the proof holds, the result supplies a concrete congruence relation for these generalized coefficients, extending classical Littlewood-Richardson theory to the Jack setting. The manuscript explicitly credits the prior definition of g^λ_μν(α) and isolates the precise obstruction for the Macdonald case, which is a useful clarification.","major_comments":[],"minor_comments":[{"comment":"The abstract refers to 'a previous work of the author's' for the conjecture; adding the precise citation (even if it is the author's own arXiv preprint) would improve traceability.","section":"Abstract"},{"comment":"Notation for the α-hook length and the 'pivot box' is used without an inline definition or forward reference to the section where it is introduced; a brief parenthetical reminder would aid readers.","section":null}],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive report and recommendation to accept the manuscript. The report contains no major comments requiring a point-by-point response.","responses":[],"tokens_in":1134,"tokens_out":47,"duration_ms":10026,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main thing here is that Mickler has proved his own prior conjecture: when two triples of partitions differ by a single box move, the shifted Jack LR coefficients are congruent modulo the alpha-hook length of the pivot box. The proof uses the fact that these coefficients are Laurent polynomials in alpha, taken from Alexandersson-Feray.\n\nThe paper does what it sets out to do. It delivers a clean argument for the Jack case and is explicit that the Macdonald extension stays open because two properties of Lassalle's shift map are still unproven. That separation is honest and avoids overclaiming.\n\nThe soft spot is the narrow scope. This is a targeted modular relation inside algebraic combinatorics with no new methods, no broader applications, and no exploration of why the congruence might matter outside the immediate setting. The result stands on the cited background but does not extend it.\n\nThe citation pattern is appropriate and points directly to the relevant prior work without padding. No circularity or invented objects appear in the claim.\n\nThis is for readers already working on Jack polynomials, shifted symmetric functions, or LR coefficients in that family. A specialist might use the congruence as a reference or building block. Others will not need it.\n\nThe paper shows clear engagement with the literature and resolves a stated conjecture with a direct proof. It deserves peer review so the details of the argument can be checked.","headline":"This note proves the author's earlier conjecture on a modular congruence for shifted Jack LR coefficients when partitions differ by one box move.","tokens_in":2176,"tokens_out":351,"would_cite":false,"duration_ms":28828,"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":"The shifted Jack Littlewood-Richardson coefficients for triples differing by a single box move are congruent modulo the α-hook length of the pivot box.","keywords":["shifted Jack Littlewood-Richardson coefficients","congruences","box moves","alpha-hook lengths","Jack parameter","partitions","Laurent polynomials","symmetric functions"],"falsifier":"A concrete counterexample would be any specific triple of partitions and box move for which the two coefficients differ by a nonzero multiple of the α-hook length when evaluated at some value of α.","tokens_in":2531,"feed_emoji":"","tokens_out":608,"duration_ms":24385,"temperature":0.7,"pith_summary":"This paper proves a conjecture stating that the shifted Jack Littlewood-Richardson coefficients g^λ_μν(α) for two triples of partitions, in which one partition differs from the other by moving a single box, are congruent modulo the α-hook length of the pivot box for that move. These coefficients are Laurent polynomials in the Jack parameter α attached to triples of partitions and generalize the classical Jack Littlewood-Richardson coefficients. A sympathetic reader would care because the congruence supplies a direct relation between values of these polynomials at different partitions, which may streamline their study or computation. The proof settles the Jack case while the analogous statement for shifted Macdonald functions is left open.","feed_headline":"Shifted Jack LR coefficients congruent mod alpha-hook for box moves","feed_subtitle":"Coefficients attached to triples that differ by one box agree modulo the pivot box alpha-hook length.","key_machinery":"The shifted Jack Littlewood-Richardson coefficients g^λ_μν(α), Laurent polynomials in α attached to triples of partitions.","core_discovery":"The central claim is that if two triples of partitions differ by a single box move in one of the partitions, then their associated shifted Jack Littlewood-Richardson coefficients are congruent modulo the α-hook length of the pivot box for that move.","pith_inferences":["The congruence might support recursive algorithms that compute the coefficients by reducing partition size one box at a time.","Specializing the parameter α to particular numbers could produce new numerical identities among ordinary Littlewood-Richardson coefficients.","The modular relation may connect to combinatorial interpretations or positivity properties that have not yet been examined."],"forward_implications":["The stated congruence holds for every single-box move between valid triples.","The result applies uniformly for all values of the Jack parameter α at which the coefficients are defined.","The Macdonald-function analogue of the congruence remains unresolved because two required properties of Lassalle's shift map are not yet established."],"fun_headline_variants":["Alpha-hook congruences for shifted Jack LR coefficients on box moves","Shifted Jack LR coeffs congruent modulo pivot box alpha-hook","Box move shifts yield congruent shifted Jack LR coefficients mod alpha-hook","Congruence modulo alpha-hook for shifted Jack LR coefficients differing by box"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The shifted Jack Littlewood-Richardson coefficients are Laurent polynomials in the Jack parameter α.","fun_headline_variants_meta":{"raw":{"variants":["Alpha-hook congruences for shifted Jack LR coefficients on box moves","Shifted Jack LR coeffs congruent modulo pivot box alpha-hook","Box move shifts yield congruent shifted Jack LR coefficients mod alpha-hook","Congruence modulo alpha-hook for shifted Jack LR coefficients differing by box"]},"model":"grok-4.3","cost_usd":0.006975,"raw_usage":{"total_tokens":3172,"prompt_tokens":549,"num_sources_used":0,"completion_tokens":71,"cost_in_usd_ticks":69749500,"prompt_tokens_details":{"text_tokens":549,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2552,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":549,"tokens_out":71,"duration_ms":28593,"temperature":1.0,"reasoning_tokens":2552,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T00:04:08.771723+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A concrete counterexample would be any specific triple of partitions and box move for which the two coefficients differ by a nonzero multiple of the α-hook length when evaluated at some value of α.","supporting_citations":[],"review_version":1}