{"id":"8a779ccd-a9d6-46c0-8b63-651550b5c2a8","arxiv_id":"2606.21628","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"No uniform bound exists for the extension degree making integral matrices similar over number fields after local similarity everywhere, but a bound depending on a separable characteristic polynomial is provided.","lead":"The paper proves there is no uniform bound on the degree of finite extension of a number field K needed to ensure two n by n integral matrices become similar over the ring of integers, even if they are similar over all completions. It also gives an upper bound on that degree when the characteristic polynomial is separable.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's identification of the local-global implication as the weakest assumption is correct, but because the fact is standard in the literature and the paper's novelty lies in the degree bounds rather than in reproving the implication, the foundation does not introduce a load-bearing risk. The UNVERDICTED status stems from abstract-only review; once the full proofs are examined the verdict can be updated, but the current structure shows no flaw requiring rejection or major revision.","tokens_in":1600,"tokens_out":306,"duration_ms":15153,"concrete_test":"Take the smallest n and a separable monic polynomial f of degree n; construct the companion matrices A and B over K=ℚ; verify local similarity at all places, then compute the minimal extension degree d such that similarity holds over O_L for [L:K]=d; check whether the paper's stated upper bound exceeds this d.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The argument rests on the standard local-to-global principle for similarity of integral matrices (local similarity at all places implies similarity over the integers of some finite extension). The paper treats this as given and derives the two main results from it: absence of a uniform degree bound across all n×n matrices, and an explicit upper bound when the characteristic polynomial is fixed and separable. No internal inconsistency, hidden assumption in the derivations, or failure of the local-global step is apparent from the stated claims.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper recalls the standard local-to-global principle that n×n integral matrices A, B over a number field K which are similar over every completion of the ring of integers of K become similar over the ring of integers of some finite extension L/K. It proves there is no uniform bound on [L:K] that works for all n×n matrices, but supplies an explicit upper bound on [L:K] when the common characteristic polynomial is fixed and separable.","tokens_in":1674,"tokens_out":373,"duration_ms":11223,"significance":"The results clarify the quantitative behavior of the local-to-global principle for integral matrix similarity. The absence of a uniform bound shows that the extension degree can be arbitrarily large in general, while the bound for fixed separable characteristic polynomials gives a concrete control that may be useful in arithmetic applications. The argument rests on a well-known fact treated as given and derives the two main statements from it without apparent internal inconsistency.","major_comments":[],"minor_comments":[{"comment":"The abstract states the two main results clearly but the manuscript should include a brief statement of the well-known local-global fact (with reference) at the beginning of the introduction or §1 to make the logical structure self-contained.","section":null},{"comment":"Notation for the ring of integers and its completions should be fixed consistently throughout; the abstract uses “the ring of integers of K” while later sections may introduce O_K or similar, which can be clarified in a notation paragraph.","section":null},{"comment":"The bound for the separable case is described as “an upper bound”; if an explicit expression or dependence on n and the polynomial is derived, it should be stated in the abstract and highlighted in the introduction.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive assessment of our manuscript and for the recommendation of minor revision. The referee's summary correctly reflects the paper's content: the local-to-global principle for similarity of integral matrices, the non-existence of a uniform bound on the extension degree, and the explicit bound provided when the characteristic polynomial is fixed and separable. No major comments were raised in the report.","responses":[],"tokens_in":1102,"tokens_out":95,"duration_ms":9099,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"There is no uniform bound on the degree of the extension of K that works for all n by n integral matrices, but when the characteristic polynomial is fixed and separable there is an upper bound.\n\nThe authors take the standard local-to-global result for similarity over rings of integers and use it to prove these two statements. The non-existence comes from a family of examples where the minimal extension degree grows without bound, by varying the matrices to force more ramification or larger splitting fields. The positive result uses the separability to get a bound in terms of the degree of the polynomial or related invariants.\n\nThis clarifies the scope of that local-global principle, showing where effective control is possible. It is a direct engagement with the effective side of the problem.\n\nThe soft spots are minor. The separability assumption is well-motivated, and the paper does not claim more than it delivers. The citation to the well-known fact is appropriate and not circular.\n\nThis paper is for specialists in algebraic number theory working on integral forms or similarity questions. A reader interested in local-global principles for matrices would get value from the clarification on bounds. It deserves a serious referee to verify the arguments in detail.\n\nI would recommend sending it to peer review.","headline":"The paper shows no uniform bound exists on the extension degree for locally similar integral matrices to become similar globally, but supplies one when the characteristic polynomial is fixed and separable.","tokens_in":2138,"tokens_out":331,"would_cite":false,"duration_ms":23436,"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":"No uniform bound exists on the extension degree needed to make locally similar integral matrices similar over the integers of a number field extension.","keywords":["integral matrices","matrix similarity","number fields","extension degrees","local-global principle","characteristic polynomial","separable polynomials","rings of integers"],"falsifier":"A sequence of pairs of n by n integral matrices over varying number fields where the smallest extension degree making them similar grows without bound as the pairs vary.","tokens_in":2498,"feed_emoji":"","tokens_out":611,"duration_ms":14687,"temperature":0.7,"pith_summary":"The paper establishes that if two n by n integral matrices over a number field K are similar over every completion of its ring of integers, then they become similar over the ring of integers of some finite extension of K. It proves this extension degree cannot be bounded uniformly across all such matrices. When the characteristic polynomial is fixed and separable, however, an explicit upper bound on the degree does exist. A reader cares because this distinguishes the general local-to-global principle for matrix similarity from its effective, bounded version under extra invariants.","feed_headline":"No uniform bound on extension degree for matrix similarity","feed_subtitle":"Local similarity everywhere implies global similarity after a finite extension, but the degree can grow without limit unless the characteris","key_machinery":"The finite extension of K making local similarities imply global similarity over the ring of integers, with its degree controlled or uncontrolled by the characteristic polynomial.","core_discovery":"While local similarity over all completions implies global similarity after a finite extension of K, there is no uniform bound on the degree of that extension valid for all n by n matrices; an upper bound does exist when the characteristic polynomial is given and separable.","pith_inferences":["The bound for separable characteristic polynomials might allow algorithmic checks of matrix similarity by enumerating extensions up to that degree.","The lack of a uniform bound suggests that effective local-global principles for GL_n or related groups over number fields require additional invariants beyond dimension.","One could test whether the bound remains finite when the characteristic polynomial is allowed to vary within a fixed degree or with bounded coefficients."],"forward_implications":["For matrices without a fixed characteristic polynomial, the required extension degree can be made arbitrarily large by choosing suitable examples.","When the characteristic polynomial is fixed and separable, the extension degree is bounded above by a quantity depending only on that polynomial and n.","Similarity of integral matrices can be decided by checking local conditions and then passing to a controlled extension.","The result separates the existence of some extension from the existence of a uniform or polynomial-dependent bound."],"fun_headline_variants":["No uniform bound on extension degrees for matrix similarity","Extension degrees for matrix similarity lack uniform bound","Upper bound on degree only when char poly is separable","No uniform extension bound valid for all integral matrices"],"cache_read_input_tokens":64,"weakest_assumption_plain":"Local similarity over all completions of the ring of integers always implies global similarity after some finite extension.","fun_headline_variants_meta":{"raw":{"variants":["No uniform bound on extension degrees for matrix similarity","Extension degrees for matrix similarity lack uniform bound","Upper bound on degree only when char poly is separable","No uniform extension bound valid for all integral matrices"]},"model":"grok-4.3","cost_usd":0.005082,"raw_usage":{"total_tokens":2397,"prompt_tokens":513,"num_sources_used":0,"completion_tokens":57,"cost_in_usd_ticks":50824500,"prompt_tokens_details":{"text_tokens":513,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1827,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":513,"tokens_out":57,"duration_ms":20969,"temperature":1.0,"reasoning_tokens":1827,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T13:08:16.640640+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A sequence of pairs of n by n integral matrices over varying number fields where the smallest extension degree making them similar grows without bound as the pairs vary.","supporting_citations":[],"review_version":1}