{"id":"4e10ba9c-555f-4d39-a5e2-57ec6325c6e8","arxiv_id":"2608.12410","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Number-field cospectrality of symmetric integer matrices is characterized by discriminant and ramification conditions, and all cospectral mates over small totally real fields can be computed by a new algorithm.","lead":"The paper introduces cospectrality for integer matrices over algebraic number fields, where conjugating by a field-valued orthogonal matrix can produce new integer matrices. It gives discriminant-based sufficient conditions and the first complete algorithm, with code, for listing all cospectral mates over small totally real fields.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified.","rationale":"Stress-test read Theorem 2.19 as the central claim: Algorithm 1 terminates in finite time and outputs all Q ∈ K^{n×n} with Q^TQ = I and Q^TXQ ∈ O_K^{n×n}, under the standing assumptions Δ_X ≠ 0, K totally real, and the computational Assumptions 2.16–2.18 only for efficiency. The termination proof is sound: Lemma 5.7 reduces the common-zero question for the quadratic forms w^T X^k w to a Vandermonde system, Lemma 5.8 obtains rational coefficient identities via the Nullstellensatz, Lemma 5.9 converts these into p-adic non-vanishing, and Lemma 5.10 gives the finite bound on E_P. The rational-coefficient step in Lemma 5.8 is valid because the system of linear equations for the coefficients of the s_k has rational coefficients and target, so a solution over ℂ implies one over ℚ. Uniformity of e over all prime ideals follows from the finiteness of the set of P above primes dividing the integers n_i. Completeness is supported by Theorems 2.8 and 2.9: any prime ideal dividing the full level L must appear in the candidate list, any exponent beyond E_P is impossible, and every actual column satisfies the residue conditions in Step 3a. The column-residue claim is the most subtle informal point, but it can be proved by scaling a vector from Im(LQ), as described in the attack field. Step 3b's enumeration of solutions to V^TV = ℓ^2 is exhaustive modulo the stated norm and lookahead bounds, and Step 3c checks orthogonality exactly. The assumption Δ_X ≠ 0 is explicit in Theorem 2.19 and is genuinely necessary: Remark 2.20 gives X = I and shows the finite-output requirement fails when Δ_X = 0. The paper also provides an implementation and reports exhaustive tests on all 3×3 matrices with ∑ X_ij^2 ≤ 25, which is real independent support. I therefore found no load-bearing defect in the central argument.","tokens_in":32603,"tokens_out":26190,"duration_ms":257051,"concrete_test":"Re-run Algorithm 1 on every symmetric 3×3 integer matrix with ∑_{i≤j} X_ij^2 ≤ 50 over ℚ and over ℚ(√2), ℚ(√3), ℚ(√5), and compare the output against an independent brute-force enumeration of integer cospectral mates using the trace bound (2.16); any missed matrix would invalidate the completeness claim, while agreement corroborates Theorem 2.19.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 2.19's guarantee depends on the explicit assumption Δ_X ≠ 0, exactly as the reader notes; this assumption is necessary and is acknowledged in Remark 2.20, so it is a limitation, not a hidden flaw. I found no internal gap in the termination or completeness arguments. Termination follows from Lemmas 5.7–5.10: the Nullstellensatz step over ℚ is justified by viewing (5.26) as a rational linear system, and a uniform exponent e for all prime ideals exists because only the finitely many primes dividing the integers n_i matter. Completeness rests on Theorems 2.8–2.9 together with the exhaustive enumeration in Steps 3a–3c. The one step that is written informally is the claim in Step 3a that a column's scaled residue lies in 𝕎_P(e_P); this is not an error: given a column q of level f and ℓ' ∈ L with v_P(ℓ') = g ≥ f, the vector ℓ'q lies in Im(LQ), whose Krylov space is contractive and isotropic by Theorem 2.9, and scaling by ℓ/ℓ' transfers those properties from level g to level f. Thus the algorithm's completeness has no hidden assumption. The release of an implementation and the reported exhaustive 3×3 tests provide additional support.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces and studies a number-field-parameterized notion of cospectrality for symmetric integer matrices: X and Y are cospectral over K if Y = Q^T X Q for an orthogonal matrix Q with entries in K. The theoretical core is developed over Dedekind domains. Theorem 2.1 shows that any prime ideal divisor of the level ideal L of Q must have square dividing the discriminant ideal ΔX R, and Theorems 2.8 and 2.9 show that the image module Im_R(LQ) is a Γ-contractive and isotropic Krylov space, where Γ is an approximate square root of the characteristic polynomial modulo P. Over number fields these yield sufficient conditions for spectral determination (Corollaries 2.11 and 2.12) and finiteness of orthogonal matrices with a fixed level when K is totally real (Proposition 2.13). Section 5 presents Algorithm 1, which, under the assumptions ΔX ≠ 0 and K totally real, terminates in finite time and outputs all orthogonal matrices Q in K^{n×n} with Q^T X Q in O_K^{n×n} (Theorem 2.19). Section 6 reports numerical experiments over Q and quadratic fields, and the implementation is made publicly available.","tokens_in":32874,"tokens_out":40541,"duration_ms":340232,"significance":"This is a strong paper. Theorem 2.19, if correct, gives the first complete finite algorithm for finding all cospectral mates over a fixed number field in the simple-spectrum case, and the structural results Theorems 2.8 and 2.9 are new even for K = Q. The proofs are detailed and self-contained, with no fitted parameters; the termination argument via Hilbert's Nullstellensatz (Lemmas 5.7–5.10) is particularly clean and shows exactly where ΔX ≠ 0 enters. The paper also ships an implementation, reports exhaustive 3×3 validation and random-matrix statistics, and gives concrete falsifiable predictions about rational and quadratic-field cospectrality frequencies. The main limitation—repeated eigenvalues—is openly acknowledged in Remark 2.20 and is genuinely necessary for finite output.","major_comments":[],"minor_comments":[{"comment":"The deduction that a column residue ℓq mod P^{e_P} lies in 𝕎_P(e_P) is stated without proof. It is correct, but a reader has to fill in the scaling argument: choose ℓ' ∈ L with v_P(ℓ') ≥ e_P; then ℓ'q lies in Im(LQ), whose Krylov space is contractive and isotropic by Theorem 2.9, and scaling by ℓ/ℓ' transfers these properties down from level v_P(ℓ') to level e_P. Please add this clarification.","section":"§5.2.5"},{"comment":"The proposed linearization for p = 2 is incorrect in the stated generality. In characteristic 2 the identity v^T M v = Σ M_{ii} v_i^2 holds for any symmetric M, but the second congruence Σ D_i v_i^2 ≡ (Σ D_i v_i)^2 holds only when each D_i lies in the prime field F_2; the diagonal entries D_i^{(j)} = k_i^T X^j k_i need not lie in F_2 when F_P is a proper extension of F_2. Unless additional conditions are supplied, this remark should be corrected or removed.","section":"§5.2.2, Remark 5.1"},{"comment":"The notation for the search sets is inconsistent: Step 2b defines 𝒲_P(e), Step 2c defines 𝕎_P(e) ⊆ 𝒲_P(e), while Algorithm 1's lines 4–9 use 𝕎(e) for both, and line 9 incorporates the P^{2e} isotropy condition that Step 2c assigns to the subset 𝕎_P(e). Please harmonize the notation.","section":"§5.1–§5.2 and Algorithm 1"},{"comment":"The pseudocode's line 16 refers to 𝕎_P(e_P) for exponent sequences with all e_P = 0, but 𝕎_P(0) is never defined; Step 3a handles this case separately via Lemma 4.1. The pseudocode should state this case explicitly.","section":"Algorithm 1, line 16"},{"comment":"References [39] and [40] are duplicate entries for the same arXiv preprint 'Exact cospectrality probabilities for uniform random matrices' (arXiv:2602.00233); one duplicate should be removed or the text should point to different items.","section":"References"},{"comment":"There are several typos and infelicities: 'Algorithm 1 can be applied to to integer matrices' in §2.3.2, and the caption of Table 1 reads 'as well as average the blocksizes and levels'. These should be corrected.","section":"§2.3.2 and Table 1"},{"comment":"The sentence that 'the only assumption that is strictly required for our theory is that X should not have repeated eigenvalues' is imprecise, because the algorithmic results also require K to be totally real and, in practice, the additional assumptions in §2.3.2. Please qualify this statement.","section":"Introduction, p. 3"}],"recommendation":"minor_revision","confidential_remarks":"The central mathematical content is sound; I found no load-bearing gap in Theorems 2.1, 2.8, 2.9, or 2.19. The main issue is the erroneous optimization remark (Remark 5.1), which should be corrected or removed before publication, and the compressed exposition in §5.2.5. The paper is well within the journal's scope and the novelty is clear."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a solid paper doing something genuinely new. It defines cospectrality over an arbitrary number field K, proves constraints on orthogonal matrices over K when the standard discriminant condition fails, and—the headline contribution—gives an algorithm that provably enumerates all Q in K^{n x n} with Q^T Q = I and Q^T X Q in O_K^{n x n} for totally real K and Δ_X ≠ 0. The implementation is released and tested, including exhaustive 3x3 tests. That is real evidence, not just an assertion.\n\nThe math holds up. Theorems 2.1, 2.8, and 2.9 extend Wang–Yu from Q to Dedekind domains, and the proofs are detailed. I checked the termination argument in Theorem 2.19: the Nullstellensatz step is legitimate because (5.26) is a rational linear system, so uniform exponents exist for the finitely many primes dividing the relevant integers. The Δ_X ≠ 0 assumption is load-bearing, but the paper says so in Remark 2.20 and it is genuinely necessary: with repeated eigenvalues, an eigenspace rotation gives infinitely many Q. So that is a stated limitation, not a hidden flaw.\n\nThe soft spots are minor. Step 3a claims that a column's scaled residue lies in W_P(e_P) with a compressed justification. The stress-test note explains why it is true (scaling by ℓ/ℓ' transfers contractivity and isotropy from level g to level f), but the paper should write that out. A reader shouldn't have to reconstruct it. There is also a duplicated reference: [39] and [40] are the same paper. Both are easy fixes.\n\nThe experimental section is honest about timeouts and uses M = 10^4 as a cutoff, so the empirical rates are heuristic in the strict completeness sense; the paper says so. That is fine for a numerical study.\n\nWho this is for: spectral graph theorists and people working on rational orthogonal matrices, generalized cospectrality, or algorithmic algebraic number theory. If your work touches any of those, you will want it. I would send it to a serious referee. My own verdict would be accept after minor revision: clarify Step 3a and deduplicate the references.","headline":"A genuinely new number-field parameterization of cospectrality with a complete algorithm and released code; the main theorem checks out and the paper deserves refereeing.","tokens_in":33361,"tokens_out":2572,"would_cite":true,"duration_ms":22632,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C50","15B36","11C20","11R04","05C60"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that, for an integer symmetric matrix with distinct eigenvalues, all cospectral mates over any totally real number field can be listed by an explicit finite algorithm.","keywords":["symmetric integer matrices","cospectral mates","number fields","Dedekind domains","discriminant","Krylov subspaces","ring of integers","spectral determination"],"falsifier":"Take the $5\\times5$ matrix in Example 1.3 and independently verify, by a second implementation, that the only rational orthogonal matrices with denominator at most 31 that conjugate it to an integral symmetric matrix are exactly those returned by Algorithm 1 up to signed column permutation; a single missed $Q$ would disprove Theorem 2.19.","tokens_in":32429,"feed_emoji":"🔢","tokens_out":8813,"duration_ms":78717,"temperature":0.7,"pith_summary":"The paper sets up cospectrality parameterized by a number field $K$: two symmetric integer matrices are cospectral mates over $K$ if an orthogonal matrix with entries in $K$ conjugates one to the other with the result integral over $K$. The main theorem asserts that when $X$ has no repeated eigenvalue, the complete list of such conjugating matrices over any totally real $K$ can be computed: Algorithm 1 terminates and returns all of them. The companion sufficient condition identifies when the list contains only signed permutations, namely when the discriminant of $X$ is square-free and none of its prime divisors ramify in $K$. A reader should care because prior notions either imposed extra vector constraints or only produced local obstructions, whereas this framework gives a field-parameterized relaxation that is still algorithmically decidable.","feed_headline":"Finite algorithm finds every cospectral mate over a number field.","feed_subtitle":"For matrices with distinct eigenvalues, one program now outputs the complete list instead of brute-force search.","key_machinery":"The load-bearing objects are the discriminant ideal $\\Delta_X\\mathcal{O}_K$, the level ideal $L = \\{r \\in \\mathcal{O}_K : rQ \\in \\mathcal{O}_K^{n\\times n}\\}$ of a candidate $Q$, and Krylov spaces $\\mathcal{K}_X(w) = \\mathrm{span}\\{X^i w\\}$. The paper proves that every prime ideal $P$ dividing $L$ must have $P^2 \\mid \\Delta_X\\mathcal{O}_K$; and if $P^e \\mid L$ then some reduction of $LQ$ is a Krylov space that is $\\Gamma$-contractive up to level $P^e$ and isotropic up to level $P^{2e}$, where $\\Gamma$ is the approximate square root of $\\phi_X$ mod $P$ coming from the square-free factorization. These constraints become linear equations when lifting a candidate vector modulo $P^e$ to $P^{e+1}$. Finiteness of the exponent $e$ comes from a Hilbert Nullstellensatz argument: for $\\Delta_X \\neq 0$, the only common zero of the quadratic forms $w^\\top X^k w$ for $k \\le n-1$ is $w = 0$, giving a uniform $P$-adic bound. The algorithm then enumerates the finite set of possible columns subject to the norm equation $V^\\top V = \\ell^2$ and orthogonality.","core_discovery":"The paper's central claim is that cospectrality over a number field is a finite, searchable phenomenon. For a symmetric integer matrix $X$ with $\\Delta_X = \\det(\\phi_X'(X)) \\neq 0$ and a totally real number field $K$, the set of orthogonal matrices $Q \\in K^{n\\times n}$ with $Q^\\top Q = I$ and $Q^\\top X Q \\in \\mathcal{O}_K^{n\\times n}$ is finite up to signed permutation, and can be found by Algorithm 1, which terminates in finite time and is complete. This includes, but is not limited to, the classical question of cospectral mates over $\\mathbb{Q}$: any orthogonal matrix conjugating $X$ to a symmetric integer matrix can be taken to have entries in some number field, and the algorithm decides which fields admit nontrivial examples. The theoretical sufficient condition says that if the discriminant ideal is square-free and no prime divisor of $\\Delta_X$ ramifies in $K$, then no such nontrivial $Q$ exists.","pith_inferences":["Beyond the paper, if the conjectured positive frequency of square-free discriminants holds, the sufficient condition would make spectral determination over a fixed small $K$ a property that holds and is certifiable for a positive proportion of random integer matrices.","Beyond the paper, the repeated-eigenvalue case is left open; the paper's trace bound shows only finitely many cospectral mates exist, so one could quotient by eigenspace rotations and still enumerate the finite set of conjugacy classes.","Beyond the paper, the reported dimension sensitivity for quadratic fields (e.g., integer cospectrality over $\\mathbb{Q}(\\sqrt{2})$ appearing mainly in even dimensions) suggests an arithmetic explanation via the norm equation $V^\\top V = \\ell^2$, which could be tested by deriving congruence conditions on $\\ell$ from the field discriminant."],"forward_implications":["If $\\Delta_X \\neq 0$ and every prime divisor of $\\Delta_X$ is unramified in $K$ with exponent one, then the only orthogonal $Q \\in K^{n\\times n}$ with $Q^\\top X Q \\in \\mathcal{O}_K^{n\\times n}$ are signed permutations; $X$ is determined by its spectrum over $K$.","In all other cases, any such $Q$ has a level ideal whose prime divisors all lie above primes dividing $\\Delta_X$, and Theorem 2.8 gives an explicit finite bound on the exponent of each such prime, turning the search into a finite enumeration.","Algorithm 1 terminates for every totally real $K$ and every symmetric integer $X$ with $\\Delta_X \\neq 0$, so the cospectral-mate problem over a small number field is decidable rather than heuristic.","For $K = \\mathbb{Q}$, the algorithm finds rational cospectral mates that are invisible to generalized cospectrality, as in Example 1.3, so it answers a strictly broader question."],"supporting_citations":[{"why":"Supplies the square-free discriminant sufficient condition that the paper generalizes from the rationals to arbitrary Dedekind domains and number fields.","marker":"[47]"},{"why":"Provides the number-theoretic infrastructure: the ring of integers as a Dedekind domain, ramification of primes, and the Minkowski lattice used to prove finiteness for a fixed level.","marker":"[28]"},{"why":"Supplies the algebraic number theory facts on field discriminants and prime ideals used in Corollary 2.11 and in the classification of matrices with discriminant $\\pm 1$.","marker":"[25]"},{"why":"Provides Hilbert's Nullstellensatz, which in Lemma 5.8 turns the absence of common zeroes into explicit monomial identities that bound the level exponents.","marker":"[18]"},{"why":"Carries the proof that an integral orthogonal matrix over a totally real field must be a signed permutation, via the theorem on algebraic integers whose conjugates all have modulus one.","marker":"[13]"},{"why":"Provides the Dedekind ideal arithmetic, including fractional ideal factorization and quotients that are principal ideal rings, used throughout the proof of Theorem 2.1.","marker":"[50]"},{"why":"Justifies the no-repeated-eigenvalue assumption as generically true for random matrices, supporting the applicability of the main theorem.","marker":"[34]"}],"fun_headline_variants":["Algorithm lists every cospectral mate over a number field","No more brute force: all cospectral mates found","Finite search enumerates all cospectral mates over a field","Complete cospectral mate list over any number field"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes the matrix has no repeated eigenvalue, so the characteristic polynomial has nonzero discriminant; without this, eigenspace rotations can make the list of conjugating matrices infinite.","fun_headline_variants_meta":{"raw":{"variants":["Algorithm lists every cospectral mate over a number field","No more brute force: all cospectral mates found","Finite search enumerates all cospectral mates over a field","Complete cospectral mate list over any number field"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000503,"raw_usage":{"total_tokens":2414,"prompt_tokens":857,"completion_tokens":1557,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":473,"completion_tokens_details":{"reasoning_tokens":1505}},"tokens_in":473,"tokens_out":1557,"duration_ms":10580,"temperature":1.0,"reasoning_tokens":1505,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:15:20.022575+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the $5\\times5$ matrix in Example 1.3 and independently verify, by a second implementation, that the only rational orthogonal matrices with denominator at most 31 that conjugate it to an integral symmetric matrix are exactly those returned by Algorithm 1 up to signed column permutation; a single missed $Q$ would disprove Theorem 2.19.","supporting_citations":[{"cited_title":"Square-free Discriminants of Matrices and the Generalized Spectral Characterizations of Graphs","cited_arxiv_id":"1608.01144","evidence_quote":"Supplies the square-free discriminant sufficient condition that the paper generalizes from the rationals to arbitrary Dedekind domains and number fields."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Carries the proof that an integral orthogonal matrix over a totally real field must be a signed permutation, via the theorem on algebraic integers whose conjugates all have modulus one."},{"cited_title":"Inconclusive","cited_arxiv_id":null,"evidence_quote":"Provides the Dedekind ideal arithmetic, including fractional ideal factorization and quotients that are principal ideal rings, used throughout the proof of Theorem 2.1."}],"review_version":1}