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Finding all cospectral mates over a number field

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2608.12410 v1 pith:S3QOGOWA submitted 2026-08-11 math.NT math.COmath.SP

Alexander Van Werde This is my paper · ORCID
classification math.NTmath.COmath.SP MSC 05C5015B3611C2011R0405C60
keywords symmetricintegermatricescospectralmatesnumberfieldsDedekinddomainsdiscriminantKrylovsubspacesringofintegersspectraldetermination
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 7 minor

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.

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.

minor comments (7)
  1. [§5.2.5] 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.
  2. [§5.2.2, Remark 5.1] 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.
  3. [§5.1–§5.2 and Algorithm 1] 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.
  4. [Algorithm 1, line 16] 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.
  5. [References] 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.
  6. [§2.3.2 and Table 1] 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.
  7. [Introduction, p. 3] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central theorems are self-contained derivations from standard Dedekind-domain and linear-algebra arguments, with self-citations confined to motivation and heuristics.

full rationale

The central claim, Theorem 2.19, is not derived from any fitted parameter or from the author's prior work. Termination is proved directly: Lemma 5.7 uses the Vandermonde determinant of the distinct eigenvalues, justified by the hypothesis Δ_X ≠ 0; Lemma 5.8 applies Hilbert's Nullstellensatz to express monomials in the ideal generated by the quadratic forms w^T X^k w; and Lemmas 5.9–5.11 convert this into a uniform p-adic bound, making the level exponents E_P finite. Completeness is proved from Theorems 2.8 and 2.9, which are themselves derived from the ideal arithmetic of Dedekind domains and the structure of the module Im_R(LQ); neither theorem is imported from elsewhere. The self-citations [24], [37], and [39] are used only for probabilistic heuristics, for the origin of the question, and for numerical conjecture context, and none of these citations carries a load-bearing step in the proofs of Theorem 2.19 or of the constraints in Section 2.2. The implementation and exhaustive 3x3 tests are external checks rather than inputs. The assumption Δ_X ≠ 0 is explicitly stated in the theorem, and its necessity is acknowledged in Remark 2.20; it is a hypothesis of the result, not a disguised circularity.

Assumptions & free parameters 0 free parameters · 11 assumptions · 0 invented entities

No free parameters are fitted; the only user-supplied numbers are the bound M in Assumption 2.17 and the field K, which are inputs rather than fitted values. No new entities are postulated. The axioms are standard facts from commutative algebra and algebraic number theory plus the two domain assumptions (nonzero discriminant and totally real field) that are explicitly part of the theorem statements.

assumptions (11)
  • standard math Fractional ideals in a Dedekind domain admit unique prime factorization and satisfy the lattice laws of Proposition 3.3.
    Invoked throughout Sections 3 and 4 for inclusion and multiplication of ideals, e.g., in Lemmas 3.2, 3.11 and Corollary 2.11.
  • standard math Every nontrivial quotient of a Dedekind domain is a principal ideal ring.
    Used in the proofs of Theorem 2.9, Lemma 3.11 and Lemma 3.12 to pick generators of P^k/P^{k+1} and to apply Smith normal form.
  • standard math Matrices over principal ideal rings admit Smith normal form.
    Used in Lemma 3.11 to reduce determinant vanishing modulo P^k to the diagonal case.
  • standard math Cayley-Hamilton theorem.
    Used to truncate Krylov space powers at n-1 and in the contractivity proof of Theorem 2.9.
  • standard math Hilbert's Nullstellensatz over Q and C.
    Used in Lemma 5.8 to obtain polynomial certificates for the fact that the quadratic forms w^T X^k w have only the zero common root when Delta_X is nonzero.
  • standard math Kronecker's theorem on algebraic integers whose embeddings all have modulus one.
    Used in Lemma 4.1 to show that integral orthogonal matrices over a totally real field are signed permutations.
  • standard math The Minkowski embedding maps O_K into a lattice, so its intersection with a compact box is finite.
    Used in Lemma 4.2 to prove finiteness of orthogonal matrices with a fixed level.
  • standard math The ring of integers O_K of a number field is a Dedekind domain; primes ramify only finitely often and are the prime divisors of disc(O_K).
    Basis for Section 2.3 and for the ramification condition in Corollaries 2.11 and 2.12.
  • domain assumption The discriminant Delta_X is nonzero, meaning X has no repeated eigenvalues.
    Required for finite termination of Algorithm 1 (Theorem 2.19); used in Lemmas 5.7 through 5.10. If false, infinitely many orthogonal matrices may exist (Remark 2.20).
  • domain assumption The number field K is totally real.
    Needed for Proposition 2.13 and for the algorithm's finite enumeration of orthogonal matrices with a given level; the numerical experiments use totally real quadratic fields.
  • domain assumption Assumptions 2.16, 2.17, and 2.18: practical computation over O_K, a known bound M on critical primes, and low degree of the polynomial Gamma.
    These are stated efficiency assumptions. They are not needed for the correctness of Theorem 2.19, but they are needed for the practical behavior reported in Section 6.

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Pith. "Pith review of Finding all cospectral mates over a number field." pith.science (2026). https://pith.science/paper/S3QOGOWA

@misc{pith2026260812410,
  author       = {Pith},
  title        = {Pith review of: Finding all cospectral mates over a number field},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S3QOGOWA}},
  note         = {Machine review of arXiv:2608.12410}
}
read the original abstract

We investigate a notion of cospectrality for integer matrices that is parameterized by algebraic number fields. Given a number field and a symmetric integer matrix, we wonder when conjugating the integer matrix by an orthogonal matrix with entries in the given field can produce new integer matrices. Our results concern sufficient conditions for the associated notion of spectral determination, and we give constraints on the orthogonal matrices when the conditions are not applicable. The results use the discriminant of the characteristic polynomial and properties of Krylov subspaces. We leverage the theory to develop an algorithm to find all cospectral mates over a given (small) field. An implementation of the algorithm is made available.

Figures

Figures reproduced from arXiv: 2608.12410 by the authors.

Figure 1
Figure 1. Two cospectral graphs and their adjacency matrices. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

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