arXiv:2507.08138v2

On Conservative Matrix Fields: Continuous Asymptotics and Arithmetic

Shachar Weinbaum, Elyasheev Leibtag, Rotem Kalisch, Michael Shalyt, Ido Kaminer

math.NTcs.SCmath.COmath.RA11J7011J8240A1533C8033C7068W30

Paper metadata

Original submission: July 10, 2025
Latest revision: July 15, 2026 · version 2
Claimed MSC: 11J70, 11J82, 40A15, 33C80, 33C70, 68W30
arXiv categories: math.NT, cs.SC, math.CO, math.RA
Paper license: CC BY-SA 4.0 ↗
Comments: Revised structure and exposition. Added a proof of a statement previously formulated as a conjecture. This version has been submitted for publication

Abstract

We present the Conservative Matrix Field (CMF) as a tool for the analysis and computation of D-finite functions. We use conservative matrix fields to establish asymptotic properties of families of linear forms in periods, including (but not limited to) multivariate Mellin integrals, via a discrete Levinson-type framework due to Benzaid and Lutz. Finally, we present an experimental analysis of the families of linear forms generated by these objects and formalize the resulting observations as conjectures on their continuous asymptotic and arithmetic properties.

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Audit summary

Audited against arXiv v2

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 15, 2026

01Statements5 reported findingsContains wrong statements

The D-finite construction of conservative matrix fields, the D-finiteness of their coefficients, and the local limiting-matrix results are correct. Two central asymptotic assertions are false under their printed hypotheses. Lemma 4.3 and Corollary 4.4 omit control of singular evaluated factors in the finite prefix, and Theorem 5.1 omits the strict spectral-modulus gap needed for convergence. Explicit rank-one and rank-two counterexamples isolate both defects. The two conjectures in Section 6 are presented as conjectures and are not treated as proved claims.

Theorem 3.1 and Lemma 3.4Correct

D-finite functions generate CMFs with D-finite coefficients

Pages 11 and 13 · Theorem 3.1 and Lemma 3.4 · arXiv:2507.08138v2

For a basis $B$ of the finite-dimensional Ore-module image, the identity $B M_v^f=\sigma_v(B)$ determines an invertible basis-change matrix. Applying two shifts gives $B M_v^f\sigma_v(M_w^f)=\sigma_{v+w}(B)$ , which is exactly the cocycle relation. For a matrix coefficient $e_i^{\mathsf T}M_v(x)e_j$ , every shift is a rational-function linear combination of the $r$ entries of $e_i^{\mathsf T}M_v(x)$ ; hence its shift-module dimension is at most $r$ . Linear combinations cover arbitrary rational vectors.

Full paper, version 2 ↗

Proposition 4.1, Corollary 4.2, and Proposition 4.5Correct

Limiting generators, trajectory expansion, and normalized spectrum

Pages 15 and 17–18 · Proposition 4.1, Corollary 4.2, and Proposition 4.5 · arXiv:2507.08138v2

Away from zeros of the leading denominator forms, every balanced rational entry has a constant term, a $1/n$ term, and an $O(n^{-2})$ remainder. The bounded inverse generators force the limiting matrices to be invertible, and the cocycle equation makes them commute. Multiplying the finitely many generator expansions gives Corollary 4.2. When the commuting limiting generators are diagonalizable, they are simultaneously diagonalizable over $\mathbb C$ ; the logarithms of the absolute values of the joint eigenvalues give the asserted continuous normalized spectrum, understood with a continuous local labeling or as an unordered spectrum.

Full paper, version 2 ↗

Lemma 4.3 and Corollary 4.4Incorrect

A singular finite factor defeats the claimed invertible factorization

Pages 16–17 · Lemma 4.3, Equations (14)–(16), and Corollary 4.4 · arXiv:2507.08138v2

In rank one, take $T(n)=(n-1)/n$ . It is a nonzero element of $\mathrm{GL}_1(\mathbb Q(n))$ , satisfies $T(n)=1-1/n$ , and its limiting eigenvalue is $1$ , but $P(n)=T(1)\cdots T(n)=0$ for every $n\geq1$ . It therefore cannot equal $B\,1^n n^{-1}(1+o(1))A$ with $A,B\in\mathrm{GL}_1(\mathbb R)$ . The same defect occurs inside the CMF class. Define the balanced rank-one CMF $M_v(x)=x/(x+v)$ . The cocycle identity holds exactly, but for $x=0$ and direction $v=1$ one has $M_n(0)=0$ for every $n\geq1$ , while the limiting trajectory eigenvalue is $1$ . Thus Corollary 4.4 is also false as stated. Repair: require the evaluated finite prefix to be defined and invertible, start the Levinson reduction after a sufficiently large index, and absorb that nonsingular prefix into $B$ .

Full paper, version 2 ↗

Theorem 5.1Incorrect

Proposition 4.5 does not supply the spectral gap required for convergence

Pages 19–20 · Theorem 5.1 and the sentence following its proof · arXiv:2507.08138v2

The theorem assumes only Proposition 4.5, which permits eigenvalues of equal modulus, but its proof invokes Equation (17), derived under the strict inequalities of Corollary 4.4. A concrete counterexample is the constant balanced rank-two CMF generated by $T=S\operatorname{diag}(1,-1)S^{-1}$ with $S=\begin{pmatrix}1&1\\1&-1\end{pmatrix}$ . It has the exact factorization $T^n=S\operatorname{diag}(1,(-1)^n)S^{-1}$ . With right vector $e_2$ , $p=(1,0)^{\mathsf T}$ , and $q=(1,2)^{\mathsf T}$ , all three displayed conditions hold with $k=1$ , $j=2$ , and $c=1/3$ , and both coordinates of $S^{-1}e_2$ are nonzero. Nevertheless, the CMF ratio equals $0$ for even $n$ and $1$ for odd $n$ , so it does not converge to $c$ . Repair: assume the strict modulus ordering from Corollary 4.4, together with its corrected evaluated-invertibility condition.

Full paper, version 2 ↗

Proposition 6.1Correct

Homogeneity and rank-two continuity under the stated existence assumptions

Pages 20–21 · Proposition 6.1 · arXiv:2507.08138v2

The identity $L_{x,kv}^{p,q}(n)=L_{x,v}^{p,q}(kn)$ gives $l(kv)=l(v)$ , $\rho(kv)=k\rho(v)$ , $\eta(kv)=k\eta(v)$ , and $\delta(kv)=\delta(v)$ directly from the definitions when the listed limits exist. In rank two, linear independence forces the leading term of the numerator-denominator cancellation to use the other eigenvalue, so the normalized convergence rate is the difference of the two normalized logarithmic moduli and is continuous on a region with the required spectral separation.

Full paper, version 2 ↗

02Proofs4 reported findingsContains incorrect or incomplete proofs

The proof of Lemma 4.3 incorrectly starts all products and transformations at index $1$ and asserts a nonzero scalar-product constant that may vanish. This creates a genuine counterexample and propagates to Corollary 4.4. Theorem 5.1 then applies Equation (17) without assuming its strict spectral-gap hypotheses. Both proof chains have verified repairs after adding the missing restrictions. A coefficient-ring notation error is reported separately in yellow.

Lemma 4.3Incorrect as written · verified repair

The proof cannot absorb a singular initial factor

Pages 16–17 · Equations (14)–(16) · arXiv:2507.08138v2

The argument uses $H(1)^{-1}$ and the diagonal product $\prod_{k=1}^n(\lambda_i+d_i/k)$ without proving that these factors are invertible. The assertion that its constant $c_i$ is nonzero fails whenever $d_i/\lambda_i$ is a negative integer; $T(n)=(n-1)/n$ realizes the failure at $k=1$ . Field invertibility in $\mathbb Q(n)$ does not imply invertibility after evaluation at every positive integer, a distinction already noted in Definition 11. The repair is complete after adding that a finite evaluated prefix is defined and invertible: choose $N$ so large that $H(n)$ and every $\lambda_i+d_i/n$ are invertible for $n\geq N$ , apply the summable-perturbation theorem to the tail product, use $\prod_{k=N}^n(1+d_i/(\lambda_i k))=c_{i,N}n^{d_i/\lambda_i}(1+o(1))$ with $c_{i,N}\neq0$ , and absorb the nonsingular prefix into $B$ .

Benzaid–Lutz asymptotic framework ↗

Corollary 4.4Incorrect as written · verified repair

Symbolic CMF invertibility is insufficient after specialization

Page 17 · Application of Lemma 4.3 immediately before Equation (17) · arXiv:2507.08138v2

A trajectory matrix belongs to $\mathrm{GL}_r(\mathbb Q(n))$ symbolically, yet an evaluated early factor may be singular or undefined. Then the finite product cannot be absorbed into an invertible $B(x,v)$ . The rank-one CMF $M_v(x)=x/(x+v)$ at $x=0$ , $v=1$ gives an explicit singular prefix and identically zero products. Add the hypothesis that the relevant evaluated trajectory factors, or equivalently one complete prefix through a sufficiently large nonsingular tail, are defined and invertible. The corrected Lemma 4.3 then proves the stated factorization.

Full paper, version 2 ↗

Theorem 5.1Incorrect as written · verified repair

Equation (17) is invoked outside its stated spectral regime

Pages 19–20 · Opening hypothesis and final use of Equation (17) · arXiv:2507.08138v2

The opening reference to Proposition 4.5 supplies diagonalizable commuting limiting generators but not $|\lambda_1|>\cdots>|\lambda_r|$ . Equation (17), on which every $\Theta$ estimate in the proof rests, was obtained only from Corollary 4.4 under exactly that strict ordering. The equal-modulus matrix counterexample in the statement finding shows that this is not an expositional omission. Replace the opening hypothesis by the corrected hypotheses of Corollary 4.4 together with those of Proposition 4.5. Then $|\lambda_j/\lambda_k|<1$ , Equation (17) applies to the denominator and the cancelled numerator, and the displayed convergence rate follows.

Full paper, version 2 ↗

Remark 3Typo

The annihilating-operator coefficient field is abbreviated incorrectly

Page 4 · First two bullets of Remark 3 · arXiv:2507.08138v2

The equivalence is printed using $K[\partial_i]\cap\operatorname{ann}(f)$ , and the next bullet likewise places the minimal shift operator in $K[S_{x_i}]$ . Replace these by $K(x,z)[\partial_i]\cap\operatorname{ann}(f)$ and $K(x,z)[S_{x_i}]$ , respectively. The displayed coefficients $p_j\in K(x,z)$ and every subsequent calculation already use the rational-function coefficient field, so the intended correction is unique and no main argument changes.

Full paper, version 2 ↗

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2507.08138v2
Authors listed: Shachar Weinbaum, Elyasheev Leibtag, Rotem Kalisch, Michael Shalyt, Ido Kaminer
Audit date: August 15, 2026

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P(delta)

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On Conservative Matrix Fields: Continuous Asymptotics and Arithmetic

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

D-finite functions generate CMFs with D-finite coefficients

Limiting generators, trajectory expansion, and normalized spectrum

A singular finite factor defeats the claimed invertible factorization

Proposition 4.5 does not supply the spectral gap required for convergence

Homogeneity and rank-two continuity under the stated existence assumptions

The proof cannot absorb a singular initial factor

Symbolic CMF invertibility is insufficient after specialization

Equation (17) is invoked outside its stated spectral regime

The annihilating-operator coefficient field is abbreviated incorrectly

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