bibkey: sra2026thetakernelcounterexample
authors: Count ex Machina (Suvrit Sra)
year: 2026
title: GPT, the Counterexample Machine
doi: null
url: https://arxiv.org/abs/2608.29595v1
claim: Theorem 5.2 reports a computer-assisted negative witness at derivative order nine and argument 1/50 for the actual Jacobi theta kernel's derivative Laguerre inequality.
strata_touched:
  - D5/S3/Zeros/Jensen/SourceThetaMomentBounds
  - D5/S3/Analytic/Fourier/XiThetaTransform
license: citation-only
triage: anchor

A reported counterexample to the original-variable derivative conjecture

Section 5, printed pp.8–9 of the pinned preprint, addresses the Coffey–Csordas conjecture restated as Csordas Open Problem 4.13. For the actual kernel

$$K(u)=\sum _{m\ge 1}\pi m^2(2\pi m^2{e}^{4u}-3)e^{5u-\pi m^2{e}^{4u}},$$

the conjecture asks for

$$J_n^K(u):=K^{(n)}(u{)}^2-K^{(n-1)}(u)K^{(n+1)}(u)>0(n\ge 1,u\in \mathbb{R}).$$

Theorem 5.2 states $J_9^K(1/50)<0$. The reported certificate uses integer derivative polynomials, outward-rounded interval summation and an explicit bound for the omitted theta tail. The paper identifies its source repository, including a high-precision evaluator and a Sage interval certificate. This note records the source’s computer-assisted result; it neither reruns that certificate nor supplies a new Lean refutation.

The source reports this witness as a refutation of the universal conjecture. Its certificate and original-variable scope must be retained in any application. Remark 5.3 does not claim verification of the entire range $n\le 8$.

Source normalization and limits of transfer

The repository’s original kernel has $\Phi (x)=2K(x/2)$. The chain rule gives the paper-level transport

$$J_n^{\Phi }(x)=2^{2-2n}{J}_n^K(x/2).$$

Thus the source’s witness corresponds to $J_9^{\Phi }(1/25)<0$ under the repository normalization. This is the actual theta source, rather than an example obtained by dropping an admissible-kernel tail condition. The scaling formula is elementary reuse, not a new counterexample.

The square-variable kernel is instead $S(T)=\Phi (2\sqrt{T})$, with $T=x^2/4$. Its derivatives mix original-variable derivatives; the density ratio $r_v(T)=A_v{e}^{T/v}S(T)$ introduces a further tilt. The reported witness consequently supplies no negative witness for the corresponding $T$-derivative expressions or the signed estimate at the actual polynomial nodes. It refutes neither RH nor Robin.

The second-level square-variable concavity theorem addresses Csordas Open Problem 4.14. These two source results must be kept separate: a positive result for the square-variable first Laguerre expression does not assert all-order positivity for original-variable derivatives.