bibkey: silva2026rvapproximants
authors: Alisson M. Silva
year: 2026
title: Finite Rodríguez-Villegas Approximants to the Riemann ξ-Function
doi: null
url: https://arxiv.org/abs/2609.25564v1
claim: Theorems 2.1 and 3.1 supply the original-theta Stieltjes profile, reciprocal finite numerators and critical-strip convergence; a directed original-source computation excludes the unit-circle numerator premise at degree four.
strata_touched: []
license: citation-only
triage: anchor

Original-theta finite RV approximants

The primary is arXiv:2609.25564v1, Alisson M. Silva, Finite Rodríguez-Villegas Approximants to the Riemann ξ-Function. Theorems 2.1, 3.1 and 4.1, their proofs and section 5 were inspected. These remain public-preprint statements; this note supplies neither a complete independent audit of the paper nor Lean verification. No source text is vendored.

Reused source and exact correspondence

Use the standard completed function $\xi$ and $\Xi (t)=\xi (1/2+it)$. The original even theta density is normalized by $\Xi (t)={\int }_{\mathbb{R}}\Phi (v)e^{itv}dv$. Equations (9)–(10) give

$$P(x)=G(x(1-x)),G(p)={\int }_0^{\infty }\frac{2\Phi (v)\cosh (v/2)}{1+4p{\sinh }^2(v/2)}dv.\text{\hspace{1em}(RV1)}$$

Theorem 2.1 supplies positivity, symmetry, the Stieltjes representation and the exact critical-strip transform. In particular the endpoint identity is $P(0)=\xi (1)=1/2$. For an even positive integer $E$, the paper’s numerator and transform are

$$U_E(q)=\sum _{j=0}^E(-1{)}^j(\genfrac{}{}{0ex}{}{E}{j})P(j/E)q^j,Z_E(s)=\sum _{j=0}^E(-1{)}^j\left(\genfrac{}{}{0ex}{}{E}{j}\right)P(j/E)\left(\genfrac{}{}{0ex}{}{E-j-s}{E}\right).\text{\hspace{1em}(RV2)}$$

Theorem 3.1 supplies reciprocity of $U_E$, $Z_E(1-s)=Z_E(s)$ and $Z_E\to \xi$ locally uniformly in $0<\Re s<1$. Theorem 4.1 interprets the same coefficients through a fermionic supertrace. These constructions are reused, not new project theorems.

The classical RV unit-circle theorem recalled in section 1 requires all numerator zeros to lie on the unit circle and $U_E(1)\ne 0$. Section 5 explicitly leaves the numerator zero locations unproved. Positivity, Stieltjes structure, reciprocity and convergence do not supply that premise.

Directed degree-four original-source obstruction

This is a repo-derived numerical application of (RV1)–(RV2), not an originality claim or a kernel-verified theorem. With $a=P(1/4)$, $b=P(1/2)$ and $p_0=1/2$, the exact numerator is

$$U_4(q)=p_0(1+q^4)-4a(q+q^3)+6b{q}^2.\text{\hspace{1em}(RV3)}$$

For $q\ne 0$ put $t=q+q^{-1}$. Then

$$q^{-2}{U}_4(q)=p_0(t^2-2)-4at+6b,\Delta =16a^2-24p_{0}b+8p_0^2=16a^2-12b+2.\text{\hspace{1em}(RV4)}$$

Directed Arb integration of the original infinite theta density gives strict rational enclosures

$$\begin{array}{rl}495681591851/10^{12}& <a<495681591854/10^{12},\\ 494287091469/10^{12}& <b<494287091472/10^{12}.\end{array}\text{\hspace{1em}(RV5)}$$

Their exact rational consequence is

$$\Delta \le -\frac{3769524708640210671}{15625000000000000000000}<0.\text{\hspace{1em}(RV6)}$$

If $|q|=1$, then $t$ is real, but the real quadratic (RV4) has no real zero. Hence $U_4$ has no unit-circle zero. Its nonzero leading and constant coefficients make all four zeros lie off that circle. The same certificate gives $U_4(1)>0$, with an enclosure centered at $0.0002698140052424128699$. The failure is the unit-circle condition, not the endpoint nonvanishing condition.

The independent experimental program and directed data are in the private experiments repository, experiments/xi-rv-quartic/. It reuses the original theta-jet and scalar-envelope implementations in docs/reports/theta-mixed-matrix. The 256-bit computation includes the infinite theta-series error. For the integration beyond $v=3$, an explicit bound is less than $1.13\cdot 10^{-544}$. Thus neither a finite theta truncation nor an unaccounted compact integration replaces the original source. The numerical enclosures and source normalization remain numerical certificate evidence, not Lean proofs.

This excludes the premise that every even-degree numerator in this specific original-source construction has unit-circle zeros. It does not exclude a sufficiently large-degree subsequence, assert an off-line zero of $Z_4$, invalidate the paper’s convergence or functional equation, or prove or refute RH. Full Robin remains unresolved. The next usable input would have to control an appropriate large-degree family or a different exact zero-location transfer; the failed universal unit-circle premise cannot be used.