A finite Guinand-Weil dictionary and archimedean tail order for the truncated Weil quadratic form

Abstract: The Connes-van Suijlekom and Connes-Consani-Moscovici truncations of the Weil quadratic form, at a prime cutoff c>1 and frequency band N, produce finite Galerkin matrices whose spectra are the finite-rank window on Weil positivity. We prove two exact finite theorems about this truncation. First, every real even Galerkin coefficient vector v determines, in closed form, a band-limited Guinand-Weil test function g_v whose zero sum over the nontrivial zeros of zeta equals the quadratic value <v, Q v> exactly: every value of the truncated form is an exact sum over the zeros. The construction factors through an exact source quotient of dimension 2N+1 and admits a non-collapsing pole-neutral subfamily. Second, beyond the Galerkin band the omitted archimedean tail is a totally positive Cauchy-Stieltjes increment. This yields a two-sided certification rule with an explicit budget B_T ~ (2N+1) rho log(T) / (pi^2 T), where T is the archimedean cutoff and rho = 2 pi / log c: finite-cutoff positivity certifies cutoff-free positivity, a finite-cutoff eigenvalue below -B_T certifies a cutoff-free negative, and a negative eigenvalue in the band [-B_T, 0) certifies nothing. Resolving a spectral scale of 10^-59 at c=100 by brute cutoff would require T of order 10^63; a cutoff-free interval LDL^T factorization resolves it directly. The dictionary is verified over the first 512 zeros of zeta and by three independent computational routes; all scripts and artifacts ship with the paper. The paper makes no Riemann Hypothesis, prime-counting, next-prime, or factoring claim.

• The paper establishes a closed-form finite Guinand–Weil dictionary linking coefficient vectors to admissible band-limited test functions.
• It derives explicit certification bounds for the archimedean tail, precisely controlling eigenvalue discrepancies in finite truncated computations.
• Extensive numerical and symbolic verifications confirm that the quadratic form exactly matches the zero sums of the Riemann zeta function.

Finite Guinand–Weil Dictionary and Archimedean Tail Order: A Detailed Summary

Introduction and Context

This paper addresses finite-dimensional truncations of the Weil quadratic form, a fundamental object in analytic number theory whose positivity is equivalent to the Riemann Hypothesis. The analysis is framed within the spectral techniques developed by Connes and collaborators, which transform the Weil quadratic form into explicit finite Galerkin matrices—constructed via cutoff at a prime $c > 1$ and frequency band $N$—giving concrete access to the spectrum of Weil positivity in a computable setting. The core contributions are twofold: the establishment of a closed-form finite Guinand–Weil dictionary associating finite coefficient vectors to admissible test functions, and a precise analysis of the archimedean tail, providing rigorous certification bounds on what finite truncations reveal about the underlying limit objects.

Construction of the Finite Dictionary

The first main result is the explicit construction of a finite Guinand–Weil dictionary. Given any real, even coefficient vector $v$ of length $N+1$, a sequence of explicit transformations yields a band-limited Guinand–Weil test function $g_v$. The transport from $v$ to $g_v$ involves: embedding $v$ into symmetric Fourier coefficients; forming a trigonometric polynomial $T_v$; computing a Volterra kernel $K_v$; and finally realizing $N$0 as a compactly supported, entire function with explicitly controlled band, support, and decay properties.

The central theorem establishes that for the associated cutoff-free matrix (i.e., without archimedean truncation), every quadratic form $N$1—assembled via the finite Connes--van Suijlekom/Connes--Consani--Moscovici explicit constructions—coincides exactly with the sum of $N$2 over the nontrivial zeros of the Riemann zeta function:

$N$3

Here, zeros are counted with multiplicity. There is no residual error due to cutoff or band limitation at this level, and every entry of the truncated quadratic form precisely measures such zero sums.

Figure 1: The construction of the dictionary at $N$4, $N$5, displaying the coefficient vector $N$6, the Volterra kernel $N$7 on $N$8, the compact Fourier weight $N$9 on $v$0, and the induced entire test function $v$1 with the first ordinates $v$2 marked.

The mapping $v$3 is shown to be injective on an explicit $v$4-dimensional source quotient, with the construction retaining non-collapsing subspaces that are pole-neutral—realizing the finite-dimensional analogue of working orthogonally to the pole in Weil’s approach. Importantly, for pole-neutral or moment-neutral subspaces, one can construct vectors $v$5 such that the pole term $v$6 vanishes identically, isolating genuine spectral contributions from the zeros.

Extensive verification is carried out numerically, summing $v$7 over the first $v$8 zeros of $v$9, with residuals at the $N+1$0 scale, and full formal verification via symbolic and interval computation.

Structure and Positivity of the Archimedean Tail

Finite computation of the quadratic form requires an archimedean cutoff $N+1$1 in the continuous part of the Weil explicit formula. The paper gives a second main result: exact analysis of the post-band archimedean tail in terms of a strictly totally positive Cauchy–Stieltjes increment. For $N+1$2 sufficiently large (specifically, $N+1$3), the tail is realized as a rank-two positive-definite Gram increment, with all minors strictly positive—a strong certificate of positivity structure.

As $N+1$4, the difference between the eigenvalues of the finite-truncated matrix $N+1$5 and those of the cutoff-free matrix is bounded above by an explicit budget $N+1$6:

$N+1$7

with

$N+1$8

for large $N+1$9. This provides a two-sided certification rule: finite-$g_v$0 positivity certifies cutoff-free positivity, eigenvalues below $g_v$1 certify a true negative eigenvalue of the full form, and eigenvalues in $g_v$2 remain inconclusive.

Figure 2: Increasing eigenvalues of $g_v$3 converging to those of the cutoff-free matrix as $g_v$4 increases, with the largest possible gap controlled by $g_v$5.

The analysis reveals the central obstruction to naive numerical certification: the required cutoff $g_v$6 to resolve deep spectral scales grows explosively. For example, to resolve eigenvalues at the $g_v$7 scale for $g_v$8 and $g_v$9, one needs $v$0 of size $v$1. The paper asserts that such depth is accessible only through closed-form non-truncated computations, not by brute-force increase of the cutoff.

Worked Example and Numerical Verification

A detailed worked example at $v$2, $v$3, for a pole- and moment-neutral vector $v$4 is presented. The sum $v$5 evaluated both via the quadratic form and directly as a sum over zeros, matches to $v$6 decimal digits when the tail correction is included. The verification package accompanying the paper demonstrates both symbolic and numeric checks, confirming dictionary identity across several independent computational routes.

Implications and Theoretical Significance

From a theoretical perspective, the closed-form finite dictionary provides a rigorous finite-dimensional correspondence: explicit finite coefficient vectors map injectively to admissible Guinand–Weil band-limited test functions, and every contracted value of the finite quadratic form coincides with a sum over the zeros of the zeta function with no approximation. The finite source quotient and pole-neutral substructure clarify the structure of the finite truncation, sharpening the spectral reading of Weil positivity.

The archimedean tail analysis has strong implications for computable number theory: it imposes clear arithmetic limitations on the certifiability of spectral positivity in finite computations, quantifies the error budget for finite truncation, and gives an explicit, monotonic certification law for positivity and negativity. The total positivity proven for the tail increment ensures robust control over sign changes and eigenvalue evolution, with direct ties to classical Cauchy kernel theory and stochastic total positivity.

Outlook and Future Directions

The results provide a precise understanding of the strengths and inherent limitations of finite truncations in the context of Weil quadratic forms and the analytic approach to the Riemann Hypothesis. This facilitates both practical certification strategies in computational mathematics and deeper theoretical exploration—including potential generalizations to other L-functions, refinement of computational event calculus, and the analysis of total positivity and operator monotonicity beyond the archimedean term. The computational and symbolic infrastructure established is anticipated to inform further studies on the fine spectral structure and observability of zeros within truncation schemes.

Conclusion

The paper delivers explicit, formally proven finite-dimensional correspondences between truncated Weil quadratic forms and band-limited Guinand–Weil test functions, and a rigorous certification law for the control of the archimedean tail in finite-truncated computations. The results both illuminate the precise information available from finite truncations and set definitive quantitative limits on their reliability for deep spectral analysis, with substantial implications for computational approaches to Weil positivity, the Riemann Hypothesis, and related problems in analytic number theory.

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