A Computable, Zeta-Free Truncation Family for the Weil Measure: Measurements on a Hilbert–Pólya Candidate

Hamann, Stefan; Rizzo, Alessandro

Note N2Working note

A Computable, Zeta-Free Truncation Family for the Weil Measure: Measurements on a Hilbert–Pólya Candidate

Finite operators from the Gaussian-E₈ Hecke tower that measurably reproduce the Weil measure — with a prediction freeze and an explicit no-proof fence

A family of finite, explicitly computable operators — truncations of one glued geometric object built from the Gaussian E₈ lattice — whose spectral data measurably reproduce the Weil measure of prime number theory without ever loading a prime table or a zeta zero. The finite places are the Hecke–commensurability tower of the rank-4 Hermitian unimodular ℤ[i]-lattice: its primitive degrees ARE the Gaussian prime norms, and a circle-free logarithm generator plus conjugation descent produce the atom comb of the Weil measure at deviation 0.0. The archimedean place is a 48-site spin cover lift whose μ₄-fixed heat trace is exactly the density behind Re ψ(1/4 + iτ/2) — the 1/4 is derived, not declared — and the two sides glue with ONE normalization: a free three-scalar fit returns (1,1,1) to below 10⁻¹², and among the dimension-8 unimodular lattices {ℤ⁸, E₈} only the Gaussian E₈ glues. On every truncation the Weil functional is a state (GNS vector state at KMS β = 1); the truncation eigenvalues, frozen by SHA-256 BEFORE any zero is loaded, hit 100% of the first 377 zeta zeros at tolerance 0.25 with ladder rate −1.61, and the matched-node nearest-neighbour statistic is 0.6178 against the zeros' own 0.6189. The finite trace formula is exact Gauss quadrature, and its term dictionary to the classical Weil explicit formula closes block by block at 10⁻¹³–10⁻¹⁶. The note then decomposes what remains: two of the three convergence steps are classical or measured, the node-capture half of the third is proof-near, and the one remaining statement — Weil positivity in the limit — is localized in four machine-verified equivalent languages (Hankel, Levinson, Fejér, Krein–Suzuki), each finitely decidable per window. No-proof fence: this note claims no theorem about the Riemann zeta function and no progress on the Riemann Hypothesis at theorem level.

Inputs

›The Gaussian E₈ lattice (the rank-4 Hermitian unimodular ℤ[i]-lattice of Note N1) and its Hecke–commensurability tower; the construction path is AST-firewalled against prime tables and zero data.

Contribution

›The zeta-free scaffold: finite matrices built from lattice counting and a cover-lift heat trace — the primitive Hecke degrees ARE the Gaussian prime norms, producing the atom comb of the Weil measure at deviation 0.0 [measured].
›The archimedean glue: the 48-site spin cover lift's μ₄-fixed heat trace is exactly the density behind Re ψ(1/4 + iτ/2) — the 1/4 derived, not declared; the free three-scalar fit returns (1,1,1) below 10⁻¹², and among {ℤ⁸, E₈} only the Gaussian E₈ glues [measured].
›The prediction-freeze methodology: truncation eigenvalues frozen by SHA-256 BEFORE any zero is loaded hit 100% of the first 377 zeta zeros at tolerance 0.25 (ladder rate −1.61); matched-node nearest-neighbour statistic 0.6178 vs the zeros' own 0.6189 [measured].
›The exact finite trace formula: Gauss quadrature, with a term dictionary to the classical Weil explicit formula closing block by block at 10⁻¹³–10⁻¹⁶.
›The four-language localization of the remaining statement (Weil positivity in the limit): Hankel, Levinson, Fejér, Krein–Suzuki — machine-verified equivalent, each finitely decidable per window; documented negatives included.

Not claimed here

›No-proof fence (stated in the abstract itself): no theorem about the Riemann zeta function and no progress on the Riemann Hypothesis at theorem level — the contributions are the computable zeta-free scaffold, the freeze methodology, the documented negatives, and the localization of the remaining statement.
›Zeta zeros enter the verification modules only as declared comparison targets loaded AFTER the printed SHA-256 freeze; every load-bearing statement carries exactly one claim tag ([proved]/[measured]/[killed]/[declared]/[cited]).

Falsification surface

›The freeze discipline is the kill switch: any construction-path access to prime tables or zero data (AST firewall), any retuning after the SHA-256 freeze, or a frozen truncation missing its declared capture rate falsifies the measurement claims.

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Highlights

Zero capture100% of 377At tolerance 0.25, SHA-256-frozen before any zero is loaded

Atom combdev 0.0Gaussian prime norms from the Hecke tower — no prime table anywhere

Glue fit(1,1,1)Free three-scalar fit below 10⁻¹²; only the Gaussian E₈ glues

Dictionary10⁻¹³–10⁻¹⁶Exact Gauss quadrature vs the classical Weil explicit formula, block by block

Modulesv714–v734The moonshot arc + keystone round (90 checks), AST-firewalled

RH claimnoneThe explicit no-proof fence: no theorem about ζ, no RH progress at theorem level

Key formulas

Zeta-free capture
$377/377 \ \text{zeros at tol } 0.25 \ \text{(SHA-256 frozen first)}$
Frozen before any zero is loaded; ladder rate −1.61. [measured]
The archimedean density
$\mathrm{Re}\,\psi\!\left(\tfrac14 + \tfrac{i\tau}{2}\right)$
The μ₄-fixed heat trace of the 48-site spin cover lift — the 1/4 derived. [measured]
The remaining wall
$\text{Weil positivity in the limit (Hankel / Levinson / Fej\'er / Krein--Suzuki)}$
Localized in four machine-verified equivalent languages, finitely decidable per window. [O]

Introduction and claim typing

The Hilbert–Pólya heuristic asks for a self-adjoint operator whose spectrum is the zeta zeros; the obstruction map is classical (Weil, Connes, Meyer, Connes–Consani, Suzuki — positivity carries the full content). This note reports a different kind of object inside that map: a computable, zeta-free truncation family, with systematic measurements of how much of the Hilbert–Pólya picture the family already exhibits at finite size, and machine-checked bookkeeping of exactly what would remain to be proved. Every load-bearing statement carries exactly one tag: [proved], [measured], [killed], [declared], or [cited].

The glued object

Finite places: the Hecke–commensurability tower of the rank-4 Hermitian unimodular ℤ[i]-lattice — its primitive degrees are the Gaussian prime norms; a circle-free logarithm generator plus conjugation descent produce the atom comb of the Weil measure at deviation 0.0. Archimedean place: a 48-site spin cover lift whose μ₄-fixed heat trace is exactly the density behind Re ψ(1/4 + iτ/2), the 1/4 derived. The glue: one normalization — the free three-scalar fit returns (1,1,1) to below 10⁻¹², and only the Gaussian E₈ glues among the dimension-8 unimodular lattices.

\mathrm{Re}\,\psi\!\left(\tfrac14 + i\tau/2\right) \ \text{— the } \tfrac14 \text{ derived, not declared}

(c_1, c_2, c_3)_{\mathrm{fit}} = (1,1,1) \ \text{to} < 10^{-12}

Measurements under the freeze

On every truncation the Weil functional is a state (GNS vector state at KMS β = 1). The truncation eigenvalues are frozen by SHA-256 before any zero is loaded; they hit 100% of the first 377 zeta zeros at tolerance 0.25 with ladder rate −1.61, and the matched-node nearest-neighbour statistic is 0.6178 against the zeros' own 0.6189. The finite trace formula is exact Gauss quadrature; its term dictionary to the classical Weil explicit formula closes block by block at 10⁻¹³–10⁻¹⁶.

\text{capture: } 377/377 \ \text{at tol } 0.25, \quad \text{ladder rate } -1.61

\text{NN statistic: } 0.6178 \ \text{vs} \ 0.6189

The decomposition and the wall in four languages

Two of the three convergence steps are classical or measured (half-plane convergence with an unconditional Chebyshev majorant, boundary exactly at s = 1/2; tightness); the node-capture half of the third is proof-near. The one remaining statement — Weil positivity in the limit — is localized in four machine-verified equivalent languages: Hankel, Levinson, Fejér, and Krein–Suzuki, each finitely decidable per window. The documented negatives and the position in the literature (Connes–Consani's atom-free regime, Suzuki's ω > 1 range, Meyer's positivity-free spectral realization) mark the proved boundary line.

Verification and reproducibility

Every number is printed by a machine-checked module of the permanent suite: the moonshot arc v714, v716–v721 and the keystone round v727–v734 (90 checks), promoted verbatim from exact-arithmetic discovery probes. Zeta zeros enter only as declared comparison targets loaded after the printed SHA-256 freeze; the construction path is AST-firewalled against prime tables and zero data.

A Computable, Zeta-Free Truncation Family for the Weil Measure: Measurements on a Hilbert–Pólya Candidate

Key formulas

Introduction and claim typing

The glued object

Measurements under the freeze

The decomposition and the wall in four languages

Verification and reproducibility

Key formulas at a glance

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