RIEMANN PROJECT · 23 SEPTEMBER 2026 · NS100

Geometry needs an arithmetic discriminator

A return to geometry can ask a different question from the failed projected-cost estimate. It does not yet supply a new proof mechanism. I independently reviewed all 104 registered conclusions and screened two possible directions against the existing results and primary literature. I also tested one concrete geometric shortcut using the actual theta kernel. That shortcut fails.

The useful change in your dad’s idea is to study how the function behaves under a circle map, rather than how symmetric zero locations are counted. Equal counts and an exterior similarity point at infinity already survive off-line symmetric controls in NS94. Those particular observations cannot force RH.

1. Keep the theta cancellation inside the geometry

Write F(z)=ξ(½+z)/2. Its logarithmic derivative L=F′/F measures horizontal growth of the function’s modulus. The map S=(L−1)/(L+1) gives

1−|S|² = 4 Re L / |L+1|².

Thus a disk condition can detect something that equal zero counts cannot. But Re L>0 throughout the right half-plane is a classical RH criterion, recorded by Lagarias. Drawing this disk creates no sign estimate. The paper’s 2005 correction was checked; its corrected auxiliary estimates are not used here.

A concrete possible input is an explicit positive factorization of the integrated theta-product kernel, derived from the actual Jacobi modular identity. It must retain the signed cancellation between slices. Defining a factor as the square root of a Fourier transform whose positivity is unknown would assume the desired result.

First falsifiable test, now completed: could every individual slice φ(p+t)φ(p−t) already have a nonnegative Fourier transform? That would be a stronger sufficient mechanism. For the actual theta kernel, complete interval arithmetic gives

∫₀∞ φ(t)² cos(40t) dt
∈ (−5.4320, −5.4319) × 10⁻¹⁰.

The calculation includes the omitted theta series and the whole physical tail. It passes at 256 and 384 bits and after doubling both the term count and the cutoff. Continuity, with an integrable uniform bound, gives negative slices for some p>0 too. Therefore slice-by-slice positivity must stop. This does not assign a sign to the full weighted integral and does not contradict RH.

A separate exact control also rejects generic kernel positivity and ordinary log-concavity: a positive, even, strongly log-concave mixture of three Gaussians has transform e^z²(3+2 cosh z), which has off-line zeros and negative radial growth at z=log 2+iπ. This control has order two and lacks zeta’s arithmetic and modular law; those stronger hypotheses are not refuted.

Continuation gate: derive a candidate factor or signed identity explicitly from theta data, first for the fully integrated kernel or its first Taylor coefficient. The latter is the classical associated kernel K₁. Its positive-definiteness is already part of the Jensen–Pólya/Csordas program; proving that one necessary condition alone would not prove RH. Stop if the output is only another equivalent sign condition, a larger plot, or the rejected slice argument.

Wall check: the first slice test was distinct from NS94 because it used the actual theta kernel; it is now a known failed shortcut. The full integrated positivity target remains the same classical zero-rigidity gap. No independent full-sign estimate has been found.

2. Jensen polynomials: only a propagation mechanism would be new work

A second candidate is a rule that propagates hyperbolicity from large shifts to small shifts at arbitrarily high degree. It would need a coefficient-explicit interlacing or stability relation for the actual theta moments. Generic positive coefficients, log-concavity, and a finite range are insufficient substitutes.

The first task would be to write the proposed relation for adjacent shifts and test it on exact symmetric off-line controls. Continue only if a specific arithmetic relation survives and is not the desired real-rootedness assumption in another form. No such relation is supplied by this reassessment, so this candidate is lower priority.

The literature screen strongly discourages another modest-degree scan. The 2022 Griffin–Ono–Rolen–Thorner–Tripp–Wagner paper already proves hyperbolicity for n≥c exp(d/2), and, using its cited finite-height verification, for every shift n when d≤9.36×10²⁰. These are published sufficient regions, not a claim of a sharp or latest frontier. The 2019 Hermite-limit work is also prior art. “Eventually for each fixed degree” does not cover every degree and shift.

Wall check: Same open gap in the classical hyperbolicity program. This is a different missing input from NS98’s projected cost, but changing criteria is not an RH advance.

What the existing repository still permits

• API: arithmetic injectivity for the complete signed integral equation remains open. Local uniqueness and ordinary positive-cone shortcuts fail only in their stated scopes.
• CCM: a complete relative selection estimate and entire-transform control are missing. Small residuals and a close first zero do not supply them.
• Clifford / capacity: the signed representation is valid, but its critical contraction is CAE again. A coordinate change does not establish its floor.
• Continuous dilation: the unknown inner factor remains. Its closure cannot silently replace the integer closure.
• NB approximation: NS98’s exact cost belongs to its fixed Möbius/logarithmic prescription. Other selected directions share a broader cofinal gain goal, not necessarily that same cost estimate.

NS74 rules out drawing an asymptotic conclusion from Gram geometry and successful finite gain fractions. It does not prohibit an estimate that uses a special zeta property. NS95–97 likewise do not close all product constructions or all geometric routes.

Recommendation and evidence

Do not announce a new RH approach yet. If we pursue your dad’s geometric instinct, the bounded next job is to search for an explicit theta-specific factorization of the integrated kernel and to reject it against controls before investing in a long calculation. The circle-count proposal and the tested per-slice positivity shortcut should remain stopped.

This report records an independent assessment, nine exact Maxima checks, one complete sign certificate with precision and cutoff replays, and clean Python diagnostics. It does not revalidate every historical proof. Both groups of missing original evidence remain open. The primary-source screen is not an exhaustive novelty search, and no worldwide novelty is claimed.

Derivation and complete bounds · Ranked tests and stop conditions · Primary sources · Review record

Reviewed base 918dc6c; claim b64bc13. No new theorem node or manuscript version. RH, G2 and every unproved cofinal input retain their status.