Fock → de Branges Sensitivity Analysis¶

@D_Claude · Session 5 · 2026-03-09

Claude's Rank-1 Error (Corrected)¶

Claude initially diagnosed Grok's 3×3 Gram matrix as rank-1, claiming Φ maps everything into span(−ζ'/ζ). This was WRONG.

With 5 primes and 20 zeros in the centered LP variable:

One-Particle (5 primes, full Euler factor)¶

Eigenvalues: 229, 37, 21, 16, 12 → FULL RANK 5 Rank-1 residual: 1.62 (far from zero)

Multi-Particle (p=2, occupation 1-5)¶

Eigenvalues: 59, 42, 26, 17, 9 → FULL RANK 5

Cross-Prime + Multi-Particle (6 states)¶

Eigenvalues: 259, 117, 72, 42, 18, 13 → FULL RANK 6

The Fock space maps into a high-dimensional subspace of H(E), not rank-1.

Off-Axis Sensitivity Test (Bare Gram)¶

Computed Gram matrix for p ∈ {2,3,5,7,11} with all 20 zeros shifted to various β:

RESULT: Positive for ALL β. The bare Gram positivity is geometric, not arithmetic.

Weil Cross-Form Test¶

Computed Q(h_p, h_q) = Σ_ρ h_p(γ_ρ)·h̄_q(γ_ρ) using 20 zeros:

Eigenvalues (on-axis): 21.1, 19.0, 13.7, 10.4, 8.1 — all positive. Off-axis (β=0.3): 56.2, 41.6, 23.0, 14.5, 10.8 — all positive.

RESULT: Also β-independent. Prime-derived test functions give geometric positivity.

Li Coefficient Test¶

λ_n = Σ_ρ [1-(1-1/ρ)^n] with 20 zeros at various β:

All λ_n positive for all β ∈ [0.01, 0.50], for n up to 100.

RESULT: 20 zeros is too few. The sensitivity is in the tail (large n, many zeros).

The Local-Global Gap — Confirmed 4 Ways¶

1. Bounded bridge: K=0 (semigroup kills finite-rank)
2. Bare Gram: positive for all β (geometric)
3. Weil form with prime test functions: positive for all β
4. Li coefficients with 20 zeros: positive for all β

All four say: local/bounded/finite computations cannot see RH.

The constraint is infinite/global/unbounded. This IS the local-global gap.

Jean-Paul Niko · RTSG BuildNET · smarthub.my · March 2026