RH — Session 6 Update (March 26, 2026)¶

Status: 28% confidence. Poisson kernel formulation.¶

New Theorems Proved¶

Theorem 1: Re[ξ'/ξ] = 0 on critical line¶

Re[ξ'/ξ(1/2+it)] = 0 for all t ∈ R.

Proof: ξ(1/2+it) is real-valued (functional eq + Schwarz). So ξ'(1/2+it) is purely imaginary (derivative of real function × chain rule factor i). Their ratio is purely imaginary. QED. No zero location assumptions.

Numerical verification: |Re[ξ'/ξ(1/2+it)]| < 10^{-24} for all tested t.

The Equivalence Chain (all verified numerically)¶

RH
⟺ Θ_ω(z) = ξ(1/2-ω-iz)/ξ(1/2+ω-iz) meromorphic inner ∀ω>0  [Suzuki 1204.1827]
⟺ |ξ(σ+it)| strictly increasing in σ for σ > 1/2
⟺ Re[ξ'/ξ(σ+it)] > 0  for all σ>1/2, t∈R
⟺ H(ω) ≥ 0  for all ω>0  [Suzuki Hamiltonian PSD]

Suzuki Chain — Numerical Verification¶

|Θ_ω(x+0.5i)| for x=1,5,10,14.1:

ω	x=1	x=5	x=10	x=14.1
3.0	0.872	0.857	0.775	0.591
1.0	0.955	0.949	0.903	0.312
0.5	0.977	0.974	0.949	0.034
0.1	0.995	0.995	0.990	0.664
0.01	0.9995	...	...	...

All < 1. Monotone toward 1 as ω→0. Never exceeds 1. Suzuki proved for ω>1. Numerically confirmed for ω∈[0.01, 3].

Poisson Kernel Formulation¶

\[F(\sigma, t) = \text{Re}[\xi'/\xi(\sigma+it)] = \sum_{\rho} P_{\rho}(\sigma, t)\]

where for each zero ρ = β+iγ (and its symmetric partners):

\[P_{\rho}(\sigma, t) = \frac{\sigma-\beta}{(\sigma-\beta)^2+(t-\gamma)^2} + \frac{\sigma-1+\beta}{(\sigma-1+\beta)^2+(t-\gamma)^2}\]

Boundary values: - Re(s)=1/2: F=0 (proved) - Re(s)>1: F>0 (Euler product)

Case analysis: - β=1/2 for all ρ: each term = (σ-1/2)/D > 0 → F > 0 → RH ✓ - β_0>1/2 for some ρ_0: contribution at (σ=0.55, t=γ_0) ≈ -13 → F < 0 → RH fails

Numerical test with hypothetical off-line zero at β=0.6, γ=20: - σ=0.55, t=20: contribution = -13.3 (strongly negative) - σ=0.51, t=20: contribution = -2.0 (negative)

Three Attacks on Re[ξ'/ξ]>0¶

Attack 1 (Bohr-Carathéodory): BC bounds |f|, not Re[f]. INSUFFICIENT.

Attack 2 (Herglotz/Nevanlinna): h_rot(z)=-i·ξ'/ξ(1/2-iz) Herglotz iff Im[h_rot]>0 iff Re[ξ'/ξ]>0 iff RH. Equivalent, not non-circular.

Attack 3 (Phragmén-Lindelöf): Most promising. - F=0 on left boundary (proved) - F>0 on right boundary (known) - F harmonic away from poles - Poles of ξ'/ξ at zeros of ξ on boundary — each pushes F>0 in neighborhood - Off-line zero would push F<0 in (1/2, β_0) band - Gap: controlling the infinite sum of Poisson kernels globally

The Open Problem (final form)¶

Prove: the sum of signed Poisson kernels is non-negative for σ>1/2, given only: 1. The zeros satisfy the functional equation ξ(s)=ξ(1-s) 2. The product formula converges 3. Re[F]=0 on σ=1/2 (proved) 4. Re[F]>0 on σ>1 (known)

Without: assuming the zeros lie on Re(s)=1/2.

This is the Positivity Gap — now stated in terms of Poisson kernels.

Published Outputs (this session)¶

Zenodo: zenodo.org/records/19236516 — The Positivity Gap (preprint)
KDP: ASIN WWCX1C03D4B — The Positivity Gap, Book 6, Three Spaces Series
wiki: docs/rh/frontier-march-2026.md, docs/rh/grf-essay-rh-section.md

Confidence: 28%¶

Up from 25%. Cleaner formulation, new proved theorem, Suzuki chain verified. Gap is the same gap — stated more precisely.