SUPERSEDED (2026-03-09)

This page predates the bounded bridge no-go theorem. See math/functional_bridge.md v5.0 and math/bounded_bridge_nogo.md for current state. RH confidence: 25%.

RH Chain Rebuild (GPT-5.4 Pro, 2026-03-08)¶

The adversarial review killed the old proof chain. GPT-5.4 rebuilt it.

What Died¶

Component	Kill	Killer
\(K^{\text{full}} = \sum_{\text{all }\chi} \theta_\chi \otimes \bar\theta_\chi\)	Divergent sum, varying levels	GPT-5.4 (V2)
\(y\partial_y(y^{1/2}) = \frac{1}{2}y^{1/2}\) as global identity	Trivial calculus, not operator algebra	Gemini (V4)
Three-line algebra on generalized eigenfunctions	Boundary terms diverge under truncation	GPT-5.4 (V3)
Cusp-local → global eigenvalue constraint	\(Bf = \mu f\) is global, commutator is cusp-local	SuperGrok
"Proves too much" for weight \(k \neq 1/2\)	Weight-4 cusp forms: K=0. Weight-4 Eisenstein: \(\langle K,f\rangle = \infty\)	REBUTTED (only weight 1/2 works)

What Survived¶

Component	Status
ζ-zeros = scattering resonances on \(\Gamma_0(N)\backslash\mathbb{H}\) for all \(N\)	✅ Verified (V1)
Character-family nonvanishing (Parseval + Hurwitz)	✅ Proved unconditionally
Weight 1/2 is the unique sweet spot	✅ Convergence + positivity + nontrivial coefficient
"Proves too much" rebuttal	✅ No other weight has all three properties

The Rebuilt Architecture¶

Fix 1: Single prime level \(4p^2\)¶

For a fixed odd prime \(p\), all even primitive characters \(\chi \bmod p\) give theta series \(\theta_\chi\) of weight \(1/2\) and level \(4p^2\). The ζ-zeros appear as resonances on \(\Gamma(4p^2)\backslash\mathbb{H}\). Everything on one surface.

Fix 2: Same-space positive operator¶

\[K_{p,Y} = J_{p,Y}^* J_{p,Y}\]

where \(J_{p,Y}: \mathcal{E}_Y \to \ell^2(\mathcal{X}_p^+)\) is the truncated Rankin-Selberg transform. Automatically positive, lives on the same Hilbert space as the resonance problem.

Fix 3: Even-character Parseval¶

For Re\((s_0) > 0\), the even-character mean square \(M_p^+(s_0) > 0\) for large \(p\). Gives a same-level visibility input.

Fix 4: Corrected intertwiner¶

\[J_{p,Y} B = (A_{p,Y} - i/4) J_{p,Y} + \mathcal{R}_{p,Y}\]

with \(A_{p,Y}\) self-adjoint and \(\mathcal{R}_{p,Y} \to 0\) in the resonance limit.

The One Remaining Theorem¶

Prove, for the truncated Eisenstein packet \(E_Y(\cdot, s_0)\):

\[\langle K_{p,Y} E_Y(\cdot,s_0), E_Y(\cdot,s_0) \rangle = c_p(s_0,Y) \sum_{\chi \in \mathcal{X}_p^+} |L(s_0,\chi)|^2 + R_p(s_0,Y)\]

with \(c_p(s_0,Y) > 0\) and \(R_p(s_0,Y)\) controlled so it doesn't cancel the main term at a resonance.

And: Prove \(\mathcal{R}_{p,Y} \to 0\) in the resonance limit.

If Both Are Proved¶

The three-line contradiction becomes legitimate: 1. Assume Re\((\rho) < 1/2\) 2. Even-character nonvanishing gives \(\langle K_{p,Y} f, f \rangle > 0\) 3. Corrected intertwiner gives Im\((\mu) = -1/4\) 4. Therefore Re\((\rho) = 1/2\). Contradiction.

RH Confidence: 78%¶

Down from 92%. The architecture is cleaner but the remaining theorem is genuinely hard — it's a Maass-Selberg / Rankin-Selberg computation on truncated packets at half-integral weight. Standard tools exist (Iwaniec Ch. 7, Shimura half-integral weight theory) but the computation has not been done.

Backup Route¶

Burnol's adelic Lax-Phillips framework (arXiv:math/0001013) may be the natural home for this computation, as the metaplectic theta theory is cleaner adelically than on a single modular surface.