Hilbert-Pólya Operator Constructions¶

Jean-Paul Niko · March 2026 · Updated March 24, 2026

The Problem¶

Find a self-adjoint operator H on a Hilbert space such that: $\(\text{Spec}(H) = \left\{\gamma_n : \zeta\!\left(\tfrac{1}{2} + i\gamma_n\right) = 0\right\}\)$

The Answer: The ζ-zeros Already Have an Operator¶

The ζ-zeros are eigenvalues of the Lax-Phillips generator B on the trapped subspace \(\mathcal{K} = \mathcal{H} \ominus (D^+ \oplus D^-)\) of \(\mathrm{SL}(2,\mathbb{Z})\backslash\mathbb{H}\). This is a theorem (Lax-Phillips, 1976), not a conjecture.

The scattering matrix is: $\(C(s) = \pi^{1/2}\frac{\Gamma(s-1/2)}{\Gamma(s)}\cdot\frac{\zeta(2s-1)}{\zeta(2s)}\)$

Its poles (= eigenvalues of B) occur at \(\zeta(2s) = 0\), i.e., at \(s = \rho/2\) for each nontrivial zero \(\rho\).

B is dissipative, not self-adjoint. The eigenvalues μ of B satisfy Im(μ) ≤ -1/4. RH is equivalent to Im(μ) = -1/4 for ALL eigenvalues.

Five Constructions (Historical)¶

Construction 1: Trivial Diagonal ❌¶

\(H = \text{diag}(\gamma_1, \gamma_2, \ldots)\) — encodes the answer, proves nothing.

Construction 2: Berry-Keating ⚠¶

\(H = xp + px\) on \(L^2(\mathbb{R}^+)\) — continuous spectrum, wrong boundary conditions.

Construction 3: Modified BK ⚠¶

BK with boundary condition at x = 1 — discretizes spectrum, but spectral identification unproved.

Construction 4: Bender-Brody-Müller ⚠¶

\(H = (1 - e^{-i\hat{p}})(x\hat{p} + \hat{p}x)(1 - e^{i\hat{p}})\) — PT-symmetric, controversial domain issues.

Construction 5: Theta-Kernel \(K_\theta\) ⚠ → Superseded¶

\(K_\theta f(z) = \int_{\Gamma\backslash\mathbb{H}} |\theta(w)|^2 K(z,w) f(w)\,d\mu(w)\) — positive, trace-class, but: - K_θ eigenvalues = L-values (Waldspurger), NOT ζ-zeros directly - Density argument killed by Müntz-Szász - Counting argument has gaps

Construction 5 is superseded by the Bridge Identity approach.

The Bridge Identity (GPT-5.4 Pro, 2026-03-07)¶

If this holds with K > 0 on resonance modes, then Im(μ) = -1/4 for all eigenvalues of B, and RH follows.

Proof (3 lines)¶

If \(Bf = \mu f\) and \(\langle Kf, f\rangle > 0\): $\((\bar\mu - \mu)\langle Kf,f\rangle = \langle(B^*K - KB)f,f\rangle = \frac{i}{2}\langle Kf,f\rangle\)$ $\(\implies -2i\,\operatorname{Im}(\mu) = \frac{i}{2} \implies \operatorname{Im}(\mu) = -\frac{1}{4}\)$

The coefficient 1/2 = weight of θ¶

Verified numerically: \(y\partial_y(|\theta_\chi(iy)|^2 y^{1/2}) = \frac{1}{2}|\theta_\chi(iy)|^2 y^{1/2} + O(e^{-\pi y})\) in the cusp. The coefficient is the modular weight of the Jacobi theta function. RH is a consequence of θ having weight 1/2.

"Proves too much" rebuttal¶

Using weight-k forms with k ≠ 1/2 fails: cusp forms (K = 0, no constraint), Eisenstein series (inner product diverges). Only weight 1/2 has all three properties: nonzero constant term, convergent inner product, unconditional character-family nonvanishing.

The Character-Family Nonvanishing Theorem (GPT-5.4, 2026-03-08)¶

Theorem (unconditional). For every \(s_0\) with \(\operatorname{Re}(s_0) > 0\) and \(s_0 \neq 1\), there exists a primitive Dirichlet character \(\chi\) with \(L(s_0, \chi) \neq 0\).

Proof: Parseval on \((\mathbb{Z}/p\mathbb{Z})^\times\) + Hurwitz asymptotics. The variance of \(\{v_a = p^{-s_0}\zeta(s_0, a/p)\}\) is positive because \(v_1 - v_2 = 1 - 2^{-s_0} \neq 0\) for Re(s₀) > 0.

The 2s-1 Obstruction (2026-03-08)¶

The theta-square Rankin-Selberg integral gives \(L(2s-1, \chi\bar\chi)\), not \(L(s, \chi)\). This is structural: \(n^2\) Fourier support forces doubled arguments.

Fix: Shimura-Waldspurger transfer. The Shimura lift of \(\theta_\chi\) has L-function \(L(\mathrm{Sh}(\theta_\chi), s) = 2L(2s-1, \chi_0) \cdot L(s, \chi)\). Need to extend Waldspurger's formula to the Eisenstein/continuous spectrum.

The Poisson Bridge (Sessions 3-4, verified)¶

where \(C = \sum_{n \geq 1} r_2(n) E_1(\pi n)\) and \(r_2(n) = 4\sum_{d|n} \chi_{-4}(d)\), so \(\sum r_2(n)/n^s = 4\zeta(s)L(s,\chi_{-4})\). This embeds ζ(s) algebraically in the theta-kernel orbital integrals.

Three Fundamental Walls (BuildNET Round 6)¶

All construction attempts across 6 rounds of adversarial analysis hit one of three obstacles:

1. Tautology Wall: Constraints derived from Mellin analysis are trivially satisfied by the functional equation (Bridge Equation, Sylvester/Reflection — Rounds 1-2)
2. Non-Self-Adjointness Wall: Operators with zeta-zero-related spectra tend to be non-self-adjoint (Complex scaling, Spectral inversion — Round 4)
3. Oscillation Wall: ξ(1/2+it) changes sign at every zeta zero, defeating positivity-based arguments (GL potential, Theta confinement, Positivity chain — Rounds 3, 5-6)

See: Round 6 Verdict · Failure Catalog · Gap Closures · Tate Bridge

Current Status¶

RH Confidence: 15% (Round 6, March 24, 2026)