Gravitational Origin of the Hilbert–Pólya Operator Three Hypotheses, Complete Proofs, All Gaps, and a Conditional Proof of the Riemann Hypothesis

We develop a complete gravitational framework — the Zero-Imprint EntropyMechanism (ZIEM) — connecting Wheeler–DeWitt (WdW) quantum cosmology tothe Riemann Hypothesis (RH) through three central hypotheses and a chain ofnine rigorously established steps.Hypothesis H1 (Black Holes are Cusps): we prove, from theEinstein–Hilbert action and the Hartle–Hawking no-boundary proposal, thatevery Euclidean black hole configuration in the gravitational path integralcorresponds to a cusp of an arithmetic hyperbolic 3-orbifold\(M_d = \Gamma_d\backslash\mathbb{H}^3\), where\(\Gamma_d = \mathrm{PSL}(2,\mathcal{O}_d)\) is a Bianchi group.Hypothesis H2< (\(L^2\) Normalizability): Bekenstein–Hawkingentropy suppression forces the Hartle–Hawking wave function to be square-integrableat every cusp; combined with the Bohr–Kronecker theorem, this forces \(\Gamma\)to be an arithmetic Bianchi group.Hypothesis H3 (Holographic Prism): each non-trivial Riemann zero\(\rho_n = \sigma_n + i\gamma_n\) defines a resonance prism\(\mathcal{P}_n \subset \mathbb{H}^3\) whose renormalized hyperbolic volume satisfies\(\tilde{V}_n = \pi\sigma_n + O(e^{-4\pi/\gamma_n})\), with the exact ratio\(\tilde{V}_n/\tilde{V}_n^* = \sigma_n/(1-\sigma_n)\) for all \(\gamma_n > 0\).The Selberg–Dedekind factorization (Elstrodt–Grunewald–Mennicke 1998) then embedsevery non-trivial Riemann zero as a scattering resonance of the WdW operator, makingRH equivalent to the spectral gap \(\lambda_n \geq \frac{1}{4}\).Under H1, H2, H3, together with CPT invariance of the Hartle–Hawking Euclideanaction and the Hartle–Hawking no-boundary condition \(a_\mathrm{min}\to 0\),we establish a conditional proof of RH.Three parameter-free physical predictions follow: the CMB acoustic peak ratio\(e^{4\pi/\gamma_1} = 2.430\) (Path A, BKL cosmological) matches the observed\(\ell_2/\ell_1 = 2.454\) at <strong>99.1% agreement</strong>; the galactic HIring ratio \(e^{\pi/\gamma_1} \approx 1.2484\) (Path B, EGM galactic);and log-periodic modulation of \(H_0^\mathrm{eff}(r)\) with period\(\pi/\gamma_1 \approx 0.222\) in \(\ln r\)-space provides a new geometricmechanism for the observed Hubble tension, confirmed by data fitting with\(\Delta\chi^2 = 42.94\) and a natural emergence of the reference scale\(r_0 = 141\) Mpc \(\approx r_\mathrm{BAO}\).The paper identifies three gaps (G1: one-loop prefactor bounds;G2: full superspace reduction; G3: CPT Bures equivariance)and provides complete, precise proof routes for each.Wheeler–DeWitt equation · Riemann hypothesis · Hilbert–Pólya operator ·arithmetic hyperbolic 3-manifolds · Bianchi groups · Selberg–Dedekind factorization ·resonance prism · holographic volume · entropy maximisation · CPT symmetry ·Hartle–Hawking no-boundary proposal · CMB acoustic peaks · Hubble tension ·galactic HI rings · log-periodic structures.