IMM Paper 13: The Riemann Hypothesis as a Spectral Problem — An Information Manifold Approach

This paper does not prove the Riemann Hypothesis, and does not claim to. It develops a spectral program: four theorems at explicit epistemic status and three precisely located open conjectures, with negative finite-volume evidence for the central identification (Conjecture 7.3) recorded in the paper itself. See the Epistemic Status Notice (v5 cover note) and IMM Paper 14 for the current frontier statement (Weil positivity / A(m) positive-definiteness).Version 4 of IMM Paper 13, comprising the unchanged v3 paper plus a labeled Erratum and Coupling-Class Addendum. The erratum withdraws the Experiment EH structural-bridge interpretation (v3 §9.1) following a pre-registered audit: the reported fit (a = 1.0000, b = 1/16, R² = 1.000) is an identity of the construction ∇Δ = (V′, Δ₀), reproduced exactly by arbitrary control potentials, and carries zero discriminating power; the corpus consequently contains no exceedance of the form-only projection-equilibrium null (Paper 16). The addendum re-derives the frozen-resonance obstruction to Conjecture 7.3 from the statistical mechanics of the primon gas (Julia, Spector), in which the nontrivial zeros are the Fisher zeros of the gas — pinned by the functional equation's Kramers–Wannier-shaped duality to the self-dual line (Knauf) — and, equivalently, collective oscillations of the level density at √N fluctuation scale. The v3 operator is a one-body, mode-coupled probe and is structurally deaf to such collective modes: frozen resonance is the generic fate of its entire coupling class. The resulting design principle — density coupling on the dilation kinetic term D = i(x∂x + ½) (Berry–Keating, Connes) — sharpens Conjecture 10.3 to a precise coupling-class requirement, and a registered synthetic-spectrum falsification test (E-RH-1), with binding criteria fixed before any code existed, is executed: the first run failed both gates (two criterion-design errors, kept on the record), and the labeled re-registered repair E-RH-1b returned a double pass — the density-coupled probe locks onto the injected collective frequency at 2e-3 relative or better across a factor of 4, while the mode-coupled v3-class probe is statistically indistinguishable from collective-free phase noise. The coupling-class principle survives its kill test on synthetic spectra. No status change to any RH-adjacent conjecture is claimed.