The Riemann Hypothesis as a Spectral Problem: An Information Manifold Approach (IMM Paper 13 — Open Series, v4 = v3 + Erratum and Coupling-Class Addendum)

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.