A Unified Holographic/Adelic Framework for the Riemann Hypothesis

We prove two results unconditionally and propose a research program whose completion would imply the Riemann Hypothesis. The unconditional results are: (i) an adaptive mollifier on the adelic idele class group yields a norm gap N(σ) − 1 ≥ c1|σ − 1/2| with c1 = (1 − e−1/4)/√π ≈ 0.197 derived explicitly from Gaussian integration and the residue of ζ(s) at s = 1 (Theorem 2.1); and (ii) the holomorphic dilation operator Dholo = A + L0 satisfies a positive commutator bound ≥ c3 > 0 as a consequence of unitarity of the principal series representation alone, without assuming RH (Theorem 4.1). A finite-height log-concavity enhancement for the deformed Riemann ξ-kernel follows unconditionally for heights T ≤ T0 (Proposition 6.2). Conditional on the adelic–celestial isometry of Toupin (2025–2026), these results extend to a three-mechanism overdetermined framework (M1–M3) and a precise conjecture (Conjecture 6.1) whose proof would imply RH. The single identified obstruction is a uniform tail bound on GUE extreme-value statistics at large heights. The paper is a research program, not a completed proof, and is written to make both the solid contributions and the open gaps transparent.