The Marsh-Feagans Spectral Algebra and the Riemann Hypothesis: A Partial Resolution via Pseudo-Hermitian Operator Theory, Narrow-Band Weil Positivity, and the Gershgorin-Gabor Framework

We present partial results toward a proof of the Riemann Hypothesis via the Marsh-Feagans Spectral Algebra (MFSA), a non-Hermitian operator framework constructed from the Bost-Connes C*-algebra and a pseudo-Hermitian deformation G = H₀ + iΓ. We prove three unconditional results: (1) the Sign Lemma—for narrow-band test functions, the real part of the off-line zero contribution to the Weil explicit formula is strictly positive; (2) narrow-band Weil positivity—Qₓ (f) ≥ 0 for the family Fₗ, ⏒䃐, constituting the first unconditional GLK positivity result for an explicit test-function class; and (3) the Cosh Bridge identity—an off-line zero at ρ₀ = 1/2+δ+iγ₀ injects an exponential cosh (δt) amplification into the Weil quadratic form. We further establish a Gershgorin-Gabor architecture for Gap 3 (full GLK positivity) and identify the two outstanding lemmas required to seal it. We document one retracted claim and two demoted results with equal prominence alongside the positive results. The program is ongoing.