Spectral Transfer Program for the Riemann Hypothesis: Proven Rigidities, Conditional Reductions, and Open Interfaces

This paper does not prove the Riemann Hypothesis (RH). Working within the Connes--Consani--Moscovici (CCM) 2025 spectral triple framework, we decompose the continuous-discrete transfer program into three classes of statements: already proven rigidity theorems, reductions conditional on unproven analytic assumptions, and genuinely open interfaces. The proven part comprises: the Toeplitz-type Carathéodory--Fejér zero-circle theorem, the Hurwitz zero-realness corollary, and the realness of spectral limits of self-adjoint operators. The conditional part has as its core: if the spectrum of the CCM truncated operators converges to the imaginary parts of the zeta-zeros, the limit objects are identified as zeros of zeta, and the imaginary parts of the nontrivial zeros are pairwise distinct, then RH follows. The logical skeleton of this reduction is rigorous, but its premises are exactly the open problems explicitly listed in the CCM paper. We further observe that the Connes--van Suijlekom 2025 continuous-kernel mechanism has already established, as a theorem, the implication "simple even lowest eigenvalue implies Fourier--Mellin zeros on the critical line." Consequently, the remaining obstacles in the CCM framework are concentrated precisely at the simplicity and evenness of the lowest eigenvalue (Gap #1) and the rate of spectral convergence (Gap #2). The value of this paper is not a claim to have completed RH; it is to write down "what is missing" as independently checkable propositions, and to disclose the boundary of AI assistance truthfully.