Partial Operator-Theoretic Results for a BRST--Mellin Framework Related to Zeta Zeros

This research note develops partial operator-theoretic results for a possible BRST--Mellin framework related to zeta zeros. The note is motivated by a formal arithmetic BRST charge built from two constraints: modular reflection \(I- W\) and multiplication by the completed Riemann zeta function \(M_ξ\). Its purpose is not to prove the Riemann Hypothesis, but to clarify which parts of the formal BRST--Mellin construction can be made rigorous and which analytic problems remain open. The main novelty is twofold. First, the note proves a previously open operator-theoretic step: in the critical-line Hilbert model, the formal arithmetic BRST charge admits a closed densely defined realization, and the associated BRST Laplacian can be constructed as a non-negative self-adjoint operator by the quadratic-form method. This gives a well-defined heat semigroup \(e^{-τ Dₐᵣᵢₜₕ}\). Second, the note introduces an explicit reproducing-kernel Hilbert-space model for zero kernels. In this RKHS setting, the adjoint action of the multiplication operator is computed on reproducing kernels:\[M_ξ^*K_ρ=ξ(ρ)̄K_ρ.\]Consequently,\[ξ(ρ)=0_ρ∈ M_ξ^*.\]Thus zeta zeros are represented not as ordinary \(L^2\)-kernel vectors of \(M_ξ\), but as reproducing-kernel vectors in the adjoint kernel of \(M_ξ\). The note also shows, under the symmetry assumptions of the RKHS model, that the modular-even projection preserves the adjoint zero-kernel sector. What remains unresolved is the Mellin boundary analysis needed to exclude off-critical zero kernels from the physical quadratic-form domain. In particular, the note identifies Mellin boundary asymptotics, non-cancellation of leading boundary coefficients, domain-level nilpotency, closed-range/Hodge decomposition, and the relation between the RKHS model and the critical-line heat-flow model as the main remaining open problems. This work should therefore be read as a partial operator-theoretic completion of a BRST--Mellin research program, not as a proof of the Riemann Hypothesis.