This paper proves an admissibility theorem for completion-locked determinant channels, in which the Archimedean completion and determinant normalization are fixed as part of the spectral realization itself rather than introduced afterward. Three operator-theoretic results are established. First, the Archimedean circle sector canonically realizes an operator-level block of supersymmetric quantum mechanics whose reduced heat trace is the Jacobi theta kernel and whose Mellin transform yields the standard Archimedean completion. A normalization-lock result shows that the required scaling is not adjustable but is forced by recovery of the standard theta kernel. Second, for any determinant-admissible nonnegative self-adjoint interior operator, the shifted determinant channel is reflection-symmetric about the critical line, and positivity alone forces all of its zeros onto that line. In particular, when a completed zeta-type object is identified with the corresponding spectral realization up to a factor that is zero-free on the open critical strip, every nontrivial zero in that strip is forced onto the critical line. Critical-line spectralization is therefore an operator-theoretic consequence of the completion-locked determinant setting. Third, when the interior block admits an odd supercharge factorization, supersymmetric quantum mechanics governs the admissible structure of the nonzero spectral sector. Positive levels are necessarily boson–fermion paired, and the graded supertrace localizes entirely to finite-dimensional kernel and index contributions. As a result, the nonzero spectrum carries no independent graded residue. Taken together, these results furnish an operator-theoretic admissibility theorem for completion-locked determinant channels. The circle sector fixes the Archimedean factor, positivity governs critical-line spectralization, and supersymmetric admissibility governs the nonzero sector by separating the spectral bulk from the residual kernel/index locus. Consequently, the nonzero spectral sector carries no residual graded content beyond boson–fermion pairing, and the only remaining obstruction is finite-dimensional kernel/index data. License note: Distributed under CC BY-NC-ND 4.0.
Salimah Meghani (Tue,) studied this question.