Route-A execution note: a consistent second-order framework for the hybrid Hamiltonian

This note executes the consistent “Route (A)” fix for an earlier draft by the author: we replace the first-order “xp” hybrid operator by the second-order Schr¨odinger operator that actually arises from the S^3 reduction, so that Weyl limit-point theory and point-interaction (delta) technology apply as stated. Within this consistent setting we prove: (i) a rigorous substitute for (P1): self-adjointness of the geometric operator plus (regularised) prime impurities via quadratic-form methods;(ii) a rigorous substitute for (P2): a convergent relative determinant for finite prime cutoffs (and a Fredholm-determinant framework under summability hypotheses).We then isolate, in a proof-checkable form, the additional statements still required to identify the resulting determinant with the Riemann ξ-function (the genuine RHbarrier).