Declarations

of the result. This manuscript establishes the X-A classical gate for one frozen regulated switched prescription. It does not begin the admission, product, or causal-product problems of a later X-B.

on prior manuscripts. The statements of VII-A, VI, III, VII-B, and IX-A quoted in Section sec:prereq are comparison background. No numbered result from them is used as a premise. In particular VII-A's identity $w[h+K\xi ]-w[h]=\xi$ for the unregulated unswitched inverse is not used as a lemma about the regulated operator; the corresponding statement is proved here as Proposition prop:preflight.

status. No successful pdflatex execution, proof-assistant check, numerical residual computation, or independent review is claimed. The source uses no external figures, bibliography database, or executable code. The nonzero witnesses are analytic.

boundary. The classical identities proved here are not advertised as a quantum Ward identity, a renormalized product, or a sharp-limit statement. The derivative kernels of Proposition prop:kernels are smooth classical kernels at fixed $\epsilon$; they are not labelled admitted to any quantum domain.