Abstract

We consider a minimally coupled scalar field on a formal metric $\mathbf{g}=\eta +\kappa h+{\kappa }^{2}k$ and specify a compactly switched scalar action together with unswitched harmonic coordinates having fixed Cauchy data. The admitted gauge transformations are supported strictly inside a region on which the interaction switch equals one and are the identity near the initial surface. For this restricted group we prove an exact first-order cancellation and a second-order classical intertwining identity. All switch derivatives are retained.

The proof uses the covariance of the switched, density-valued Euler operator under plateau-supported transformations and uniqueness of the coefficientwise Cauchy problems. We give the complete second-order relational coefficient, verify its transformation directly, and show that the construction restricts to a precisely defined plateau Einstein–scalar coupling jet. We also derive the first-order defect for larger gauge supports and explain why it cannot be repaired by adding a higher-order coefficient without changing the first-order construction.

A causal support bound and a classical comparison map between switches are proved. Neither equality for switches sharing only an observation plateau nor a quantum algebraic cutoff comparison is asserted. Renormalized products, quantum Ward identities, quantum responses, and counterterm separation are not constructed here. The result closes the classical gate of the finite-time programme, not its radiative gates.