Abstract

We develop a self-contained companion to the supplied Family 282-01 record, Scale and conformal symmetry in four-dimensional operational quantum field theory, dated 26 September 2026. Its conditional scale-to-local-conformal theorem is recorded, not independently verified and not used as a premise.

Our mathematical input is instead a declared stress tensor, virial field, improvement scalar, common field domain, and specified Ward-insertion distributions. We prove the flat-space improvement identities, conservation of conformal-Killing currents, and conditional preservation of local affine Ward actions. We then give an exact finite local-Ward certificate with an explicit contact-defect ledger. A finite admissible correction space has a complete linear-algebraic feasibility test; extension from finite tests requires an independently proved coverage condition. An optional bounded-score validation theorem certifies only the declared finite residuals and tolerances.

Explicit examples distinguish current conservation from Ward contact identities, infinitesimal actions from global implementation, modular standardness from invariant-vector uniqueness, and fixed scaling or modular records from full-code recovery and entropy histories. The free massless scalar improvement is calculated with its sign and normalization fixed. No new improvement scalar is reconstructed from the Family 282 hypotheses.

All stages of the proposed comparison are addressed. No global conformal action on the entire bounded net, gravitational dictionary, Einstein dynamics, ultraviolet completion of the supplied series, Page curve, or inherited characteristic quantum coefficient is constructed.