Abstract

We construct a polynomial quantum carrier for the finite-ramp Einstein–scalar multiplet $(\gamma ,\varphi ,c,\bar{c},b)$ at fixed $\epsilon >0$. The equation operator, the action Hessian, and the Hadamard product matrix have distinct roles. Reduced solution coordinates provide a componentwise wave-propagator separation lemma. An oriented Hadamard smoothing lemma then proves that polynomials vanishing on the complete gauge-fixed free shell form a two-sided quantum ideal, without requiring microcausal equation primitives or a mollification argument.

The regular quotient recovers the causal free-field algebra. Smooth admissible changes of product data yield continuous ordering isomorphisms. Their corrections on the quadratic insertion and the scalar-cubic tree are computed explicitly and lie in the regular subalgebra; consequently the named, formally completed composite sector is ordering-stable. The frozen free differential is checked in both ghost orderings and transported by conjugation.

Continuity is stated at the established level: fixed-support kernel continuity, stagewise product hypocontinuity, a Hausdorff shell quotient, and global separate continuity. Global hypocontinuity and completeness are not released. A regular time-slice theorem identifies ambient and interior-slab free algebras.

Finally, taking XI-G's observation-qualified obstruction as an explicit input theorem, we exclude the detected cubic shell classes for every admissible ordering. Continuous detection functionals make the detected class open and dense on every fixed-support test stage on which a detector is nonzero. The undetected set, the Einstein subshell, and any enlargement remain undecided.