Abstract

We examine the passage from a finite-time relational scalar to a quantum insertion, compatible products, and an interacting physical observable. The fixed classical prescription uses coordinate-time Cauchy derivatives; replacing them by derivatives along a metric-dependent unit normal would change the observable.

All three proposed repair routes are developed. Field-argument convolution gives a regular free algebra at positive width, but an explicit positive first-order defect excludes covariance under the unchanged plateau gauge action. A smooth divergence-form density preparation gives compact microcausal polynomial insertions through second coordinate order. A tree-composition argument supplies their free products, adjoints, and ordering-stable generated sector. Its sharp limit, however, has a calculated nonzero transmission velocity and is not the original zero-Cauchy prescription.

For the sharp route we construct a minimal boundary-sensitive classical carrier and prove that unchanged regular approximants do not converge in its topology. Separately, the VII-C/VII-D numerical asymptotic gives a quantitative incompatibility between old-product agreement, convergent sharp admission, bounded-set hypocontinuity, and continuous extraction of the triangle coefficient. A finite part cannot satisfy all those requirements.

Auxiliary coordinates are classically equivalent to the prescribed Cauchy construction when both equations and data are imposed. Quantum elimination requires inherited cross-contractions; an independent central-coordinate prescription fails that test. Linear sharp coordinates do have exact smooth representatives on the free shell, yielding a genuine linear free quantum sector.

Presentation descent, associativity, and relative classical gauge closure are proved at their stated levels. The sharp nonlinear interacting quantum observable, its quantum differential, and its complete normalization catalogue remain unresolved. No missing characteristic composition is supplied by a classical carrier, an internal tree kernel, or a numerical subtraction.