Abstract

We prove a quantitative extension of the scalar-only first plateau coefficient studied in the supplied Em06-A draft. The metric coefficient remains a prescribed classical input. We independently derive the density variation, zero-coordinate-time Cauchy prescription, compact-test representation, and restricted gauge identity. On common compact supports we prove an explicit, nonoptimal estimate using six derivatives of the observation test and metric coefficient and three derivatives of the switch.

For displaced Fock-normal states with finite residual number expectation $n_{ϵ}$, the condition $ϵn_{ϵ}\to 0$ gives uniform mean, variance, mean-square, and spectral Wasserstein concentration on bounded input families. Observable-specific second moments give the exact necessary and sufficient condition, so the number condition is sufficient but not necessary. Finite commuting families possess joint spectral laws; general families are treated by ordered Weyl expectations, not by an assumed joint probability distribution.

Normalized fluctuations retain the declared, possibly degenerate CCR form. Their laws are exactly the laws of the undisplaced fields in the residual states, and concentration alone does not determine a fluctuation limit. We prove input-stability estimates and evaluate nonzero success, bias-failure, amplified-variance, and nonconvergent-fluctuation examples. An explicitly defined additive-noise POVM separates intrinsic from measured concentration.

The release comprises the scalar-only gates B1–B5. The optional finite-mode metric gate B6 is not entered. No interacting gravitational product, regulator removal, backreaction, metric reconstruction, Einstein dynamics, holographic identification, or inherited gravitational coefficient admission is obtained.