Theorem: Em06-B analytical release

For the scalar-only first coefficient and the frozen conventions of this manuscript:

1. B1: classical and test audit. The density, coordinate Cauchy, compact-test, support, and restricted gauge identities are proved, together with the explicit nonoptimal estimate (eq:main-estimate).

2. B2: finite families. Symmetric covariance and commutation are separated. Commuting families have the joint spectral theorem (eq:joint-general)–(eq:joint-vacuum); arbitrary families have the ordered Weyl theorem (eq:ordered-general)–(eq:ordered-bound).

3. B3: displaced-state concentration. The exact observable-specific criterion is $ϵ⟨\Phi (v{)}^2{⟩}_{{\rho }_{ϵ}}\to 0$. Finite residual number moments with $ϵn_{ϵ}\to 0$ give uniform mean, variance, mean-square, and spectral $W_2$ concentration.

4. B4: input control. Metric, switch, observation-test, and preparation changes have the allowances (eq:input-difference)–(eq:combined). Exact gauge equality stays inside the common plateau; the outside-plateau defect has a separate error estimate.

5. B5: fluctuation scale. Normalized fluctuations have the exact residual-state representation (eq:fluctuation-conjugation) and CCR (eq:fluctuation-CCR). A fluctuation limit requires further state information; concentration alone is insufficient.

A nonzero relational target, non-Gaussian success, distinct bias and variance failures, and failure of a unique fluctuation limit are explicitly evaluated. The additive-noise POVM has the exact measurement allowance (eq:measurement-budget).

The ending is Constructed: the scalar-only B1–B5 package. The optional metric gate B6 is not entered.

Proof. Propositions prop:test–prop:gauge, Lemma lem:mass-shell, Theorem thm:estimate, and Appendix app:PDE establish B1. Proposition prop:covariance and Theorems thm:commuting–thm:ordered establish B2. Theorem thm:states and Corollary cor:uniform establish B3. Theorem thm:input-stability, Corollary cor:combined, and (eq:outside-gauge)–(eq:gauge-error) establish B4. Theorem thm:fluctuations and Propositions prop:uncertainty–prop:fluctuation-limit establish B5. Section sec:examples gives the stated examples, and Proposition prop:noise proves the measurement allowance.