Theorem: X-J-ST mathematical release

Under Assumption ass:host:

1. the second charged coefficient is the compact ghost-number-zero scalar

   $${\mathcal{C}}_2[\varphi ]=\int w_2(\varphi -x^1){\mathrm{d}}^{4}x,$$

   obtained by the finite action polarization (eq:polarization);

2. its value is independent of the first moment, and the charge matrix is the identity on the declared two-parameter comparison family;

3. a local common charge is realized exactly, while whole-relative-algebra matching fails on an explicit independent direction;

4. the selective amplitude instrument has the exact positive gap (eq:TV), with finite-dictionary formula (eq:dictionaryTV) and the exact $3/16$ calibration instance;

5. the clock, signal, task, and optional phase instruments are complete on arbitrary finite inputs and all untouched references;

6. the higher-order encoding test and the robust gap margins hold with their stated hypotheses;

7. finite adaptive composition, record-retaining continuations, scheduled locality, and regular report transport have the proved scopes;

8. the X-I-ST action is unchanged, its free-limit qualification remains, and the earlier tree and obstruction diagnostics are neither recounted nor repaired.

This is a local Grade S selective coefficient package, not a whole-algebra invariant convolution reference, not quantum BRST, and not a characteristic coefficient.

Proof. Theorem thm:coefficient proves (i). Theorem thm:localmatching proves (ii) and the positive part of (iii); Theorem thm:wholealgebra proves its boundary. Theorems thm:TV and thm:dictionaryTV, with Corollary cor:benchmark, prove (iv). Theorems thm:signal, thm:clock, and thm:relational prove (v). Theorems thm:secondorder and thm:robust prove (vi). Theorem thm:composition, Corollary cor:gapcontinuation, and Theorems thm:locality–thm:route prove (vii). Propositions prop:recovery and prop:noninterference, together with Section sec:diagnostics, prove (viii).