Theorem: XX-E mathematical release and native Q4 verdict

For the identical printed original-rod variant, this manuscript proves:

1. the output-density identification restoring the XX-A smeared target;

2. the complete coupled classical Ward calculation and the exact compact-source-cutoff residual;

3. the triviality of compact gauge-fixed residual parameters and the failure of the naive compact gauge translation on the free equation quotient;

4. the central-second-jet derivation test;

5. the full spatial Newton images of the displayed contact family and the classical Ward images of the declared comparison correction family;

6. the conditional simultaneous actual-image criterion for insertion and subsidiary repair;

7. formal classical-symbol recovery, corrected product-order bookkeeping, and the identical-prescription refinement counterexample.

The supplied XX-A–XX-D data do not verify Definition def:package or Assumption ass:repair. In particular, no actual native quantum breaking $B$, complete admissible quantum image map $L$, or solution of their cancellation equation is constructed.

Native Q3 therefore remains open. Under the revised prerequisite rule, native Q4 and Q5 are unentered and unclosed. No universal nonexistence theorem for a future completion is asserted.

Proof. Item (i) is Proposition prop:density. Item (ii) is Theorem thm:classicalWard and Theorem thm:cutoff. Item (iii) is Proposition prop:residual and Theorem thm:translation. Item (iv) is Proposition prop:central. Item (v) is Theorem thm:contactimage and Proposition prop:counterimage. Item (vi) is Theorem thm:repair. Item (vii) follows from Lemma lem:formal, Propositions prop:symbol and prop:productorder, and Theorem thm:escaping.

The contact family was not proved exhaustive or quantum-admissible. The conditional repair theorem assumes a constructed quantum defect and operation. The ordinary symbol calculation does not supply them. The attached XX-D explicitly retains unverified native infrared hypotheses. Thus none of the new mathematical results discharges the absent application data, and the stated gate ending follows.