Theorem: XX-C mathematical release

On the identical frozen XX-B target, this manuscript proves:

1. an explicit same-class normalization that is negative on a nonzero gauge-invariant compact tensor observable;

2. a correctly normalized local subtraction parametrix with smooth near-diagonal Feynman and Hadamard remainders;

3. uniform inverse-square original-rod tails for compact-source retarded and advanced responses and for the stated classical second-rod domain;

4. a sufficient uniform infrared hypothesis for one-sided covariance tails and preservation under fixed admissible classical pullback;

5. a same-operator smooth covariance ambiguity for which the decaying rod domain cannot be inferred;

6. an unavoidable inverse-distance term in the native two-rod commutator;

7. compatibility with the unchanged original elliptic rod equation on the proved domains;

8. smoothness of the subsidiary contraction defect and its exact actual-image correction criterion.

These results do not select a positive full physical state or prove the additional infrared condition for a chosen native representative. Accordingly, the requested Q2 state-and-tail package is not closed.

Proof. Item (i) is Theorem thm:nonpositive. Item (ii) is Theorem thm:subtraction. Item (iii) follows from Theorems thm:dipole and thm:GreenTail and the second-rod source calculation. Item (iv) is Theorem thm:conditional and Proposition prop:pullback. Item (v) is Theorem thm:IRfailure. Item (vi) is Theorem thm:tworod and Corollary cor:tailobstruction. Item (vii) is Theorem thm:compatibility. Item (viii) is Propositions prop:smoothdefect and prop:repair.

Neither the physical state nor the unconditional one-sided covariance-tail hypothesis is supplied by these results. The final statement therefore retains the actual deliverable boundary.