Theorem: XIV-G mathematical release

For the frozen retained instance:

1. The chosen common space and characteristic test locus are $W$ and $N={\tilde{\mathcal{N}}}_{\mathrm{att}}$, of dimensions sixteen and eight, respectively.

2. The positive conormal cancellation is diagnostic input data, not a joint symbol section.

3. For a smooth residual test $G$, the full differentiated source ${\mathcal{J}}_{\mathrm{D}}^{\prime }[G]$ is a differential operator of order at most two with the local symbol (eq:Qsymbol).

4. Smooth residual testing can hide nonzero attachment-diagonal wavefront directions even for compact balanced kernels.

5. Individually defined conormal factors of the retained numerical incidence still fail the ordinary final product test at the selected characteristic covectors.

6. No complete internal symbol relation, joint section, controlled remainder, or characteristic prescription for (eq:target) is constructed.

The ending for the grouped target is Unresolved at G2, within F2/E1/D2/J2.

No universal incompatibility, admitted ${R_1}^{(1)}$, or Ward defect is asserted.

Proof. The copied lift geometry and (eq:chosenN) give (i). Proposition prop:zerosum gives (ii). Proposition prop:sourceoperator gives (iii). Proposition prop:testing gives (iv). Lemma lem:UVexist, Proposition prop:factorsymbol, and Theorem thm:pullbackfailure give (v).

Theorem thm:G2 identifies the first missing common composition data. Proposition prop:nocancellation separates the proved diagnostics from a cancellation verdict. No prescription is supplied in Section sec:G5, proving (vi) and the stated ending.