[GoGoKo699]

Nonmonomial Clifford images alone do not complete the quartic lookup face. This revision resolves the concrete trials, pays the successful fixed-coefficient circuit, and proves why an arbitrary fixed final decoder cannot repair low-degree Pauli transport. The general R96 bound and selected Result D/R79 are unchanged.

• Add the exact Galois condition on the combined two-outer-conjugation word, including the even displacement-rank restriction for fourth-root outer phases and an odd-linear counterexample.
• Prove a 1/16 complete error floor for the specified Hadamard-pair/CCZ first coupling with arbitrary later Clifford walls and cubic-sign last phases. Weighted phases escape that proof; the explicit T-pair word still has a 3/16 final-Clifford floor.
• Construct the complete fixed-top-coefficient Fourier-swap face, then reduce it to a quadratic finite difference. Its direct phase uses 28 T gates, at most eight bands, 84 total gates and no helpers. A four-corner product proves a 1/2 error floor for the common held-address extension to arbitrary top coefficient.
• Prove the syndrome-covering bound for low-degree signed-Pauli preprocessing followed by any fixed carrier/helper unitary. Affine-query stages require q >= p exactly and q >= p/2 - O(sqrt(p)) at fixed small complete error. This is a source-grammar bound, not a native-depth lower bound.
• Update the continuing checkpoint, route decisions, proof catalogue and finite-evidence catalogue.

All proofs retain arbitrary dirty/reference inputs and distinguish exact identities, approximate gaps, finite circuits and unrestricted questions. The next construction must introduce genuine later address participation or leave the stated Pauli/coupling grammar; the successful finite difference supplies no all-size depth improvement.

Validation completed on final head 927fcb438efbecf6ac86d2f0d071cd02103abf3f, tree 4062f7b468468bbffe5de202c3966062f2e4c486:

• All ten local WORKSPACE gate commands returned zero in 1250.123 seconds. Python 3.12.14 / NumPy 2.3.5: 1,907 tests, OK (1217.915 seconds). Initial and final tree hashes agree; no tracked changes occurred during validation.
• Local gate receipt SHA-256: 81fd6e4f5944b4f46c6d390ee2cc90056019821264df14b55f7cc89debfbeca3.
• Final PR validation: Python 3.11 job 121554407555 and Python 3.13 job 121554407661 both passed all steps. Decoded complete logs independently report 1,907 tests, OK, in 1155.815 and 1326.942 seconds respectively.
• Both CI logs checked out synthetic merge 280fc9dd147b8afa9bfcbac006c4547bfb7eceb8, independently verified to have the same final tree.
• Final full-corpus presentation: all nine steps passed. Artifact 12647743673, SHA-256 5ba296dca545352544ff3037edef011fe3030445857cf449ed633b86ebc7e0e0.
• Local changed-page presentation passed all eight pages at desktop/narrow widths; 18 focused views were inspected. Final prose-variable polish was rerendered and checked.
• Twenty-one scientific tests were added. Focused scientific, navigation, typography and independent internal proof reviews passed.
• Offline synchronization validation passed locally and in both numerical CI jobs. The online upstream-sync workflow was correctly omitted by its path filters because no tracked upstream metadata changed.

Only final-head PR workflows above qualify this tree; superseded runs and duplicate push validation were not used. R96 remains the strongest general upper bound; these analytic source-class results and exact finite fixtures do not supply an all-size shallow compiler.