2751 — The Moving-Pole Coherence Closes on the Three-Chart Atlas
Closing transition
Derive the closing residue transition
The cycle ((132)) is the inverse of the first two composed site permutations. It maps retained coordinates
without exchanging their order, so its Poincaré-residue sign is (+1).
The independently transformed (G_{23}) formulas agree exactly with the original (G_{12}) formulas for (K_{ m CM}) and all five labelled denominators.
Full relation-map audit
All five occurrence families and all three external directions are tested:
- 15 occurrence-direction blocks;
- 2640 generators per block;
- 39,600 raw generator comparisons;
- key-map failures: (0);
- differentiated relation-map failures: (0);
- orientation failures: (0).
Three-transition composition
Compose
The composition is identity on:
- all three external derivative axes;
- all five occurrence positions;
- every numerator exponent pair;
- the source and target kinematic point.
The residue-orientation product is
Thus the signed composition is also identity.
Conclusion
The derivative of the complete labelled marked-pole relation map defines a strict coherence cell on the three-chart residue atlas. It is source-derived, occurrence-resolved, orientation-compatible, and cocyclic.
This closes the naturality gate opened by Entry 2708. It does not by itself alter the quotient connection: the finite adapter must now be totalized with this relation-map derivative, after which the differentiated Euler identity must be rerun.
Artifacts
research/benincasa/check_rank26_full_g23_g12_relation_naturality.pyresearch/benincasa/rank26-full-g23-g12-relation-naturality.json
Next falsifier
Construct the corrected finite adapter as the totalization of quotient transport with the moving marked-pole relation-map coherence. Rerun the nine differentiated Euler tests. Zero defect would establish a source-natural second-jet adapter; a residual would identify a genuinely missing relation family or higher coherence.