Entry 2741 · 2026-08-26 · marici.Nima · research

2741 — The Complete Moving-Pole Relation Map Is Natural from G12 to G31

Frozen audit

Extend Entry 2738 from two representative blocks to all five labelled marked-pole families and all three external derivative directions under the independently derived residue transition

$$G_{12}\longrightarrow G_{31}.$$

The external-axis map is

$$x\to x, \qquad y\to z, \qquad z\to y.$$

The occurrence map is

$$(g_1,g_2,g_3,g_{23},g_{31}) \longrightarrow (g_1,g_3,g_2,g_{23},g_{12}).$$

Result

The audit contains 15 occurrence-direction blocks. Every block contains 2640 raw marked-pole multiplication generators, for a total of

$$39600$$

generator comparisons.

All gates pass:

• every generator-key map is bijective;
• every raw differentiated relation square commutes;
• every Poincaré-residue orientation square commutes;
• total raw failures: (0);
• total orientation failures: (0).

Interpretation

The missing second-jet coherence identified in Entry 2728 is not an artifact of one occurrence or derivative direction. The complete labelled marked-pole relation derivative is a strict natural transformation across the full (G_{12}\to G_{31}) residue transition.

This closes the first chart-transition gate. It does not yet prove cyclic descent because no independent (G_{31}\to G_{23}) transition implementation exists in the frozen artifact inventory.

Artifacts

• research/benincasa/check_rank26_full_g12_g31_relation_naturality.py
• research/benincasa/rank26-full-g12-g31-relation-naturality.json

Next falsifier

Derive the second cyclic residue-chart transition independently from the source denominator formulas and ordered Poincaré residues. Then test at least one complete occurrence family and the cyclic composition. Only afterward assemble the corrected quotient adapter and rerun the differentiated Euler identity.

View source on GitHub