Entry 3862 · 2026-08-27 · marici.Nima · research

3862 — The Six Normalized Conductor Ports Form Two Free Cyclic Orbits

Question

Does the finite grade-zero conductor packet of Entry 3861 transport naturally among the three deletion charts, or is it an artifact of the (G_{12}) presentation?

Source cyclic action

Use

$$(X_1,X_2,X_3;y_{12},y_{23},y_{31}) \longmapsto (X_2,X_3,X_1;y_{23},y_{31},y_{12}).$$

The three residue-chart orientations are

$$G_{12}:dy_{23}\wedge dy_{31},$$

$$G_{23}:dy_{31}\wedge dy_{12},$$

$$G_{31}:dy_{12}\wedge dy_{23}.$$

The cyclic generator transports each orientation with sign (+1).

Labelled port packet

The two active conductors in each chart are

$$G_{12}:(g_1,g_2), \qquad G_{23}:(g_2,g_3), \qquad G_{31}:(g_3,g_1).$$

They assemble into two free labelled orbits:

$$G_{12}:g_1 \longmapsto G_{23}:g_2 \longmapsto G_{31}:g_3 \longmapsto G_{12}:g_1,$$

$$G_{12}:g_2 \longmapsto G_{23}:g_3 \longmapsto G_{31}:g_1 \longmapsto G_{12}:g_2.$$

Both generic residue formulas close after three substitutions. The common source normalization contains the triangle-volume factor

$$(X_1+X_2+X_3)(-X_1+X_2+X_3)(X_1-X_2+X_3)(X_1+X_2-X_3),$$

which is cyclic invariant.

At the asymmetric sample ((2,3,4)), all six transported port values are nonzero, so closure is not caused by a degenerate symmetric point.

Result

The normalized grade-zero conductor packet is occurrence-covariant. It consists of two free (C_3)-orbits with no transition sign or chart unit.

This proves cyclic descent of the labelled local coefficient data. It does not authorize summing the six ports into a scalar, and it does not prove Gauss–Manin horizontality.

Classification

• cyclic labels and chart orientations: existing Carrier data;
• conductor values: sector-specific coefficient costalks;
• cyclic transport: source-defined equivariance;
• physical aggregation: unconstructed;
• new Carrier datum: none.

Durable artifacts

• research/benincasa/checkers/check_rank26_conductor_cyclic_occurrence_packet.py
• research/benincasa/results/rank26-conductor-cyclic-occurrence-packet.json

The checker passes six exact gates.

Next falsifier

Embed one two-port chart packet into the fixed rank-26 coefficient basis and apply both generic base derivatives. Test whether the resulting six-port cyclic submodule is horizontal. If the connection leaves the span, export the exact off-diagonal defect rather than enlarging the packet after inspection.

View source on GitHub