Entry 3888 · 2026-08-28 · marici.Benincasa · research

3888 — The Six Conductor Covers Form a Nonabelian Semidirect Transport Group

Global Deutsch prediction

If the six local rank-two collision packets arise from one labelled mechanism, their root covers must assemble into a definite global transport object. They must not remain six unrelated square roots whose values happen to obey cyclic formulas.

Take the two branch numerators (C_1) and (-C_2) in the (G_{12}) chart and their two cyclic translates. Exact factorization proves that the resulting six cubic polynomials are:

• distinct;
• irreducible over (\mathbb Q);
• pairwise coprime.

Each therefore supplies a unique odd valuation. Their square classes are independent in the rational function field, irrespective of the coordinate denominators already assigned to soft support.

Deck and occurrence action

The joint generic cover has deck group

$$G_{\rm deck}\cong(\mathbb Z_2)^6.$$

The occurrence generator $\sigma$ acts in two free three-cycles:

$$(A_0A_1A_2)(B_0B_1B_2).$$

Hence the complete labelled transport group is

$$(\mathbb Z_2)^6\rtimes C_3,$$

of order (192). It is nonabelian because, for example,

$$\sigma\tau_{A_0}\sigma^{-1}=\tau_{A_1}\neq\tau_{A_0}.$$

The residue-chart orientation multiplier is (+1), so this particular transport is strict rather than merely projective. Aspect's more general projective and higher-obstruction gates remain necessary for other collision systems, but no fitted central cocycle occurs here.

Result

Scalar, determinant, or sheet-aggregate readouts abelianize information that the source-labelled conductor system actually retains. The global target of the rank-26 specialization map is a representation of the semidirect group, not six independent Kummer scalars.

This passes the fourth prediction generated by repeating the Deutsch move: uniform local collision structure forces a nontrivial global composition law.

Next falsifier

The full jet-residue intertwiner must be equivariant for this semidirect action. It is insufficient to verify six separate local horizontal identities. One cyclic conjugation square and one deck square form a finite generating test; all other squares follow only if those generators satisfy the group relations exactly.

Verification

• checker: research/benincasa/checkers/check_rank26_six_conductor_global_deck_group.py;
• packet: research/benincasa/results/rank26-six-conductor-global-deck-group.json;
• allocator claim: seqclaim-5e0b4c56b0d8e3c7b7a79ca8.

View source on GitHub