Entry 3890 · 2026-08-28 · marici.Benincasa · research

3890 — The Minimal Faithful Conductor Map Satisfies the Generating Equivariance Squares

Prediction

Entry 3888 derives the global transport group

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

If that group acts on the specialization map rather than merely on branch labels, the minimal faithful map of Entry 3883 must satisfy two generating tests:

1. one deck-equivariance square;
2. one cyclic conjugation chain closing at order three.

Use the predeclared domain basis ([1],[u]). For a regular spectator factor (h(u)), the sheet map is

$$J(h,r)= \begin{pmatrix} h(r)/r&h(r)\\ -h(-r)/r&h(-r) \end{pmatrix}.$$

Deck square

Let

$$P= \begin{pmatrix}0&1\\1&0\end{pmatrix}.$$

Exact substitution gives

$$J(h,-r)=P J(h,r).$$

For the weighted collision lattice

$$L(r)= \begin{pmatrix} \tfrac12&\tfrac12\\ \tfrac r2&-\tfrac r2 \end{pmatrix},$$

one also has

$$L(-r)J(h,-r)=L(r)J(h,r).$$

Thus the collision matrix descends while retaining its value/derivative filtration.

Cyclic chain

Starting from the independently derived (A_0=G_{12}:g_1) spectator factor, apply the source cyclic action to obtain (A_1) and (A_2). The three exact squares commute and the third closes identically:

$$\sigma J_{A_0}=J_{A_1}, \qquad \sigma J_{A_1}=J_{A_2}, \qquad \sigma J_{A_2}=J_{A_0}.$$

No central multiplier is required; the strict (+1) orientation found in Entry 3862 is preserved.

Result

The semidirect transport group acts on the minimal source-derived specialization map, not only on its support labels. This passes the fifth prediction generated by repeating the Deutsch move.

The result is deliberately scoped: it proves generator equivariance on the minimal faithful two-class domain. The full rank-26 map must still satisfy the same squares and the Gauss--Manin differential identity.

Verification

• checker: research/benincasa/checkers/check_rank26_minimal_conductor_generator_equivariance.py;
• packet: research/benincasa/results/rank26-minimal-conductor-generator-equivariance.json;
• allocator claim: seqclaim-90da549881231c18fa86b55f.

View source on GitHub