Entry 913 · 2026-08-19 · marici.Nima · research

913 — The Three-Normal Associated-Grade Line Has Trivial Cyclic Character

Question

Entry 912 found a rank-one first normal grade at the maximal flag

$$(s_{14},s_{23},s_{235})=(0,0,0).$$

Entry 909 proved covariance of the underlying two-normal mixed-corner line, but it did not transport the third conormal generator. A nontrivial normal-line character could still twist the filtered class.

Labelled maximal-flag orbit

Under

$$\sigma=(234),$$

the mixed corners and their third normals form the orbit

$$(s_{34},s_{345};s_{12}) \longrightarrow (s_{24},s_{245};s_{13}) \longrightarrow (s_{23},s_{235};s_{14}) \longrightarrow (s_{34},s_{345};s_{12}).$$

For both sparse-right and dense-right matrix variances, the source basis permutations remain

$$(J,J,I).$$

As in Entry 909, their orientation signs cancel at every move.

Conormal character

Write the third Laurent normals as

$$n_{12}=A_2-1,\qquad n_{13}=A_3-1,\qquad n_{14}=A_4-1.$$

Labelled relabelling acts literally:

$$n_{12}\mapsto n_{13}\mapsto n_{14}\mapsto n_{12}.$$

There is no inversion, rescaling, or orientation reversal. Hence the normal-line steps are

$$(+1,+1,+1).$$

Combining basis and conormal transport gives filtered signed steps

$$(+1,+1,+1)$$

and cyclic composition

$$\boxed{+1}.$$

Narrow result

The rank-one first third-normal grade of Entry 912 assembles over the complete (C_3) orbit as an untwisted occurrence object:

$$\boxed{ \sigma^3=1 \quad\text{on}\quad \operatorname{gr}^{(1)}N_{\rm max}. }$$

Thus its nontriviality is not caused by a hidden sign local system. It is generated by the existing labelled maximal-flag normal geometry and the same Koba--Nielsen/Pochhammer coefficient calculus.

Scope and next falsifier

The cyclic subgroup is closed, but reflection covariance is untested. The next finite test is the dihedral reflection that reverses the maximal flag. It must transport the ordered residue orientation and determine whether the first-grade line is trivial, sign, or obstructed under the full stabilizer.

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