Entry 914 · 2026-08-19 · marici.Nima · research

914 — The Three-Normal Associated-Grade Line Is Dihedrally Untwisted

Reflection gate

Entries 912–913 construct a rank-one first third-normal grade and prove its trivial cyclic character. To determine the full stabilizer representation, freeze the reflection

$$\tau=(24).$$

It transports the labelled maximal flag by

$$(s_{14},s_{23},s_{235}) \longmapsto (s_{12},s_{34},s_{345}).$$

Four possible signs are retained independently.

Basis variances

The dense-right source block transforms as

$$(423,432) \stackrel{\tau}{\longmapsto} (243,234),$$

which is the reversed (C_0) basis. The sparse-right block likewise transforms as

$$(152364,153264) \stackrel{\tau}{\longmapsto} (154362,153462),$$

again reversed relative to (C_0). Hence

$$\chi_B(\tau)=-1, \qquad \chi_D(\tau)=-1,$$

and their matrix-variance contribution is (+1).

Residue and conormal orientations

The reflection preserves the ordered normal slots:

$$(s_{14},s_{23},s_{235}) \mapsto (s_{12},s_{34},s_{345}).$$

It does not permute the positions of the one-, two-, and three-particle normals. Thus the ordered residue orientation contributes (+1).

The first-grade generator transforms literally:

$$A_4-1\mapsto A_2-1,$$

without inversion or rescaling, so the conormal character is also (+1).

Result

The complete filtered reflection character is

$$\chi_{\rm filt}(\tau) =(-1)(-1)(+1)(+1) =+1.$$

Combining this with Entry 913 gives

$$\boxed{ \operatorname{gr}^{(1)}N_{\rm max} \text{ carries the trivial }D_3\text{ character}. }$$

The first three-normal string class is therefore globally occurrence-covariant over this maximal-flag orbit, not merely cyclically covariant.

Scope and next falsifier

This closes the symmetry character of one rank-one maximal-flag class. It does not determine whether all maximal six-point flags yield the same filtered rank. The next test is an orbit census of maximal compatible channel triples, followed by one exact representative calculation per (D_3)-orbit. A rank jump would falsify uniformity of the local coefficient packet without requiring a new carrier.

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