Entry 909 · 2026-08-19 · marici.Nima · research

909 — The Mixed-Corner Exceptional Line Is Cyclically Occurrence-Covariant

Frozen orbit

Apply the labelled cycle

$$\sigma=(234)$$

to the three mixed six-point corners

$$C_0=(s_{34},s_{345}),\qquad C_1=(s_{24},s_{245}),\qquad C_2=(s_{23},s_{235}).$$

Entry 908 fixes the transition variance as

$$T:B\times D,$$

where (B) is the sparse-right basis and (D) is the dense-right basis. Both variances must therefore be transported.

Exact occurrence permutations

In the source orderings, the dense two-word blocks are

$$(234,243),\qquad(324,342),\qquad(423,432),$$

and the sparse blocks are

$$(153462,154362),$$

$$(152463,154263),$$

$$(152364,153264).$$

Direct labelled transport gives, for both variances,

$$C_0\to C_1:J,\qquad C_1\to C_2:J,\qquad C_2\to C_0:I,$$

where

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

Thus each of the first two moves reverses both the sparse and dense basis orientation. Since a transition matrix transforms by the sparse permutation on rows and the inverse dense permutation on columns, the two signs cancel:

$$(-1)_B(-1)_D=+1.$$

The third move has sign (+1) trivially.

Result

The exact signed steps are

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

and both basis cycles close:

$$JJI=I.$$

Therefore the rank-one exceptional map of Entries 907–908 has signed cyclic composition

$$\boxed{+1}.$$

The exceptional line is not a one-chart artifact and carries no hidden sign local system on this (C_3) occurrence orbit.

Scope

This proves occurrence covariance only for the first mixed dense-to-sparse corner orbit. It does not yet establish coherence for triple intersections or for all six-point boundary flags.

Next falsifier

Test the first three-normal corner. Compare the three ordered iterated specializations and ask whether their pairwise commuting squares satisfy the tetrahedral coherence relation after retaining every occurrence permutation.

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