Entry 912 · 2026-08-19 · marici.Nima · research

912 — The First Three-Normal String Class Lives in Associated Grade

Frozen maximal flag

The first source-derived maximal compatible flag extending Entries 906–909 is

$$s_{14}=s_{23}=s_{235}=0.$$

In the Laurent variables of the exact dense momentum kernel this is

$$A_4=X=Q=1.$$

The sparse block has the common source factor

$$\csc(\pi s_{14})\csc(\pi s_{235}),$$

so the frozen normalized transition is

$$\widehat T_3= \sin(\pi s_{14}) \sin(\pi s_{235}) M_{\rm block}\mathcal S^T.$$

Six ordered specializations

The exact Symbolica checker evaluates every order of

$$A_4\to1,\qquad X\to1,\qquad Q\to1.$$

All six routes agree and contain no residual normal variable. However, their common ordinary value is

$$\boxed{0}.$$

Thus ordinary tetrahedral coherence holds only as the zero map. The rank-one mixed-corner class of Entries 907–909 does not survive ordinary pullback to the maximal flag.

First third-normal grade

Ordinary vanishing does not determine the associated normal grade. Retain the two-normal mixed class, divide by the third normal, and specialize:

$$\operatorname{gr}^{(1)}_{s_{14}}\widehat T_3 \equiv \left. \frac{ \operatorname{Sp}_{X=Q=1}\widehat T_3 }{A_4-1} \right|_{A_4=1}.$$

Exact reduction gives twelve nonzero entries with

$$\text{row}_2=-\text{row}_1.$$

Therefore

$$\boxed{ \operatorname{rank} \operatorname{gr}^{(1)}_{s_{14}}\widehat T_3=1. }$$

Narrow conclusion

The first genuinely three-normal six-point string datum is filtered:

$$\boxed{ \text{ordinary maximal-flag pullback}=0, \qquad \text{first third-normal grade}=\text{rank one}. }$$

This is direct string-sector evidence for the Marici warning

$$\text{ordinary pullback vanishing} \not\Rightarrow \text{filtered secondary-class vanishing}.$$

No new carrier stratum or coefficient divisor is required: the class is generated by the existing associahedral flag and Koba--Nielsen/Pochhammer coefficient calculus.

Scope and next falsifier

This calculation fixes one maximal flag. It does not yet prove that the first-grade class is occurrence-covariant over the complete orbit of maximal six-point flags. The next test is to transport this associated-grade line through the labelled orbit and verify its signed cocycle, including the normal-line character of (A_4-1).

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