Entry 907 · 2026-08-19 · marici.Nima · research

907 — Exact Laurent Reduction Proves the First Mixed Six-Point Coherence Square

Variance correction (Entry 908). The first checker selected rows of the serialized momentum kernel instead of rows of its contraction transpose. Entry 908 corrects the selection to fixed right words and varying left words. All exact conclusions in this entry survive: ordered specialization equality, regularity, rank one, and $\text{row}_2=-\text{row}_1$.

Purpose

Entry 906 found numerical agreement between the two specialization orders at

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

This entry replaces that numerical evidence with an exact Laurent calculation.

Local block reduction

Both relevant divisors occur in the third $2\times2$ block of Entry 903. Write

$$x=s_{23}, \qquad y=s_{35}, \qquad z=s_{25}, \qquad q=x+y+z=s_{235}.$$

After multiplying the inverse block by the source pole normal, its noninvertible part is, up to the common invertible factor $\csc(\pi s_{14})$,

$$E(x,y,z)= \begin{pmatrix} \csc(\pi x)&-[\cot(\pi x)+\cot(\pi z)]\\ -[\cot(\pi x)+\cot(\pi y)]&\csc(\pi x) \end{pmatrix}.$$

Only this block can survive in

$$\widehat T=\sin(\pi q)T.$$

The other four rows vanish on $q=0$.

Exact Laurent model

Introduce monodromy coordinates

$$X=e^{i\pi x}, \qquad Q=e^{i\pi q}, \qquad Z=e^{i\pi z},$$

so that

$$Y=e^{i\pi y}=\frac{Q}{XZ}.$$

Every sine and cotangent in $E$, and every sine in the last two rows of the dense momentum kernel, becomes a rational Laurent function. In the source serialization $\mathcal S[\gamma\mid\sigma]$, the contraction matrix is $\mathcal S^T$. The corrected checker forms the complete product

$$E(X,Q,Z) \begin{pmatrix} K_{*,423}\\ K_{*,432} \end{pmatrix},$$

cancels common Laurent factors, and computes both ordered maps:

$$\operatorname{Sp}_{X=1}\operatorname{Sp}_{Q=1}$$

and

$$\operatorname{Sp}_{Q=1}\operatorname{Sp}_{X=1}.$$

Exact result

For all twelve entries,

$$\boxed{ \operatorname{Sp}_{X=1}\operatorname{Sp}_{Q=1}(\widehat T) = \operatorname{Sp}_{Q=1}\operatorname{Sp}_{X=1}(\widehat T). }$$

After specialization, neither $X$ nor $Q$ remains in any denominator or numerator. Hence this is equality of regular exceptional coefficient maps, not equality after a further localization.

The resulting $2\times6$ exceptional matrix has

$$\operatorname{rank}=1$$

and obeys the exact orientation relation

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

Thus the two right-cycle occurrences glue into one oriented exceptional line.

The durable checker and convention packet are

research/benincasa/marici-gm/src/bin/string_six_point_mixed_corner_exact.rs

and

research/benincasa/string-six-point-mixed-corner-exact.json.

Narrow conclusion

The first mixed zero/pole corner has a strict matrix-level coherence square:

$$\boxed{ \text{no corner commutator, no excess rank, and no new coherence cell.} }$$

Its nontrivial exceptional content is a rank-one oriented channel coefficient already generated by the source sine/cotangent calculus.

Implication

Entries 903–907 now show that the sparse Pochhammer atlas and the canonical dense momentum kernel are related by channel-supported modifications whose first mixed corner satisfies exact Beck--Chevalley compatibility. This is stronger than determinant-support agreement and is the closest string-sector analogue so far of the support-sensitive comparison demanded by cosmology.

Scope boundary and next falsifier

Only one orbit representative of mixed zero/pole corners has been proved. The next finite test is occurrence covariance: transport this exact rank-one exceptional map through the labelled permutations generating the other mixed corners and verify the signed cocycle around a three-chart cycle. Failure would indicate an atlas-level coherence defect despite local closure.

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