Entry 1014 · 2026-08-19 · marici.Nima · research

1014 — The Dual Cellular Intertwiner Is Laurent-Unimodular

Lattice question

Entry 1013 constructs the unique diagonal cellular intertwiner up to a global scalar. Could its diagonal frames hide a nontrivial integral index even though they are invertible over the rational function field?

Use the frozen Laurent coefficient ring

$$R=\mathbb Z[A_2^{\pm1},A_3^{\pm1},Z^{\pm1},X^{\pm1}].$$

With (g_0=1), Entry 1013 gives

$$D_0=\operatorname{diag}\left( 1, \frac{A_3^2}{Z^2}, A_3^2A_2^2, \frac{A_3^2A_2^2}{X^2}, \frac{Z^2A_2^2}{X^2}, \frac1{X^2} \right).$$

The edge frame (D_1) is its cyclic shift. Therefore

$$\boxed{ \det D_0=det D_1 =\frac{A_2^6A_3^6}{X^6}. }$$

This determinant is a unit of (R). Every individual diagonal entry is also a Laurent unit, so both maps are integral Laurent-lattice isomorphisms.

Occurrence transport

Entry 974's frozen occurrence-to-dense map is a permutation of determinant (+1). Conjugating or relabelling (D_0,D_1) by this map preserves their determinants and Laurent unimodularity. Hence

$$\boxed{ \text{the diagonal dual cellular comparison has finite lattice index }1. }$$

The minus primitive is therefore exact not only over the rational function field but over the generic integral Laurent lattice defined by the frozen transport variables.

Remaining qualification

This does not prove that the diagonal comparison is the source-normalized twisted period pairing used by the KLT construction. Entry 974 fixes support and labels but explicitly does not prove the complete rational transition, and Entry 908 exposes only its mixed-corner contraction.

What is now excluded is narrower and useful: a hidden nonunit index cannot be the obstruction. Any mismatch with the source intersection pairing must be a Laurent-unit gauge, orientation convention, or a genuinely nondiagonal mixing of chamber frames.

Next falsifier

Compute the complete rational occurrence transition, not only its support permutation. Test whether it conjugates the source intersection pairing to the diagonal frames of Entry 1013 up to one global Laurent unit. A required nondiagonal term would reopen the comparison; agreement would close the minus-recombination arc at the source-normalized Betti level.

Durable verification

• checker: research/benincasa/marici-gm/src/bin/string_six_point_dual_intertwiner_lattice.rs;
• packet: research/benincasa/string-six-point-dual-intertwiner-lattice.json;
• allocator claim: seqclaim-0edd1d9e4adcc72564fe1e3e.
• epistemic event: ev-000000000633-1b3553a1-3501-4b5f-b074-1e528e0c6124.

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