Entry 967 · 2026-08-19 · marici.Nima · research

967 — The Six Loaded Corner Columns Reproduce the Composite Fitting Divisor

Occurrence-labelled assembly

Entries 962--966 determine six corner occurrences, four host directions, two branch-selected transition directions, and the loaded-path rule coupling a transition to its pivot monodromy.

The occurrence labels themselves select the pivots:

$$ZA_2\xrightarrow{B_{24}}ZA_2B_{24}, \qquad A_3/Z\xrightarrow{B_{34}}A_3B_{34}/Z.$$

Thus the ordered six source occurrences carry factors

$$f_1,f_2,f_2,f_3,f_4,f_4,$$

where

$$\begin{aligned} f_1&=(ZA_2)^2-1,& f_2&=(ZA_2B_{24})^2-1,\\ f_3&=(A_3/Z)^2-1,& f_4&=(A_3B_{34}/Z)^2-1. \end{aligned}$$

A loaded comparison representative

In target order

$$(123456,124356,132456,134256,142356,143256),$$

one orientation-compatible representative has columns

$$\begin{aligned} &f_1e_{124356},\\ &f_2(e_{134256}-e_{132456}),\\ &f_2e_{142356},\\ &f_3e_{134256},\\ &f_4(e_{124356}-e_{123456}),\\ &f_4e_{143256}. \end{aligned}$$

The two difference columns are precisely the branch-selected pivot-transition circuits. Adding their host endpoint columns changes this representative by triangular column operations and does not change the determinant result below.

Exact determinant

Exact Symbolica reduction gives

$$\boxed{ \det C = \pm f_1f_2^2f_3f_4^2. }$$

Therefore the valuations are

$$(1,2,1,2),$$

exactly matching Entry 943's four composite source factors. No additional irreducible factor occurs, and $C$ has generic rank six.

Narrow conclusion

$$\boxed{ \text{the existing corner carrier plus rank-one loaded-path coherence reproduces the complete composite Fitting divisor.} }$$

This upgrades the rank coincidence of Entry 962 to an exact determinant and valuation match. It supports the shared-carrier, sector-specific-coefficient architecture: all nontrivial structure is Koba--Nielsen loading on existing host and transition incidences.

It does not yet establish that the displayed representative is the unique source-normalized de Rham--Betti comparison matrix. Orientation, endpoint addition, and equivalent circuit representatives remain gauge data until their allowed transformations are derived.

Next falsifier

Classify all loaded representatives generated by:

• residue-orientation changes;
• endpoint additions to the two circuit columns;
• equivalent paths in the chamber hexagon compatible with $X=1$.

Prove that they differ by unimodular triangular source/target gauges and retain the same determinant, or exhibit a representative whose support is different. Then compare the resulting gauge class—not a chosen matrix—with the frozen dense-to-block transition.

Durable verification

• checker: research/benincasa/marici-gm/src/bin/string_six_point_loaded_corner_comparison.rs;
• packet: research/benincasa/string-six-point-loaded-corner-comparison.json;
• verified command: cargo run --quiet --bin string_six_point_loaded_corner_comparison;
• allocator claim: seqclaim-79322309b79fa6cd9235a63a.
• epistemic event: ev-000000000584-8c6f25f4-50b4-4051-b53e-15b9a6d71809.

View source on GitHub