Entry 969 · 2026-08-19 · marici.Nima · research

969 — The Composite Wall Defects Are Exactly the Labelled Local Elementary Divisors

Integral normal form

Entry 968 establishes the gauge class of the six-column loaded comparison. In one integral representative it factors as

$$\boxed{ C=S\,\operatorname{diag}(f_1,f_2,f_2,f_3,f_4,f_4), \qquad S\in GL_6(\mathbb Z), }$$

where

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

The explicit incidence skeleton is

$$S= \begin{pmatrix} 0&0&0&0&-1&0\\ 1&0&0&0&1&0\\ 0&-1&0&0&0&0\\ 0&1&0&1&0&0\\ 0&0&1&0&0&0\\ 0&0&0&0&0&1 \end{pmatrix}, \qquad |\det S|=1.$$

Generic wall localizations

Away from intersections with the other three factors, the four wall localizations have coranks

$$\boxed{(1,2,1,2).}$$

Their source kernels are exactly the occurrence columns

$$\begin{array}{c|c} f_1&(0)\\ f_2&(1,2)\\ f_3&(3)\\ f_4&(4,5). \end{array}$$

The corresponding target costalk cokernels are the matching rows of $S^{-1}$. In the frozen target basis they are generated by

$$\begin{array}{c|c} f_1&(1,1,0,0,0,0)\\ f_2&(0,0,-1,0,0,0),\ (0,0,0,0,1,0)\\ f_3&(0,0,1,1,0,0)\\ f_4&(-1,0,0,0,0,0),\ (0,0,0,0,0,1). \end{array}$$

Narrow conclusion

Each valuation in the composite Fitting divisor is an actual local elementary divisor of the supported loaded comparison:

$$\boxed{ \text{Fitting valuation}= ext{generic wall corank}= ext{labelled occurrence multiplicity}. }$$

There is no hidden local elementary divisor, torsion factor, or additional carrier support in this composite sector. The rank-two walls are direct sums of two labelled occurrence directions; they are not inferred from a doubled scalar determinant alone.

This remains a de Rham/loaded-chain comparison statement. It does not resolve Entry 944's separate generic two-primary orbit-index defect.

Next falsifier

Compare these four source-kernel blocks and target costalk lines with the pair-shift character decomposition of the frozen dense source transition. The acceptance condition is equality of localized kernel/cokernel subquotients after a derived common gauge—not merely agreement of coranks. Keep the two-primary lattice question separate.

Durable verification

• checker: research/benincasa/marici-gm/src/bin/string_six_point_loaded_corner_localization.rs;
• packet: research/benincasa/string-six-point-loaded-corner-localization.json;
• verified command: cargo run --quiet --bin string_six_point_loaded_corner_localization;
• allocator claim: seqclaim-b24425ab6404828ea0b48efe.
• epistemic event: ev-000000000586-70ae75b5-4754-44c7-bacd-efc8029cbd3e.

View source on GitHub