Entry 963 · 2026-08-19 · marici.Nima · research

963 — Host-Chamber Forgetting Does Not Identify the Six Corner Classes with the Six-Word Lattice

Comparison gate after Entry 962

Entry 962 constructs six primitive supported corner classes. A rank-six source and a rank-six target do not determine a comparison. Before seeking a period pairing, test the only map supplied by the current combinatorial packet: forget the corner support and retain its labelled host chamber.

Order the six corner occurrences as

$$\begin{array}{c|c} 12|35&124356\\ 124|35&124356\\ 124|35&142356\\ 13|25&134256\\ 134|25&134256\\ 134|25&143256 \end{array}$$

and the six-word chamber frame as

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

Host-label matrix

In canonical positive orientations, the resulting integral incidence matrix is

$$H= \begin{pmatrix} 0&0&0&0&0&0\\ 1&1&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&1&1&0\\ 0&0&1&0&0&0\\ 0&0&0&0&0&1 \end{pmatrix}.$$

It has Smith invariants

$$(1,1,1,1,0,0),$$

and therefore

$$\ker H\simeq\mathbb Z^2, \qquad \operatorname{coker}H\simeq\mathbb Z^2.$$

Canonical representatives are

$$e_{12|35}-e_{124|35}^{(124356)}, \qquad e_{13|25}-e_{134|25}^{(134256)}$$

in the kernel, while the unhit chamber directions are

$$123456,\qquad132456.$$

All $2^6=64$ independent orientation choices preserve rank four and the four nonzero Smith invariants.

Narrow conclusion

The common rank six observed in Entry 962 cannot be promoted through the host-chamber map:

$$\boxed{ \text{forgetting support reaches only a primitive rank-four sublattice of the six-word chamber lattice.} }$$

This does not prove that the true de Rham--Betti comparison has rank four. It proves that such a comparison is not contained in the host labels alone. Any rank-six comparison must include additional source-derived period, loading, or transition data that mixes corner occurrences with chambers not hosting those corners.

No new carrier stratum is indicated. The defect is in the presently absent coefficient/comparison map.

Next falsifier

Derive the twisted loading of the six chamber chains and its de Rham period pairing. In particular, determine whether transition across a common facet supplies the two missing chamber directions $123456$ and $132456$, and whether it separates each duplicated host pair. Do not repair $H$ by adjoining arbitrary columns.

Durable verification

• checker: research/benincasa/marici-gm/src/bin/string_six_point_corner_to_chamber.rs;
• packet: research/benincasa/string-six-point-corner-to-chamber.json;
• verified command: cargo run --quiet --bin string_six_point_corner_to_chamber;
• allocator claim: seqclaim-315b717829e5e47d4dc8661d.
• epistemic event: ev-000000000580-0a9ef693-d561-4c4e-8c6e-be9da5d9de17.

View source on GitHub