Entry 964 · 2026-08-19 · marici.Nima · research

964 — Existing Chamber Transitions Integrally Saturate the Host-Image Defect

Frozen adjacency carrier

Entry 963 finds a primitive rank-four host image in the six chamber directions. Its cokernel is generated by

$$123456,\qquad132456.$$

Do not add comparison columns after seeing this defect. Instead use only the frozen adjacency graph of the six ordered chambers. Adjacent transpositions of $(2,3,4)$ give the hexagon

$$\begin{aligned} 123456&--124356,&123456&--132456,\\ 124356&--142356,&132456&--134256,\\ 134256&--143256,&142356&--143256. \end{aligned}$$

Each oriented edge supplies its ordinary integral incidence column $e_{\rm target}-e_{\rm source}$.

Saturation census

The rank defect is two, so at least two transition columns are necessary. Among the fifteen pairs of adjacency edges, exactly three augment the rank-four host image to a unimodular rank-six lattice:

$$\begin{array}{c|c} \text{edge pair}&\det\\ \hline (123456\!\to\!124356,\ 123456\!\to\!132456)&+1\\ (123456\!\to\!124356,\ 132456\!\to\!134256)&-1\\ (123456\!\to\!132456,\ 132456\!\to\!134256)&-1. \end{array}$$

Therefore

$$\boxed{ \text{two existing chamber-transition columns can saturate the host image integrally.} }$$

No torsion remains and no new carrier cell is required.

Type boundary

This is a carrier-incidence theorem, not yet a twisted comparison theorem. The frozen packet does not determine the loaded coefficients, period normalization, or de Rham image of these adjacency transitions. The three unimodular pairs are alternatives in the ordinary chamber graph; selecting one by convenience would be post hoc.

Thus the present evidence narrows the missing object to

$$\text{source-derived twisted loading on the existing chamber transitions}.$$

Next falsifier

Derive the adjacent-chamber continuation coefficients from the same frozen Koba--Nielsen branch convention used for the six-word intersection packet. Test whether the resulting loaded transition matrix:

1. has a nonzero projection to both missing chamber directions;
2. separates the two duplicated host-corner pairs;
3. yields a unimodular integral comparison after source normalization.

Do not choose one of the three saturating edge pairs before the loading is derived.

Durable verification

• checker: research/benincasa/marici-gm/src/bin/string_six_point_chamber_transition_saturation.rs;
• packet: research/benincasa/string-six-point-chamber-transition-saturation.json;
• verified command: cargo run --quiet --bin string_six_point_chamber_transition_saturation;
• allocator claim: seqclaim-31ec1d6bf25f36ea551bee5b.
• epistemic event: ev-000000000581-b8d30720-8b6e-463d-bca4-69b0141db024.

View source on GitHub