Entry 958 · 2026-08-19 · marici.Nima · research

958 — The Four Composite Resonances Are Existing Codimension-Two Chamber Corners

Question from Entry 957

Entry 957 separated four direct chamber facets from four $Z$-dependent source Fitting factors. The first admissible provenance test for the latter is whether each is the monodromy of two compatible facets meeting in an existing associahedral corner.

Use the six frozen ordered chambers and the exact quotient lattice of Entry 957. Two facet partitions are admitted only when they are compatible splits and occur in a common chamber. Test signed sums of their channel vectors modulo momentum conservation and the three branch normals.

Exact corner identities

All four composite factors are generated:

$$\begin{aligned} ZA_2&\longleftrightarrow s_{12}+s_{35},\\ ZA_2B_{24}&\longleftrightarrow s_{124}+s_{35},\\ A_3/Z&\longleftrightarrow s_{13}+s_{25},\\ A_3B_{34}/Z&\longleftrightarrow s_{134}+s_{25}. \end{aligned}$$

For the ratio factors, the frozen equations give

$$s_{23}=0,\qquad s_{235}=0 \quad\Longrightarrow\quad s_{25}=-s_{35},$$

so the last two identities are precisely the additive forms of $A_3/Z$ and $A_3B_{34}/Z$.

Labelled chamber occurrences

The compatible pairs occur in:

$$\begin{array}{c|c|c} \text{factor}&\text{ordered chambers}&\text{count}\\ \hline ZA_2&(124356)&1\\ ZA_2B_{24}&(124356),(142356)&2\\ A_3/Z&(134256)&1\\ A_3B_{34}/Z&(134256),(143256)&2. \end{array}$$

These counts exactly reproduce their valuations in Entry 943's source Fitting minor:

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

This agreement was not used to select the corners: the corner list was generated from the six cyclic words before comparison with the Fitting valuation.

Narrow conclusion

The four factors left unmatched by the codimension-one census are not new carrier walls. They are supported on existing codimension-two chamber incidences:

$$\boxed{ \text{four direct facets} + \text{four existing two-facet corners} = \text{all eight source Fitting factors}. }$$

This materially strengthens the common-carrier interpretation of the six-point source lattice. It does not yet construct the integral twisted boundary comparison: the ordered corner orientations and loaded iterated boundary coefficients remain uncomputed.

Next falsifier

For the four displayed pairs, derive the two ordered boundary routes

$$\partial_{F_2}\partial_{F_1}\gamma, \qquad \partial_{F_1}\partial_{F_2}\gamma,$$

including associahedral incidence signs and the loaded factors $(M_{F_1}-1)(M_{F_2}-1)$. Verify the Koszul anticommutation and compare the resulting primitive corner columns with the corresponding source branch blocks. A multiplicity match without the signed matrix comparison is not yet an integral de Rham--Betti theorem.

Durable verification

• checker: research/benincasa/marici-gm/src/bin/string_six_point_branch_chamber_facets.rs;
• packet: research/benincasa/string-six-point-branch-chamber-facets.json;
• verified command: cargo run --quiet --bin string_six_point_branch_chamber_facets;
• allocator claim: seqclaim-ee71554feb54966a928bb393.
• epistemic event: ev-000000000575-6b180142-52f9-4ae1-b792-50fa3691d465.

View source on GitHub