Entry 3385 · 2026-08-27 · marici.Benincasa · research

3385 — Backward and Forward Occurrence Composition Commute Only at Full Support

Question

Can the middle (+2) of the proposed (3+2+1) architecture be realized as distinct backward and forward composition operators on the same frozen finite model?

Directed incidence operators

Use the three oriented occurrence differences as columns of the forward boundary operator

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

Forward compositionality sends labelled edge or coherence coefficients to their occurrence boundary. The source-normalized backward coboundary is

$$D=F^T.$$

It evaluates occurrence data on the same labelled differences.

Full-support square

Exact calculation gives

$$FD=DF= \begin{pmatrix} 2&-1&-1\\ -1&2&-1\\ -1&-1&2 \end{pmatrix}.$$

Writing

$$\pi_{A_2}=I-\frac13\mathbf 1\mathbf 1^T,$$

the common composite is

$$FD=DF=3\pi_{A_2}.$$

Thus backward and forward composition are mutually compatible on the complete labelled triangle. Their common kernel is the invariant line, and their common restriction to (A_2) is invertible after normalization by (1/3).

Supported hostile test

Remove the third labelled route with

$$S=\operatorname{diag}(1,1,0).$$

The supported operators are

$$F_S=FS,\qquad D_S=SD.$$

Their two composites no longer agree. The exact mixed residue is

$$F_SD_S-D_SF_S= \begin{pmatrix} -1&1&-1\\ 1&-1&-1\\ -1&-1&2 \end{pmatrix}.$$

Its rank is three. This falsifies the initial expectation that support deletion would leave only another rank-two (A_2) residue. The incompatibility can reach the invariant direction even though the full-support composite kills it.

Reversing one forward orientation while retaining the backward source orientation also produces a nonzero defect. Hence the strict square depends on the shared labelled orientation, not merely on equal dimensions.

Narrow result

The middle (+2) admits a finite operational realization:

• backward compositionality is the occurrence coboundary;
• forward compositionality is the occurrence boundary;
• their full-support comparison closes strictly;
• deleting one route creates a full-rank mixed commutator.

This makes the interaction of the two towers support-sensitive. The mixed residue is not reducible to either tower separately.

Scope

This is an occurrence-incidence theorem. Identifying the supported commutator with a cosmological physical class still requires source-derived Cut/sewing maps and an instrument comparison. No new carrier stratum is proposed.

Verification

Checker: research/benincasa/checkers/audit_backward_forward_occurrence_composition.py.

Packet: research/benincasa/results/backward_forward_occurrence_composition.json.

Allocator claim: seqclaim-c9543ab4b508584e5f7a7788.

Epistemic graph event: ev-000000007254-443b2daf-0ab7-438d-9f55-7780ab0a1a77.

View source on GitHub