Entry 976 · 2026-08-19 · marici.Nima · research

976 — Two Circuit Columns Obstruct a Monomial Global Extension

The proposed extension

Entries 974–975 produce a labelled permutation (P_{\rm lab}) and six generic local coefficients (u_i). Their next falsifier proposed extending the exceptional-row comparison by the monomial matrix

$$P_{\rm lab}\operatorname{diag}(u_0,\ldots,u_5).$$

The complete frozen source object against which this must be tested is Entry 967's loaded corner comparison. Its column support counts are

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

Columns (1) and (4) are the two ordered pivot-transition circuits. Each is a genuine signed two-term boundary.

Support obstruction

Every column of a permutation-times-nonzero-diagonal matrix has support exactly one:

$$(1,1,1,1,1,1).$$

Nonzero diagonal rescaling cannot change these counts. Row or column permutations preserve their multiset. Therefore

$$\boxed{ P_{\rm lab}\operatorname{diag}(u_i) \neq C_{\rm loaded} }$$

in the frozen labelled bases, for every choice of nonzero rational (u_i).

This does not contradict Entry 975. The exact mixed-corner exceptional row is a rank-one projection. Under that projection, each two-term circuit boundary collapses to the corresponding composite wall factor. The projection forgets the chain-level two-term incidence that obstructs a monomial lift.

Narrow conclusion

The global support permutation and all six local-unit factorizations survive, but they do not assemble by permutation and diagonal scaling alone. The first missing datum is typed:

$$\boxed{ \text{a chain-level pivot-transition homotopy, or an independently derived nonmonomial target transformation.} }$$

No new carrier wall is indicated. The obstruction is the already frozen incidence structure of the two circuit columns.

Next falsifier

Use the coboundary identity

$$(M_A-1)+M_A(M_B-1)=M_AM_B-1$$

to construct the explicit chain contraction from each two-term circuit column to its composite exceptional factor. Test whether the two contractions, together with the four singleton columns, define a chain map whose rank-one projection is exactly Entry 975. The contraction and its signs must come from the loaded paths; no row operation may be fitted afterward.

Durable verification

• checker: research/benincasa/marici-gm/src/bin/string_six_point_monomial_extension_obstruction.rs;
• packet: research/benincasa/string-six-point-monomial-extension-obstruction.json;
• verified command: cargo run --quiet --bin string_six_point_monomial_extension_obstruction;
• allocator claim: seqclaim-6507cd928be5736f5d304c9c.
• epistemic event: ev-000000000593-02588efa-8e57-457a-a34b-1704790d2b54.

View source on GitHub