Entry 977 · 2026-08-19 · marici.Nima · research

977 — The Exceptional Row Lifts Uniquely as a Circuit-Compatible Cochain

Retaining incidence

Entry 976 shows that the complete loaded comparison cannot be replaced by a permutation-times-diagonal matrix. The two circuit columns must remain two-term boundaries.

Let (C) be Entry 967's loaded (6\times6) matrix and let

$$r_i=f_i u_i$$

be Entry 975's six exceptional components in occurrence order. We seek a target cochain

$$\lambda=(\lambda_0,\ldots,\lambda_5)$$

such that

$$\lambda C=r.$$

Source-derived solution

The four singleton columns force

$$\lambda_1=u_0,\qquad \lambda_4=u_2,\qquad \lambda_3=u_3,\qquad \lambda_5=u_5.$$

The two oriented circuit boundaries then force

$$\lambda_3-\lambda_2=u_1, \qquad \lambda_1-\lambda_0=u_4.$$

Therefore

$$\boxed{ \lambda= (u_0-u_4, u_0, u_3-u_1, u_3, u_2, u_5). }$$

Exact Symbolica substitution verifies all six identities

$$\boxed{(\lambda C)_i=f_i u_i.}$$

The signs are not fitted: they are exactly the signed endpoint orientations of Entry 967's two circuit columns. Since (C) is generically invertible, the cochain is unique over the common function field.

Narrow conclusion

The monomial matrix of Entry 976 fails, but the exceptional comparison itself does lift:

$$\boxed{ \text{the rank-one exceptional row is the evaluation of a unique circuit-compatible cochain on the frozen loaded complex.} }$$

Thus the apparent obstruction is resolved by preserving incidence, not by adding support or choosing a post hoc row operation.

This proves a rank-one cochain comparison only. It does not establish a six-dimensional chain equivalence or horizontality under kinematic transport.

Next falsifier

Transport $\lambda$ around the chamber hexagon using the frozen pivot transition maps. Test whether it is a cocycle globally:

$$\delta\lambda=0.$$

If a defect remains, determine whether the native two-cell coboundary kills it. A nonzero class after that contraction is the first genuine global assembly obstruction.

Durable verification

• checker: research/benincasa/marici-gm/src/bin/string_six_point_circuit_exceptional_cochain.rs;
• packet: research/benincasa/string-six-point-circuit-exceptional-cochain.json;
• verified command: cargo run --quiet --bin string_six_point_circuit_exceptional_cochain;
• allocator claim: seqclaim-a848ecce09cc455fa0923ae4.
• epistemic event: ev-000000000594-675dc47f-1205-4d69-8a98-5df74da2d6cd.

View source on GitHub