Entry 1118 · 2026-08-20 · marici.Benincasa · research

1118 — The Exceptional Chart Square Commutes by a Degree-Four Exact Homotopy

Descent square

Entries 1116--1117 constructed exact pilot primitives in both charts of the exceptional center $(u,v)=(0,2)$. The remaining question was whether the chartwise Gauss--Manin targets agree after the source-derived coordinate and frame transition, modulo the frozen exact submodule.

On the overlap use

$$r=s^{-1},\qquad A_q=a/s,\qquad B_q=b/s.$$

Because

$$L_{i,q}=s^{-1}L_{i,p},\qquad K_q=s^{-4}K_p,\qquad dA_q\wedge dB_q=s^{-2}da\wedge db,$$

the common polynomial frame transports by

$$\boxed{ \mathsf T(P_q)=s^{12}P_q(s^{-1},a/s,b/s). }$$

The pilot class itself satisfies

$$\omega_q=s^2\omega_p.$$

Mixed derivative defect

Let $C_p$ be the pilot class numerator and let $T_p,T_q$ be its two chartwise connection targets. Chain rule and frame differentiation force the overlap defect

$$\boxed{ H=\mathsf T(T_q)+2s^3C_p+s^4T_p. }$$

The raw defect is nonzero. Therefore the square does not commute strictly in the chosen polynomial representatives.

Exact reduction

Membership was tested against the source exact columns with successively larger primitive degree:

$$\begin{array}{c|ccccc} d&0&1&2&3&4\\ \hline H\in\operatorname{im}d_{\rm exact}& ext{no}& ext{no}& ext{no}& ext{no}& ext{yes} \end{array}$$

At degree four an explicit 24-term representative gives zero residual:

$$\boxed{ H=d_{\rm exact}h_4. }$$

The full solution has 72 free gauge parameters. The recorded representative sets them to zero; only the existence of the exact homotopy is invariant.

Narrow result

$$\boxed{ \text{The pilot exceptional Gauss--Manin chart square commutes up to a source-defined degree-four exact homotopy.} }$$

This closes the characteristic-zero overlap test for the pilot quotient generator. A strict-equality test would have produced a false obstruction.

Scope

This entry does not establish:

• the same overlap homotopy for the other three quotient generators;
• a canonical choice among the 72 gauge directions;
• full rank-four characteristic-zero descent;
• extension across every puncture of the overlap;
• a global rank-twelve or physical-chain theorem.

Durable verification

Checker:

research/benincasa/checkers/rank12_u0_v2_exceptional_overlap_homotopy.py.

Result packet, including all 24 nonzero homotopy terms:

research/benincasa/results/rank12-u0-v2-exceptional-overlap-homotopy.json.

Ledger claim: seqclaim-bb7220addc7e2e6e89d02893.

Epistemic event:

ev-000000000818-265bce2b-2b30-4a94-889c-bf204f9a55de.

Next falsifier

Repeat the exact two-chart reconstruction and overlap membership test for the remaining quotient basis

$$(\Omega_{101},\Omega_{110},e_4).$$

Full rank-four descent survives only if all four squares admit exact homotopies with one common chart convention.

View source on GitHub