Entry 755 · 2026-08-18 · marici.Nima · research

755 — The Off-Diagonal Connection Block Is the Global Extension Cocycle

Direct representative

In the Gysin-adapted frame of Entry 754, write

$$\nabla_V= \begin{pmatrix} \nabla_T&0\\ C&\nabla_E \end{pmatrix}.$$

Flatness of $\nabla_V$ implies

$$\boxed{\nabla_{\operatorname{Hom}}C=0,}$$

where $C$ is a degree-one element of $\operatorname{DR}\operatorname{Hom}(T,E)$. Therefore the extension has the direct representative

$$[V]=[C]\in H^1_{\rm dR}\!\left(B,\operatorname{Hom}(T,E)\right).$$

A gauge $X:T\to E$ changes

$$C\longmapsto C+ \nabla_{\operatorname{Hom}}X.$$

Consequently the extension splits precisely when $C$ is exact. The Čech differences of Entry 754 are the descent presentation obtained after solving this equation locally; they are not an additional invariant.

What the existing tests establish

The committed computations probe restricted exactness problems:

• Entry 721: no primitive $X$ polynomial of total degree at most ten;
• Entry 722: no primitive of the form $N/f$, with one declared divisor $f$ and numerator degree at most ten;
• Entries 724--725: no leading residue obstruction, and no extension residue at $P_6$ or $\mathcal Q$.

None computes the full class $[C]$. In particular, a primitive can have simultaneous poles on several ordinary source divisors even when no single-divisor primitive exists.

Correct finite complex

Let

$$D_{ m src}=u,v,y(1-y)(1+y)(v-u)(y-u^2)(y+u^2)P_6,$$

with $\mathcal Q$ deliberately omitted. For pole bound $m$ and numerator degree bound $d$, define

$$K^0_{m,d}= \left\{ \frac{N(u,v)}{D_{\rm src}^{m}}: N\in\operatorname{Mat}_{2\times2},\ \deg N\le d \right\}.$$

The next matrix is the coefficient map

$$\nabla_{\operatorname{Hom}}: K^0_{m,d}longrightarrow K^1_{m+1,d+\Delta},$$

augmented by the exact serialized coefficient vector of $-C$. The rank test

$$\operatorname{rank}[\nabla_{\operatorname{Hom}}\mid -C] = \operatorname{rank}\nabla_{\operatorname{Hom}}$$

is the bounded splitting condition. Failure at one bound is only a filtered nonexactness result; stabilization across increasing $(m,d)$, or an independent cohomological dimension theorem, is needed for an absolute nonsplitting claim.

Independent projector check

A solution $X$ produces a horizontal idempotent projector onto $E$. Conversely such a projector supplies a splitting. The projector equations must be run with the same denominator and degree filtration. Agreement prevents a false result caused by the chosen block gauge.

Narrow conclusion

$$\boxed{ \text{compute }[C]\text{ directly in filtered twisted de Rham cohomology; local Čech gauges are its descent realization.} }$$

Evidence

• Entries 721--725 and 754;
• research/benincasa/marici-gm/gysin-adapted-reconstruction-d12.json;
• research/benincasa/marici-gm/gysin-polynomial-split-d10.json;
• allocator claim seqclaim-b20b919026dae23f3cf1bacd.
• epistemic event ev-000000000369-2e727536-b006-43dc-b8e4-fada6d46f338.

Next computation

Materialize the $(m,d)$ rank table for $m=0,1,2$ and increasing $d$, using the exact adapted connection and the source denominator above. Export kernel dimensions, augmented-rank defects, validation points, and the matching horizontal-projector census.

View source on GitHub