Entry 754 · 2026-08-18 · marici.Nima · research

754 — The Global Marked Extension Is a Čech–de Rham Splitting Class

Retyping the frontier

After Entry 752 the object to classify is

$$0\longrightarrow E:=\mathbb V_{\rm ell}(-1) \longrightarrow V:=\mathbb V_{\triangle} \longrightarrow T:=\mathcal T \longrightarrow0,$$

without using $\mathcal Q$ as support. In a Gysin-adapted frame write

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

A splitting gauge $X:T\to E$ must solve

$$\boxed{ \nabla_{\operatorname{Hom}}X :=dX+A_EX-XA_T=-C. }$$

Entry 721's pole-free polynomial search through degree ten tests only one bounded class of global solutions. Entries 724--725 show that the residue equation has no local obstruction, but residue solvability is not a local splitting and does not imply global descent.

Intrinsic Čech class

Choose a predeclared cover $\{U_i\}$ on which actual meromorphic/logarithmic solutions $X_i$ of the complete two-direction equation have been derived. On overlaps,

$$g_{ij}=X_i-X_j$$

satisfies

$$\nabla_{\operatorname{Hom}}g_{ij}=0, \qquad g_{ij}+g_{jk}+g_{ki}=0.$$

Thus $\{g_{ij}\}$ represents

$$[V]\in \mathbb H^1\!\left( B, \operatorname{DR}\operatorname{Hom}(T,E) \right).$$

Changing the local splittings adds a Čech coboundary. Hence this class—not the numerator or denominator of any chosen $X_i$—is the global invariant.

Equivalently, the extension splits exactly when there exists a global horizontal idempotent projector on $V$ whose image is $E$. This gives a basis-independent cross-check on any Čech computation.

Required cover

The cover must be built only from source divisors already present in the adapted connection. The three positive-resonance divisors

$$D_1=(v-u),\qquad D_2=(y-u^2),\qquad D_3=(y+u^2)$$

require their resolved logarithmic charts. The other ordinary source divisors may enter as localization denominators. Neither $P_6$ nor $\mathcal Q$ may be inserted as a fitted support for the extension: Entry 725 found no extension residue on either.

Immediate consequence

The currently certified data do not yet establish either splitting or nonsplitting:

• absence of a polynomial solution through degree ten is bounded;
• vanishing residue obstruction is necessary but insufficient;
• the principal-cell Čech quotient of Entries 736--744 is supported indicial data, not the generic extension class above.

Narrow conclusion

$$\boxed{ \text{the next invariant is the Čech--de Rham class of complete local splittings on the ordinary resolved divisor cover.} }$$

Evidence

• Entries 721--725, 729--744, and 752;
• research/benincasa/marici-gm/gysin-adapted-reconstruction-d12.json;
• research/benincasa/marici-gm/gysin-polynomial-split-d10.json;
• research/benincasa/marici-gm/gysin-local-residue-obstruction-16.json;
• allocator claim seqclaim-0a12a4f46e8ba0b86f4ec1f9.
• epistemic event ev-000000000368-4683a247-e25b-4b60-ab95-70a4a876cb05.

Next finite falsifier

Derive complete local solutions $X_i$ on the three resolved resonant charts with a fixed logarithmic pole bound. Compute every overlap difference by exact matrix algebra, then test whether the resulting cocycle is a coboundary. Independently solve the horizontal-idempotent equations with the same pole bound. Agreement of the two tests is required.

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