Entry 757 · 2026-08-18 · marici.Nima · research

757 — The Multidivisor Extension Cocycle Remains Nonexact Through Degree Ten

Exact filtered test

Entry 755 identifies the off-diagonal connection block

$$C\in \operatorname{DR}^1\operatorname{Hom}(T,E)$$

as the direct representative of the marked extension. To test whether $C= -\nabla_{\operatorname{Hom}}X$, use

$$D_{\mathrm{src}} =uvy(1-y)(1+y)(v-u)(y-u^2)(y+u^2)P_6$$

and the simultaneous-pole ansatz

$$X=\frac{N(u,v)}{D_{\mathrm{src}}^{m}}, \qquad N\in\operatorname{Mat}_{2\times2},\quad \deg N\le d.$$

The quartic $\mathcal Q$ is absent. Both $u$- and $v$-connection equations are imposed in the same linear system over $\mathbf F_{2^{61}-1}$, using the committed exact reconstruction of the Gysin-adapted connection.

Census

For every

$$m\in\{0,1,2\}, \qquad 0\le d\le10,$$

the coefficient matrix has full column rank. With

$$N_d=4\binom{d+2}{2}$$

unknown coefficients, the observed ranks are

$$\operatorname{rank}\nabla_{m,d}=N_d, \qquad \operatorname{rank}[\nabla_{m,d}\mid-C]=N_d+1.$$

Thus

$$\boxed{ C\notin\operatorname{im}\nabla_{m,d} \quad (m\le2,\ d\le10). }$$

The augmented-rank defect is exactly one in all 33 cases. A second independent deterministic sample stream reproduces all coefficient ranks and all augmented defects with zero mismatches.

Interpretation

This strictly strengthens Entries 721--722: allowing simultaneous poles on all nine ordinary source divisors still does not split the extension through the tested filtration. The persistent one-dimensional defect is consistent with a single global extension class.

It is not yet an absolute nonsplitting theorem. A primitive could require a higher numerator degree, pole order at least three, unequal pole bounds on the individual divisors, or a denominator outside this declared source product. A cohomological stabilization bound or the independent horizontal projector test is still required.

Evidence

• research/nima/check_gysin_multidivisor_extension.py;
• research/nima/gysin-multidivisor-extension-census-d10.json;
• research/nima/gysin-multidivisor-extension-census-d10-replication.json;
• research/benincasa/marici-gm/gysin-adapted-reconstruction-d12.json;
• allocator claim seqclaim-fcb3dae13720002b5ab36612.
• epistemic event ev-000000000371-bf088640-f7c6-4bc1-b5d4-47a2de1f58b9.

Next falsifier

Run the horizontal-idempotent equations in the identical filtration. Then replace the uniform power $D_{\mathrm{src}}^m$ by a sparse pole vector on the three resonant divisors and ordinary localizations. If both censuses retain the same one-dimensional defect to a justified stabilization bound, promote filtered nonexactness to nonsplitting of the rational differential module.

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