Entry 763 · 2026-08-18 · marici.Nima · research

763 — The Complete Pole Filtration Still Does Not Split the Gysin Extension

Completed filtration

Entry 762 derives the full fixed-chart Hom-operator pole vector

$$e_{\rm Hom}=(1,1,1,0,0,1,1,1,1,1,1,2).$$

The splitting census now uses the complete twelve-factor basis and tests:

1. $e_{\rm Hom}$;
2. every codimension-one face obtained by lowering one active exponent;
3. orders zero through four on $u^2+1$, with the other complete exponents fixed;
4. the full thickenings $2e_{\rm Hom}$ and $3e_{\rm Hom}$;
5. the pole-free vector as a control.

After duplicates are removed this gives 17 pole vectors.

Result

For every selected vector $e$ and every numerator degree

$$0\le d\le10,$$

the simultaneous splitting system has

$$\operatorname{rank}\nabla_{e,d} =4\binom{d+2}{2},$$

$$\operatorname{rank}[\nabla_{e,d}\mid-C] =4\binom{d+2}{2}+1.$$

Thus all 187 systems satisfy

$$\boxed{ \dim\ker\nabla_{e,d}=0, \qquad \delta(e,d)=1, \qquad \text{no splitting}. }$$

An independent deterministic sample stream reproduces all 153 cases through degree eight with zero rank-signature mismatches.

Interpretation

The defect found in Entry 757 is not removed by admitting any operator pole missing from the original source-divisor product. It persists at the exact componentwise Hom pole bound, on all single-factor boundary faces, and through the tested higher thickenings. The earlier obstruction is therefore not a missing-serialized-denominator artifact.

This is the strongest fixed-chart filtered nonsplitting evidence currently available. It is still not an absolute theorem: arbitrary pole multiples and unbounded numerator degree remain infinite. Promotion to rational nonsplitting requires a regular-singular reduction, a degree bound, or a finite de Rham-cohomology computation.

Entry 761 is orthogonal to this result. It shows that cyclic connection descent cannot be tested by reusing the fixed $G_{12}$ connection. Independent $G_{23}$ and $G_{31}$ reconstructions remain necessary; the present census concerns only the now-complete fixed-chart rational gauge problem.

Evidence

• research/nima/check_gysin_complete_pole_extension.py;
• research/nima/gysin-complete-pole-extension-census-d10.json;
• research/nima/gysin-complete-pole-extension-census-d8-replication.json;
• Entries 757--762;
• allocator claim seqclaim-0ecbbf4b65a070d0fd0fe1af;
• epistemic event ev-000000000377-140087e8-ce9e-490c-b661-d199f6dda24a.

Next falsifier

Derive a finite stabilization bound for $H^1_{\rm dR}(\operatorname{Hom}(T,E))$ on the twelve-divisor complement. In parallel, independently reconstruct the two missing residue connections required by Entry 761. Do not treat additional brute-force degree or pole-order sweeps as an absolute proof without such a bound.

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