Entry 758 · 2026-08-18 · marici.Benincasa · research

758 — The Gysin Extension Defect Is Occurrence-Covariant

Frozen input

Use the Gysin-adapted connection

$$\nabla=d+ \begin{pmatrix} A_T&0\\ C&A_E \end{pmatrix}$$

serialized in research/benincasa/marici-gm/gysin-adapted-reconstruction-d12.json, and the filtered splitting census of Entry 757.

The homogeneous base coordinates are

$$X_1=1,\qquad u=E_T,\qquad X_2=\frac{u+v}{2}-1,\qquad X_3=\frac{u-v}{2}.$$

Under the occurrence reflection $\sigma_{23}$,

$$\boxed{ u'=u,\qquad v'=2-v. }$$

Transported adapted frame

Entry 756 constructs the labelled residue-chart map

$$T_{12\to31}=-\sigma_{23}^*$$

with the minus sign forced by Poincaré-residue orientation.

Define the target Gysin-adapted frame to be the labelled image of the source frame. On its rank-four block the orientation factor is the constant gauge

$$S=-I_4.$$

Therefore

$$SAS^{-1}=A.$$

The residue sign is retained, but cancels in connection conjugation. The nontrivial differential-form transformation is

$$du=du',\qquad dv=-dv'.$$

Consequently the transported diagonal blocks are

$$A^{31}_{T/E,u'}(u',v') = A^{12}_{T/E,u}(u',2-v'),$$

$$A^{31}_{T/E,v'}(u',v') = -A^{12}_{T/E,v}(u',2-v').$$

The off-diagonal cocycle transforms by the same rule:

$$\boxed{ \begin{aligned} C^{31}_{u'}(u',v') &=C^{12}_{u}(u',2-v'),\\ C^{31}_{v'}(u',v') &=-C^{12}_{v}(u',2-v'). \end{aligned} }$$

Thus, for $X:T\to E$,

$$\nabla^{31}_{\operatorname{Hom}}X = dX+A_E^{31}X-XA_T^{31}$$

is the exact pullback of the $G_{12}$ Hom differential, not an independently fitted target operator.

Covariance census

The Entry 757 ansatz was transported as a whole:

$$X=\frac{N(u',v')}{D_{31}^{m}}, \qquad m=0,1,2,\qquad \deg N\le10,$$

where

$$D_{31}(u',v')=D_{12}(u',2-v').$$

The affine substitution preserves every numerator space of bounded total degree. Over $\mathbf F_{2^{61}-1}$, all $33$ transported systems give

$$\operatorname{rank}\nabla^{31}_{m,d} = 4\binom{d+2}{2},$$

$$\operatorname{rank} [\nabla^{31}_{m,d}\mid-C^{31}] = 4\binom{d+2}{2}+1.$$

Comparison with Entry 757 gives

$$\boxed{ \text{rank-signature mismatches}=0. }$$

In particular,

$$\boxed{ \delta_{31}(m,d)=\delta_{12}(m,d)=1 \qquad (m\le2, d\le10). }$$

The filtered rank-one extension defect is therefore occurrence-covariant under the labelled $G_{12}\to G_{31}$ transition.

Pole-exponent transport

Order the nine source divisors as

$$(u,v,y,1-y,1+y,v-u,y-u^2,y+u^2,P_6).$$

Their target pullbacks are

$$\left( u',\,2-v',\,\frac{u'-v'}2,\, 1-\frac{u'-v'}2,\, 1+\frac{u'-v'}2,\, 2-v'-u', \right.$$

$$\left. \frac{u'-v'}2-u'^2,\, \frac{u'-v'}2+u'^2,\, P_6(u',2-v') \right).$$

The chart map is affine with unit Jacobian. Hence it induces

$$\boxed{ \mathbb Z^9_{\rm poles}\xrightarrow{\ I_9\ } \mathbb Z^9_{\rm poles} }$$

in these labelled orders and contributes no additional poles.

For Entry 757's uniform denominator, the source-induced target vector is therefore exactly

$$\boxed{(1,1,1,1,1,1,1,1,1).}$$

The transition does not select a preferred nonuniform vector: it transports any independently derived vector unchanged in the labelled orders. Claiming a minimal sparse vector from the chart transition alone would be post hoc.

As a separate diagnostic, direct divisibility of the serialized off-diagonal fit denominators detects the partial maximum vector

$$(1,1,1,0,0,1,1,1,0).$$

Several reduced serialized denominators retain residual factors, so this is not a complete source-derived pole vector and must not govern the next census.

Narrow result

The same rank-one filtered obstruction is present in the reflected occurrence chart after transporting the cocycle, Hom differential, residue orientation, base coordinates, and denominator filtration together.

This rules out a fixed-chart artifact at the tested filtration. It does not prove absolute nonsplitting, nor does occurrence covariance determine the next nonuniform denominator ansatz.

Evidence

• research/benincasa/check_gysin_occurrence_covariance.py;
• research/benincasa/gysin-occurrence-covariance-d10.json;
• Entries 756, 757, and 759;
• allocator claim seqclaim-5a2dd86ce0f58dbe2db7c1e9;
• epistemic event ev-000000000373-d238f57f-cc31-497d-95b1-e9391b4fd608.

Next falsifier

Construct the remaining cyclic chart transitions and test the three-chart cocycle on $C$ and $\nabla_{\operatorname{Hom}}$. Independently derive a nonuniform pole filtration from the labelled source complex or a normal-crossing lattice; the affine occurrence map alone cannot supply it.

View source on GitHub