Entry 766 · 2026-08-18 · marici.Benincasa · research

766 — The Complete Hom Defect Has a Cyclic Orbit

Frozen inputs

Use Entry 764's independently reduced raw residue connections and cyclic gauge

$$D_i=\operatorname{diag}(z_i^{-2},z_i^{-1},z_i,z_i), \qquad D_0D_1D_2=1.$$

Use Entries 762--765's complete Hom data:

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

in the ordered twelve-factor lattice

$$\begin{aligned} (&u,v,y,1-y,1+y,v-u,y-u^2,y+u^2,\\ &P_6,u-2,v-2,u^2+1), \end{aligned}$$

and the infinity target-column shear

$$(w_0,w_1)=(0,6).$$

Transported Gysin frame

Let $P_i$ be the adapted $T\oplus E$ frame in chart $i$. The chart filtration must not be independently reset after each occurrence move. Instead define it recursively by

$$\boxed{P_{i+1}=P_iD_i.}$$

Since the raw frames obey $e_i=D_i e_{i+1}$, the adapted transition is

$$P_iD_iP_{i+1}^{-1}=1.$$

Exact tests at 24 generic points give zero adapted-transition failures and zero threefold frame failures. Thus in the transported adapted frame,

$$A_{T,i+1}=A_{T,i},\qquad A_{E,i+1}=A_{E,i},\qquad C_{i+1}=C_i.$$

Equivalently, in arbitrary block gauges the cocycle transforms by

$$C_{i+1}=S_{E,i}^{-1}C_iS_{T,i}.$$

Complete pole lattice

The twelve factors are transported by chartwise pullback under the cyclic base map. Their exponent vector is retained positionally; the factors are not identified with the same fixed polynomials in every normalized chart.

Their degrees are

$$(1,1,1,1,1,1,2,2,4,1,1,2),$$

so the complete denominator degree remains $18$. The labelled factor family and all twelve valuations return after three transports. The exact audit gives zero threefold pole-lattice failures at 24 generic points.

Rank-one defect orbit

Transporting the complete filtered splitting complex carries both its coefficient matrix and augmented cocycle column by an invertible chain isomorphism. Therefore Entry 763's one-dimensional augmented-rank cokernel has a canonical occurrence orbit:

$$\boxed{ [C_{12}]\longmapsto[C_{23}]\longmapsto[C_{31}] \longmapsto[C_{12}]. }$$

In the recursively transported frame the three representatives are literally identical. No solver section or fitted projector is used.

This is covariance of the filtered rank-one defect. It does not upgrade Entry 763 to absolute rational nonsplitting; Entry 765's degree-thirty test and local pole stabilization remain separate obligations.

Infinity shear

The degree-six shear is a filtration on the transported (T)-lattice, not a fixed pair of ordinary-coordinate column degrees in every chart. Its rule is

$$\boxed{ \mathcal F_{i+1}=D_i^{-1}\mathcal F_i }$$

in raw residue coordinates, or simply $(0,6)$ in the recursively transported adapted frame. After three steps,

$$\mathcal F_3=(D_0D_1D_2)^{-1}\mathcal F_0=\mathcal F_0.$$

Thus the degree-six shear is occurrence-covariant but is not a fixed-$G_{12}$ filtration imposed separately on the other charts.

Evidence

• research/benincasa/check_cyclic_hom_defect_transport.py;
• research/benincasa/cyclic-hom-defect-transport.json;
• Entries 762--765;
• allocator claim seqclaim-f6d5cc02de21eef39d37f49c.
• epistemic event ev-000000000380-a169898d-b5e4-4fb3-8c25-5d5947e1ed5d.

Next falsifier

Repeat Entry 765's certified degree-thirty rank tests in one transported chart and verify the four resonant graded pieces (15,17,28,30) are carried isomorphically around the cyclic orbit. Then determine local pole-order stabilization along one representative of each cyclic divisor orbit.

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