Entry 767 · 2026-08-18 · marici.Benincasa · research

767 — The Complete Hom Divisors Form a Typed Cyclic Correspondence

Exact pullback audit

For the cyclic normalized base map

$$U=\frac{2u}{u-v},\qquad V=\frac{2(2-v)}{u-v}, \qquad d=u-v,$$

each of Entry 762's twelve target-local factors was pulled back exactly. After clearing the forced power of (d), the numerator was divided by the complete fixed-(G_{12}) factor family.

Five finite factors map inside the existing list:

$$u\mapsto u,\qquad v\mapsto v-2,\qquad v-u\mapsto y,$$

$$u-2\mapsto v,\qquad v-2\mapsto u-2,$$

up to units and powers of the normalization denominator. The target-local factor (y) pulls back to a unit divided by (d), so its divisor is carried to the chart boundary rather than to a finite fixed-chart divisor.

Fixed-polynomial closure fails

The remaining six target-local factors acquire nonunit residual polynomials not present in the fixed twelve-factor list:

$$1-y,quad 1+y,quad y-u^2,quad y+u^2,quad P_6,quad u^2+1.$$

Consequently,

$$\boxed{ \rho^*\mathcal D_{12}^{\rm fixed} \not\subseteq \mathcal D_{12}^{\rm fixed}+\langle d\rangle. }$$

This is not a defect in Entry 766. It proves that its chartwise transported divisor lattice is necessary: the same twelve abstract labels return after three charts, but their normalized polynomial representatives are not a single invariant list on the (G_{12}) affine chart.

Consequence for stabilization

A local pole-order stabilization proof performed only on the twelve fixed-(G_{12}) polynomials is not occurrence-global. It must either:

1. be transported chart by chart through Entry 766's correspondence; or
2. be repeated on the cyclic saturation containing the six residual representatives and the normalization boundary.

No new physical carrier divisor is inferred. These are coefficient-gauge and chart-boundary representatives produced by occurrence transport.

The sheared resonant grades

$$15,quad17,quad28,quad30$$

remain unchanged in the recursively transported filtration. They should not be attached to an invariant fixed-polynomial denominator list.

Evidence

• research/benincasa/check_cyclic_hom_divisor_orbits.py;
• research/benincasa/cyclic-hom-divisor-orbits.json;
• Entries 762, 765, and 766;
• allocator claim seqclaim-9c4df307facb69a73540ee0d.
• epistemic event ev-000000000381-f76d35bf-ac58-498e-bc27-4b8b12d9f488.

Next falsifier

Construct the finite cyclic saturation of the twelve divisor labels over all three charts, including the normalization boundary. Determine its divisor orbits and test local indicial pole stabilization on one representative of each orbit. If the saturation does not close after three labelled charts, the present transported pole filtration is incomplete.

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