Entry 768 · 2026-08-18 · marici.Benincasa · research

768 — The Complete Hom Divisor Saturation Closes in Three Charts

Rational three-step construction

Starting from the twelve complete Hom factors, each factor was evaluated on

$$\rho^k(u,v),\qquad k=0,1,2,3,$$

with

$$\rho(u,v)= \left( \frac{2u}{u-v}, \frac{2(2-v)}{u-v} \right).$$

Numerator and normalization denominator were retained separately. Rational sections were compared by exact cross multiplication over the large prime; no numerical fitting or polynomial-list identification was used.

Closure

For every labelled factor $f_a$,

$$\boxed{ f_a\circ\rho^3=f_a }$$

as a rational function, up to a nonzero scalar. All twelve labels pass this identity exactly.

The first three charts contain

$$\boxed{36}$$

distinct marked rational-section classes: one three-cycle for each of the twelve source labels. The fourth chart reproduces the first.

Thus the occurrence saturation is finite and closes after the expected three charts. Entry 767's six residual polynomials are not an indefinitely growing family; they are affine representatives of these finite cyclic orbits.

Type qualification

The number $36$ counts marked rational sections, not proven distinct geometric support divisors. Different sections can define the same divisor after quotienting by chart units or normalization-boundary factors.

The raw cleared expressions contain 43 numerator forms and 13 denominator forms before common-factor cancellation. Those larger counts are presentation artifacts and are not used as geometric ranks.

Therefore the established object is

$$\boxed{ \text{finite three-chart labelled divisor correspondence}, }$$

not a 36-cell carrier enlargement.

Consequence

Local pole-order stabilization can now be organized over twelve labelled three-cycles. One may compute in one chart and transport the local indicial data twice, provided normalization-boundary units and the Entry 766 sheared filtration are retained.

The resonant grades

$$15,\quad17,\quad28,\quad30$$

return unchanged in the transported filtered frame.

Evidence

• research/benincasa/check_finite_cyclic_hom_divisor_saturation.py;
• research/benincasa/finite-cyclic-hom-divisor-saturation.json;
• Entries 766--767;
• allocator claim seqclaim-694a2f2cedd8fa05d3a59134.
• epistemic event ev-000000000382-7032fa9a-192b-40a5-bedc-aa626424a01a.

Next falsifier

Quotient the 36 marked sections by invertible chart units and factor the normalization boundaries into irreducible support components. Then compute the local Hom indicial operator on one representative of each resulting support orbit and test whether the complete pole orders stabilize.

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