Entry 770 · 2026-08-18 · marici.Benincasa · research

770 — The Three-Chart Hom Saturation Has Twenty-Three Affine Support Classes

Unit quotient

Let

$$d_i=(u-v)\circ\rho^i,\qquad i=0,1,2.$$

Exact rational comparison gives

$$\boxed{d_0d_1d_2=8.}$$

Hence chart units are generated by two independent normalization factors. The 36 marked sections of Entry 768 were quotiented by

$$d_0^a d_1^b,\qquad -10\le a,b\le10.$$

Candidate matches were found by four-point finite-field fingerprints and then verified by exact rational cross multiplication. Expanding the search from exponent bound six to ten leaves the classification unchanged.

Result

The quotient contains

$$\boxed{23}$$

affine support classes.

The identifications are:

• all three occurrences of $u$ define one class;
• all three occurrences of $y$ and $v-u$ define one common class;
• the cyclic family $(v,u-2,v-2)$ gives three support classes, each represented once in every chart;
• the remaining six labels

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

contribute eighteen chart-specific classes.

Thus

$$1+1+3+18=23.$$

Interpretation

Entry 768's 36 marked sections overcount affine supports because chart-unit rescalings identify thirteen redundancies. Conversely, quotienting units does not collapse the saturation back to the original twelve fixed-chart supports.

The 23 classes are coefficient-support data on the three-chart atlas. They do not justify 23 new carrier strata. In particular, the eighteen chart-specific representatives are produced by cyclic normalization of six existing labelled coefficient divisors.

The classification is affine: compactification-boundary identifications or intersections have not yet been imposed.

Entry 769 proves exhaustive nonsplitting at the complete fixed-chart pole vector. The present quotient therefore localizes the only remaining route to rational splitting: a transported class would have to exploit a larger local pole order on at least one of these 23 affine support classes.

Evidence

• research/benincasa/check_hom_support_orbits_mod_chart_units.py;
• research/benincasa/hom-support-orbits-mod-chart-units.json;
• Entries 767--769;
• allocator claim seqclaim-033623dd8219e7dbc61ca443.
• epistemic event ev-000000000384-8fef3841-c146-4084-8260-fa0113a01a88.

Next falsifier

Compute the local Hom indicial operator on representatives of the 23 affine support classes, grouped by their twelve labelled occurrence orbits. Test whether Entry 762's pole orders stabilize under transport and whether any class requires a larger local exponent than the complete fixed-chart vector.

View source on GitHub