Integer-polynomial lifting and the classification of algebraic extrapolation sets

Henry Zweiman — September 9, 2026

Full Markdown manuscript · Nine-page PDF · LaTeX source

This manuscript gives an exact arithmetic description of finite extrapolation sets for algebraic-integer parameters. Integer-polynomial lifting supplies the central argument: an integer polynomial residue class has a representative strictly between zero and one on the unit interval precisely when it satisfies the internal-root inequalities and two endpoint congruences.

The consequences include the strong-PV discreteness classification, a model-set description with a finite congruence coordinate, an exact criterion for equality with the full internal cube, and finite generation of that cube. They address the questions in Section 11 of Fenner–Green–Homer, Discrete & Computational Geometry 76 (2026). All of these related consequences form one paper. Finite-generation assertions explicitly exclude the degenerate parameters zero and one.

The source assessment distinguishes the new claim from classical approximation and known special cases. The proof audit records the internal review and external theorem dependencies. This is an internally checked proposed solution, with historical priority and significance subject to scholarly review; it has not received human referee review.

Reproduction

The general arguments are in the manuscript. Optional finite consistency checks use only Python's standard library:

python3 check.py

The saved check results cover Bernstein identities, endpoint arithmetic, published quadratic cases, and bounded complex examples. Finite examples do not prove the general theorems. Compile manuscript.tex with Tectonic or a compatible LaTeX installation. The artifact record documents the inspection of all nine PDF pages and the full Markdown conversion. The manifest provides hashes of the publication files.