A complete classification of a primitive cubic family

Henry Zweiman · September 9, 2026

Complete Markdown paper · Seven-page PDF · LaTeX source

For every prime power q, this paper proposes that a primitive polynomial X^3+X^2+X+lambda over F_(q^2), with lambda primitive, exists exactly when the characteristic is not three. This completely determines the surviving cases of Awasthi--Sharma Conjecture 9.1. The already published q=3 counterexample is credited. Counting estimates distinguish odd and even characteristic.

The proof uses an explicit Kummer parametrization, the Fu--Wan incomplete-character-sum theorem with its geometric hypotheses checked, and an exact sieve. The infinite argument covers q>=1511. Explicit certificates cover all 261 remaining prime powers outside characteristic three.

Run python3 check.py to verify the witnesses, cubic irreducibility and exact root orders, complete finite coverage, all rational sieve inequalities, and the base case of the infinite-tail induction. The verifier uses only the Python standard library. Stored results and the optional C++ witness generator are provided. To regenerate witnesses, compile the generator with c++ -std=c++17 -O2 search.cpp -o search and run ./search > regenerated-witnesses.json.

The source assessment and proof review explain the established dependencies, prior counterexample, and scope. Artifact QA records seven inspected PDF pages and complete Markdown checks; the manifest records SHA-256 hashes.

This is one AI-generated, internally checked proposed manuscript. No human expert review, separate-agent review, journal acceptance, or exhaustive priority certification is claimed. It does not settle the higher-degree subfield-coefficient problem or automatically certify a significant solution toward the 45-paper goal.