The Mendez–Pinto–Villarreal conjecture in unmixed height three

Henry Zweiman, September 9, 2026. Proposed research preprint, seven pages.

For a monomial ideal whose minimal primary components are irreducible and whose associated primes all have height three, the manuscript proves that equality of every symbolic and ordinary power forces a single weight for each variable across its primary components. The ideal is therefore a standard weighting of its radical, and the radical is Simis. Exponents, number of variables, and generator support sizes are unrestricted.

This is a proposed resolution of the unmixed height-three case of Méndez–Vaz Pinto–Villarreal Conjecture 5.7. It does not resolve the full conjecture, mixed heights, or the classification of squarefree Simis ideals. The assessment identifies prior results and the scoped contribution. The review records the proof audit and checks.

Run the self-contained exact supplementary checks with Python 3:

python3 check.py

The checks use only the standard library and are not a replacement for the arbitrary-exponent proof. The manuscript and computations were prepared with OpenAI Codex and audited by the originating agent. No human or separate-agent verification is claimed. Novelty, significance, and journal readiness remain provisional. This is one manuscript.