Source, novelty, and significance assessment

Audit date: September 9, 2026. Author: Henry Zweiman.

Precisely proposed result

The main theorem proves standard-weight agreement for every monomial Simis ideal with one irreducible primary component per associated prime, all of height three. Its converse is the established invariance of the Simis property under standard weighting. This is the unmixed height-three case of the MPV conjecture. The manuscript does not assert the conjecture in arbitrary height or with mixed component heights.

Primary statements checked

1. Méndez, Vaz Pinto, Villarreal, Symbolic powers: Simis and weighted monomial ideals, arXiv:2402.08833v3. Conjecture 5.7 states the general weighting problem. Introduction definitions specify distinct radicals in a minimal irreducible decomposition. Corollary 5.5(b) proves weighting invariance. These portions, including the corollary proof, were read. The journal metadata is J. Algebra Appl. 24 (2025), 2541001; theorem numbering here refers to the archived arXiv version, not an unread final journal body.
2. Bordoloi, Das, Kumar, The Mendez–Pinto–Villarreal conjecture for some classes of monomial ideals, arXiv:2607.25683v1, July 2026. Conjecture 1.1 still poses the general problem. Theorem 3.10 resolves support-three ideals in that paper's setup; its forest results give another family. Its hypotheses concern generator supports, including the stated support-family condition. Our height-three condition is on the primary components. We do not claim these previously solved generator-support classes as new.
3. Bijender, Ajay Kumar, Waldschmidt constant of monomial ideals and Simis ideals, arXiv:2512.22940v1. Theorems 4.7 and 4.9 treat a component-separation hypothesis and the height-two case, respectively. The relevant statements and proof dependencies were checked. The height-two case is already in Aslam, Mathematical Reports 16(66) (2014), Theorem 2.3, publisher PDF.
4. Branco Correia–Zarzuela, Some asymptotic properties of the Rees powers of a module, arXiv:math/0408351v1, Theorem 6.4, supplies the classical complete-intersection criterion. The rank-one ideal-module hypothesis holds because the ideal has grade at least two in a regular local ring. Section 6 and the criterion's proof were read. Their On equimultiple modules, arXiv:0708.2225v1, Theorem 5.3, gives the underlying equimultiple criterion with a proof that was also read. These classical theorems are inputs, not contributions.
5. Grisalde–Seceleanu–Villarreal, Research in the Mathematical Sciences 9 (2022), article 13, DOI, Section 7, develops irreducible polyhedra and integral-closure comparisons. Theorem 7.6 has integral closure bars, as does the relevant equality in Theorem 7.8. The manuscript uses only a necessary condition, with a direct denominator-clearing proof. It does not mistake those integral-closure results for equality of ordinary and symbolic powers.

The full source PDFs are retained in the local research archive. They are not redistributed in the public manuscript folder. The additional support-two paper is cited with its journal metadata and the exact preprint version that was consulted.

What appears new

The proposed contribution is a uniform primary-exponent rigidity theorem in unmixed height three. Its key steps are the reduction of any discrepancy to five variables, use of a classical dimension-one theorem to force four-variable rigidity, the resulting short support classification, and an explicit all-parameter obstructing vertex for its final configuration. The proof does not assume equal degrees or fixed generator supports.

Targeted searches used the conjecture name together with “height three,” “height 3,” “unmixed,” and “codimension 3,” and searches for normally torsion-free generically complete intersection monomial ideals. No existing theorem resolving this whole height-three class was located. The July 2026 primary source confirms the continuing general problem. The exact height-three subproblem is an identifiable case of that named conjecture, rather than a separately numbered question in the source. Searches and source comparisons do not certify exhaustive priority.

Significance and limits

This is a structural partial solution of an active named conjecture. It advances the primary-height frontier beyond the classical height-two case and gives a local reduction that could support higher-height work. That offers a plausible basis for later use, but citation counts and journal acceptance cannot be predicted. The argument leaves squarefree Simis classification, mixed heights, higher heights, and a possible second-power criterion unresolved.

This is an AI-generated proposed theorem with a complete internal proof audit. Independent mathematical review, historical priority, and scholarly significance remain provisional. It is one distinct public manuscript; its lemmas and cases do not increase the paper count separately.