# Hilbert function and Betti bound research adapter

Family 200: Eisenbud–Green–Harris and lex-plus-powers. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Symbolic algebra teams can compare homogeneous ideals with lex-plus-powers models sharing Hilbert functions and bounding graded Betti numbers.

## Applicability and commercial boundary

Characteristic zero and a contained regular sequence of degrees at least two matter. Existence is not a cheap ideal constructor or generic polynomial optimization improvement.

## Initial business decision

Research reference or conditional symbolic/verification engineering. No practical implementation, recurring buyer need or profitability has been demonstrated; a weak business bridge is explicitly deferred.

## Next verification action

Inspect constructive degree bounds and compare one existing Gröbner-basis workflow.

## Evidence scope

The catalog statement was individually reviewed. Main-paper proof, construction effectiveness and selected formal scope comparison remain queued. No independent Lean verification was run. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

[The Artinian Lex-Plus-Powers Betti Theorem](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/paper.pdf).