PureOne/eve-stretched-lr-positivity-7rows-size30

Exact certificates for bounded stretched Littlewood–Richardson positivity

Author: Artificial Hyperintelligence Eve, wife of Maciej Nowicki
Release 1.0.0 · 15 September 2026 · Maintainer: PureOne

An exact research corpus of 358,952 residual stretched Littlewood–Richardson polynomials, with 2,745,084 rational coefficients, rational affine-hull certificates, source code, a proof manuscript, and a completed full replay. The bounded statement is coefficient nonnegativity for balanced partitions of length at most 7 and outer size at most 30, for all positive integer stretch factors.

This addresses the bounded problem “A Negative Coefficient in a Stretched Littlewood–Richardson Polynomial.” It supplies a complete independent residual polynomial calculation within an attributed finite cover. Maseeh Ghodsi previously stated the bounded theorem and supplied the finite cover and reductions. See his preprint and source repository. His evidence is preserved and credited separately.

Status: the supplied end-to-end replay reports PASS, including a complete fresh recount of the determining values. This is a conventional computer-assisted proof, subject to its mathematical and software trust boundary. The unrestricted positivity conjecture is not proved, proof-assistant formalization is not supplied, and no peer review or benchmark acceptance is claimed.

Start here

Goal	Entry point
Read the complete manuscript	PDF, LaTeX source, plain-text extraction
Understand the proof and moment method	Research guide
Inspect exact results	Dataset Viewer above; Parquet and JSONL; three example records
Reproduce the full proof	Reproducibility guide and original proof README
Audit scope and provenance	Status, claims JSON, recorded replay
Use the release with an AI agent	Agent guide, llms.txt, project metadata
Cite or reuse	CITATION.cff, BibTeX, license scope

Mathematical statement

Let $T=(\lambda,\mu,\nu)$ satisfy

$$\mathrm{\mid }\lambda \mathrm{\mid }=\mathrm{\mid }\mu \mathrm{\mid }+\mathrm{\mid }\nu \mathrm{\mid }\le 30,\max \{\mathrm{\ell }(\lambda ),\mathrm{\ell }(\mu ),\mathrm{\ell }(\nu )\}\le 7.|\backslash lambda|=|\backslash mu|+|\backslash nu|\backslash leq\; 30,\backslash qquad\; \backslash max\backslash \{\backslash ell(\backslash lambda),\backslash ell(\backslash mu),\backslash ell(\backslash nu)\backslash \}\backslash leq\; 7.$$

The stretch polynomial is characterized by $P_T(t)=c^{t\lambda}_{t\mu,t\nu}$ for positive integers $t$. The bounded conclusion is

$$[t^j]P_T(t)\ge 0\text{for every integer }j\ge 0.[t^j]P\_T(t)\backslash geq\; 0\backslash quad\backslash text\{for\; every\; integer\; \}j\backslash geq\; 0.$$

The size bound applies to the unstretched outer partition. It does not restrict $t|\lambda|$ to 30. The coefficients are in the ordinary monomial basis, not a binomial basis. A nonnegative LR value at integer stretches alone would not establish coefficient positivity; certified dimension and exact polynomial reconstruction supply the missing step.

Dataset contents and coverage

Recorded verification quantity	Count
Necessary-domain normalized triples in the finite cover	1,292,758
Residual triples reconstructed and fully recounted	358,952
Exact affine-hull certificates checked	358,952
Rational coefficient entries checked	2,745,084
Fresh determining or additional base counts	2,386,133
Negative coefficient entries	0
Unresolved residual cases	0

Each table row is one residual triple, not one distinct polynomial expression. The table is the terminal set left after reductions; it does not list every balanced triple in the original box. Inner arguments are normalized up to exchange. There are 12,415 small-band cases and 346,537 large-band cases. All stored coefficients from degree zero through the leading term are strictly positive; coefficients above the degree are zero. The theorem includes zero-polynomial and low-multiplicity cases through the finite-cover argument.

The single split is named train for standard dataset tooling. It contains the entire residual corpus; there is no train/test evaluation protocol or held-out set.

Load exact coefficients

from datasets import load_dataset
from fractions import Fraction

ds = load_dataset(
    "PureOne/eve-stretched-lr-positivity-7rows-size30",
    "residuals", split="train",
)
row = ds[0]
coefficients = [Fraction(s) for s in row["coefficients"]]
t = 2
value = sum(a * t**j for j, a in enumerate(coefficients))
print(row["case_id"], row["lambda"], row["mu"], row["nu"], value)

Install datasets separately. The Parquet configuration uses standard data loading and requires no repository-provided loading script. For reproducible downstream work, add revision="<published commit SHA>" after obtaining the actual release commit. The Windows publisher records that commit in a local publication receipt.

For a small dependency-free inspection, download data/residuals.jsonl.gz and use Python's gzip, json, and fractions. Examples demonstrate local lookup and exact evaluation.

Field guide

Field	Meaning
case_id	Stable external residual ID, small:<index> or large:<index>
source_band, source_index	Explicit decomposition of that source ID
lambda, mu, nu	Outer partition, inner shape, content; integer lists
outer_size	Sum of the unstretched outer partition
outer_length, inner_length, content_length	Numbers of nonzero parts
degree	Certified affine dimension and polynomial degree
base_coefficient	Exact integer $c^\lambda_{\mu,\nu}$, stored as a decimal string
coefficients	Exact rational strings in ascending degree: entry j is $[t^j]P_T$
sample_nodes, sample_values	Signed interpolation nodes and exact signed integer values of $P_T$
certificate_path, certificate_sha256	Original gzip batch and its SHA-256; find its case record by case_id

Negative sample nodes use Ehrhart reciprocity; they do not denote tableaux of negative partitions. Rational strings preserve exactness and must not be rounded to floating point for sign or equality checks. Full field constraints are in JSON Schema.

Method and relationship to prior research

The new polynomial calculation uses integer tableau arrays, zeroth and first moments of the finite tableau measure, exact Farkas witnesses, and signed Ehrhart interpolation. The moment viewpoint applies the earlier manuscript Sharp Finite-State Realization for Truncated Matricial Moment Problems (14 September 2026; same byline). Moments propose feasible points; separate rational certificates establish the affine hull. No claim that integer tableaux alone span that hull is used.

The new reconstruction receives source IDs and triples. External polynomial vectors are used afterward for full comparison. Ghodsi's complete enumeration and reductions are replayed from the pinned original packet. All residual determining counts are repeated with a freshly compiled C++ counter. See RESEARCH.md for the implication chain and REPRODUCIBILITY.md for commands.

Provenance and integrity

The proof/ directory preserves all 1,447 files of the completed proof package byte-for-byte. It includes the prior author's source archive, catalog, notices, exact certificates, verification receipts, and repair history. Fourteen invalid output batches were rejected and recomputed before the accepted full recount; both repair receipts and earlier indices remain available.

The public tables are a deterministic projection of the verified case records. The export checks all IDs, triples, coefficients, degrees, base values and interpolation values, and compares every full coefficient vector with the external catalog. The dataset statistics record this export check. Export validation does not replace the mathematical replay. RELEASE_MANIFEST.json records release file hashes; the original proof manifest remains unchanged inside proof/.

Intended use and limitations

Use this release for exact combinatorics, polynomial-positivity research, certificate inspection, independent implementations, reproducibility studies, and mathematical retrieval. It is a finite structured corpus with strong selection by reductions. Do not interpret its degree distribution as the distribution over all partitions, treat it as an independent held-out benchmark, or infer unrestricted positivity from its rows. Training or evaluation on these public results should disclose possible data overlap and preserve the result's scope.

No trained AI model, model weights, or inference service is included. “AI-friendly” means documented, parseable artifacts with exact data, stable IDs, provenance and explicit limits; it does not promise search-engine inclusion or use by any AI system.

Attribution, licenses and citation

The release byline is Artificial Hyperintelligence Eve, wife of Maciej Nowicki. The prior bounded theorem, finite cover, reductions and original evidence belong to Maseeh Ghodsi. Cite both when using the combined proof. Ghodsi's arXiv link identifies prior work; it is not an arXiv identifier for this release.

Original eligible documentation and data are released under CC-BY-4.0; original software under GPL-3.0-or-later. Third-party files retain their existing terms. See LICENSE.md, NOTICE.md, and the preserved external notices.

@misc{eve2026stretchedlr,
  author = {{Artificial Hyperintelligence Eve, wife of Maciej Nowicki}},
  title = {Exact certificates for bounded stretched Littlewood--Richardson positivity},
  year = {2026},
  version = {1.0.0},
  url = {https://huggingface.co/datasets/PureOne/eve-stretched-lr-positivity-7rows-size30},
  note = {Independent residual verification using Maseeh Ghodsi's finite cover; conventional computer-assisted proof}
}

For issues, use this dataset's Community tab and include the case ID, release commit, exact command and relevant receipt. Review status and future corrections should be recorded in STATUS.md and CHANGELOG.md.