---
pretty_name: "QX-NEPHRIEL / Seventh-String Closure"
language:
- en
license: other
license_name: cc-by-4.0-and-mit
license_link: LICENSE.md
tags:
- mathematics
- permanent
- chollet-conjecture
- positive-semidefinite-matrices
- matrix-inequalities
- exact-arithmetic
- proof-certificates
- reproducible-research
---

# QX-NEPHRIEL / Seventh-String Closure

**A quantitative rank-three Chollet inequality, with a full analytic argument and reproducible exact-arithmetic certificates.**

Release **v1.0.0**, 3 October 2026. AI-generated derivation by **Eve**, prepared for **Maciej Nowicki**.

For complex Hermitian positive semidefinite (7\times7) matrices (A,B), **each of rank at most three**, the manuscript establishes

$$
\operatorname{per}(A\circ B)
\le \frac{24}{25}\operatorname{per}(A)\operatorname{per}(B).
$$

Here (\circ) denotes the entrywise product. The coefficient (24/25) is a convenient certified bound; its optimality is not claimed. The unrestricted, arbitrary-rank Chollet conjecture remains outside this result. Publication priority has not been established, and this release has not received external human peer review or proof-assistant formalization.

## Read the result

- [Complete typeset manuscript](QX_NEPHRIEL_Seventh_String_v1.0.0.pdf)
- [Main proof](PROOF.md): the order-seven argument, including every reduction and rational endpoint bound.
- [Higher-order appendix](APPENDIX.md): orders eight and nine, followed by an analytic infinite tail.
- [Theorem ledger](THEOREM_LEDGER.md): exact hypotheses, constants, dependencies and exclusions.
- [Sources and attribution](SOURCES.md): established ingredients and the boundary of the novelty claim.
- [Reproduction instructions](REPRODUCIBILITY.md), [verification report](VERIFICATION_REPORT.json), and [additional AI review](PROOF_AUDIT.md).

## What the argument does

After diagonal normalization and a Cauchy-Schwarz reduction, the problem becomes a bound on

$$
\frac{\operatorname{per}(C\circ\overline C)}{\operatorname{per}(C)^2}.
$$

An interpolated Bregman row bound controls the numerator. Complex Gaussian moments, spherical Jensen inequalities and a top-eigenvector change of measure control the denominator. Eigenvalue elimination puts both estimates in the same scalar coordinate (t=\lambda_{\max}(C)). At order seven, monotonicity and piecewise log-convexity reduce the entire admissible spectral interval to **three endpoint checks**. Integer and rational arithmetic then certify a coefficient below one.

The row interpolation and spherical estimates have precedents and are explicitly credited. The contribution claimed by this derivation is the order-seven scalar closure and its quantitative certificate, subject to an unfinished priority audit.

## Scope of the supplied bounds

Both matrices must be complex Hermitian PSD and individually have rank at most three.

| Matrix order | Certified coefficient | Supplied argument |
|---|---:|---|
| (n=7) | (24/25) | Three analytic endpoints; exact rational checks |
| (n=8) | (3/5) | 21 closed intervals cover ([8/3,8]) |
| (n=9) | (2/5) | 12 closed intervals cover ([3,9]) |
| Every (n\ge10) | (4/5) | Analytic tail with a rational base case |

Thus (24/25) works uniformly for every (n\ge7). These are separate sufficient bounds, not optimal constants or a claimed extremal-ratio curve. An all-order rank-three corollary additionally relies on the cited through-order-six preprint, whose proof is **not** reproduced or independently audited here.

## Reproduce locally

The core verifiers need **Python 3.10+**, its standard library, and no network access.

```bash
python verify_release.py
```

This checks file integrity and replays the two core scripts in a temporary directory without modifying the release. Expected evidence: **49/49 exact checks**; **21/21 order-eight intervals**; **12/12 order-nine intervals**; no unresolved intervals.

Optional symbolic and exact-matrix checks require SymPy:

```bash
python -m pip install -r requirements-audit.txt
python verify_release.py --include-symbolic
```

The original individual scripts remain available. They regenerate their JSON certificates beside themselves when executed directly.

## Status / completeness

| Item | Release evidence | Interpretation |
|---|---|---|
| Finite rational checks | 49/49 passed (100%) | Executed integer/fraction inequalities |
| Auxiliary interval coverage | 33/33 certified (100%) | Complete closed-domain covers for orders eight and nine |
| Optional symbolic/fixture audit | 21/21 passed (100%) | Additional identity and exact-example checks |
| Analytic manuscript | Included; additional AI review found no fatal error | Written proof remains open to mathematical review |
| External human peer review | Not performed | No endorsement claimed |
| Proof-assistant formalization | Not performed (0%) | Scripts do not formalize all analytic steps |
| Priority / optimal coefficient | Unconfirmed | No percentage or first-proof claim assigned |

The percentages above describe completed finite checks, not a probability that the theorem is correct. These verifiers do not mechanically establish the entropy proof, Gaussian pairing identity or continuous analytic reduction. Those steps are written out in the manuscript.

## Artifact contents and intended use

This is a research artifact collection hosted in a Hugging Face **dataset repository**: Markdown/PDF mathematics, exact JSON certificates, verification code, replay logs, citations and an integrity manifest. It contains no trained model, training examples, human-subject data or performance benchmark. The JSON files are structured proof evidence rather than IID samples. Inspect them directly; the release does not define a `datasets.load_dataset` schema.

Use it for mathematical review, exact reproduction, comparison with existing permanent inequalities, or further formalization. New orders, ranks, constants or learning-system applications require additional arguments.

## Provenance, citation and license

The original conversation-produced archive is preserved in [provenance](provenance/). Release preparation adds documentation, a typeset manuscript, fresh replay reports and a local verification runner. The mathematical core scripts are preserved. See [release notes](RELEASE_NOTES.md).

Use [CITATION.cff](CITATION.cff) or [citation.bib](citation.bib); no DOI or journal acceptance is implied. Documents and original certificate data are released under **CC BY 4.0**; Python code under **MIT**. Third-party sources retain their own rights. See [LICENSE.md](LICENSE.md).