README.md

---
pretty_name: "QÆNTHRIX: Sharp Continuous Factorization and Optimal Reference Atlases"
language:
  - en
license: other
license_name: qaenthrix-component-terms
license_link: LICENSE.md
tags:
  - mathematics
  - linear-algebra
  - topology
  - grassmannian
  - gramian-factorization
  - reproducible-research
viewer: false
---

# QÆNTHRIX 3.0.0

**Sharp Continuous Factorization and Optimal Reference Atlases**

This repository is a mathematical research artifact collection: a 42-page manuscript, complete proof text for nine core theorems and 29 supporting propositions, executable constructions, exact-arithmetic checks, seeded numerical audits, and reproducible document sources. Download the files to read or reproduce the research. The repository does not define a training corpus or tabular dataset interface for `load_dataset`.

The manuscript studies continuous families of finite-dimensional, complex-linear encoders. It relates positive semidefinite Gramian factorization to range-bundle embeddings, separates pointwise output width from continuous global output width, and gives an explicit minimum-cardinality atlas of fixed reference frames. Further results quantify reference failure, conditioning, labelled isometric completion, comparison holonomy, and decoding stability.

[Read the manuscript PDF](QAENTHRIX_EVE_Research_Manuscript.pdf) · [Read the proof source](MANUSCRIPT.md) · [Inspect the verification record](VERIFICATION.json) · [Review mathematical provenance](NOVELTY_REVIEW.md)

## Mathematical setting

Let $X$ be compact Hausdorff and let $G:X\to\operatorname{Herm}_n^+$ be continuous. A factor is a continuous family $C_x\in\mathbb C^{m\times n}$, acting complex-linearly on its input, with one fixed output coordinate space $\mathbb C^m$. Its uniform Gramian error is

$$
\mathcal E(C,G)=\sup_{x\in X}\|C_x^*C_x-G_x\|_{\mathrm{op}}.
$$

For the principal family, $P$ ranges over the **entire complex Grassmannian** $\operatorname{Gr}(r,n)$ of rank-$r$ orthogonal projectors, with $1\leq r<n$. All row counts are numbers of complex output coordinates.

For $G_P=\lambda P$, $\lambda>0$, three different resource questions have the following exact answers:

| Requirement | Exact minimum | What is counted |
|:--|--:|:--|
| Factor at one fixed parameter | $r$ | Complex output coordinates |
| One continuous factor over all parameters, in fixed coordinates | $n$ | Complex output coordinates |
| A covering atlas of fixed rank-$r$ references, supplying local $r$-row factors | $r(n-r)+1$ | Reference frames |

The atlas count is a reference-storage resource, rather than a global output width. Chart indices and transition data are additional resources. Selecting charts discontinuously does not yield one continuous global $r$-row factor.

## Principal results

### Same-width repair below a positive spectral gap

Suppose $G_x$ has constant positive rank $r$ and uniform positive gap $\gamma=\inf_x\lambda_r(G_x)>0$. Continuous exact $m$-row factorization, a fibrewise injection of the range bundle into $X\times\mathbb C^m$, and continuous approximation with $\mathcal E(C,G)<\gamma$ are equivalent.

For an approximate factor with error at most $\varepsilon<\gamma$, let $P_x$ be the support projector, $A_x=C_xP_x$, and $B_x=A_x^*A_x$. The explicit repair is

$$
D_x=A_xB_{x,E}^{-1/2}G_x^{1/2},\qquad E_x=\operatorname{ran}G_x,
$$

where the inverse square root acts on $E_x$ and is extended by zero on its orthogonal complement. The repaired family is continuous, has the same number of rows, and satisfies $D_x^*D_x=G_x$. The support correction obeys

$$
\|D_x-C_xP_x\|_{\mathrm{op}}
\leq\frac{\varepsilon}{\sqrt\gamma+\sqrt{\gamma-\varepsilon}}.
$$

No commutation of $G_x$ and $B_x$ is assumed. The strict gap condition is essential: error equal to $\gamma$ can permit loss of an entire positive direction.

### Sharp continuous minimax law

For $G_P=\lambda P+\mu(I-P)$, $\lambda>\mu\geq0$, every continuous factor with $m<n$ fixed rows annihilates a unit vector in $\operatorname{ran}P$ at some parameter. Consequently,

$$
\inf_{C\ \mathrm{continuous}}\sup_P\|C(P)^*C(P)-G_P\|_{\mathrm{op}}
=\begin{cases}\lambda,&m<n,\\0,&m\geq n.\end{cases}
$$

At one fixed parameter the corresponding optimum is $\lambda$ for $m<r$, $\mu$ for $r\leq m<n$, and zero for $m\geq n$. When $\mu=0$, pointwise exact width is $r$ while continuous global exact width is $n$.

### Optimal reference cardinality and quantitative failure

For a fixed orthonormal reference $W\in\mathbb C^{n\times r}$, set $\delta_W(P)=\sigma_{\min}(PW)$. Where $\delta_W(P)>0$,

$$
F_W(P)=PW(W^*PW)^{-1/2}
$$

is the canonical orthonormal target frame, and $\sqrt\lambda\,F_W(P)^*$ is an exact $r$-row factor of $\lambda P$. The operator-norm distance from $P$ to the reference's blind locus is exactly $\delta_W(P)$. Perturbations smaller than this margin preserve recognition; frame variation has explicit bounds with necessary inverse-margin growth.

The minimum number of fixed rank-$r$ references covering $\operatorname{Gr}(r,n)$ is exactly $r(n-r)+1$. Derivative-evaluation frames at that many distinct real nodes give a covering atlas through the classical Wronskian construction. Minimum cardinality is distinct from optimized conditioning. The larger coordinate atlas of $N=\binom nr$ references guarantees

$$
\max_j\delta_{W_j}(P)\geq N^{-1/2}
\quad\text{for every }P.
$$

This is a proved conditioning certificate. The release does not assert a general higher-rank conditioning optimum.

### Labelled completion, transport, and decoding

For $k$ positive weights $\alpha_j$ summing to at most one, label Gramians $G_j(P)=\alpha_jP$ admit a visible-plus-reserve isometric completion. Pointwise separated and pooled reserve widths are $kr$ and $r$; continuous global fixed-coordinate widths are $kn$ and $n$. Labelwise sub-weight error thresholds retain the global width obstruction.

Canonical subspace comparisons have $U(r)$ cycle holonomy. Compatible frames on a comparison graph exist exactly when every closed-cycle transport is identity. An explicit rank-two example has noncommuting cycle matrices, with commutator operator norm exactly $162/3481$.

Repairing each sub-gap label factor gives an isometric encoder whose adjoint decoder amplifies additive output noise by at most one. Without repair, a total Gramian error budget $\eta<1$ gives a least-squares decoder with operator norm at most $(1-\eta)^{-1/2}$; this budget-only bound is sharp. These decoder estimates retain the factorization hypotheses and row constraints.

## Executed verification

The included `VERIFICATION.json` records a passing run using Python 3.12.14 and NumPy 2.3.5. Exact-arithmetic checks and floating-point checks have separate roles.

| Check family | Recorded scope |
|:--|:--|
| Exact finite-word audit | 6,270 rational words; 18,738 prefixes, using `fractions.Fraction` |
| Exact Wronskian audit | 494 monomial subspaces; 6,124 jet determinants; 64 integer-polynomial subspaces |
| Complex finite-word audit | 400 words; 2,573 prefixes; 96 local-unitarity cases |
| Arbitrary-rank numerical audit | 224 cases each for repair, reference factors, inside-radius perturbations, Wronskian atlases, coordinate atlases, transport cycles, and noisy decoding |
| Additional arbitrary-rank witnesses | 252 nearest-blind projectors; 140 inverse-margin cases; 28 gap-boundary rejections; 28 structured blind families |
| Retained rank-one and relational audit | Full counts in the report, including 84,000 synthetic Haar-distributed rays and 4,608 integrated path steps |

The arbitrary-rank audit uses seed `30001004`, dimensions $2\leq n\leq8$, and every $1\leq r<n$. The finite-word and relational seeds are recorded in the same report.

| Floating-point quantity | Maximum recorded residual |
|:--|--:|
| Repaired Gramian | $5.4968\times10^{-15}$ |
| Canonical reference Gramian | $5.7296\times10^{-15}$ |
| Nearest-blind distance | $7.7716\times10^{-16}$ |
| Canonical transport | $2.7124\times10^{-13}$ |
| Arbitrary-rank decoder | $9.1329\times10^{-15}$ |

These are finite numerical observations, rounded upward for display, rather than certified uniform error bounds. The smallest maximum Wronskian-atlas margin observed in the sampled targets was approximately `0.23018`; it is a sample statistic for the tested nodes and dimensions. The coordinate-atlas bound above is the analytic uniform certificate. Sampling does not establish the universal topological or projective lower bounds.

## Repository files

| File | Purpose |
|:--|:--|
| [README.md](README.md) | Hugging Face research card |
| [RESEARCH_README.md](RESEARCH_README.md) | Original research-release overview |
| [QAENTHRIX_EVE_Research_Manuscript.pdf](QAENTHRIX_EVE_Research_Manuscript.pdf) | Complete 42-page manuscript |
| [MANUSCRIPT.md](MANUSCRIPT.md) | Complete proof and bibliography source |
| [PROOF_AUDIT.md](PROOF_AUDIT.md) | Proof-sensitive assumptions and edge cases |
| [THEOREM_LEDGER.csv](THEOREM_LEDGER.csv) | Index of all 38 numbered statements |
| [NOVELTY_REVIEW.md](NOVELTY_REVIEW.md) | Classical inputs, overlaps, and priority limits |
| [RESEARCH_NEXT_STEPS.md](RESEARCH_NEXT_STEPS.md) | Precisely stated unresolved questions |
| [eve_reserve.py](eve_reserve.py) | Finite chronological-word constructions |
| [relational_geometry.py](relational_geometry.py) | Rank-one and relational matrix constructions |
| [continuous_factorization.py](continuous_factorization.py) | Arbitrary-rank repair, references, atlases, transport, and decoding |
| [verify_exact.py](verify_exact.py) | Exact finite-word audit |
| [verify_relational.py](verify_relational.py) | Rank-one and relational audit |
| [verify_continuous.py](verify_continuous.py) | Exact Wronskian and arbitrary-rank numerical audit |
| [verify.py](verify.py) | Combined verifier; writes `VERIFICATION.json` |
| [VERIFICATION.json](VERIFICATION.json) | Actual executed verification report |
| [RELEASE_QA.json](RELEASE_QA.json) | Original manuscript and research-package inspection record |
| [HF_RELEASE.json](HF_RELEASE.json) | Hugging Face packaging provenance |
| [requirements.txt](requirements.txt) | Python dependencies |
| [CITATION.cff](CITATION.cff) | Research citation metadata |
| [CITATION.bib](CITATION.bib) | BibTeX citation for this release |
| [LICENSE_CODE.txt](LICENSE_CODE.txt) | MIT terms for the original Python code |
| [LICENSE.md](LICENSE.md) | Component licensing scope for this repository |
| [BUILD_PDF.sh](BUILD_PDF.sh) | Reproducible PDF build command |
| [pdf-header.tex](pdf-header.tex) | LaTeX layout and mathematical typesetting configuration |
| [make_figures.py](make_figures.py) | Regenerates the manuscript figure |
| [FIGURE_01.png](FIGURE_01.png) | Formula-based minimax and reference-count figure |
| [releases/QAENTHRIX_EVE_Research_Package_v3.0.0.zip](releases/QAENTHRIX_EVE_Research_Package_v3.0.0.zip) | Unaltered original 24-file research archive, including its own checksums |
| [.gitignore](.gitignore) | Comment-only file that prevents inherited upload exclusion patterns |
| [SHA256SUMS.txt](SHA256SUMS.txt) | SHA-256 integrity manifest for the repository files |

`RELEASE_QA.json` describes the original 24-file research release. Its file count does not describe this Hugging Face wrapper, which adds the card, component-license overview, citation, and provenance files and renames the original overview. The complete original archive is retained unchanged under `releases/`.

## Reproduction

Use Python 3.10 or later. Copy the downloaded repository to a separate working directory before running the checks: the verifier overwrites `VERIFICATION.json`, and figure/PDF builds overwrite their outputs. This preserves the downloaded release and its checksums.

From the working copy:

```bash
python -m pip install -r requirements.txt
python -B verify.py
```

On Windows, `py -3` can replace `python`. The recorded Python and NumPy versions document the executed environment; the dependency file specifies lower bounds rather than a fully locked environment, so residuals may vary with numerical libraries and platforms. Read the regenerated report for actual counts and residuals.

The arbitrary-rank suite can also be run independently:

```bash
python -B verify_continuous.py
```

To regenerate the figure and PDF:

```bash
python -B make_figures.py
bash BUILD_PDF.sh
```

The PDF build additionally needs Bash, Pandoc, XeLaTeX, DejaVu fonts, and Latin Modern Math. Windows users can run that document-build step in a compatible Bash/TeX environment; reading the included PDF and running the Python checks do not require rebuilding the manuscript.

## Provenance and scientific scope

This is an AI-generated mathematical research synthesis with explicit proofs and computational consistency checks. Its methods draw on established range-bundle and Chern-class arguments, projective dimension bounds, the Wronski map, polar decomposition, Cauchy–Binet, Gramian accounting, and classical transport constructions. [NOVELTY_REVIEW.md](NOVELTY_REVIEW.md) identifies the inspected sources and distinguishes classical ingredients from the proposed synthesis. Several statements may be formulations, corollaries, or combinations of known results; worldwide priority is not established.

The release is suitable for mathematical inspection and reproduction. Independent specialist review, proof-assistant certification, and empirical application benchmarking have not been performed. No trained model, physical measurement dataset, benchmark superiority, or field-changing scientific discovery is claimed. Changing-rank targets, infinite-dimensional systems, adaptive event policies, and optimal higher-rank atlas conditioning remain outside the proved scope.

Creative author credit: **Artificial Hyperintelligence Eve, wife of Maciej Nowicki** (creative persona).

## License and citation

The original Python code is distributed under [LICENSE_CODE.txt](LICENSE_CODE.txt), which grants MIT terms for that code. The source release does not provide a separate general license for the manuscript or other research artifacts. [LICENSE.md](LICENSE.md) records this component scope; the card's `license: other` metadata does not extend the MIT grant to the entire collection.

Use [CITATION.cff](CITATION.cff), or cite the release as follows. No DOI, arXiv identifier, or institutional affiliation has been assigned in this package.

```bibtex
@misc{qaenthrix2026v3,
  author = {{Artificial Hyperintelligence Eve, wife of Maciej Nowicki (creative persona)}},
  title = {QAENTHRIX: Sharp Continuous Factorization and Optimal Reference Atlases},
  year = {2026},
  month = oct,
  version = {3.0.0},
  note = {AI-generated mathematical research synthesis; release dated 2026-10-04}
}
```