PureOne/qaenthrix-eve

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 · Read the proof source · Inspect the verification record · Review mathematical provenance

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

$$\mathcal{E}(C,G)=\underset{x\in X}{\sup }\mathrm{\parallel }{C}_x^{\ast }{C}_x-G_x{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\mathrm{.}\backslash mathcal\; E(C,G)=\backslash sup\_\{x\backslash in\; X\}\backslash |C\_x^*C\_x-G\_x\backslash |\_\{\backslash mathrm\{op\}\}.$$

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

$$D_x=A_x{B}_{x,E}^{-1\mathrm{/}2}{G}_x^{1\mathrm{/}2},E_x=\mathrm{ran}{G}_x,D\_x=A\_xB\_\{x,E\}^\{-1/2\}G\_x^\{1/2\},\backslash qquad\; E\_x=\backslash operatorname\{ran\}G\_x,$$

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

$$\mathrm{\parallel }{D}_x-C_x{P}_x{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le \frac{\epsilon }{\sqrt{\gamma }+\sqrt{\gamma -\epsilon }}\mathrm{.}\backslash |D\_x-C\_xP\_x\backslash |\_\{\backslash mathrm\{op\}\}\; \backslash leq\backslash frac\{\backslash varepsilon\}\{\backslash sqrt\backslash gamma+\backslash sqrt\{\backslash gamma-\backslash varepsilon\}\}.$$

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,

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 $\delta_W(P)>0$,

$$F_W(P)=PW(W^{\ast }PW{)}^{-1\mathrm{/}2}F\_W(P)=PW(W^*PW)^\{-1/2\}$$

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

$$\underset{j}{\max }{\delta }_{W_j}(P)\ge N^{-1\mathrm{/}2}\text{for every }P\mathrm{.}\backslash max\_j\backslash delta\_\{W\_j\}(P)\backslash geq\; N^\{-1/2\}\; \backslash quad\backslash text\{for\; every\; \}P.$$

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

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:

python -B verify_continuous.py

To regenerate the figure and PDF:

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 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, 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 records this component scope; the card's license: other metadata does not extend the MIT grant to the entire collection.

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

@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}
}