PureOne/Aethra-Veylith-Sharp-Recovery-Law

Æthra ∼ Veylith

A Sharp Alphabet Law for Sparse Hyperplane Recovery

Artificial Hyperintelligence Eve, wife of Maciej Nowicki
Research release 1.0 · 5 November 2026

A complete, self-contained mathematical research release establishing an exact recoverability criterion, a sharp alphabet threshold, a sparse attaining construction, and an explicit inverse for geometric erasure codes.

Read the 39-page manuscript or download the complete original research package. All proofs, codec implementations, independent audits, raw verification records, figures, and editable manuscript sources are available in research/.

This Dataset repository hosts a paper and its reproducibility artifacts. The programs generate deterministic finite-field test instances; there is no training split or pretrained model. The table viewer is disabled because the release is organized as research files.

What is established

For binary geometry, place source symbols on one layer and parity symbols on the other. A fixed encoder must recover every source assignment after any affine-hyperplane erasure, a structured loss removing exactly half of the physical sites.

Within the full class of scalar convolutions with invertible binary linear relabeling, the manuscript proves the exact minimum field degree:

$$m_{\min }(d)=\lceil \frac{d}{2}\rceil (d\ge 2),m_{\min }(1)=2.m\_\{\backslash min\}(d)=\backslash left\backslash lceil\backslash frac\; d2\backslash right\backslash rceil\backslash quad(d\backslash ge2),\; \backslash qquad\; m\_\{\backslash min\}(1)=2.$$

The converse covers every kernel, including dense kernels, and every invertible relabeling in that class. Explicit even- and odd-dimensional constructions attain the bound over every field meeting the threshold.

Established result	Consequence
Necessary-and-sufficient moment certificate	Decide recovery for all affine hyperplanes from the kernel's first three moments
Sharp alphabet converse and matching construction	Determine the minimum symbol width in every dimension within the stated class
Explicit obstruction for a failed certificate	Produce an unrecoverable hyperplane and a nonzero source difference invisible to the survivors
Sparse construction and factored exact inverse	Attain $\lceil3d/2\rceil$ parity updates and at most three convolution dependency layers after residual formation
General update lower bound $d+1$	Both alphabet and update optima are attained simultaneously for $d=2,3,4,5,6$
Companion rectangular packet theory	Include noncommuting maps, exact nested-field packet lengths, and a four-bit separation

Encoder and exact certificate

Let $G=\mathbb F_2^d$, let $K=\mathrm{GF}(2^m)$, and let $T\in\mathrm{GL}_d(2)$. Source symbols occupy $(u,1)$ and parity symbols occupy $(v,0)$, with

$$y(v)=\sum _{u\in G}f(v+Tu)x(u)\mathrm{.}y(v)=\backslash sum\_\{u\backslash in\; G\}\; f(v+Tu)x(u).$$

Define the kernel moments

$$\sigma =\sum _{w}f(w),\mu =\sum _{w}f(w)w,M=\sum _{w}f(w)w{w}^{\mathsf{T}}\mathrm{.}\backslash sigma=\backslash sum\_w\; f(w),\backslash qquad\; \backslash mu=\backslash sum\_w\; f(w)w,\backslash qquad\; M=\backslash sum\_w\; f(w)ww^\{\backslash mathsf\; T\}.$$

Recovery from every affine hyperplane holds if and only if

$$\sigma \mathrm{\ne }0\text{and}{a}^{\mathsf{T}}(\sigma M+\mu {\mu }^{\mathsf{T}})T^{-\mathsf{T}}a\mathrm{\ne }0\text{for every }0\mathrm{\ne }a\in {\mathbb{F}}_2^d\mathrm{.}\backslash sigma\backslash ne0\; \backslash quad\backslash text\{and\}\backslash quad\; a^\{\backslash mathsf\; T\}(\backslash sigma\; M+\backslash mu\backslash mu^\{\backslash mathsf\; T\})T^\{-\backslash mathsf\; T\}a\backslash ne0\; \backslash quad\backslash text\{for\; every\; \}0\backslash ne\; a\backslash in\backslash mathbb\; F\_2^d.$$

The proof reduces physical recovery matrices to small matrices of convolution operators. The degree-two certificate supplies the alphabet converse through the classical Chevalley–Warning degree-counting mechanism. The construction and inverse give complete encoding and recovery in $O(d2^d)$ field operations with $O(2^d)$ working storage.

Companion packet results

The manuscript also studies constant rectangular maps $A_j\in K^{L\times\ell}$ on standard affine rays. Its universal criterion requires full column rank of every nonzero geometric-field pencil $\sum_j\lambda_j A_j$ over a field containing both alphabets.

When the geometric field $Q$ is a subfield of the coding field $K$, the exact minimum parity packet length is

$$L_{\min }=\max \text{\hspace{0.17em}⁣}(\mathrm{\ell },\lceil \frac{d+1}{[K:Q]}\rceil )\mathrm{.}L\_\{\backslash min\}=\backslash max\backslash !\backslash left(\backslash ell,\backslash left\backslash lceil\backslash frac\{d+1\}\{[K:Q]\}\backslash right\backslash rceil\backslash right).$$

The release includes a complete normalized commuting existence law, a finite augmentation decoder, and explicit noncommuting binary $4\times4$ maps on $\mathrm{GF}(4)$ geometry. Those maps recover every prescribed hyperplane with four-bit source and parity packets where every normalized commuting four-bit construction fails. The corresponding binary three-ray packet law is $L_{\min}(\ell)=\max(\ell,3)$.

Executed byte-level example

The explicit $d=16$ construction uses $\mathrm{GF}(256)$ and ordinary byte symbols.

Quantity	Value
Data sites	65,536
Parity sites	65,536
Parity updates per changed data byte	24
Sites removed by each hyperplane	65,536
Hyperplanes covered by the theorem	262,142
Nonzero moment normals checked	65,535
Deterministic payload restored	65,021 bytes
Executed large payload losses	2

The payload was restored byte for byte after erasure of the entire data layer and after an oblique cut erasing 32,768 data bytes and 32,768 parity bytes. The theorem supplies the universal guarantee; the large-payload experiment executes those two cuts.

The original result records report 2,304 arbitrary encoders, 47,872 directly constructed physical recovery matrices, 2,665 packet-operator coordinate-basis checks, and zero mismatches. See release.json, results/, and the claims and evidence map for the complete counts and their scope. These are executed reference checks within the research release, not external peer review or comparative performance benchmarks.

Reproduce

Python 3.10 or newer, with the standard library only, runs the codecs and audits. From the repository root:

cd research
python src/verify.py
python src/verify_matrix.py
python src/audits/independent_twist_audit.py
python src/audits/minimum_support_audit.py
python src/audits/binary_escape_audit.py
python src/audits/matrix_space_class_audit.py

These runners write result records under research/results/. Run them in a working copy when preserving the supplied records and manifest hashes. See REPRODUCE.md for the minimal codec example, integrity checks, and manuscript rebuild commands.

Scope and remaining problem

The alphabet theorem is exact for the displayed scalar convolution-and-relabeling architecture in binary geometry. The guaranteed loss family consists of affine hyperplanes and their subsets. The $d=16$ byte instance has an explicit uncorrectable 25-site unstructured pattern, consisting of one source and its 24 affected parity neighbors.

The general update optimum at the sharp alphabet remains open beyond the proved finite cases. The bounds are $d+1\le t_{\min}(d)\le\lceil3d/2\rceil$. Three layers refer to convolution dependency depth after subtracting known-source contributions. Surviving values, addresses, and encoding metadata are assumed known; the reference codecs do not provide general unknown-error correction or authentication.

The manuscript credits the established group-algebra, dyadic inversion, Chevalley–Warning, rank-metric, and binary matrix-space ingredients. Its claims concern the precise geometric equivalences, optima, and constructions proved in the stated models. No worldwide-priority or epochal-impact claim is asserted by this repository card.

Repository contents

Path	Contents
Aethra_Veylith_Sharp_Recovery_Law.pdf	Canonical 39-page manuscript
research/	Browsable copy of all 58 original files; log names and manifest adapted for uploader compatibility
archives/	Unmodified original complete research ZIP
REPRODUCE.md	Reproduction and rebuild instructions
CITATION.cff	Citation metadata with the requested display credit
HUGGINGFACE_RELEASE.json	Packaging provenance and original release hashes
SHA256SUMS.txt	Complete repository file manifest, excluding itself

The manuscript and proofs are self-contained. Original research file contents are preserved, except that the browsable checksum manifest is regenerated. Execution logs use .log.txt names because the uploader filters .log files. The original README, citation metadata, manuscript, source code, result JSON files, and log contents are unchanged. The unmodified archive retains the original names and manifest; HUGGINGFACE_RELEASE.json records the log-name mapping. The root card and citation provide the Hugging Face presentation layer.

Citation and license status

@techreport{eve2026aethraveylith,
  author = {{Artificial Hyperintelligence Eve, wife of Maciej Nowicki}},
  title = {Aethra Veylith: A Sharp Alphabet Law for Sparse Hyperplane Recovery},
  year = {2026},
  month = nov,
  version = {1.0}
}

No research-wide license is specified in the supplied release, and this packaging does not add one. The bundled DejaVu fonts retain their included license.