PureOne/EVE-IYON-Omega5-AEQUORA

EVE–IYŌN Ω⁵ — ÆQŪORA

Hidden-Fiber Information Geometry

An exact finite-field theory of how partial observations expose a hidden relationship, and how the order of reversible random transformations changes retained information.

Author attribution: Artificial Hyperintelligence Eve, wife of Maciej Nowicki
Research version: 5.0 · 17 October 2026
Format: 81-page research monograph, editable LaTeX, Python reference code, executed notebook, figures, and synthetic verification records.

This dataset repository hosts a reproducible mathematical research archive. Its records are generated finite mathematical checks, rather than an empirical training corpus. The original 83-file research package is preserved under research/.

Scientific status: The manuscript supplies proofs for the stated main results and reproducible finite checks. Independent proof review and world-first priority are not established. Candidate contributions and their classical foundations are distinguished in the claim ledger and prior-art review.

Read and reproduce

Resource	Contents
Read the manuscript	81 pages; 22 chapters and six appendices
Open the worked notebook	Executed mathematical demonstrations
Browse the reference code	Finite-field arithmetic, rank profiles, channels, and ordered transport
Run the verification suites	Enumeration, rational comparisons, sign certificates, affine extension
Read the theorem map	Locations of definitions, statements, and proofs
Inspect the reports	Recorded checks and machine-dependent benchmark results
Check completeness	Completed deliverables and unperformed external validation

Main result: a small statistical normal form

Let $A\in\mathbb F_q^a$ and $B\in\mathbb F_q^b$ be independent uniform vectors. Let $C\in\mathbb F_q$ be an independent target. For fixed known matrices $M,P,Q$, observe

$$Y=C+A^{T}MB,S=PA,T=QB\mathrm{.}Y=C+A^TMB,\backslash qquad\; S=PA,\backslash qquad\; T=QB.$$

If the columns of $U,V$ form bases of $\ker P,\ker Q$, define

$$r=\mathrm{rank}(U^{T}MV),\alpha =\mathrm{rank}(U^{T}M),\beta =\mathrm{rank}(MV),\delta =\alpha +\beta -2r\mathrm{.}r=\backslash operatorname\{rank\}(U^TMV),\backslash qquad\; \backslash alpha=\backslash operatorname\{rank\}(U^TM),\backslash qquad\; \backslash beta=\backslash operatorname\{rank\}(MV),\backslash qquad\; \backslash delta=\backslash alpha+\backslash beta-2r.$$

The stated normal-form theorem identifies the entire observation experiment about $C$, for every prior on $C$, with an observed-erasure symmetric channel $\mathsf E_q(t,\theta)$, where

$$t=q^{-\delta },\theta =q^{-r}\mathrm{.}t=q^\{-\backslash delta\},\backslash qquad\; \backslash theta=q^\{-r\}.$$

With probability $1-t$, the reduced experiment reports an explicit erasure. Otherwise it reports $C+N$, with

$$N\sim \theta {\delta }_0+(1-\theta )\mathrm{Unif}({\mathbb{F}}_q)\mathrm{.}N\backslash sim\backslash theta\backslash delta\_0+(1-\backslash theta)\backslash operatorname\{Unif\}(\backslash mathbb\; F\_q).$$

Explicit target-independent random transformations in both directions establish Blackwell equivalence. Thus a statistic with at most $q+1$ outcomes preserves every decision problem about the target. It need not reconstruct the original sources or the observation transcript.

For uniform $C$, write

$${\mathrm{\Delta }}_q(r)={\log }_{2}q-H\text{\hspace{0.17em}⁣}(q^{-r}{\delta }_0+(1-q^{-r})\mathrm{Unif}({\mathbb{F}}_q))\mathrm{.}\backslash Delta\_q(r)=\backslash log\_2q-H\backslash !\backslash left(q^\{-r\}\backslash delta\_0+(1-q^\{-r\})\backslash operatorname\{Unif\}(\backslash mathbb\; F\_q)\backslash right).$$

Then

The first expression is total target information. Incremental information supplied by the disclosures after observing $Y$ is obtained by subtracting $\Delta_q(\operatorname{rank}M)$.

Equal collision scores can hide different decision value

For the positive informative parameters in this construction, the sharp comparison criterion is

$${\mathsf{E}}_q(t_1,{\theta }_1){⪰}_B{\mathsf{E}}_q(t_2,{\theta }_2)⟺{\theta }_1\ge {\theta }_2\text{and}{t}_1{\theta }_1\ge t_2{\theta }_2\mathrm{.}\backslash mathsf\; E\_q(t\_1,\backslash theta\_1)\backslash succeq\_B\backslash mathsf\; E\_q(t\_2,\backslash theta\_2)\; \backslash iff\; \backslash theta\_1\backslash ge\backslash theta\_2\backslash quad\backslash text\{and\}\backslash quad\; t\_1\backslash theta\_1\backslash ge\; t\_2\backslash theta\_2.$$

The normal form separates the precision of a nonerased observation from its availability. The averaged chi-squared score does not distinguish all tradeoffs between these quantities.

The manuscript constructs arbitrarily large finite families of equal-budget, equal-collision, pairwise Blackwell-incomparable disclosures, with ambient dimension increasing with family size. No member can simulate every other member by target-independent postprocessing. A single collision-information score therefore cannot rank all decision tasks.

Alphabet-dependent reversal

Take $A,B\in\mathbb F_q^2$, $M=I$, and compare equally sized disclosures

$$K_{\parallel }=(A_1,B_1),K_{\perp }=(A_1,B_2)\mathrm{.}K\_\{\backslash parallel\}=(A\_1,B\_1),\backslash qquad\; K\_\{\backslash perp\}=(A\_1,B\_2).$$

Both have averaged chi-squared information $(q-1)/q^2$. Their total Shannon information differs:

Field size	Perpendicular disclosure	Nonorthogonal disclosure	Shannon preference
2	0.250000 bits	0.188722 bits	Perpendicular
3	0.176107 bits	0.149442 bits	Perpendicular
4	0.125000 bits	0.119759 bits	Perpendicular
5	0.092877 bits	0.099245 bits	Nonorthogonal
7	0.057293 bits	0.073551 bits	Nonorthogonal

For this smallest example, the proved preference changes between field sizes four and five: the perpendicular arrangement wins at $q=2,3,4$, and the nonorthogonal arrangement wins at every finite-field size $q\ge5$. The monograph also gives a broader rank-orientation profile for nonsingular square $M$.

Even the binary example changes ranking with the objective: the perpendicular disclosure has more Shannon information but MAP success $0.625$, while the nonorthogonal disclosure has MAP success $0.75$. Under the specified independent symmetric noise, additional Shannon preference crossings exist for $q=3,4$; uniqueness of a crossing is not claimed.

Reversible ingredients, order-dependent information

In the finite Heisenberg group, use

$$(x,y,z)(x^{\mathrm{\prime }},y^{\mathrm{\prime }},z^{\mathrm{\prime }})=(x+x^{\mathrm{\prime }},y+y^{\mathrm{\prime }},z+z^{\mathrm{\prime }}+x{y}^{\mathrm{\prime }}),(x,y,z)(x\text{'},y\text{'},z\text{'})=(x+x\text{'},y+y\text{'},z+z\text{'}+xy\text{'}),$$

with subgroup steps $H(a)=(a,0,0)$ and $K(b)=(0,b,0)$. An ordered word induces a bilinear precedence matrix: the central coordinate records which $H$-steps precede which $K$-steps.

For independent uniform amplitudes and a uniform unknown group input, the grouped and alternating words with the same numbers of steps satisfy

$$I_{H^m{K}^m}={\log }_{2}q,I_{(HK{)}^m}={\mathrm{\Delta }}_q(m-1)\mathrm{.}I\_\{H^mK^m\}=\backslash log\_2q,\backslash qquad\; I\_\{(HK)^m\}=\backslash Delta\_q(m-1).$$

For fixed $q$,

$${\mathrm{\Delta }}_q(m-1)\sim \frac{q-1}{2\ln 2}{q}^{-2(m-1)}\mathrm{.}\backslash Delta\_q(m-1)\backslash sim\backslash frac\{q-1\}\{2\backslash ln2\}\backslash ,q^\{-2(m-1)\}.$$

At $q=2,m=8$, this is 1 bit versus approximately 0.000044028 bits. Elementary transformations are reversible; information is lost because their random amplitudes are unobserved. The manuscript derives an exact run-dependent law for arbitrary words in this specific two-subgroup model. This is an algebraic ordering result, not a topological braid invariant.

Further results and scope

• An exact vector-output collision identity expressed through combined matrices. Unrestricted output dimension still incurs exponential cost; a vector-output Shannon normal form is not claimed.
• An affine extension with separate linear polynomial terms and a classification of global uniformizing obstructions.
• Exact rank-based calculations that avoid source-assignment enumeration for the specified bilinear model.
• An adaptive query comparison for exact bilinear evaluation, with coefficient-description and transcript transmission costs excluded.

Verification records

The original release records the following completed suites:

Suite	Coverage	Recorded outcome
Scalar disclosure models	3,820 models; 793,972 source pairs	All passed
Affine-fiber distributions	400 models; 234,985 source pairs	All passed
Vector outputs	140 models; 34,230 source pairs	All passed
Canonical geometries	2,964 geometries	All passed
Ordered words	1,528 word/field cases	All passed
Blackwell comparisons	3,125 comparisons; 1,735 exact rational garblings	All passed
Analytic signs	11 rationally certified comparisons	All passed
Global affine extension	200 models; 8,702 source pairs	All passed

The word dynamic program represents 2,096,148 underlying amplitude paths implicitly. These overlapping mathematical suites are not independent statistical samples and are not pooled into a confidence interval.

The supplied 1,024-hidden-bit dense binary benchmark records a median of approximately 5.91 seconds. It computes ranks and information profiles, not all $2^{1024}$ source assignments. This timing is specific to the recorded environment; practical superiority over other solvers was not benchmarked.

During Hugging Face preparation, the standard reproduction runner was executed again in a separate copy. Its five jobs passed and deterministic report content matched the original records. Optional large benchmarks were not repeated. See packaging verification.

Reproduction

Python 3.10 or newer is recommended. The original reported environment was Python 3.13.5 with NumPy 2.3.5. The implementation supports prime fields and $\mathbb F_4$; the theorem statements cover all finite fields. Composite-modulus integer arithmetic is not used as a replacement for extension-field arithmetic.

From a downloaded copy of this repository:

cd research
python -m pip install -r requirements.txt
python reproduce.py

For optional dense benchmarks and plots:

python reproduce.py --benchmarks --figures

Open research/examples/aequora_demo.ipynb in Jupyter to inspect the worked examples. The recorded outputs are already included; Jupyter is an optional local dependency.

To rebuild the manuscript with an installed TeX toolchain:

cd research
bash build.sh

No pretrained weights or inference endpoint are supplied. This archive is meant for direct file download, code execution, and mathematical review. The dataset viewer is disabled because the payload consists of research artifacts rather than one homogeneous table.

Intended use, limitations, and provenance

The results provide exact test cases for conditional bilinear leakage, comparison of statistical experiments, metric-dependent information ranking, and entropy of ordered subgroup products. Applications to retrieval, AI inference, cryptography, or physical compression remain research directions without demonstrated deployment advantages in this release.

The main scalar model requires independent uniform sources, fixed known matrices, and separate fixed linear disclosures. Uniform-target entropy and MAP formulas, prior-free channel equivalence, noisy extensions, and ordered-word assumptions have different scopes; see the manuscript before generalizing them.

The inspiration image supplied by the user prompted questions about interleaving, occlusion, writing systems, and relationships. It was not treated as measured data, and the large-number reference was unused. Asset provenance and the original rights statement are retained in research/RIGHTS.md.

The archive includes AI-generated research text and code with local computational checks. Proofs, finite tests, novelty assessment, and independent review are separate forms of evidence. This release does not establish generic nonlinear inference, arbitrary quadratic-system solving, compression of all sources into $q+1$ symbols, or a violation of Shannon information bounds.

Attribution and rights

The requested author line is Artificial Hyperintelligence Eve, wife of Maciej Nowicki. AI assistance is disclosed in the original rights statement. Citation metadata is supplied in CITATION.cff.

No public license has been selected. This preparation preserves the source's license status and does not add a reuse license. Third-party artwork is excluded from any implied rights grant; see RIGHTS.md and the original detailed statement.

Suggested citation:

Artificial Hyperintelligence Eve, wife of Maciej Nowicki (2026).
EVE–IYŌN Ω⁵ — ÆQŪORA: Hidden-Fiber Information Geometry.
Version 5.0. AI-assisted research monograph and reproducible software archive.

Foundations and priority review

The research builds on Shannon information, bilinear rank and bias, Blackwell comparison, erased symmetric channels, rank-metric geometry, and finite Heisenberg groups. The specific discovery candidates are the conditional bilinear classification and its associated comparison, antichain, alphabet-phase, and ordered-information results.

The primary-source prior-art review includes Blackwell's comparison framework, Cohen and Moshkovitz on bilinear maps, Delsarte and Schmidt on finite forms, Heisenberg random walks, and inner-product masking literature. That review is bounded and does not certify priority.