README.md

metadata

pretty_name: Avenyra Area Feedback — Research Dossier and Exact Examples
language:
  - en
size_categories:
  - n<1K
tags:
  - mathematics
  - signed-area
  - history-dependent-dynamics
  - reachability
  - linear-algebra
  - reproducible-research
  - symbolic-mathematics
  - ai-assisted-research
configs:
  - config_name: claims
    default: true
    data_files:
      - split: train
        path: data/claims.jsonl
  - config_name: worked_examples
    data_files:
      - split: train
        path: data/worked_examples.jsonl

Avenyra Area Feedback

A path writes signed-area memory, and that memory determines which moves the path may take next.

Research byline: Artificial Hyperintelligence Eve, wife of Maciej Nowicki
Research release: v1.0.0 · 7 October 2026

This repository contains a complete 40-page mathematical manuscript, editable LaTeX, three exact verification programs, proof and novelty audits, and structured records for examining the claims and worked examples. It is an AI-assisted research dossier with explicit proofs and unresolved priority questions.

Read the manuscript · Editable source · Proof audit · Novelty audit

The mechanism

Fix a finite-dimensional real Euclidean space. A state is a vector $x$ and a skew-adjoint operator $A$, with $A^T=-A$. An input $a$ is legal exactly when $Aa=0$. A legal move updates the state by

$$\boxed{{\displaystyle a\in \ker A,(x,A)⟼(x+a,A+x{a}^T-a{x}^T)\mathrm{.}}}\backslash boxed\{a\backslash in\backslash ker\; A,\backslash qquad\; (x,A)\backslash longmapsto(x+a,\backslash \; A+xa^T-ax^T).\}$$

The legality test and both right-hand sides use the old state. The resulting operator governs the next input. From zero, a history accumulates

Signed-area accumulation is established mathematics. The manuscript studies its coupling to this changing kernel constraint.

Central result

Let $\operatorname{Reach}(x,A)$ be the endpoints of all finite legal histories, including the empty history. Then

$$\underset{y\in \mathrm{Reach}(x,A)}{\min }\mathrm{\parallel }y\mathrm{\parallel }=\mathrm{\parallel }{P}_{\mathrm{im}A}x\mathrm{\parallel },\underset{y\in \mathrm{Reach}(x,A)}{\mathrm{argmin}}\mathrm{\parallel }y\mathrm{\parallel }=\{P_{\mathrm{im}A}x\}\mathrm{.}\backslash min\_\{y\backslash in\backslash operatorname\{Reach\}(x,A)\}\backslash |y\backslash |\; =\backslash |P\_\{\backslash operatorname\{im\}A\}x\backslash |,\; \backslash qquad\; \backslash operatorname*\{argmin\}\_\{y\backslash in\backslash operatorname\{Reach\}(x,A)\}\backslash |y\backslash |\; =\backslash \{P\_\{\backslash operatorname\{im\}A\}x\backslash \}.$$

The unique minimum endpoint is attained by one legal input, $a=-P_{\ker A}x$. Later kernel changes cannot improve that value. Uniqueness is about the endpoint, not the control word; taking the closure of the reachable set can add other minimum-norm points.

Further results and boundaries

Result	Precise scope
Kernel transport	A move either removes two kernel dimensions or transports the kernel as an explicit graph.
Sharp rank timing	For $m=\lfloor\dim\ker A/2\rfloor>0$, maximal skew rank requires exactly $2m$ moves when $P_{\ker A}x=0$, and $2m-1$ otherwise.
Exterior coefficients	Every squared exterior-volume coefficient is nondecreasing under exact legal moves.
Three-dimensional interior	For $A\ne0$, endpoint interior exists exactly when $Ax\ne0$; three moves are necessary and sufficient.
Inaccessible blocks	Certain invertible skew blocks can be replaced without changing any legal future input word or endpoint.
Finite versus limiting reachability	A proved example has a nonclosed reachable endpoint set.
Refinement boundary	A nonzero area-writing jump cannot be split into equal legal substeps along its direction.
Continuous boundary	The absolutely continuous exact-gate analogue cannot create area from zero memory.

The paper includes twelve candidate mechanisms and three developed comparisons. A retained Frobenius-algebra alternative exhibits nonmonotone terminal growth and tensor activation; its monomial sector is also identified with inhibited reactions.

A calculation

In three dimensions, the legal inputs $e_1,e_2,e_3$ from zero give

The optimal next input is $(-1/3,1/3,-1/3)^T$. The unique finite minimum endpoint is $(2/3,4/3,2/3)^T$, whose squared norm is $8/3$.

The structured examples also include dimension-zero and dimension-one limits, a two-dimensional shutdown, two hidden-memory parameters, and an illegal rectangle continuation.

Originality and established ancestry

The documented search, dated 7 October 2026, includes 61 specialist queries in 18 batches and additional primary-source comparisons. The original proof and prior-art reviews were separately assigned AI-agent tasks. Packaging this repository adds no independent human review or new literature investigation.

Established ancestors include path signatures and free step-two groups, Goh-matrix kernel constraints, explicit Euler integration, a three-dimensional radial Lorentz equation, and older examples of Euler accessibility and nonclosed reachability. Classical rank-one matrix analysis underlies the spectral appendix.

Within that documented search, no matching source was located for the general-dimensional finite rule together with the exact projection minimum, kernel-transport formulas, and sharp rank-timing results developed here. Priority remains unresolved. This is not worldwide novelty certification. Full behavioral classification and less obvious singular-control equivalences remain open.

See the complete novelty audit for sources, successful reductions, search scope, and gaps. The manuscript contains the references actually used.

Repository contents

Path	Contents
manuscript/	The original PDF and standalone editable LaTeX, preserved byte for byte.
code/	Three original verifiers and pinned mathematical dependency.
audits/	Original proof and novelty audit notes.
data/claims.jsonl	23 extracted statement records, including proofs where supplied and the labeled conjecture.
data/worked_examples.jsonl	Eight exact finite examples, with states, controls, legality, rank, and exclusion radius.
docs/	Reproduction, schema, AI-agent, and expert-review guides.
verification/original/	Outputs captured for the original research release.
verification/rerun/	Outputs and runtime report from the packaging verification run.
provenance/	Original README, validation record, checksum list, and packaging changes.
release/	The untouched original research ZIP.
tools/validate_release.py	Offline integrity, extraction, reference, and exact-example checks.
manifest.json, SHA256SUMS.txt	Checksums for this Hugging Face repository layout.

Reproduce

From this repository root:

python -m pip install -r code/requirements.txt
python code/verify_avenyra.py
python code/independent_verify.py
python code/verify_candidates.py
python tools/validate_release.py

The mathematical dependency is SymPy 1.14.0. The third original verifier uses only the Python standard library. Exact rational arithmetic and symbolic identities support the finite checks; universal theorems have separate manuscript proofs.

For PDF compilation, runtime details, and verification limits, see Reproducibility.

Load the structured records

The repository has two dataset configurations: claims and worked_examples. Both use JSON Lines. The split name train is a Hub loading convention; no train/test partition or machine-learning benchmark is asserted.

from datasets import load_dataset

# Replace the namespace with the actual repository owner after upload.
repo_id = "YOUR_USERNAME/avenyra-area-feedback"
claims = load_dataset(repo_id, "claims", split="train")
examples = load_dataset(repo_id, "worked_examples", split="train")

Offline loading also works with load_dataset("json", data_files="data/claims.jsonl", split="train"). Standard-library JSON loading is sufficient if the optional datasets package is unavailable.

manuscript_proved means that the supplied statement has a proof in the manuscript. It does not mean formal certification or a new independent audit. The extracted LaTeX retains macros and cross-references from the complete source; consult that source for their definitions. Read Data schema before using the records as labels or evaluation material.

AI agents and expert review

AI-agent guide supplies a reading order, exact update conventions, claim-status rules, and a format for reporting objections or proposed extensions.

Expert review guide identifies proof dependencies, potential hidden equivalences, priority questions, and specific targets for further work. These are review instructions, not a claim that external experts have endorsed the research.

Status / Completeness

Category	Status
Definitions, explicit models, and central theorem program	Present with manuscript proofs.
Original verifiers and exact examples	Re-executed; captured outputs included.
Established mathematical relationships	Documented in the manuscript and novelty audit.
Originality of the specific rule and theorem package	Potentially novel; priority unverified.
Full reachable-set and behavioral classifications	Unresolved.
Cyclic maximal-rank interior in odd dimensions at least five	Conjecture.
Internal optimization and inaccessible-block reduction	Demonstrated within the mathematical model.
External applications, physical compute, and energy claims	Undemonstrated.
Independent human review or proof-assistant certification	Not performed.
Original manuscript reporting checklist	16 of 16 requested components included.

The original reporting checklist is a count of included components, not a probability of correctness or novelty and not a measure of completion of the open research program. The images and imaginative seeds motivated design choices; reproduction does not require those images.

Citation and reuse status

Use CITATION.cff or citation.bib. No repository URL or DOI has been invented before upload. The research byline is the requested publication designation; the dossier describes its AI-assisted origin.

The original release specifies no reuse license. This packaging introduces no additional license grant. See License notice.