Datasets:

PureOne
/

avenyra-area-feedback

---
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](manuscript/Avenyra_Area_Feedback_v1.0.0.pdf)** · **[Editable source](manuscript/Avenyra_Area_Feedback_v1.0.0.tex)** · **[Proof audit](audits/PROOF_AUDIT.txt)** · **[Novelty audit](audits/NOVELTY_AUDIT.txt)**

## 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

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

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

$$
x_n=\sum_i a_i,\qquad A_n=\sum_{i<j}(a_i a_j^T-a_j a_i^T).
$$
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
$$
\min_{y\in\operatorname{Reach}(x,A)}\|y\|
=\|P_{\operatorname{im}A}x\|,
\qquad
\operatorname*{argmin}_{y\in\operatorname{Reach}(x,A)}\|y\|
=\{P_{\operatorname{im}A}x\}.
$$
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
$$
x=(1,1,1)^T,\quad
A=\begin{pmatrix}0&1&1\\-1&0&1\\-1&-1&0\end{pmatrix},\quad
\ker A=\operatorname{span}\{(1,-1,1)^T\}.
$$
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](audits/NOVELTY_AUDIT.txt) 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:

```bash
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](docs/REPRODUCIBILITY.md).

## 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.

```python
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](docs/DATA_SCHEMA.md) before using the records as labels or evaluation material.

## AI agents and expert review

**[AI-agent guide](docs/AI_AGENT_GUIDE.md)** supplies a reading order, exact update conventions, claim-status rules, and a format for reporting objections or proposed extensions.

**[Expert review guide](docs/EXPERT_REVIEW.md)** 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](CITATION.cff) or [citation.bib](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](LICENSE_NOTICE.md).

	---
	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](manuscript/Avenyra_Area_Feedback_v1.0.0.pdf) · [Editable source](manuscript/Avenyra_Area_Feedback_v1.0.0.tex) · [Proof audit](audits/PROOF_AUDIT.txt) · [Novelty audit](audits/NOVELTY_AUDIT.txt)

	## 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

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

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

	$$
	x_n=\sum_i a_i,\qquad A_n=\sum_{i<j}(a_i a_j^T-a_j a_i^T).
	$$

	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

	$$
	\min_{y\in\operatorname{Reach}(x,A)}\\|y\\|
	=\\|P_{\operatorname{im}A}x\\|,
	\qquad
	\operatorname*{argmin}_{y\in\operatorname{Reach}(x,A)}\\|y\\|
	=\{P_{\operatorname{im}A}x\}.
	$$

	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

	$$
	x=(1,1,1)^T,\quad
	A=\begin{pmatrix}0&1&1\\-1&0&1\\-1&-1&0\end{pmatrix},\quad
	\ker A=\operatorname{span}\{(1,-1,1)^T\}.
	$$

	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](audits/NOVELTY_AUDIT.txt) 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:

	```bash
	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](docs/REPRODUCIBILITY.md).

	## 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.

	```python
	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](docs/DATA_SCHEMA.md) before using the records as labels or evaluation material.

	## AI agents and expert review

	[AI-agent guide](docs/AI_AGENT_GUIDE.md) supplies a reading order, exact update conventions, claim-status rules, and a format for reporting objections or proposed extensions.

	[Expert review guide](docs/EXPERT_REVIEW.md) 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](CITATION.cff) or [citation.bib](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](LICENSE_NOTICE.md).