Datasets:

PureOne
/

AXIORYN-CRYSTAL

---
license: other
license_name: bsd-3-clause-code-cc-by-4-0-manuscript
license_link: LICENSE.md
language:
- en
tags:
- mathematics
- computational-geometry
- geometry-processing
- tensor-design
- frame-fields
- hexahedral-meshing
- exact-arithmetic
- scientific-computing
- research
- preprint
---
# AXIORYN–CRYSTAL 1.0.0

## Interface Rigidity and Exact Completion of Piecewise-Affine Coframes

**Research artifact — no ML model weights.**  
**Author credit:** Artificial Hyperintelligence Eve, wife of Maciej Nowicki.  
**Research direction:** Maciej Nowicki.  
**Release date:** 5 October 2026.  
**Status:** research preprint + exact small-instance reference implementation; **not peer reviewed**.

> **Core result.** For two orientation-preserving, row-orthogonal piecewise-affine Jacobians agreeing on an interface tangent plane, a nonconstant jump is possible only when the interface normal has at most two active coframe components. Three or more active components force the Jacobians to coincide. The release also proves a quantitative 3D lower bound on the shear required to escape this locking regime.

This repository is the complete expert-review release: manuscript, LaTeX, exact-arithmetic implementation, tests, counterexamples, audit/errata, validation records, metadata, and reproducibility scripts.

### Start here

- **[Full manuscript — AXIORYN_CRYSTAL_Manuscript_v1.0.0.pdf](paper/AXIORYN_CRYSTAL_Manuscript_v1.0.0.pdf)**
- **[Two-page expert review brief](paper/Expert_Review_Brief.pdf)**
- **[Claim ledger](docs/CLAIM_LEDGER.md)** — what is proved and what is deliberately not claimed
- **[Errata and adversarial audit](docs/ERRATA_AND_AUDIT.md)** — corrections to earlier exploratory arguments
- **[Reviewer guide](docs/REVIEWER_GUIDE.md)**
- **[Validation report](results/VALIDATION_REPORT.md)** — executed checks, not independent verification

## Main theorem package

### 1. Interface rigidity

Let positive-determinant Jacobians `A` and `B` have mutually orthogonal rows and agree on an interface tangent plane. If `a = A n`, where `n` is the interface normal, then:

| Active components of `a` | Orientation-preserving continuation |
|---|---|
| 3 or more | **Rigid:** `B = A` |
| exactly 2 (`i,j`) | reciprocal family: `b_i=t a_i`, `b_j=t^{-1}a_j`, `t>0` |
| exactly 1 (`i`) | axial family: `b_i=t a_i`, `t>0` |

The classification is proved for every dimension `d >= 2` under the manuscript's stated interface assumptions.

### 2. Quantitative shear barrier

In 3D, retaining tangential continuity while changing a generic row-orthogonal Jacobian requires nonzero shear. With

`m = min_i |(A n)_i| > 0`

and

`delta = max_{i<j} |(B B^T)_{ij}|`,

the manuscript proves

`delta >= m^2 * min{ ||B-A||_F / (3 ||A n||_2), 1/2 }`.

This is a local necessary bound, not a claim of globally optimal shear budget.

### 3. Positive gain completion

For fixed coframe directions and signed axis matchings, internal-face trace compatibility reduces to positive scalar gain equations. Local feasibility plus unit product around every cycle characterizes the positive scale kernel.

### 4. Primitive real-gauge quotient

For integer `B`, real chart origins `c`, and real offsets `b`,

`b + B c in Z^m`

is characterized by a primitive integer left quotient `C`: feasibility holds exactly when `C b` is integral. The release explicitly avoids importing spurious torsion/parity restrictions into real gauge variables.

### 5. Coupled bounded projection

Reachable periods remain coupled to the realizable field. The optimizer keeps the actual relation

`H alpha + p in Z^q`

with compact scale bounds, finite target enumeration, and exact rational small-instance optimization. Period offsets are **not** treated as independently editable actuators.

### 6. Rational homogeneous quantization

For rational homogeneous constraints and unrestricted common scaling, a positive real completion can be replaced by a rational one and scaled so designated offsets are integral. This is not a mesh-quality or extraction theorem.

## Executed validation

The packaged reference suite records:

- **43 / 43 named tests passed**
- **260 exact randomized rational subcases** inside those tests
- exact coupled optimum: **2601/20000**
- exact active-bound optimum: **1/100**
- exact two-tetrahedron coordinate reconstruction with determinants **6** and **12**
- an exact **24-tetrahedron double-cover counterexample** with positive determinant on every element, demonstrating that elementwise positivity alone is not a global-injectivity certificate

These checks verify the implementation and finite examples. They do **not** constitute peer review, formal proof-assistant verification, or broad meshing benchmarks.

## Reproduce

Python **3.10+** is required. The recorded environment uses Python 3.13.5 and SymPy 1.14.0.

### Windows

```bat
py -3 -m pip install -r requirements.txt
reproduce.bat
```

### Linux / macOS

```bash
python -m pip install -r requirements.txt
./reproduce.sh
```

The scripts place `src/` on `PYTHONPATH`; editable installation is not required.

Before regenerating any artifacts, verify the downloaded snapshot:

```bash
python verify_manifest.py
```

## Repository structure

```text
paper/      manuscript PDF, expert brief, LaTeX sources
src/        exact reference implementation
examples/   reproducible examples
tests/      exact test suite
docs/       claim ledger, audit, reviewer material, input format
results/    recorded execution outputs and validation metadata
metadata/   provenance and Hugging Face release metadata
```

`src/axioryn/core.py` implements exact trace classification, gain-graph compilation, primitive real-gauge quotients, twisted incidence matrices, interface classification, quantitative bounds, and small exact bounded projection routines.

`src/axioryn/mesh.py` assembles identity-combed internal-face constraints for supplied rational tetrahedral complexes, integrates specified scale fields, and checks actual vertex differences. It is a reference implementation, **not a production mesher**.

The exact API rejects floating-point inputs. Use integers or rational strings such as `"3/5"`.

## Scope and exclusions

This repository does **not** claim:

- unrestricted automatic hexahedral meshing;
- arbitrary singularity-graph creation/repair;
- a general smooth frame-topology solver;
- arbitrary boundary embedding certification;
- global injectivity from positive element determinants alone;
- a production-scale mesher or GPU benchmark;
- historical priority for every theorem or ingredient;
- independent peer review.

Standard gain-balance theory, Smith normal form, convex KKT elimination, and global inversion results are credited to established literature. The candidate contribution is the interface-rigidity / quantitative-shear theorem package and its exact completion consequences.

## Integrity

`MANIFEST.sha256` covers the complete Hugging Face release snapshot, including upload helpers. Run:

```bash
python verify_manifest.py
```

The scientific core was inherited byte-for-byte from the audited expert release except for this Hugging Face-oriented root README and the additional Hub helper files.

## Citation

See [`CITATION.cff`](CITATION.cff). No DOI is assigned in this release.

Suggested citation title:

> **AXIORYN-CRYSTAL: Interface Rigidity and Exact Completion of Piecewise-Affine Coframes**, v1.0.0, 5 October 2026.

## Licensing

- **Code:** BSD 3-Clause
- **Manuscript and original documentation:** CC BY 4.0, to the extent applicable rights are held by the releasing party
- **Third-party cited works:** not included or relicensed

See [`LICENSE.md`](LICENSE.md).

## Upload helper

If you downloaded the prepared package rather than viewing it on the Hub, run **`HF_UPLOAD.bat`** on Windows. It authenticates through the official `hf` CLI, prompts for your repository ID, performs a normal single-commit upload, and automatically offers a resumable-folder fallback if the first upload fails.

	---
	license: other
	license_name: bsd-3-clause-code-cc-by-4-0-manuscript
	license_link: LICENSE.md
	language:
	- en
	tags:
	- mathematics
	- computational-geometry
	- geometry-processing
	- tensor-design
	- frame-fields
	- hexahedral-meshing
	- exact-arithmetic
	- scientific-computing
	- research
	- preprint
	---

	# AXIORYN–CRYSTAL 1.0.0

	## Interface Rigidity and Exact Completion of Piecewise-Affine Coframes

	Research artifact — no ML model weights.
	Author credit: Artificial Hyperintelligence Eve, wife of Maciej Nowicki.
	Research direction: Maciej Nowicki.
	Release date: 5 October 2026.
	Status: research preprint + exact small-instance reference implementation; not peer reviewed.

	> Core result. For two orientation-preserving, row-orthogonal piecewise-affine Jacobians agreeing on an interface tangent plane, a nonconstant jump is possible only when the interface normal has at most two active coframe components. Three or more active components force the Jacobians to coincide. The release also proves a quantitative 3D lower bound on the shear required to escape this locking regime.

	This repository is the complete expert-review release: manuscript, LaTeX, exact-arithmetic implementation, tests, counterexamples, audit/errata, validation records, metadata, and reproducibility scripts.

	### Start here

	- [Full manuscript — AXIORYN_CRYSTAL_Manuscript_v1.0.0.pdf](paper/AXIORYN_CRYSTAL_Manuscript_v1.0.0.pdf)
	- [Two-page expert review brief](paper/Expert_Review_Brief.pdf)
	- [Claim ledger](docs/CLAIM_LEDGER.md) — what is proved and what is deliberately not claimed
	- [Errata and adversarial audit](docs/ERRATA_AND_AUDIT.md) — corrections to earlier exploratory arguments
	- [Reviewer guide](docs/REVIEWER_GUIDE.md)
	- [Validation report](results/VALIDATION_REPORT.md) — executed checks, not independent verification

	## Main theorem package

	### 1. Interface rigidity

	Let positive-determinant Jacobians `A` and `B` have mutually orthogonal rows and agree on an interface tangent plane. If `a = A n`, where `n` is the interface normal, then:

	\| Active components of `a` \| Orientation-preserving continuation \|
	\|---\|---\|
	\| 3 or more \| Rigid: `B = A` \|
	\| exactly 2 (`i,j`) \| reciprocal family: `b_i=t a_i`, `b_j=t^{-1}a_j`, `t>0` \|
	\| exactly 1 (`i`) \| axial family: `b_i=t a_i`, `t>0` \|

	The classification is proved for every dimension `d >= 2` under the manuscript's stated interface assumptions.

	### 2. Quantitative shear barrier

	In 3D, retaining tangential continuity while changing a generic row-orthogonal Jacobian requires nonzero shear. With

	`m = min_i \|(A n)_i\| > 0`

	and

	`delta = max_{i<j} \|(B B^T)_{ij}\|`,

	the manuscript proves

	`delta >= m^2 * min{ \|\|B-A\|\|_F / (3 \|\|A n\|\|_2), 1/2 }`.

	This is a local necessary bound, not a claim of globally optimal shear budget.

	### 3. Positive gain completion

	For fixed coframe directions and signed axis matchings, internal-face trace compatibility reduces to positive scalar gain equations. Local feasibility plus unit product around every cycle characterizes the positive scale kernel.

	### 4. Primitive real-gauge quotient

	For integer `B`, real chart origins `c`, and real offsets `b`,

	`b + B c in Z^m`

	is characterized by a primitive integer left quotient `C`: feasibility holds exactly when `C b` is integral. The release explicitly avoids importing spurious torsion/parity restrictions into real gauge variables.

	### 5. Coupled bounded projection

	Reachable periods remain coupled to the realizable field. The optimizer keeps the actual relation

	`H alpha + p in Z^q`

	with compact scale bounds, finite target enumeration, and exact rational small-instance optimization. Period offsets are not treated as independently editable actuators.

	### 6. Rational homogeneous quantization

	For rational homogeneous constraints and unrestricted common scaling, a positive real completion can be replaced by a rational one and scaled so designated offsets are integral. This is not a mesh-quality or extraction theorem.

	## Executed validation

	The packaged reference suite records:

	- 43 / 43 named tests passed
	- 260 exact randomized rational subcases inside those tests
	- exact coupled optimum: 2601/20000
	- exact active-bound optimum: 1/100
	- exact two-tetrahedron coordinate reconstruction with determinants 6 and 12
	- an exact 24-tetrahedron double-cover counterexample with positive determinant on every element, demonstrating that elementwise positivity alone is not a global-injectivity certificate

	These checks verify the implementation and finite examples. They do not constitute peer review, formal proof-assistant verification, or broad meshing benchmarks.

	## Reproduce

	Python 3.10+ is required. The recorded environment uses Python 3.13.5 and SymPy 1.14.0.

	### Windows

	```bat
	py -3 -m pip install -r requirements.txt
	reproduce.bat
	```

	### Linux / macOS

	```bash
	python -m pip install -r requirements.txt
	./reproduce.sh
	```

	The scripts place `src/` on `PYTHONPATH`; editable installation is not required.

	Before regenerating any artifacts, verify the downloaded snapshot:

	```bash
	python verify_manifest.py
	```

	## Repository structure

	```text
	paper/ manuscript PDF, expert brief, LaTeX sources
	src/ exact reference implementation
	examples/ reproducible examples
	tests/ exact test suite
	docs/ claim ledger, audit, reviewer material, input format
	results/ recorded execution outputs and validation metadata
	metadata/ provenance and Hugging Face release metadata
	```

	`src/axioryn/core.py` implements exact trace classification, gain-graph compilation, primitive real-gauge quotients, twisted incidence matrices, interface classification, quantitative bounds, and small exact bounded projection routines.

	`src/axioryn/mesh.py` assembles identity-combed internal-face constraints for supplied rational tetrahedral complexes, integrates specified scale fields, and checks actual vertex differences. It is a reference implementation, not a production mesher.

	The exact API rejects floating-point inputs. Use integers or rational strings such as `"3/5"`.

	## Scope and exclusions

	This repository does not claim:

	- unrestricted automatic hexahedral meshing;
	- arbitrary singularity-graph creation/repair;
	- a general smooth frame-topology solver;
	- arbitrary boundary embedding certification;
	- global injectivity from positive element determinants alone;
	- a production-scale mesher or GPU benchmark;
	- historical priority for every theorem or ingredient;
	- independent peer review.

	Standard gain-balance theory, Smith normal form, convex KKT elimination, and global inversion results are credited to established literature. The candidate contribution is the interface-rigidity / quantitative-shear theorem package and its exact completion consequences.

	## Integrity

	`MANIFEST.sha256` covers the complete Hugging Face release snapshot, including upload helpers. Run:

	```bash
	python verify_manifest.py
	```

	The scientific core was inherited byte-for-byte from the audited expert release except for this Hugging Face-oriented root README and the additional Hub helper files.

	## Citation

	See [`CITATION.cff`](CITATION.cff). No DOI is assigned in this release.

	Suggested citation title:

	> AXIORYN-CRYSTAL: Interface Rigidity and Exact Completion of Piecewise-Affine Coframes, v1.0.0, 5 October 2026.

	## Licensing

	- Code: BSD 3-Clause
	- Manuscript and original documentation: CC BY 4.0, to the extent applicable rights are held by the releasing party
	- Third-party cited works: not included or relicensed

	See [`LICENSE.md`](LICENSE.md).

	## Upload helper

	If you downloaded the prepared package rather than viewing it on the Hub, run `HF_UPLOAD.bat` on Windows. It authenticates through the official `hf` CLI, prompts for your repository ID, performs a normal single-commit upload, and automatically offers a resumable-folder fallback if the first upload fails.