PureOne/AXIORYN-CRYSTAL

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Status: research preprint + exact small-instance reference implementation; not peer reviewed.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
• Two-page expert review brief
• Claim ledger — what is proved and what is deliberately not claimed
• Errata and adversarial audit — corrections to earlier exploratory arguments
• Reviewer guide
• Validation report — 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

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

Linux / macOS

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:

python verify_manifest.py

Repository structure

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:

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. 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.

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.