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Status: research preprint + exact small-instance reference implementation; not peer reviewed.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
- 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.