| ---
|
| license: other
|
| license_name: ahti-mixed-mit-cc-by-4.0
|
| license_link: https://huggingface.co/PureOne/AHTI-NEXIFORM-1.0.0/blob/main/LICENSE
|
| tags:
|
| - mathematics
|
| - computational-geometry
|
| - geometry-processing
|
| - tensor-fields
|
| - differential-geometry
|
| - topology
|
| - cohomology
|
| - volumetric-parameterization
|
| - hexahedral-meshing
|
| - reproducible-research
|
| - scientific-computing
|
| pretty_name: "AHTI / NEXIFORM 1.0.0 — Certified Tensor-to-Coordinate Synthesis"
|
| ---
|
|
|
| # AHTI / NEXIFORM 1.0.0
|
| ## Certified Tensor-to-Coordinate Synthesis
|
|
|
| **Release date:** 2026-10-05
|
| **Author credit:** Artificial Hyperintelligence Eve, wife of Maciej Nowicki. This is requested creative AI attribution.
|
| **Status:** self-contained research/software release; not independently peer reviewed.
|
| **Primary artifact:** [`paper/AHTI_1.0.0.pdf`](paper/AHTI_1.0.0.pdf)
|
|
|
| AHTI / NEXIFORM develops a finite, executable framework connecting nonsingular tensor moments, local frame compatibility, fixed-sector affine/cohomological structure, exact minimum repair, integer-period quantization, and geometric validity certificates. The implemented scope is finite complexes with a specified flat local system. It is **not** presented as a universal hexahedral mesh generator or as a proof of unrestricted meshability.
|
|
|
| ## Main result
|
|
|
| For a finite cochain complex
|
|
|
| \[
|
| C^0 \xrightarrow{B} C^1 \xrightarrow{C} C^2,
|
| \qquad CB=0,
|
| \]
|
|
|
| with geometric positive-definite weight \(W\), every increment field admits the weighted orthogonal decomposition
|
|
|
| \[
|
| a=e+h+\ell,
|
| \]
|
|
|
| where \(e\in\operatorname{im}B\) is exact, \(h\) carries the global harmonic/cohomological component, and \(\ell\) is the local incompatibility component.
|
|
|
| If \(L\mathbb Z^r\) is the constructed lattice of integral harmonic periods, then the release proves
|
|
|
| \[
|
| \boxed{
|
| \min_{Cb=0,\ [b]\text{ integral}}
|
| \|a-b\|_W^2
|
| =
|
| \|\ell\|_W^2+
|
| \min_{k\in\mathbb Z^r}\|h-Lk\|_W^2
|
| }
|
| \]
|
|
|
| and reconstructs an optimal repaired field from an exact closest-vector solution. When that algebraic minimizer also passes the declared geometric validity conditions, it is automatically globally optimal over the corresponding geometrically admissible subset because it attains the unconstrained algebraic lower bound.
|
|
|
| ## Additional results implemented in this release
|
|
|
| The package includes:
|
|
|
| - deterministic reconstruction of a nonsingular 3-frame from its second- and fourth-order symmetric moments, using at most thirteen contraction probes in exact arithmetic;
|
| - optional sixth-order validation;
|
| - exact rational moment witnesses when bounded-denominator reconstruction succeeds;
|
| - a moment-coordinate pullback metric reproducing ordinary Frobenius frame variation on the regular moment manifold;
|
| - exact proper face matching and a regular ordinary-frame adapter;
|
| - weighted rational Hodge decomposition into exact, harmonic, and local-incompatibility components;
|
| - saturated integral cohomology computation with torsion via verified Smith transformations;
|
| - exact finite closest-vector search with explicit resource-limit behavior;
|
| - boundary-constrained Schur/lattice reduction;
|
| - tetrahedral Jacobian metrics;
|
| - rational no-flip certificates for an entire straight repair path;
|
| - a sufficient global injectivity certificate relative to a known convex bijective reference map.
|
|
|
| The full theorem statements, hypotheses, proofs, and scope restrictions are in the manuscript.
|
|
|
| ## Quick start
|
|
|
| Python 3.11 or later is required.
|
|
|
| ```bash
|
| python -m pip install -r requirements.txt
|
| python -m unittest discover -s tests -v
|
| python examples/run_demo.py
|
| python -m ahti examples/twisted_circle.json --output certificate.json
|
| ```
|
|
|
| Windows users can double-click `RUN_ALL.bat` to create a local virtual environment, install the pinned dependencies, run the test suite, and generate the examples.
|
|
|
| ## Verification status
|
|
|
| The recorded reference run passed **35/35 declared tests**. One test performs **80 seeded numerical frame reconstructions**; the largest observed orbit error in the recorded run is approximately `3.462e-15`. That value is a floating-point test result, not a universal stability theorem.
|
|
|
| Exact included examples include:
|
|
|
| - twisted-circle integral cohomology `Z/2 + Z/2 + Z`;
|
| - nearest integral-period repair cost `4/25`;
|
| - noisy two-tetrahedron zero-period repair cost `1/48000`;
|
| - positive rational no-flip minors for that repair;
|
| - certified global lower Lipschitz bound `99/100` on the explicit convex example;
|
| - boundary-constrained repair cost `3/50`.
|
|
|
| Machine-readable outputs are in [`reports/`](reports/).
|
|
|
| ## What the certificates mean
|
|
|
| `exact_algebraic_optimum_for_declared_cochain_problem` means the finite rational cochain/lattice problem was solved and verified for the stated model. It does **not** by itself certify mesh injectivity.
|
|
|
| `floating_point_reconstruction_checked` means the numerical moment/frame reconstruction residual checks passed. It does **not** convert noisy floating-point input into an exact theorem.
|
|
|
| `exact_rational_moment_witness` means every supplied exact moment entry agrees with the reconstructed rational frame. Failure of the bounded-denominator search does not exclude irrational or larger-denominator solutions.
|
|
|
| `ResourceLimit` means the exact closest-vector search was not completed under the configured work cap; no optimality claim is returned in that case.
|
|
|
| A failed sufficient geometric inequality means **not certified**, not necessarily geometrically invalid.
|
|
|
| ## Scientific scope and claim boundary
|
|
|
| The release deliberately separates proved finite-dimensional statements from broader open problems.
|
|
|
| Established within the stated hypotheses are the cochain repair decomposition, the fixed-sector integer-period optimization, the reconstruction and metric statements for nonsingular frames, and the implemented sufficient geometric certificates.
|
|
|
| Not established are unrestricted hex-meshability, automatic globally optimal singularity design, arbitrary-topology injectivity, a universal singularity-surgery engine, a complete IGM feature compiler, industrial-scale performance superiority, or worldwide historical priority for the integrated formulation.
|
|
|
| Whitening, tensor contraction, Hodge decomposition, Smith normal form, bundle/holonomy language, and monotonicity methods have substantial prior art. The release's novelty claim is therefore limited to the integrated formulation and the deductions explicitly proved in the manuscript. See [`docs/SOURCE_LEDGER.md`](docs/SOURCE_LEDGER.md) and [`docs/CLAIM_AUDIT.md`](docs/CLAIM_AUDIT.md).
|
|
|
| ## Repository layout
|
|
|
| ```text
|
| paper/ self-contained PDF manuscript and LaTeX source
|
| ahti/ Python reference library and CLI
|
| tests/ reproducibility tests
|
| examples/ runnable examples and sample inputs
|
| reports/ recorded machine-readable certificates and logs
|
| docs/ claim audit, source ledger, API/reproducibility notes
|
| ```
|
|
|
| `MANIFEST.sha256` contains SHA-256 hashes for the repository files.
|
|
|
| ## Citation
|
|
|
| Citation metadata is provided in [`CITATION.cff`](CITATION.cff). A BibTeX template is also included in [`CITATION.bib`](CITATION.bib).
|
|
|
| Suggested citation:
|
|
|
| > Artificial Hyperintelligence Eve, wife of Maciej Nowicki. **AHTI: Affine-Holonomy Tensor Integrability — Certified Tensor-to-Coordinate Synthesis.** Version 1.0.0, 2026. AI-generated research/software release; not independently peer reviewed.
|
|
|
| ## Licensing
|
|
|
| The Python code is licensed under the MIT License; see [`LICENSE-CODE.txt`](LICENSE-CODE.txt).
|
|
|
| The original AHTI manuscript and original documentation are licensed under Creative Commons Attribution 4.0 International; see [`LICENSE-DOCUMENTS.txt`](LICENSE-DOCUMENTS.txt).
|
|
|
| Third-party dependencies and cited external works retain their own licenses and are not relicensed by this repository.
|
|
|
| ## Reproduce everything
|
|
|
| ```bash
|
| python -m pip install -r requirements.txt
|
| python -m unittest discover -s tests -v
|
| python examples/run_demo.py
|
| ```
|
|
|
| For Windows, use:
|
|
|
| ```text
|
| RUN_ALL.bat
|
| ```
|
|
|
| For Hugging Face publication/update instructions, see [`HUGGINGFACE_PUBLISHING.md`](HUGGINGFACE_PUBLISHING.md).
|
|
|
