| # API and reproducibility | |
| ## Exact finite model | |
| `HodgeModel.build(B,C,W)` accepts rational SymPy matrices with `C*B == 0` and symmetric positive-definite `W`. Integers and exact fraction strings are preferred. Floating input is rejected by the exact layer. | |
| `model.split(a)` returns `(exact, harmonic, local)`. | |
| `model.integral_lattice()` returns integer representatives, a harmonic lattice basis, torsion invariant factors, and the free rank. It uses saturated integer kernels. Individually clearing denominators of a rational kernel basis is not an accepted substitute. | |
| `model.repair(a, target='exact')` returns the nearest exact cochain. With `target='integral'`, it solves the complete finite closest-vector search enclosure, up to the declared work cap. The repaired cochain is unique only when the lattice minimizer is unique. Ties are reported. | |
| `constrained_quantize(model,a,M,N,d)` minimizes `||a-(B*u+Z*k)||_W^2` under `M*u+N*k=d` and integral `k`, where `Z` is the chosen free integer cohomology representative matrix. The meaning of these coordinate constraints is part of the input model. This is not an automatic translator for every geometric boundary condition. | |
| ## Moments | |
| `encode(J)` uses the rows of `J` as physical frame vectors. It returns `L2`, `L4`, and, by default, `L6`. | |
| `decode(L2,L4,L6=None)` uses whitening and thirteen contractions, selecting the best measured eigenvalue separation. It checks reconstruction residuals. Ill-conditioned, singular, or inconsistent inputs are rejected. | |
| `exact_moments(J)` constructs rational tensor arrays. `decode_rational(...)` proposes a bounded-denominator rational candidate using the numerical decoder, then rechecks every exact tensor entry. An exact witness is stronger than a small residual; failure is not an impossibility certificate. | |
| `exact_moment_metric(J)` returns the 21-by-9 moment differential and an ambient semidefinite metric whose pullback is exactly the 9-by-9 identity. It is positive definite on the moment-manifold tangent space only. | |
| ## Geometry | |
| `tetrahedral_operators(vertices,tetrahedra,transports=None)` creates the cochain complex, all-root geometric metric, and local reconstruction operators. All physical tetrahedra must be nondegenerate. Supplied 3D transports must be proper signed permutations and flat on triangles. | |
| `compile_ordinary_frames(vertices,tetrahedra,frames)` gauges exact proper face matches along a dual spanning tree, rejects nontrivial rotational graph holonomy, checks shared edge increments, and constructs the ordinary cochain. It requires a face-connected regular mesh; arbitrary singular graph processing is not implemented. | |
| `no_flip_certificate(J0,J1)` checks, in rational arithmetic, positivity of all leading principal minors of `I - D.T*D`, where `D=(J1-J0)*J0.inv()`. Passing proves positive determinant along the entire segment between the two Jacobians. Failure only means this sufficient criterion did not pass. | |
| `monotonicity_certificate(J0,J1,epsilon)` checks all principal minors of `sym(J1*J0.inv()) - epsilon*I`. Passing every element check implies global injectivity only with the manuscript's additional hypothesis: J0 must come from a known continuous bijective PL reference map onto a convex domain. | |
| ## Complexity and resource limits | |
| The exact implementation is dense. Matrix arithmetic can develop large rational numerators. The lattice search is exhaustive within a proved finite box and can be exponential in harmonic dimension. A resource cap causes a distinct failure, not an approximate certificate. No GPU acceleration, industrial sparse benchmark, or general SDP optimization is included. | |
| ## Rebuilding the manuscript | |
| A standard LaTeX installation with `pdflatex`, Latin Modern, amsmath, amsthm, geometry, hyperref, fancyhdr, booktabs, microtype, and listings is sufficient. From `paper/`, run `pdflatex AHTI_1.0.0.tex` at least twice, until cross-reference warnings disappear. Font files themselves are not distributed. | |
| ## Reproducing outputs | |
| Run the commands in the root README. The JSON examples contain exact fractions as strings. Numerical decoder values can vary slightly across BLAS/platform versions; the exact rational identities should not. Unit-test wall-clock time is not a performance benchmark. `MANIFEST.sha256` verifies the delivered file contents, not mathematical correctness. | |