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.