TMD-Hash v9: Arithmetization-Oriented Cryptographic Hashing via Rigorous DEC Laplacians and ZK-Optimized Partial Rounds

We present TMD-Hash v9, an arithmetization-oriented cryptographic hash function engineered as a hardware-software co-design for Zero-Knowledge proof systems (ZK-SNARKs/STARKs) and native CPU execution. TMD-Hash v9 resolves the paradox between hardware execution speed and ZK constraint efficiency by combining a sparse graph-theoretic linear diffusion layer with minimized state geometries (|V| ∈ {12, 24}) and a Partial-Round Feistel execution strategy. Linear mixing is driven by a strictly regularized Shifted Hodge-Laplacian operator evaluated on explicit bipartite Ramanujan graphs. We formally prove absolute non-singularity over F_p (p = 2^64 - 59) using bounding spectral determinant analysis, eliminating heuristic assumptions. To enforce wide-trail active branch bounds while mitigating Sparse Gröbner basis attacks, local 4x4 Maximum Distance Separable (MDS) Cauchy matrices induce rapid algebraic densification. All structural matrices and round constants are generated via a rigorous Nothing-Up-My-Sleeve (NUMS) methodology utilizing SHAKE-256. Non-linear substitution utilizes unconditionally bijective tri-monomial maps (x^3, x^5, x^7 (mod p)). By integrating Mixed-Integer Linear Programming (MILP) validated branch bounds, TMD-Hash v9 achieves an overwhelming security margin against differential, linear, and algebraic cryptanalysis while maintaining a minimal non-linear constraint footprint.