Identifying the unalterable structures governing natural phenomena requires explaining the presence or absence of transformations in complex systems. While continuous approximations like Singular Value Decomposition (SVD) are highly effective for extracting continuous features from noisy data, structural invariants in discrete physical, biological, and socio-technical systems manifest strictly as integer vectors. Extracting these structurally meaningful invariants necessitates finding short basis vectors over an integer lattice — a computationally hard problem analogous to solving homogeneous Diophantine equations. In this paper, we formalize this heuristic search to a rigorous algebraic framework. By modeling discrete state transitions as a general chain complex over Z, we reframe the discovery of conservation laws as the exact computation of the integer zeroth cohomology group (H⁰). We formulate a polynomial-time algorithmic pipeline utilizing Hermite Normal Form (HNF) and Lenstra–Lenstra–Lovász (LLL) lattice basis reduction to extract exact topological generators with provably short coefficient vectors, without arbitrary floating-point rounding. Furthermore, we establish how this cohomological extraction provides the formal mathematical foundation for Topoethics: demonstrating that in complex non-linear dynamical systems, the deletion of a single state variable is not a continuous trajectory shift, but an irreversible structural mutation of the system's geometric constraints.
Yusuke Yokota (Sun,) studied this question.