The Minimal Stable Shell: A Parity-to-Duality Unification of Discrete Structures — A ℤ2-Lower-Bound Framework Bridging Mathematics and Engineering

This work develops a minimal unifying framework for discrete structures by progressively removing assumptions until structural collapse is imminent but avoided.Rather than attempting to unify all of discrete mathematics, it identifies a lowest stable layer at which structural coherence is preserved. All constructions are carried out over Z2, without introducing order, metric, weight, orientation, or exact counting.Within this setting, symmetric difference, parity projection, boundary operators, kernel–image decomposition, homology, and orthogonal duality arise necessarily and canonically. Combinatorial parity arguments, graph-theoretic cycle–cut structure, mod-2 homology, Kirchhoff-type conservation laws, and system-level engineering constraints are shown to be isomorphic manifestations of the same underlying structure. The result is not a comprehensive theory of discrete mathematics, but a minimal stable shell:the last level at which decomposition, invariants, duality, and algorithmic decidability coexist without auxiliary assumptions. This framework is intended both as a structural reference for discrete mathematics and as a diagnostic tool for engineered systems where controllability, consistency, and rollback depend on topological constraints.