Nilpotent Flux as a Governance Law: Falsification, Square-Zero Operators, and Robust Near-Closure

We present a falsification-driven evaluation of the Nilpotent Flux hypothesis within a governance-first framework for adaptive and self-modifying systems. Using explicit diagnostics, we show that nilpotent closure does not emerge generically from unconstrained dynamics, soft projection, or manifold-based smoothing, thereby falsifying strong non-constructive formulations. A hard-clamp construction passes as a control but does not establish natural emergence. We identify a precise operator-level formulation that survives falsification: when the governing update is generated by a square-zero linear operator N satisfying \mathrm{Im}(N)\subseteq\mathrm{Ker}(N), nilpotency (N^2=0) holds exactly by algebra. Under this construction, governed increments evolve within a square-zero flux subspace. Empirical tests across multiple seeds, noise levels, and step sizes demonstrate a broad regime of robust near-closure, defined by bounded increment variation, without hard clamping or state-space extension. These results reframe Nilpotent Flux as a designable governance law rather than a generic emergent phenomenon. The work provides explicit negative results, control constructions, and a minimal surviving formulation, clarifying the scope, limitations, and applicability of nilpotent governance in long-horizon adaptive systems. governed evolution nilpotent operators falsification square-zero operators stability constraints adaptive systems AI governance theoretical framework This release presents a falsification-driven analysis of a nilpotent operator formulation for constrained dynamical evolution. The objective is to determine whether nilpotent closure arises generically or only under specific algebraic conditions. Negative Results We show that nilpotent closure does not emerge generically from unconstrained dynamics, smooth projection, or manifold-based regularization. These regimes fail to suppress second-order accumulation, falsifying interpretations in which nilpotent behavior is assumed to arise naturally. Surviving Formulation A minimal formulation survives falsification: when the evolution increment is generated by a square-zero linear operator N satisfying \mathrm{Im}(N)\subseteq\mathrm{Ker}(N), nilpotency (N^2=0) holds exactly by algebra. Under this construction, the dynamics evolve within a square-zero flux subspace. Stability Numerical tests demonstrate a broad parameter regime in which near-closure of increments is observed under bounded noise and discretization step size. Degradation occurs smoothly outside this regime, indicating that the result is robust rather than fine-tuned. Interpretation Nilpotent behavior is therefore identified as an algebraically enforced property of the evolution operator, not a generic feature of dynamical systems. The included negative results delimit the scope of applicability and provide a precise reference for further theoretical or numerical study. “We falsify generic nilpotent closure in dynamical systems and identify a minimal square-zero operator formulation that enforces exact nilpotency and robust near-closure under bounded perturbations.”