Second-Order Constraint Geometry

Second-Order Constraint Geometry (Σ₂) introduces a diagnostic framework for identifying when state-based descriptions are structurally insufficient, even though they remain predictively accurate. The paper shows that in certain physically realizable regimes, admissibility of future operations depends on ordered history rather than instantaneous state alone. A symmetry-based definition of faithful state representation is introduced, and a geometric constraint space over admissible histories (Σ₂) is formalized. A fully worked two-qubit Hamiltonian example demonstrates physical realizability within standard unitary quantum mechanics: two operationally indistinguishable boundary states yield divergent post-boundary diagnostics under identical evolution. Any state augmentation restoring prediction necessarily breaks operationally verified symmetries. The framework is non-dynamical and non-mechanistic. It does not modify quantum mechanics or introduce hidden variables. Instead, it classifies when trajectory-level constraints become unavoidable and provides falsifiable operational diagnostics. Applications include non-Markovian dynamics, decoherence, measurement asymmetries, and irreversibility.