Anticipatory Mathematics: Linear Collapse, Certificates, and Canonical Normal Forms

Anticipatory Mathematics is a pre-dynamic framework for determining whether a system of constraints can exist at all, prior to optimization, simulation, control, or measurement. Rather than studying system behavior, it isolates the structural conditions that must be satisfied before any dynamics are meaningful. This work develops the complete linear theory of Anticipatory Mathematics for mixed systems of linear equalities and inequalities. Linear collapse (infeasibility) is shown to be finitely witnessed, algebraically certified, and canonically classifiable. Using Helly-type boundedness, any infeasible system in ℝⁿ is proven to admit a minimal infeasible subsystem of size at most n + 1. A mixed form of Farkas’ lemma provides exact algebraic certificates of collapse, and a one-to-one correspondence is established between minimal infeasible subsystems and support-minimal dual certificates. The theory derives sharp dimension and rank-dependent bounds, showing how equality constraints reduce effective dimensionality and strictly limit the number of inequalities required to witness collapse. Every mixed linear core is further shown to admit a canonical normal form under affine equivalence, decomposing collapse into an affine subspace restriction and a minimal cone-sum contradiction. Explicit constructions demonstrate that all bounds are tight. By unifying bounded collapse, certificate minimality, rank-aware bounds, and canonical normal forms, this paper establishes a complete pre-dynamic foundation for linear system coherence. Anticipatory Mathematics provides a structural layer that precedes and constrains downstream theories in optimization, control, and scientific modeling.