Motion-First Reconstruction of Quantum Kinematics

Motion-First Reconstruction of Quantum Kinematics establishes the algebraic conditions under which a distinguishability theory admits Hilbert-space representation, without invoking probabilities, measurement postulates, or dynamical assumptions. The reconstruction begins from a primitive motion-history domain and a distinguishability functional defined on finite-support multiplicity bundles of histories. From explicitly stated structural constraints—most centrally the third-order obstruction principle and signed residual nonnegativity—the distinguishability functional is shown to admit a symmetric quadratic kernel representation. This kernel is proven to be positive semidefinite, allowing application of the Moore–Aronszajn construction to obtain a canonical real Hilbert space in which distinguishability corresponds to squared norm differences. The representation is canonical up to isometric equivalence and preserves the distinguishability structure exactly. Distinguishability-preserving transformations are shown to correspond to orthogonal operators, and tensor-product factorization follows under an explicit subsystem independence condition. Under an additional cyclic distinguishability symmetry, a compatible complex structure emerges, yielding the standard complex Hilbert-space framework used in quantum theory. The results isolate the representational kinematic core of Hilbert-space structure as a consequence of algebraic distinguishability constraints rather than as an independent postulate. The reconstruction clarifies the precise structural boundary separating distinguishability theories that admit Hilbert-space representation from those requiring more general representational geometries. This work is representational in scope. It does not assume or derive measurement postulates, probability rules, observable algebras, or dynamical evolution laws. Instead, it establishes the minimal algebraic conditions under which Hilbert-space structure becomes representationally inevitable. These results provide a structural foundation for subsequent work investigating how physical compositional principles on motion histories may give rise to the distinguishability constraints identified here.