From Relation to Hilbert Space. Reduction Before Quantization in Bilocal Geometry.

Previous works have shown that bilocal constraint geometry naturally gives rise to twodistinct geometric structures.The radial sector generates a distinguished one-dimensional characteristic flowinterpreted as relational time, whereas the angular sector reduces to a symplecticorbit carrying an internal Hamiltonian symmetry. The present work investigates the next natural step of this construction.Instead of interpreting the reduced orbit merely as a classical internal state space,we propose that it should be regarded as the classical precursor of a quantum statespace. Starting from the reduced symplectic orbit obtained by presymplectic reduction,we show that its geometric quantization naturally leads to a Hilbert space associatedwith every bilocal relation.After localization over spacetime, these Hilbert spaces form a bundle of quantumstates whose sections play the role of wave functions. Within this framework, neither time, gauge fields, nor Hilbert space are introducedas fundamental structures.All three emerge from the geometry of bilocal constraint orbits through differentstages of the same relational construction.