PAPER-DCQ1: A Discrete-Continuous-Quantum Correspondence

We construct a kinematic correspondence between a finite binary configuration space, a continuous Grassmannian geometry, and associated quantum-geometric carrier spaces. The core construction starts from the six-bit spaceH6 = {±1}6,with 64 configurations, and embeds it into the complex Grassmannian Gr(3, 6) by a phaseencoding map. A key point is that the phase assignment is chosen by a Gray-code rule. Thus changing one bit inside a bit-pair changes the corresponding phase by exactly π/2, while changing both bits changes it by π. This ensures metric compatibility: a pairwise Hamming-style metric on H6 is reproduced exactly by the Grassmannian principal-angle distance between the embedded 3-planes. The Pl¨ucker/Fock carrier naturally associated with the Grassmannian embedding is Λ3(C6), a 20-dimensional vector space. Separately, the three-pair structure of the six-bit space admits a threefold C4-tensor readout (C4)⊗3, whose fully symmetric and fully antisymmetric permutation sectors form the pure Bose–Fermi readout carrierRBF := Sym3(C4) ⊕ Λ3(C4), dimRBF = 20 + 4 = 24. This 24-dimensional readout space is not identified with Λ3(C6); it is a separate statistical readout layer. The determinant line bundle on Gr(3, 6) equips the construction with a Berry connection whose curvature represents an integral first Chern class. Moreover, the finite binary phase data embed into the fourth-root phase subgroup μ34⊂ U(1)3, so that continuous phase data restrict consistently to a finite discrete phase sector. The resulting framework is a kinematic foundation for a discrete–continuous–quantum correspondence. Dynamical, gravitational, and phenomenological interpretations are left to laterwork.