PAPER-DCQ3: The Double Readout Space and Categorical Statistics in the Discrete–Continuous–Quantum Correspondence

Papers DCQ1 and DCQ2 established the kinematic and phase-geometric foundations of the Discrete–Continuous–Quantum correspondence. DCQ1 introduced an isometric phaseencodedembedding H6 = ±16 ↩→ Gr (3, 6), together with its natural Plucker/Fock carrier Λ3 (C6), dimΛ3 (C6) = 20. In its revised form, DCQ1 also distinguished this 20-dimensional Grassmannian carrier from a separate pure Bose–Fermi readout carrier RBF: = Sym3 (C4) ⊕ Λ3 (C4), dimRBF = 20 + 4 = 24, arising from the threefold tensor readout (C4) ⊗3. DCQ2 then developed the phase-orbit geometry N ≃ (CP1) 3, the Berry–Chern class (1, 1, 1), an effective diagonal U (1) reduction to a formal four-dimensional configuration space, and a Morse–thimble ansatz for semiclassical organization. The present paper does not introduce a new fundamental geometry. Instead, it organizes the structures of DCQ1–DCQ2 into a double readout space VDR = VB ⊕ VF, whose two components are modeled on the pure symmetric and antisymmetric readout sectors VB ≃ Sym3 (C4), VF ≃ Λ3 (C4). A Z2-grading τ is used to encode the distinction between Bose-type and Fermi-type readout. This grading is not claimed to derive all spin-statistics physics; rather, it provides a controlled categorical bookkeeping device for separating symmetric and antisymmetric readout channels inside the DCQ framework. We then formulate a finite effective amplitude ansatz on the reduced phase-geometric background. This ansatz should not be read as a full physical quantum-field-theoretic path integral. It is a finite-dimensional, compactly supported effective-amplitude framework designed to organize later dynamical models. Finally, we record possible programmatic extensions, including a future index-theoretic interpretation of spectral and Chern-type quantities. These extensions are explicitly treated as conjectural outlook rather than as results of thepresent paper.