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 = {±1}6 ↩→ 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 the
present paper.