PAPER-FBT01A: Symplectic Pairing, Connection Curvature, and Dual–Phase Readout

The present paper reconstructs the curvature-readout layer of the FBT framework. The reconstruction follows the removal of the primitive physical 4+2 claim from FBT0A and the single-carrier relative-torus architecture of FBT0B v4.x. The foundational coherent carrier remains Bcoh6 = (CP1)3, with product Fubini–Study symplectic form ΩB and a Hamiltonian T3-phase frame. Removing the diagonal common phase defines the abstract contrast quotient Qrel = T3/ΔU(1). This quotient does not automatically act on the original carrier. An actual relative two-torus Krel ⊂ T3 must be selected. On a regular branch where the chosen action is free and proper, it defines the unique active four-dimensional orbit base Krel ↪→ X(6)Regπ −−→ B(4)Reg, B(4)Reg = X(6)Reg/Krel. For the symmetric choice KA2 = ker(z1z2z3), one has KA2/μ3 ≃ Qrel; the acting torus and the abstract quotient are isogenous rather than canonically identical. The principal geometric result of the present paper is that the relative-phase orbit distribution is isotropic with respect to the parent symplectic form: ΩB|V = 0, V = ker dπ. After choosing a Krel-invariant connection A ∈ Ω1(X(6)Reg; krel), one obtains TX(6)Reg = HA ⊕ V, HA = ker A. Relative to this connection-induced splitting, the parent form has the reduced bidegree decomposition ΩB = ωH + η, with no independent vertical symplectic block. The mixed term η ∈ H∗A ∧ V ∗ records the canonical action–phase pairing relative to the chosen polarization. It is a kinematic symplectic pairing and should not by itself be identified with dynamical interaction, bundle twisting, or physical backreaction. The isotropic-fibre geometry also reveals an exact structural identity. Because ΩB = ωH + η, ωH ∈ Λ2,0, η ∈ Λ1,1, and because the horizontal and vertical dimensions are respectively four and two, the Liouville volume satisfies 1/3!Ω∧3B = 1/2 ωH ∧ η ∧ η. Thus the six-dimensional Liouville volume is completely supported by one horizontal pairing and two horizontal–vertical pairings. The ratio previously denoted by ρmix therefore equals unity identically wherever the Liouville volume is nonzero. It is not an independent coupling-strength observable. Instead, it is reclassified as the Liouville mixed-volume saturation identity. The nondegeneracy of ΩB further impliesωH ∧ η ∧ η ̸= 0, so a mixed symplectic pairing of sufficient rank is structurally necessary in any connectioninduced isotropic-fibre 4+2 realization of the six-dimensional carrier. The revision further separates linear mixed readability from quadratic top-order readout. The contraction map η♭V : V −→ H∗A, v ↦−→ ιvη|HA, is injective on every regular symplectic branch and hence has rank two. By contrast, η ↦−→1/2ωH ∧ η∧2 is quadratic and generally non-injective; in particular it is even under η ↦→ −η. This supplies the concrete mixed-geometry interface used by FBT0C: higher-order readout may lose signor orientation-sensitive reconstructibility without deleting a geometric dimension. The revised paper instead distinguishes four geometric layers: ΩB, A, FA = dA, Ωeffphase. They represent, respectively, carrier symplectic pairing, the horizontal comparison rule, its curvature and holonomy, and the effective Berry/prequantum curvature assigned through a specified phase-readout bridge. A nontrivial internal–external physical transduction requires additional constitutive, Hamiltonian, QGT, central-channel, or dynamical data; it is not generated by the mixed component alone. The scalar readout layer is correspondingly reorganized. The global Liouville capacity and reduced phase flux are retained. A genuinely nontrivial local twisting density may be defined only after specifying a four-dimensional readout volume, for example through F(1)A ∧ F(2)A = ρA volread4 . Alternatively, after choosing a QGT-compatible metric, one may use the norm ∥η∥2g read as a metric-dependent measure of kinematic action–phase pairing. Accordingly, the former mixed-volume ratio is not discarded without geometric content. Its identically unit value expresses an intrinsic Liouville-volume reconstruction: ΩB ↦−→ (ωH, η) ↦−→1/2ωH ∧ η ∧ η = 1/3!Ω∧3B . This reconstruction is a kinematic volume-factorisation theorem, not a measure of interaction strength or a derivation of physical spacetime. The resulting paper is therefore not a mixed-density theory. It is a geometric separation theorem and transduction bridge: the carrier form, torus connection, connection curvature, effective phase curvature, and downstream physical readout must be kept at distinct logical levels.