PAPER-FBT21: An Effective Sink-Field Extension of the DCQ Morse Network

This paper develops an effective bridge layer between the affine-central S-sector of the FBT framework, the thermodynamic Morse-network structure of DCQ10, and the fourth temporal layer of PAPER-Ω. The first input is the affine-central picture of FBT02A, where the residual S-sector is realised as a strict affine central-charge sector of the reduced dual-phase current algebra, with microscopic level k =1/2πZΣ2Ωphase. The second input is the thermodynamic reading of the adapted Morse landscape developed in DCQ10 on the six-dimensional phase-orbit manifold N ≃ (CP1) 3. In DCQ10, the 64 finite phase-sector states inherited from H6 = ±16 are prescribed as nondegenerate Morse minima of an adapted landscape. After interpreting the Morse function as an effective energy, the compact partition function Z (β) =ZNe−βf dμN has a low-temperature regime governed by this 64-fold ground-sector geometry. The purpose of the present paper is to connect these structures through an effective coarse-grained sink-field layer. We introduce a local effective field ϕ (x) ∼ ⟨k⟩coarse (x), interpreted as a coarse-grained readout of affine S-level density. The 64 Morse minima are organised as nodes of an effective Morse network, and ϕ (x) is used as a local stability variable controlling the strength of unresolved fluctuation suppression. The paper further clarifies the relation of this construction to PAPER-Ω. In the fourlayer temporal classification of PAPER-Ω, the fourth temporal layer is entropy-ordered cosmological temporality. The present paper supplies the sink-field coarse-graining and entropy-stability side of that layer. It introduces an entropy functional Sϕ =ZCδs (ϕ (x) ) − D2|∇ϕ (x) |2dμ (x), shows that, under strict concavity and fixed total occupancy, uniform ϕ is the unconstrained near-equilibrium reference maximiser of the simplified entropy functional, and records a conditional entropy-monotonicity statement for entropy-compatible gradient-flow evolution. In addition, this version identifies how the third step of the fourth temporal layer is geometrically implemented. DCQ10 supplies a 64-basin Morse ground-sector geometry. A branch-conditioned low-temperature initial state localised near one selected Morse minimum gives a non-uniform, low-entropy sink-field configuration. This provides a geometric pasthypothesis sector for FBT21, while the microscopic reason for selecting one basin and the full escape dynamics through the Morse network remain open problems. This does not by itself prove the full cosmological arrow of time. A complete arrow still requires a microscopic justification of the effective dissipative closure and a derivation of the branch-selection mechanism. Thus FBT21 should be read as the sink-field coarse-graining, geometric-past-hypothesis, and entropy-stability bridge underlying the fourth temporal layer of PAPER-Ω, not as a complete derivation of cosmological irreversibility from affine central charge alone. A further qualification is added in light of FBT16A. Since the underlying DCQ–FBT background is symplectic, the fourth temporal layer should not be interpreted as an absolute one-way law of the microscopic carrier. The microscopic Hamiltonian, phase, or Berrytransport layers may remain reversible, recurrent, or holonomy-rich. The entropy arrow appears only after projection to a finite-resolution coarse-grained branch. Moreover, entropy growth need not imply complete homogenisation: structurally admissible branches may remain inhomogeneous, sectorised, or hierarchically organised while still becoming more entropy-stable. Thus FBT21 should be read as a theory of branchwise coarse-grained entropy orientation, not as a proof of universal relaxation to perfect uniformity. This version also clarifies a further structural point. The fourth temporal layer should not be modelled as a merely closed periodic clock. Local operational carriers, such as the residual S-gate circle of FBT01B and the local Hermitian-line U (1) of FBT05D, are naturally closed: they provide stable, repeatable, and quantisable phase units. By contrast, a global history flow through the full relative dual-phase torus may be non-closed. In particular, the dense weak-resonance Kronecker flows studied in DCQ7B provide a geometric model for stable but non-periodic traversal through closed local phase carriers. Thus the fourth temporal layer is interpreted here according to the principle closed carrier, non-closed trajectory. This principle explains how local gauge closure can coexist with global non-recurrence of the entropy-oriented cosmological branch. The paper remains a bridge paper in a precise sense: it does not replace the microscopic affine-central construction of FBT02A, nor the thermodynamic Morse reading of DCQ10, nor the temporal classification of PAPER-Ω. It connects them through an effective coarsegrained ϕ-field layer.