Part IV — Composites

Abstract

Part IV derives the composite sector, family replication, and the pre-fit mass architecture of the Dual–Flux (DF) framework using only the operator content fixed in Parts I–III. No new kernels, parameters, internal quantum numbers, or geometric postulates are introduced. Working entirely on the present surface P₀, we show that sector tensoring combined with present-closure and coercivity of the DF quadratic form leaves a unique admissible and stable three-centre class in the charge-capable sector: the closed antisymmetric tripole subspace inside P_C^⊗3. All other three-body sectorings are either annihilated by closure or fail the stability (gap) criterion.

On this surviving support, the ℤ₃^t torsion cycle enforces exactly three closed, non-degenerate branches (n = 0, 1, 2) for a single composite defect, providing a structural mechanism for three-family replication. Masses are then defined as branch-restricted expectation values of a fixed operator inherited from Parts I–III; restricting this functional to the three-branch torsion orbit yields a canonical real Fourier three-level pattern. In an ideal DF-symmetric limit—where the restricted mass functional is isotropic between the uniform branch mode and the torsion modulation—this pattern reduces to the trigonometric Foot/Koide locus. No numerical fit is performed in this part: the three-branch hierarchy follows from closure and torsion structure alone.

Quark tripoles are expected to deviate strongly from the symmetric limit when the highest torsion branch approaches the boundary of coercivity, producing a non-circulant closed operator and large splittings. Finally, meson- and baryon-like composites, confinement as a closure constraint, and three-branch replication are obtained as structural consequences of the DF projector/closure architecture, without invoking color or interaction-level dynamics. Numerical predictions and coupling identifications are deferred to Part V.