Part V — Interactions

Abstract

Part V develops the interaction sector of the Dual–Flux (DF) framework exclusively from the operator and sector structure established in Parts I–IV. No new kernels, scales, or sector-specific parameters are introduced beyond the universal leakage bound ε_leak. All interaction effects are realised as closure-induced operators on the present surface P₀, acting on the closed sectors selected by DF projectors.

We first show that multi-centre closure generically fails to be additive: projected sums of admissible single-centre excitations generate bilinear and trilinear couplings on P₀ as the non-additive remainder of closure. This yields a minimal and universal vertex family governed by the same operator ingredients as the single-centre quadratic form. Within this trace-preserving closed-sector formalism, Ward-type identities arise as projector relations, encoding charge and current conservation without postulating gauge symmetry. Sector redundancies then induce a unique effective compact action on closed states with (1,2,3) block structure, naturally identified with U(1) × SU(2) × SU(3). Color singlets coincide with antisymmetric tripole closure; non-singlet configurations have vanishing closed support on P₀, giving confinement as a support property rather than a postulated potential.

Electric charge is defined intrinsically as coherent absorption of the flux Φ⁺ on P₀, and the decomposition Q = T₃ + Y/2 follows as a projector identity. Anomaly-like cancellation conditions appear as exact trace equalities on admissible closed sectors. In the bosonic sector, the scalar W₀ and the neutral metric-long spin–2 family arise as spectral branches of the same DF kernel; their mass ratios are fixed by restricted spectra and the long-memory branch partition determined in Part III.

Electroweak vector masses and the scalar curvature scale are anchored by a single stiffness normalization (e.g. G_F) in the numerical implementation. Mixing matrices emerge from small inter-sector leakage on P₀, while observable CP phases require the combined effect of torsion and leakage and scale as φ_CP = O(Δt ε_leak). Finally, the memory-depth response of the kernel induces a running pattern compatible with asymptotic freedom in the color face and screening in the Abelian face, together with parameter-free inequalities among effective strengths.

Overall, Part V shows that DF interactions, effective gauge structure, and bosonic spectra are not independent ingredients but consequences of present-closure and sector geometry, tightly linked to the scales already fixed in the mass sector.