Torsion, Chirality, and Transport in Phase-Spatial Closure From Constitutive Deformation to Covariant Relational Transport

Paper II in the reconstructed coherence-hyperelastic transport sequence Paper I: D = UP → Paper II: 𝒟μD This manuscript develops the transport layer that follows the constitutive closure theory of Paper I. Its central methodological rule is that constitutive strain, connection curvature, relative transport curvature, torsion, and chirality remain distinct mathematical structures even when they are dynamically coupled. Paper I developed a nonlinear constitutive theory of relational closure by decomposing a local relational state as D = UP, with U representing closure-preserving orientation and P representing positive constitutive deformation. The present work develops the next layer: covariant transport of closure. A local closure connection 𝒜μ compares relational states across a base domain. The transported polar state separates into an orientation transport field Ωμ = (𝒟μU)U⁻¹ and a constitutive transport field Lμ = (𝒟μP)P⁻¹. Their sum Ξμ = Ωμ + Lμ transforms covariantly but is not itself a connection. The effective connection is 𝒜eff = 𝒜 + Ξ. Its curvature separates exactly as ℛeff = ℛ + 𝒦, where ℛ is the curvature of the baseline closure connection and 𝒦 = D𝒜Ξ + Ξ∧Ξ is the relative polar-transport curvature. Holonomy of 𝒜eff gives finite path memory. Torsion is introduced only after a coframe or soldering structure is specified, while chirality is represented by an independent parity-odd order parameter χC. A typed coupled action yields mixed stability, inter-sector transfer, chiral branching, and rotation-induced closure branching. The theory does not identify closure curvature with spacetime curvature, torsion with Einstein-Cartan torsion, chirality with weak parity violation, or closure dragging with gravitational frame dragging; it supplies the transport mathematics required before such physical disclosure claims can be posed rigorously. Keywords: closure transport; torsion; chirality; holonomy; phase-spatial closure; partial closure; PSOC4(3); covariant transport; relational curvature; polar decomposition; closure dragging.