Get started
Work

Compositional Split-Complex Rapidity Residue and the Minimal SO(1,1)+ Factor in Complexified Synthetic Differential Geometry

Year: 2026
Type: preprint
Abstract: We develop the noncompact rapidity-sector analog of the U(1) phase result of the companion paper: in synthetic differential geometry (SDG) with first-order nilpotent infinitesimals over the split-complexified line C_sp = R[j]/(j² - 1), a microlinear residue homomorphism from the based closed-traversal monoid of a microlinear pointed substrate (T, e_T) with tangent module a microlinear C_sp-module V, satisfying compositionality, microlinear continuity, invertibility, nontriviality, noncompact rapidity closure in the identity component, second-order commutator-triviality, and infinitesimal generation, factors through a microlinear Lie group whose minimal nontrivial noncompact connected projected factor is (R, +) ≅ SO(1,1)+ in split-complex unit-hyperbola representation. Two scope cautions are foregrounded throughout. First, SO(1,1)+ is placed in the post-geometric rapidity / boost layer, downstream of P-geometric admission — NOT alongside the U(1) phase residue of the companion paper at the source-side H-phase layer. The CT 4.0 ordering "U(1) → SO/Lorentz/Spin" is an ordering of layers, not an identification of value-spaces. Second, SO(1,1)+ is the one-dimensional boost group; full Lorentz groups in higher dimensions are non-abelian (Thomas–Wigner rotation obstruction). The 1-D rapidity-sector cocycle closure given here is a toy model for the Lorentz layer, NOT a derivation of the full multi-subject Lorentz cocycle theorem of CT 4.0 §10. The paper exhibits the substrate-grounded proof pattern across two members of the 2D real algebra family R[x]/(x² - c) for c ∈ {-1, +1}, both supported by the dual-number (c = 0) substrate structure. The compact U(1) case is developed in the companion paper (P1, DOI 10.5281/zenodo.20326522); this paper develops the noncompact (R, +) ≅ SO(1,1)+ case in the post-geometric rapidity sector.
DOI: 10.5281/zenodo.20326530
Language: English
Open Access status: green
OpenAlex ID: W7162002806
Cites:
Cited by:
Related to: