A Pregeometric ℤ₂-Deck Residue Formalism via Whitney-Immersed Klein Bottles

Abstract

We develop a pregeometric topological formalism in which four-dimensional observables are not postulated as fundamental dynamical variables, but defined as the residue of a Whitney immersion followed by a Deck projection. The pregeometric layer consists of a zero-dimensional Euler-spiral limit point ��₀ whose automorphism group is ℤ₂, generated by an involution g with g² = id. From this logical-algebraic seed we construct, in succession, a non-orientable Klein-bottle carrier K via Whitney immersion ι: K ↪ ℝ⁴, a Deck double cover π_g: M₄ → M₄/⟨g⟩, and a projection residue δO = ιO − gιO. The electromagnetic winding number W_EM^K is explicitly not introduced as an axiom; it is elevated to an open mathematical lemma (Ω) whose establishment would close the chain from pregeometry to the fine-structure constant. As a zero-order corollary conditional on Lemma Ω, α⁻¹ = π(4π² + π + 1) ≈ 137.0363037759 deviates from the CODATA-2018 central value 137.035999177(21) by 2.22 ppm, an offset we attribute to higher-order Deck projection residues not captured at zero order. Five falsifiable experimental thresholds are stated, including a hard > 10 ppm veto line on α⁻¹. Two genuine prior-predictive signatures—collective excitation near 21.87 GeV and cosmic-void negative weak-lensing—are presented together with symbolic scaling relations and well-defined observational exclusion criteria.