The Two Closure Regimes of the Displaced Cut in the F_N Möbius Strip Family: Koppa Variation, N-Doubling, and the Cosmological Parameter s*

Abstract:When a Möbius strip F_N is cut at a displaced position s* ≠ 0.5, the two sides of the cut do not naturally coincide after one revolution of 2π. Two distinct physical regimes resolve this incompatibility. In Regime 1 (koppa variation), the field intensity ϙ(t) varies along the cycle such that ϙ(2π)/ϙ(0) = (1−s*)/s*, forcing closure in the first revolution while preserving the original topology N. In Regime 2 (constant koppa), ϙ remains fixed and the system requires a second revolution (4π total) to close, producing an effective doubled torsion N_eff = 2N.These two regimes are extended systematically to all cases N = 1/2, 1, 2, 3, 4, with experimental cross-references for each. For N=1/2 (Singlet Point): Regime 2 generates N_eff=1, providing a geometric mechanism for the transition from the arrival pole to the stable fermion — consistent with N=1/2 never appearing as a free state. For N=1 (Möbius/SU(2)): Regime 1 recovers the Zitterbewegung of the free electron, and a new result emerges — the fine structure constant α⁻¹ = 137 equals the koppa ratio a₀/λ_C of the hydrogen atom in Regime 1. Regime 2 generates N_eff=2, providing a geometric derivation of the spinor double cover without algebraic postulate, verified by Bitter & Dubbers (1987). For N=2 (SU(3)/color): Regime 1 models the running of the strong coupling α_s and the gluon string tension; Regime 2 (N_eff=4) provides a testable hypothesis — the graviton as a photon that did not close in the first revolution, explaining its spin-2 character, masslessness, and non-detection. For N=3 (CP violation): Regime 1 gives s* = 0.5 ± ε_CP/2 = 0.5011, with ϙ(2π)/ϙ(0) = 0.9978 for matter and 1.0022 for antimatter, providing the geometric mechanism for matter survival — consistent with ε_CP(kaon) = 2.228×10⁻³ (PDG 2024), Sangiri & Sarkar (2024), and LHCb baryon CP violation (March 2025, 5.2σ). For N=4 (Higgs/U(1)): Regime 1 models the Higgs mechanism with ϙ_Higgs = v/√2 = 174.1 GeV/c².A central result emerges: the displacement parameter s* is mathematically equivalent to the cosmological matter density parameter Ω_matter, with s* = Ω_m/(Ω_m + Ω_Λ) = 0.315 (Planck 2018). The koppa ratio ϙ(2π)/ϙ(0) = (1−s*)/s* = 2.175 corresponds to cosmic expansion, and the extreme limits s→0 (Big Bang) and s→1 (Big Crunch) follow naturally. The N-doubling chain of Regime 2 (N=1/2→1→2→3→4 generating N_eff=1→2→4→6→8) provides a unified geometric account of why elementary particles with spin > 2 are not observed: N_eff ≥ 6 states require energies beyond current experimental reach.Keywords:displaced cut; koppa variation; N-doubling; closure regimes; Möbius strip; F_N family; spinor double cover; fine structure constant; Zitterbewegung; graviton hypothesis; CP violation; Berry phase; cosmological parameter; Higgs mechanism; Schwinger critical field; SU(2); SU(3); U(1); Standard Model; Unified Theory of Fields and Universal Dynamics