Non-orientable Möbius Algebra and the Computable Origin of the Standard Model

Abstract

Starting from the intuition that a Möbius strip has "one side marked 0 and the other marked 1," this paper sews zero and infinity together along a meridian via 0 ≅ ∞, and explicitly relaxes four implicit assumptions on which Hurwitz's theorem relies — that zero is a global absorbing element, that inverses are unique and global, that multiplication is associative (or alternative), and that the base space is orientable. Within the S¹ × ℤ₂ non-orientable meridian algebra, dimension, zero, and inverse are redefined. Consequently, the sedenion 𝕊 is no longer a pathological extension in the Cayley–Dickson sequence, but a legitimate non-orientable cross-section algebra: its zero-divisor pairs are produced by the local V₄ = ℤ₂ × ℤ₂ symmetry sewn by τ, and counting them yields the twist characteristic number χ_tw = N/2. Working in the twisted de Rham framework of Bott–Tu, we give an explicit realization of the τ-compensated derivative D, the Leibniz correction L_flip, and the deg_M-weighted integral, deriving the non-orientable Stokes theorem and the chiral anomaly coefficient. Taking N = 16 gives χ_tw = 8; under the G₂ → SU(3)_c splitting this forces a three-generation fermion quota plus a right-handed neutrino. The τ-flip amplitude of the real-pair yields the Higgs twist field φ_M, whose potential is locked by non-orientable Stokes to V = Λ⁴(1 − cos 2α). The ratio χ_tw/N = 1/2, fed into the SU(2)_L RGE running, outputs the electroweak scale v ≃ 245 GeV. The paper shows that the algebraic origin of the Standard Model and its vacuum structure are uniquely determined by the Möbius topology via χ_tw.