Unification of Topological Phase Transitions: Subatomic Creation, Gravitational Eversions and Non-Bizarre Field Mechanics

This article presents a framework that links non-trivial topological geometries - the Mobius strip and its generalizations with N twists - to phase transitions, particle creation and the eversions of gravitational collapse. The length-to-width ratio of the strip, L/W, organizes three thresholds: sqrt(3), 3 sqrt(3) and 4 sqrt(3). The first is the infimum of existence of the paper Mobius band, proved by Schwartz (2023), and corresponds to the triangular fundamental domain; the second coincides with the critical impact parameter of light around a black hole, b_c = 3 sqrt(3) GM/c^2; the third marks the supercritical regime. The 90-degree phase quadrature (i = exp(i pi/2)), distributed over the physical twist ramp, turns the abrupt coordinate reversal (X to -X) into continuous 720-degree spinor trajectories. The topological origin of spin and statistics is derived (phase exp(i pi N) for N half-twists), and it is shown that a wave confined to the Mobius strip has, on the same manifold, modes with integer and half-integer quantum numbers, separated by transverse parity - the joint generation of the bosonic and fermionic families. It is also shown that the tapered throat of the triangular domain acts as a spectral sieve: each frequency stops at its own turning point, deeper for higher frequencies. The geometry is extended to stellar collapse: above a threshold, a topological half-eversion (a fold in Mercator projection) connects the central region to the exterior vacuum, in a bounce model without singularity, consistent with recent results on regular black-hole interiors. Pair creation, the relation between the mass and the radius of the proton (m_p r_p c / hbar approximately equal to 4) and the one-electron topology of Wheeler and Feynman are discussed. .Status of the results. Each section carries a label: [A] derived result, numerically verified resultor known theorem; [B] hypothesis with mechanism, prediction and a way to fail; [C] conjecture.Section 9 collects the predictions.Keywords: Möbius topology; phase transitions; quantum gravity; gravitational eversion; one-electron universe; phase quadrature; vacuum energy; optimal Möbius band; spin-statistics.