The Oscillating Worldline: Symplectic Involution, Topological Mass Generation, and the Geometric Resolution of Baryon Asymmetry

Abstract

By introducing the Extended Equivalence Principle, this paper proposes a topological framework that reinterprets the kinematic null boundary in Special Relativity (v → c) and the gravitational event horizon in General Relativity (r → rs) as isomorphic manifestations of a universal Null Topological Seam (ds2Σ → 0). We posit that elementary particles do not terminate at these boundaries, but rather undergo a discrete topological transformation of their proper causal flow. By elevating the geometry to the complexified cotangent bundle, we demonstrate that intersecting this null boundary triggers a symplectic involution (Ω → −Ω) governed by the topological Maslov index. This rigorously reverses the canonical Hamiltonian vector flow, acting as an exact geometric CPT inversion into a “Black Mirror” bipartite topology while preserving the background metric (gμν ). Applied to gravitational collapse, the algebraic inversion of the Raychaudhuri expansion scalar (θ → −θ) gracefully bypasses the Penrose singularity theorems, generating exact relativistic repulsive gravity. By preserving global state purity across the conjugate phase sectors, the framework geometrically resolves the eternal Black Hole Information Paradox, predicting anomalous, maximally squeezed pair correlators in the Hawking flux. Crucially, this framework natively derives the Standard Model mass hierarchy and fundamental gauge parameters. Path-integral evaluations of the boundary’s chiral projection dictate a strict trans-phasic transition probability, while evaluating the Bohr-Sommerfeld winding numbers on the doubled SU (3)k × SU (3)−k boundary yields the exact Koide formula vacuum angle (δ = 2/9), shown to hold for on-shell pole masses. Its stability against radiative corrections is formulated as a rigorous 6-point constraint problem for future bulk dynamics. Furthermore, the boundary geometry inherently models the mass-suppressed, chiral-restricted neutrino as a topological zero-mode. Applying Călugăreanu’s theorem (Lk = T w + W r) to the boundary’s chiral projection strictly requires the antichronous return trajectories to generate Writhe, self-braiding into stable SU (3)C Trefoil knots (31). The topological self-energy of this knot natively derives the hadronic mass gap (∼ 1836me). Finally, a WKB expansion of the global wave functional across this bipartite manifold recovers the local Schrödinger equation while satisfying the timeless Wheeler-DeWitt constraint (ˆH_totΨ = 0) as a strict topological symmetry, resolving the Problem of Time. Ultimately, we establish that the requisite cosmological antimatter is not missing, but is geometrically sequestered within this stable hadronic topology, rigorously resolving the Baryon Asymmetry. <p>The chemical formula for water is H₂O.</p>

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