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 (dsΣ² → 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 (ĤₜₒₜΨ = 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.

Rights & Licensing

License: Creative Commons Attribution 4.0 International (CC BY 4.0)

Copyright: Copyright © 2026 Ashim Nath (ORCiD: 0009-0009-6278-8247)

This manuscript is cryptographically registered via CERN Zenodo repository, preprint: 10.5281/zenodo.21231438 (DOI)
This manuscript is identical to the formally indexed Springer Nature's Research Square preprint: 10.21203/rs.3.rs-8838027/v3 (DOI)

All derivative work or academic application of this framework must provide explicit citation to the author and primary DOIs.

Subject Classifications & Indexing

Mathematical Physics & Topology

Symplectic Geometry | Topological Quantum Field Theory | Knot Theory | Cotangent Bundle Topology | Bipartite Phase Space | Null Boundary Topology | Călugăreanu Theorem | Chern-Simons Theory | Clifford Algebra

High Energy & Particle Physics

Standard Model Mass Hierarchy | Neutrino Mass Generation | Koide Formula | Muon g-2 | Antimatter & CPT Symmetry | Feynman-Stueckelberg Mechanism | Chiral Zero-Modes | Fine Structure Constant

Astrophysics, Cosmology & Gravity

Baryon Asymmetry | Black Hole Information Paradox | Wheeler-DeWitt Equation | The Problem of Time | Geodesic Completeness | Dark Matter | Black Holes & White Holes

Foundational Quantum & Classical Mechanics

Symplectic Involution | Extended Equivalence Principle | One-Fermion Universe | Hamiltonian Mechanics | Quantum Entanglement | Bell's Theorem | WKB Expansion

Cite this work

Manuscript

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Files ashimnath_the_oscillating_worldline_2026_07_07-eA8Mh6VkHFswdtqs.pdf

External Links

Unified Version 7.0: Current Baseline Architecture

July 2026

Status: Active Draft | Ongoing Expansion

• Zenodo: Indexed as v3 (July 7, 2026)

• Research Square: Indexed as v3 (July 8, 2026)

• Notes: Refinement of the geometric resolution. Forms the baseline for forthcoming structural integrations.

Versions 5.0 & 6.0: Permanent Archival Ledger

June – July 2026

Status: Archival Ledgers

• Zenodo: v5 indexed as v1 (June 16, 2026) | v6 indexed as v2 (July 2, 2026)

• Notes: Migration of timeline provenance to CERN infrastructure to establish CC BY 4.0 licensing and ORCiD anchoring.

Version 4.0: The Geometric Transition

May 2026

Status: Major Architectural Upgrade

• Research Square: v4 indexed as v2 (May 19, 2026)

• viXra Redundancy: v4 (May 10, 2026)

• Notes: A fundamental shift in the framework. The core concept transitioned from a purely kinematical description to a formalized geometric and topological architecture.

Versions 1.0, 2.0 & 3.0: Initial Kinematical Framework

December 2025 - February 2026

Status: Conceptual Registration & Initial Indexing

• Research Square: v3 indexed as v1 (February 11, 2026)

• viXra: v1 (December 23, 2025) | v2 (December 29, 2025) | v3 (February 8, 2026)

• Notes: Initial immutable timestamping and development of the core oscillating concept using a kinematical framework prior to full topological formalization.

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