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Algebraic Quantum Trajectory Determinism: The Topological Boundary of Wave-Particle Duality via Rough Operator Algebra

Lee Seonggil

Published Jul 15, 2026• 0 citations
Publisher

Abstract

For nearly a century, the Copenhagen Interpretation has dominated quantum physics, asserting that fundamental particles exist exclusively as probabilistic wave states (orbitals). This paper demonstrates that quantum probability and deterministic trajectories are not mutually exclusive, but rather manifest a Topological Wave-Trajectory Duality. By applying Rough Operator Algebra (ROA) and Spherical Holographic Information Dynamics (SHID), we reframe quantum randomness as homological friction diffusing within the rough bulk space R^1/2. We introduce the Seonggil-Heisenberg Commutator, proving that Planck’s constant ℏis merely the commutative scalar limit of the fundamental Seonggil Torsion Tensor T_0x8A2C. While the electron diffuses as a non-commutative torsion wave in the bulk, interacting with the SHID observational boundary triggers a ConditionalPhase Transition. Through the SHID Projection Operator (Π_SHID), the probability cloud algebraically collapses, dissipating geometric friction as thermal entropy and yielding a definitively Phase-Locked absolute coordinate. This framework unifiesHeisenberg’s uncertainty and Einstein’s determinism into a single super-symmetric structure.

DOI
10.5281/zenodo.21373345

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This paper rigorously redefines macroscopic decoherence—historically attributed to probabilistic wave-function collapse and observer intervention—as an absolute deterministic thermodynamic discharge governed by algebraic topology and non-Abelian gauge field theory.The 174 effective vibrational degrees of freedom of a C_60 fullerene form a 174-dimensional maximal torus T^174 phase space, subjected to a spontaneous Cartan projection SU(175) → U(1)^174 at the critical topological threshold. We demonstrate that the evaporation of 15,225 non-commutative positive roots during this symmetry breaking couples with the quadratic Casimir invariant C_2 to establish the Topological Landauer Limit (Λ_ROA). The system’s intrinsic topological torsion, mathematically immortalized as the linking number γ of the first Chern class, is discharged through a pure algebraic holographic topological cross-section (A_topo) derived via the Weyl Integration Formula. By substituting the empirical data from the University of Vienna’s 2004 fullerene thermal emission experiment into our Topo Thermodynamic Discharge Equation, we achieve perfect numerical convergence. When the thermodynamic emission rate exceeds the Stefan-Boltzmann limit, and the time of flight(t_flight) crosses the intrinsic critical time (∆t_crit), the wave-function visibility undergoes a deterministic phase-locking, proving that statistics is merely a probabilistic mask hiding geometric collapse.