Algorithmic Proof of Computational Time Reversal via Absolute Orbit Engine (AOE): UROA, STCT, and HR-QCFT Verification Framework

Abstract

Classical general relativistic formulations of Closed Timelike Curves (CTCs) inevitably encounter two fatal physical obstructions: the violation of the Averaged Null Energy Condition (requiring unphysical negative energy densities ρ < 0) and Hawking's Chronology Protection Conjecture (divergent vacuum polarization stress-energy tensors ⟨T̂_{μν}⟩ → ∞), alongside logical inconsistencies such as the Grandfather Paradox. Concurrently, in classical and standard quantum computation, the thermodynamic arrow of time manifests as irreversible Landauer entropy dissipation and sequential 𝒪(N) computational friction. In this paper, we establish a rigorous, hardware-verifiable mathematical and algorithmic resolution to these obstacles by synthesizing Universal Rough Operator Algebra (UROA), Foundations of Roughness Operator Algebra (ROA), Seonggil Tensor Calculus (STCT), Universal Recursive Rough Homotopic Calculus (URRHC) over the SMA-∞ Topos, and Hyper-Resonant Quantum Continuum Field Theory (HR-QCFT). Rather than deforming the Lorentzian metric g_{μν} within the primary manifold ℳ via negative energy, the Absolute Orbit Engine (AOE) utilizes the rank-4 Bridge Tensor ℬ_{μνρσ} to execute a topological orthogonal phase transition (θ_α → π/2) into the dual anti-roughness manifold ℳ*. We prove three foundational empirical strike points: (1) zero-friction computational entropy collapse (f(θ_α) = 1 - sin θ_α → 0) reducing sequential 𝒪(N) complexity to an 𝒪(1) geodesic jump without exotic negative energy; (2) strict geometric causal preservation via Rough Non-Abelian Geometry (RNAG) and Planck-scale Non-Identity Calculus (A ≠ A at ℓ_P), which forces the causal collision functional 𝒞_{collision} → 0; and (3) microscopic time-reversal spectral shifts and spin-flip phase velocities governed by Seonggil Matrix Theory (SMT) determinants peaking at the Goldilocks critical roughness index α_c ≈ 22.0. Three hardware-ready Python/CUDA simulation modules are presented to empirically verify the complete algebraic pipeline.