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MS2: Phase Synchronization and Matrix Mechanics in a Discrete TEGR Lattice

Jonathan Fisher

Published Jul 1, 2026• 0 citations
Publisher

Abstract

While continuous field formulations of the Teleparallel Equivalent of General Relativity (TEGR) successfully describe macroscopic, localized classical gravity (as detailed in Manuscript 1), simulating quantum entanglement requires a computational architecture capable of bypassing local causal boundaries ($v \le c$). In this paper, we extend the local Weitzenböck computational engine by introducing the "Super-Matrix"—a non-local graph-theory extension utilizing an Entanglement Adjacency Tensor ($W_{ij}$). By applying a Kuramoto phase-synchronization algorithm across this tensor, we demonstrate how distant topological defects can maintain instantaneous internal clock coherence (EPR) while manifesting geometric "rigid rod" connections via spin-vorticity coupling (ER). Furthermore, we detail the dynamic decoherence threshold that severs this tensor connection under high mechanical strain, providing a computational mechanism for wavefunction collapse and non-local topological severance without spatial curvature.

DOI
10.5281/zenodo.21252170

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While continuous field formulations of the Teleparallel Equivalent of General Relativity (TEGR) successfully describe macroscopic, localized classical gravity (as detailed in Manuscript 1), simulating quantum entanglement requires a computational architecture capable of bypassing local causal boundaries (v≤cv \le cv≤c). In this paper, we extend the local Weitzenböck computational engine by introducing the "Super-Matrix"—a non-local graph-theory extension utilizing an Entanglement Adjacency Tensor (WijW_{ij}Wij​). By applying a Kuramoto phase-synchronization algorithm across this tensor, we demonstrate how distant topological defects can maintain instantaneous internal clock coherence (EPR) while manifesting geometric "rigid rod" connections via spin-vorticity coupling (ER). Furthermore, we detail the dynamic decoherence threshold that severs this tensor connection under high mechanical strain, providing a computational mechanism for wavefunction collapse and non-local topological severance without spatial curvature.
The formulation of a background-independent quantum gravity framework necessitates the reconciliation of classical geometric locality with quantum non-locality. Building upon the Z-Canvas framework—where macroscopic 3D spacetime and Newtonian inertia strictly emerge from the real, symmetric subgraph governed by the non-Markovian Master Equation of Z—this Letter investigates the fundamental topological substrate of quantum mechanics. We explicitly introduce the complete complex Hermitian adjacency matrix, dedicating our analysis entirely to the antisymmetric, imaginary quantum phase-graph. By decoupling topological phase correlations from the causal transport limits of the real geometric scaffold, we provide an exact graph-theoretic derivation of quantum non-locality. We demonstrate that apparent "spooky action at a distance" is a natural feature of dual-topology propagation, formally bridging Bell inequality violations without breaking macroscopic relativistic causality. Furthermore, we establish a mathematically rigorous realization of the Susskind-Maldacena ER=EPR conjecture, proving that highly entangled bipartite states intrinsically deform the local real geometry. Finally, we derive an objective mechanism for quantum decoherence yielding globally unitary, falsifiable bounds for macroscopic superpositions via the Joos-Zeh scattering limit.
Abstract Standard quantum mechanics often describes entanglement as a non-local correlation, presenting persistent conceptual conflicts with relativistic causality. This paper proposes a heuristic geometric hypothesis within a constrained 3+1+2 multitemporal manifold, reinterpreting entanglement not as superluminal information transfer, but as a deterministic consequence of topological continuity across extra transverse-time dimensions. Within this theoretical framework, we hypothesize two distinct geometric mechanisms depending on particle topology. For massless bosons, modeled as open helical solitons, entanglement is proposed as the continuous bifurcation of a single phase-wavefront. This macroscopic splitting conceptually accounts for the energy bisection and wavelength doubling observed in spontaneous parametric down-conversion (SPDC), as the overarching topological tension is distributed across two macroscopic projections while remaining unbroken in the extra dimensions. Conversely, for massive fermions, which are modeled as closed topological rigid bodies, we propose that macroscopic spatial separation (Δx ≠ 0) can mathematically coexist with coordinate overlap in the 2D transverse-time plane (Δτ = 0). It is hypothesized that these rigid particles are strictly meshed together at a singular coordinate within the transverse-time dimension. This direct interlocking is strictly governed by the Pauli exclusion principle, which mandates that the two topological "gears" must mesh with opposite, complementary rotations to avoid metric rupture. Furthermore, SL(4,ℝ) tensor projections mathematically guarantee that this shared geometric state persists independently of any macroscopic spatial distancing. Finally, we suggest that "wave-function collapse" can be reinterpreted as a forced geometric alignment induced by macroscopic mass ("phase-anchors"). We hypothesize that quantum superposition arises from the transverse-time fluctuations of low-mass particles. Upon contact with a phase-anchor, the Pauli exclusion principle dictates a temporary structural meshing that synchronizes the particle's mass with the macroscopic cluster. This instantaneously forces the fluctuating particle to align with the overarching Meta-Time axis, deterministically halting the superposition and severing the entanglement.
This manuscript establishes the microscopic dynamics (Phase B) of the Information-Topological Register Model. Building upon the macroscopic derivation of gravity, mass-energy equivalence, and the Schwarzschild metric (Works 1-3), this paper resolves the conceptual conflict between continuous classical kinematic forces and the discrete quantum probability current. By formalizing physical matter as an objective, phase-encoded information amplitude within a 1D topological lattice, this framework moves beyond the epistemic limitations of the Copenhagen interpretation, aligning with an advanced, deterministically derivable formulation of pilot-wave dynamics. Key Derivations and Theoretical Milestones: • Phase-Encoded Kinematics: We demonstrate that the topological register does not operate via classical Newtonian forces (F=ma), but executes spatial address updates via phase-gradient descent: ∇ Im(ln Ψ). • The Topological Guidance Equation: We derive the exact dynamical law for multiparticle states. The quantum mechanical momentum is redefined as an emergent, guided vector field, deterministically regulated by a topological correlation scalar. • The Decoherence Limit: The model smoothly recovers classical, local field theory. As topological entanglement collapses (η → 0), the guidance momentum decays exponentially beyond the system's intrinsic coherence length. • Deterministic Resolution of the EPR Paradox: The framework provides a rigorous geometric solution to quantum non-locality. We analytically prove that at extreme entanglement (η → 1), the geometric distance parameter (d_geo) is mathematically annihilated from the dynamic equation. The "spooky action at a distance" is thus explained not as a physical anomaly, but as a systematic geometric requirement of the 1D information-topological background manifold. This paper formally concludes the synthesis of Information Topology and modern physics, providing an analytic bridge between local macroscopic isolation and non-local microscopic entanglement.
Quantum entanglement presents a fundamental challenge to relativistic causality, as measurements on one particle instantaneously affect its entangled partner regard- less of spatial separation. We propose a complete theoretical framework—Dynamical Scalar Field Theory (DSFT)—in which entanglement is not a nonlocal correlation but a topological remnant of the pre-Big Bang Euclidean phase. In this framework, two entangled particles reside at a single point in the scalar field configuration space, connected by a static topological soliton (phase-locking filament) where the classi- cal time dimension remains frozen (Ω = 0). We derive the full Lagrangian from first principles, including non-minimal coupling to matter and curvature, and demon- strate that the decoherence process corresponds to a sudden geometric phase tran- sition (Ω : 0 → 1) occurring within ∆tjump = 85.2 ± 0.4 attoseconds—independent of spatial distance. Our numerical simulations yield precise predictions: a thermal pulse of power Pthermal = 2.36 × 10−4 W, an entropy jump ∆S = 1.46 × 10−22 J/K, and a high-frequency gravitational wave signature at f ∼ 106 Hz with strain ampli- tude h ∼ 10−38. These predictions align quantitatively with unexplained noise in superconducting qubits (IBM/Google) and attosecond time delays in photoionization experiments (Max Planck 2024). We propose a definitive experimental protocol us- ing Ramsey interferometry and TES bolometry to test the distance-independence of the decoherence time—a unique and falsifiable prediction of DSFT. This work estab- lishes entanglement as a manifestation of un-crystallized spacetime geometry, resolvable through attosecond and quantum thermodynamics experiments
Abstract Within the Generalized Mass as Twisted Time (G-MaTT) framework, we propose a candidate geometric mechanism for quantum entanglement. Systems that share the same braid topology and phase synchronization within a coherent domain exhibit non-local correlations, not through superluminal signaling, but through their common geometric substrate in the G₂ root lattice. We derive a characteristic entanglement relaxation time from the TUT window dynamics. This timescale is bounded below by the Planck time and is independent of spatial separation. The resulting picture is fully consistent with standard quantum entanglement observations while providing a deeper ontological foundation: entanglement as shared geometric identity rather than mysterious action at a distance. This mechanism does not violate causality and reframes non-locality as a natural consequence of the pre-spacetime G₂ geometry.
The phenomenon of quantum entanglement and its apparent non-locality have challenged the boundaries of classical physics and Einstein’s special relativity for nearly a century. Mainstream interpretations often rely on the ambiguous concept of ”spooky action at a distance” or superluminal communication. This paper completely dismantles the non-locality paradox by shifting the paradigm from a 3dimensional spatial background to an infinite-dimensional non-commutative supermanifold governed by Rough Operator Algebra (ROA) and the Seonggil Theory of Composite Torsion (STCT). We prove that entangled particles are not distinct entities separated by macroscopic distance, but are geometrically bound as a single ”Topological Knot” at a continuous fractional dynamic rank (ν) within the Hypertensor space. Furthermore, we demonstrate that wave-function collapse uponmeasurement is not a transmission of information, but a simultaneous geometric shadow cast by the Tensor Projection Operator (P_4D), triggered by a topological shockwave from the collision of macroscopic and microscopic roughness indices(α). Consequently, absolute quantum unitarity and local causality are flawlessly preserved without violating the speed of light limit.