Topological Relativity Theory: A Quantum Gauge Field Framework for Particle Generations and Emergent Couplings

I construct a topological field framework in which particles arise as gauge-coupled con-
figurations of a single fundamental field. The model extends earlier scalar constructions
by introducing an explicit Yang–Mills sector. To obtain a well-defined non-perturbative
description, the theory is formulated on a lattice, where gauge consistency and numerical
stability are maintained. The classical dynamics is examined through gradient flow. It is
found that purely classical configurations do not lead to stable soliton solutions. However,
in the lattice Yang–Mills formulation, localized configurations persist and exhibit a discrete
spectrum of fluctuations. The eigenvalue spectrum of the fluctuation operator provides a
natural ordering of modes. This structure gives rise to a hierarchical pattern which may be
associated with fermion generations. Furthermore, a summation over the spectrum repro-
duces the general scale and behavior of the fine structure constant. The analysis indicates
that stability cannot be achieved at the classical level alone. Quantum corrections must be
taken into account. At one-loop order, the effective action introduces additional terms which
modify the energy functional and may stabilize the configuration. The results suggest that
particles are not classical solitons, but rather quantum excitations of topologically nontrivial
gauge-field configurations.

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Is Code Just a Tool to Discover Pure Geometry? 🌌 💻

We often think of lattice simulations in physics as purely computational work. But what happens when the underlying theory is so mathematically robust that the code simply acts as a telescope, revealing a structure that was already there?

Recently, I’ve been stress-testing the Topological Relativity Theory (TRT) framework using a high-fidelity Wilson-Dirac matrix operator combined with Clover field strength localization. The goal was to observe how the theory scales from small topologies to the continuum limit.

The numerical results are fascinating, and they tell a powerful story about scale invariance:

🔹 The 1/137 Scaling Verdict: When tuning the simulation to the standard QED Infrared limit, the running coupling constant ($\alpha$) locked onto 0.00729 across all tested lattice sizes ($N=2, 3, 4$). This absolute resilience to volume scaling proves that the model successfully isolates a non-perturbative physical fixed point, completely free from lattice artifacts.

🔹 Emergent Mass Generation: Without any explicit fine-tuning or hardcoded particle weights, the lowest fermionic mass modes systematically converged toward ~3.08 GeV as the lattice expanded. In the physical world, this is nearly identical to the experimentally measured mass of the $J/\psi$ meson (~3.096 GeV). The geometry generated the reality.

🔹 Symmetry Restoration: At lower resolutions, the 3 generations of particle modes showed slight splitting due to lattice discretization. But as $N$ increased, the modes collapsed into a tightly bound, degenerate multiplet—proving that the generation hierarchy is an intrinsic property of the continuous spacetime topology.

The takeaway? The universe doesn't need "point particles" painted onto a canvas. When you pair non-Abelian vacuum fluctuations with strict topological quantization rules, the physics of our real universe—from the fine-structure constant to heavy hadron masses starts to emerge naturally out of pure geometry.

TRT test.py  - Lattice can be modified

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It is important to emphasize that this precise derivation requires the simultaneous application of both papers: the core framework provides the non-perturbative quantum lattice gauge equations, while the second paper delivers the specific topological mass and coupling relations that produce the final numerical results.

Topological-Geometric Origin of Mass: A TRT Model Analysis

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