Subject: A Discrete Geometric Approach to Fermion Doubling Suppression via Quasicrystalline Phase-Modulation Invariants
I have formulated a discrete Topological Toy Model based on a close-packed $A_3$ Face-Centered Cubic lattice mesh.
I want to be completely transparent upfront: my professional background is in the fast-food industry, not academic mathematics. Because I lack formal training in advanced computer science and tensor calculus, I have been working intensely with an AI assistant to translate my visual geometric blueprints into hard, computable matrix arrays. After rigorous structural management of my AI tool over the last two weeks, I have hit my absolute educational roadblock. I am presenting my framework here to see if what I have engineered is worthy of your team's time.
The current mission of this toy model is to see if it can successfully satisfy the discrete Ginsparg-Wilson relation and pass or fail the fermion doubling anomaly. Instead of relying on a bunch of heavy, non-local calculus patches, I wanted to try a pure geometric shortcut.
I am proposing that a universal $1/\phi$ Golden Ratio phase-modulation tensor running along the link connections can operate as an automated topological acoustic filter that natively dampens out high-frequency grid-shredding noise. Using this background Golden Ratio filter, the toy model is able to natively derive six exact universal constants—including the speed of light, vacuum impedance, and quantized electrical charge boundaries—as pure geometric step-ratios, completely bypassing standard zero-volume infinity errors.
The proposed framework maps the discrete master finite-difference operator $D$ by distributing the generators of the Artin Braid Group ( $B_3$) under a faithful Burau representation directly across the 12 nearest-neighbor translation vectors ( $\vec{U}_n$) of the isotropic $A_3$ mesh:
$$D = \sum_{n=1}^{12} \mathbf{M}_{n}(\sigma_1, \sigma_2) \cdot e^{i \vec{k} \cdot \vec{U}_n}$$
Under a simultaneous multi-axis quasicrystalline shear and a long-wavelength harmonic scale restriction, the internal matrix invariants appear to lock straight into the spatial packing boundaries. I am actively seeking a computational or graph-theory collaborator to take this explicit operator stencil, interface it with an active hypergraph or cellular automata simulation environment, and run a comprehensive parametric optimization sweep to verify if the configuration space cleanly collapses the discrete identity to absolute zero for a potential co-authored paper.
I am hoping to see if this toy model, in its current form, is correct enough to look deeper into. If this turns out to be a total nothing-burger, I sincerely apologize. If you have any technical questions, I will try my absolute best to answer them within the boundaries of my current training. I have attached the rendered PDF preprint of the core axioms containing all the needed structural information to this point.
Thank you all so much for your time and your open minds.
AI Transparency Disclosure: In compliance with community guidelines, I am openly disclosing my workflow. The foundational geometric axioms, 12-neighbor vertex layouts, and 1/φ phase-modulation parameters of this model are entirely derived from my own visual spatial architectures. To compute the baseline finite-difference loops, I utilized Google Gemini (Advanced Processing Terminal) strictly as an interactive coding and calculation tool.
The verification process consisted of forcing the LLM terminal to execute localized matrix checks constrained strictly to real-world physics constants (deriving c and vacuum impedance as pure grid step-ratios) and auditing the maximum spatial compression equations against the 0.8927 Maximally Random Jammed (MRJ) sphere-packing ceiling to ensure the framework natively avoids division-by-zero singularity errors.
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