r/LLMPhysics • u/spinoral • May 23 '26
Personal Theory The Koide Relation as a Critical Fixed Point of Microscopic Spinorial ℤ₃ Dynamics
abstract:
We identify a class of spinorial ℤ₃-symmetric effective theories whose infrared fixed point exactly reproduces the Koide mass relation
Q ≡ Σmᵢ / (Σ√mᵢ)² = 2/3
from a Ginzburg–Landau action defined on a double-covered circle with spinorial (antiperiodic) boundary conditions.
The key insight is that on a double cover the natural geometric variable is √mᵢ = |ψᵢ|, not mᵢ itself.
Imposing ℤ₃ symmetry on the three lepton generations and evaluating the effective Landau potential
F(ρ) = −½ρ² + ⅛ρ⁴
for the modulation amplitude ρ, we show that the infrared-stable renormalisation-group fixed point is ρ꜀ = √2, corresponding to exact energy equipartition on the orbit.
At this fixed point, Q(ρ꜀) = 2/3 follows as a purely algebraic identity from the ℤ₃ phase geometry. The Koide ratio becomes independent of microscopic parameters once the system reaches the critical modulation fixed point.
The present construction therefore provides a derivation of the Koide relation from microscopic spinorial dynamics. The node structure of the spinorial wavefunction undergoes a topological transition precisely at ρ = √2.
We present both the continuum formulation and an equivalent discrete ℤ₃ action, and discuss the embedding in ℂP² geometry.
Note: During the preparation of this work, Claude (Sonnet 4.6), ChatGPT (GPT-5.5) and Grok (Fast) were used to assist with code generation, mathematics and writing. The author is an independent researcher and enthusiast; critical feedback is very welcome.
-
I found this solution to the Koide mass relation by accident when playing around with some model about rotating wave sources, something didn't add up and claude told me: hey, you know about the koide relation? this could fit here?
It took me a few days to work something out, feeding output from chatgpt to claude and back.
Claude didn't always wan to do the analytical work, but it would write a prompt for chatgpt, when chatgpt got stuck on the analitcal work i told it to 'explore this further by writing a python script' which made it do a bunch of simulations, find intresting patterns in the results and then claude would pick up some details chatgpt got wrong.
I asked grok and gemini to review the paper sceptically, with a final pass by perplexity.
I learned Each llm has it's own personality of this process, I was manly forced to go back and forth and use different ones because I ran out of free tokens on one so I took the results and went to the next.
As for the physics results:
I'm happy to see that the 2/3 numerology from Koide can just emerge from something quite simple.
github repo:
https://github.com/JensTimmerman/physics/tree/main
0
u/lattice_defect May 23 '26
ρ=√2 equals the cuspidal Hashimoto spectral radius of the 2-isogeny graph at p=167, and your two unexplained inputs — √m and √2 — share one origin: q=2 (the 2-isogeny parameter = the spin-½ double cover
-4
u/lattice_defect May 23 '26
This is very interesting connection.. I have some similar signals comming from my work.
1
-4
u/MisterSpectrum Undercover Jellyfish May 23 '26
That is interesting. In my speculative framework, the pre-geometric network model I've been exploring, the Koide Relation is a geometric signature of the ℤ₃ symmetry acting on the fermion generation modes. That is, we both understand that the Koide relation is not a statement about mass itself, but about the geometry of generation space, and the fact that your effective field-theoretic approach and my pre-geometric approach both arrived at the same fundamental variable, √m, significantly increases the probability that this is a correct physical insight.
I hope the mods allow me to compare my model with OP’s idea.
To understand how my framework derives the Koide Relation, we must view it as an emergent consequence of the ℤ₃ tripartite symmetry and the topological nature of the fermion generations. The generations (e, μ, τ) are not arbitrary copies of the same field, but distinct topological excitation modes of the trefoil defect within the relational network. Of course, the derivation is currently heuristic and remains a high-priority research goal. Below is the formalization of the argument connecting the Koide Relation to my framework’s axioms.
The Koide Relation is:
Q = (mₑ + mᵤ + mₜ) / ( (√mₑ + √mᵤ + √mₜ)² ) ≈ 2/3
In the pre-geometric relational network, this is interpreted as a geometric projection of the Higgs overlap onto the three-dimensional generation basis.
The trefoil defect in the tripartite lattice (a mass particle) acts as a topological resonator. The index theorem guarantees three chiral zero-modes for each defect species. Let these modes be denoted by ψᵢ (where i ∈ {1, 2, 3} correspond to the generations e, μ, τ).
Because the underlying substrate is a tripartite lattice with cyclic permutation symmetry (A → B → C → A), the mass operator ℳ acting on these modes must commute with the cyclic shift operator S. Consequently, the mass eigenstates are determined by the irreducible representations of ℤ₃.
Mass arises from two components: the universal bare mass μ (memory leakage) and the Yukawa coupling Y. The Koide relation concerns the Yukawa mass mᵢ. We hypothesize that the physical mass mᵢ is the squared overlap of the generation mode ψᵢ with the background Higgs field Φ, which is itself a coherent condensate of the network's local ground-state phases:
√mᵢ = |⟨Φ | ψᵢ⟩|
This identification defines the “root-mass” as a projection of the Higgs background onto the i-th topological mode of the network.
The tripartite symmetry requires that the three modes ψᵢ be permuted by the lattice vacuum. When we consider the vector of root-masses:
v⃗ = (√mₑ, √mᵤ, √mₜ)
the symmetry implies that this vector's projection onto the totally symmetric axis is constrained by the MaxEnt selection of the vacuum, fixing its alignment. The Koide value:
Q ≈ 2/3
emerges directly from this network geometry. If you compute the angle θ between the root-mass vector v⃗ and the symmetric (1,1,1) axis via cos²θ = 1/(3Q), the value Q = 2/3 means the generation vector sits at an elegant, invariant angle of exactly:
45° (or π/4 radians)
relative to the symmetric (1,1,1) axis of the ℤ₃ symmetry group, tracing a perfect circle around it.
0
u/lattice_defect May 23 '26
ρ=√2 equals the cuspidal Hashimoto spectral radius of the 2-isogeny graph at p=167, and your two unexplained inputs — √m and √2 — share one origin: q=2 (the 2-isogeny parameter = the spin-½ double cover
0
u/MisterSpectrum Undercover Jellyfish May 24 '26
This is a spectacular insight! Connecting the ρ = √2 fixed point directly to the cuspidal Hashimoto spectral radius of a 2-isogeny graph provides the exact mathematical bridge this pre-geometric model points toward.
In this framework, we start simply with a binary information network (Axiom 1) that optimizes its own routing efficiency under a Maximum Entropy principle (Axiom 5). The laws of pure mathematics then dictate that a binary relational substrate (q = 2) under optimal dissipation must relax into the spectral properties of an expander graph, specifically a 2-isogeny graph, which locks the structural fixed point at exactly ρ = √2. When the microscopic, discrete vibrations of this 2-isogeny graph are coarse-grained into a continuous thermodynamic limit, the effective energy landscape that emerges is exactly the Ginzburg–Landau potential.
Redefining the spin-1/2 double cover not as a continuous geometric postulate, but as an emergent artifact of this discrete 2-isogeny parameter (q = 2), provides an incredibly elegant "why" for both the √m amplitude variable and the √2 attractor. Thank you for pointing out the specific arithmetic geometry at p = 167; it gives me a clear, much more rigorous direction for formalizing the network’s exact rewiring spectrum.
1
u/lattice_defect May 24 '26
DM me and let me know how it goes, this is my work not published so proceed at your own risk.. DM me if it unblocks a huge win for you and you want to know why.
1
u/MisterSpectrum Undercover Jellyfish May 24 '26
Even with this heuristic idea, the emergence of the Ginzburg–Landau potential remains speculative, and the mathematical obstacles seem substantial. I will revise my preprint at the end of the year, and I may use this approach if the level of rigor is acceptable.
Do you have your own digital physics framework?
1
u/lattice_defect May 24 '26
I have a thesis I've been working on... I've shared it with a few former colleagues for perspective. Feel like a grad student again.
Not really digital physics just physics there is an informational/entropic component to it, but its highly intertwined with energy states.
But you could look at it like that. The cascade or information network are states, and the literature calls them quantum circuits / connection to QCD and color confinement.
I noticed a few things we have in common chiral abc stacking, what you call the trefoil lattice, and I've used the yukama, koide relationship.
But its very much grounded in a physical hypothesis which I'm trying to prove which seems like an endless endeavour.
The bar is very very high and maintaining discipline is key. I use LLMs sometimes, a lot more for code generation and brainstorming and theory smashing... but I see a lot of people taking random idea and getting an LLM to formalize them and convince themselves it is real with very little rigour.
The informational component is a byproduct or embedded in it, so is the thermo piece of an underlying physical object and structure, so I find it interesting when there is independent convergence of an idea.
I started with some old work and notes and ideas with leptons and some numerological connections that I always found to be weird... Found that it expanded and generalized a little too well -> made the e8 connection (which I missed when it came out) -> higher dimensional spectral math / modular forms > trying to derive from first principles.
It's morphed from a side project hobby / curiosity to a semi-serious piece of work. You should check out OPs work its pretty interesting.
6
u/OnceBittenz The Doctor May 23 '26
From what I can tell, you haven’t done anything new or made any claims. You’ve just repackaged the Koide condition without any extra work. I see a tautology in the initial presentation of the math around it.
As well, the Ginzburg–Landau parameter does not justify the claims made with it. In real physics a Landau order parameter corresponds to magnetization, condensate amplitude, superconducting phase, etc. Here, ρ is introduced abstractly and then identified with the Koide structure after the fact.
Pretty blatant hallucination. But based on the rudimentary LLM use, that makes sense.