Dist Babble: TFP

Emergent Particle Physics from Recursive Binary Dynamics

John Gavel

Abstract

We present a substrate theory where spacetime geometry, gauge symmetries, and particle properties emerge from recursive binary flow dynamics on a discrete causal network. Starting from minimal axioms—binary states, local adjacency, and tension minimization—we derive:

3+1 dimensional geometry from correlation structure
U(1)×SU(2)×SU(3) gauge groups from holonomy of multi-path recursions
Fermionic matter as 4-path causal multiplets
Quantitative mass and charge ratios without free parameters

The framework makes testable predictions including specific mass scaling laws and coupling constant flows.

1. Introduction

1.1 Motivation

The Standard Model contains ~19 free parameters whose values are determined empirically rather than derived from first principles. Notable unexplained features include:

Fractional quark charges (±1/3, ±2/3) e
Three generations with hierarchical masses
Gauge group U(1)×SU(2)×SU(3) without deeper origin
Spacetime dimensionality (3+1)

Various approaches have attempted unification through extended gauge theories, extra dimensions, or string constructions. We pursue a different route: what if particles and spacetime geometry emerge together from simpler dynamics?

1.2 Core Premise

We propose that reality is fundamentally a recursive information network where:

Nodes carry binary states \(F_i \in \{+1, -1\}\)
States update according to local tension minimization
Time emerges from irreversible state evolution
Space emerges from correlation structure
Particles are stable recursive patterns

This is not a discretization of known physics—it's a derivation of known physics from discrete foundations.

1.3 Key Results

From these axioms alone, we obtain:

Geometric emergence:

Operational distance \(d_{ij} = -\ln|C_{ij}(\tau)|\) from correlations
Dimensional reduction to \(d_\text{eff} \approx 3\) from coordination structure
Curvature from correlation holonomy

Particle emergence:

Electron, muon, tau from recursion depths \(d = 1, 2, 3\)
Quarks from 4-path causal multiplets with tetrahedral/octahedral geometry
Charge quantization: ±1/3 = 4/12, ±2/3 = 8/12 from 12-neighbor kissing number

Gauge structure:

U(1): single-path phase accumulation
SU(2): 2-path doublet rephasing
SU(3): 3-path triplet with higher harmonics

Testable predictions:

Mass scaling \(M(N) = E_0 \exp(K_\text{EXP} N^{1.25})(1-\beta) + \Delta M_\text{spin}\)
Speed of light \(c = \frac{a}{\tau_0} \sqrt{12} \rho_I \frac{4}{9} \frac{5}{4}\) from geometric factors
Running coupling flows with specific fixed points

2. Substrate Axioms

We postulate a minimal mathematical structure:

Axiom A2 (Binary Dynamics): Each node \(i\) carries state \(F_i(t) \in \{+1, -1\}

Axiom A3 (Local Adjacency): Nodes have neighbors \(j \sim i\) with coordination number \(k_i\)

Axiom A6 (Tension Minimization): Updates minimize local tension \(T_i = 2 n_i^-\), where \(n_i^-\) counts misaligned neighbors

Axiom A9 (Stochastic Updates): State flips occur with probability \(p(F_i \to -F_i) = \frac{1 + \tanh(\beta T_i)}{2}\)

No spacetime manifold, Lagrangian, or quantum Hilbert space is assumed. These emerge.

2.1 Update Dynamics


\(T_i = \sum_{j\sim i} \frac{1 - F_i F_j}{2} = n_i^-\)

Higher tension increases flip probability, driving the system toward local alignment. This generates non-trivial temporal correlations:


\(C_{ij}(\tau) = \langle F_i(t) F_j(t+\tau) \rangle_t\)

2.2 Coarse-Grained Description


\(\frac{\partial A}{\partial t} = D \nabla^2 A - \kappa A + \eta\)

where:

\(D = k_\text{avg} a^2 / (T_\text{eff} \tau_0)\): diffusion from neighbor averaging
\(\kappa = \mu_\text{eff}^2 / \tau_0\): decay from tension-driven relaxation
\(\eta\): substrate noise

The coherence length \(L_c = \sqrt{D/\kappa} = a \sqrt{k_\text{avg}/(T_\text{eff} \mu_\text{eff}^2)}\) sets the scale where substrate discreteness becomes visible.

3. Emergent Geometry

3.1 Distance from Information


\(d_{ij} = -\ln(\max_\tau |C_{ij}(\tau)|)\)

Properties:

\(d_{ii} = 0\) (identity)
\(d_{ij} = d_{ji}\) (symmetry)
\(d_{ik} \lesssim d_{ij} + d_{jk}\) (triangle inequality)

For exponentially decaying correlations \(C_{ij} \propto e^{-r/L_c}\), this recovers geometric distance: \(d_{ij} = r_{ij}/L_c\).

3.2 Dimension from Coordination


\(d_\text{eff} = \frac{\log k_i}{\log 2}\)

For typical coordination \(k \approx 6-12\), we get \(d_\text{eff} \approx 2.6-3.6\), naturally selecting \(d \approx 3\).


d_\text{eff}^\text{(corr)} = \frac{\log N_c}{\log L_c}

3.3 Curvature from Holonomy


τ_{i→j} = argmax_τ C_{ij}(τ)

H_p = τ_{i→j} + τ_{j→k} + τ_{k→i}

R(i) ∝ ⟨H_p⟩ / ⟨A_p⟩

3.4 Lorentz Signature

Time direction: Irreversible coarsening, \(C_{ii}(\tau)\) decays monotonically
Space directions: Reversible correlations, \(C_{ij}\) symmetric under reflection

This produces signature \(\eta^{\mu\nu} = \text{diag}(+1, -1, -1, -1)\) without postulating relativity.

4. Particle Emergence

4.1 Stability Criterion


δF(t + τ_0) = J · δF(t)

Eigenvalues λ_j of Jacobian J determine stability:

|λ_j| < 1: stable particle
|λ_j| ≥ 1: unstable, decays

Define spectral efficiency β = max_j |λ_j|.

4.2 Mass from Coherence Resistance


M ∝ exp[-π(Ω - 1)]

Ω_eff = Ω_0(d) + ΔΩ_int + ΔΩ_conf

4.3 Mass Hierarchy


m_A/m_B = exp[π(Ω_B - Ω_A)]

Particle	d	Interpretation	Predicted Ω
Electron	1	Single-node oscillation	Ω_e ≈ 1.0
Muon	2	Path resonance	Ω_μ ≈ 1.5
Tau	3	Surface mode	Ω_τ ≈ 2.0

Contact: greenbug@gmail.com

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