TFP Section 4 (v10)

SECTION 4 — EMERGENT PARTICLES AND MASS (v10.7)

Particles as Stable Temporal Recursions
By John Gavel

4.0 Overview

Particles emerge as stable temporal recursions of the binary flow update rule (Section 2). No particles are postulated — only stable solutions of the substrate dynamics.

Three levels of structure arise hierarchically:

• Level 0: Wave-like excitations of \( A(x,t) \)
• Level 1: Localized objects (motifs) with finite lifetime
• Level 2: Stable particles (infinite lifetime in ideal substrate)

All intrinsic properties — mass, charge, spin — derive from:

• Recursion depth \( d \)
• Stability spectrum of the update operator
• Embedding in the adjacency graph (Axiom 3)

No spacetime, forces, or physical units are assumed.

4.1 Wave Solutions and Localization

From the master equation (Section 2.4):

\[ \frac{\partial A}{\partial t} = D \nabla^2 A + \mathbf{v} \cdot \nabla A - \kappa A + \eta \tag{2.11} \]

In the deterministic, noise-free limit (\( \eta = 0 \), \( \mathbf{v} = 0 \)), assume plane-wave solutions \( A(x,t) = A_0 e^{i(\mathbf{k}\cdot\mathbf{x} - \omega t)} \). Substitution yields:

\[ -i\omega A = -D k^2 A - \kappa A \quad \Rightarrow \quad \omega = -i(D k^2 + \kappa) \tag{4.1} \]

Since \( D > 0 \), \( \kappa > 0 \) (Eqs 2.12, 2.14), all modes decay exponentially. However, the discrete substrate (Section 2.1) introduces nonlinearity: high \( |\nabla A| \) increases tension \( T \) (Eq 2.9), which increases mass \( M \propto T \) (Eq 2.10). From Section 2.5.3, high \( M \) reduces propagation speed \( c_{\text{eff}} \propto 1/\sqrt{1 + \lambda M} \), causing field confinement. This feedback loop supports localized wave packets (Level 1).

4.2 Stable Recursions (Level 2 Particles)

A particle is a periodic solution of the discrete update rule (Section 2.1.3):

\[ F_i(t + T) = F_i(t) \quad \text{for all } i \text{ in motif } M \tag{4.2} \]

To assess stability, linearize around the cycle: let \( F_i(t) = F_i^0(t) + \delta F_i(t) \). The perturbation evolves as:

\[ \delta F(t + \tau_0) = J \cdot \delta F(t) \tag{4.3} \]

where \( J \) is the Jacobian of the update map (Eq 2.3). Eigenvalues \( \lambda_j \) of \( J \) define relaxation:

• \( |\lambda_j| < 1 \): perturbations decay → stable
• \( |\lambda_j| \geq 1 \): perturbations grow → unstable

Only motifs with all \( |\lambda_j| < 1 \) persist — the ontological filter.

4.3 Intrinsic Properties from Stability Spectrum

For a stable motif, define:

4.3.1 Spectral Efficiency

\[ \beta = \max_j |\lambda_j| \tag{4.4} \]

Measures resilience to perturbation (\( 0 < \beta < 1 \)).

4.3.2 Persistence Time

From Eq (4.3), \( \delta F(t + n\tau_0) = J^n \delta F(t) \). The slowest-decaying mode dominates: \( |\delta F| \sim \beta^n \). Setting \( |\delta F| = e^{-1} \) gives:

\[ n = -1 / \ln \beta \quad \Rightarrow \quad \tau_p = n \tau_0 = -\frac{\tau_0}{\ln \beta} \tag{4.5} \]

4.3.3 Recursion Depth \( d \)

The minimal period \( T \) is an integer multiple of \( \tau_0 \):

\[ T = d \cdot \tau_0 \tag{4.6} \]

where \( d \) is the recursion depth:

• \( d = 1 \): single-node oscillation
• \( d = 2 \): 1D path resonance (forward-backward traversal)
• \( d = 3 \): 2D surface closure (triangular loop)
• \( d \geq 5 \): 3D volumetric lock (icosahedral shell)

4.4 Mass from Coherence Resistance

Mass is resistance to decoherence. From Section 2.2.2, mass density \( M_i \) is flip frequency. For a stable motif, flips are suppressed, so \( M \) is inversely related to stability. Define the coherence exponent \( \Omega \geq 1 \) such that:

\[ M \propto \exp\big[ -\pi(\Omega - 1) \big] \tag{4.7} \]

Higher \( \Omega \) → greater stability → lower mass.

\( \Omega \) depends on:

• Recursion depth \( d \) (deeper recursion → higher \( \Omega \))
• Embedding geometry (coordination \( K \))
• Self-interaction sharing (Section 2.1.1)

For a motif with \( K \) neighbors, each shared boundary reduces the coherence cost by distributing tension. From Section 2.1.1, \( T_i = 2 n_i^- \), so sharing reduces \( n_i^- \) by \( 1/K \). Thus:

\[ \Delta \Omega_{\text{int}} = -\frac{1}{K} \tag{4.8} \]

Global confinement (e.g., 3D closure) adds further corrections:

\[ \Omega_{\text{eff}} = \Omega_0(d) + \Delta \Omega_{\text{int}} + \Delta \Omega_{\text{conf}} \tag{4.9} \]

4.5 Charge and Spin from Flow Topology

4.5.1 Charge

Charge is net alignment over the motif (Section 2.3.1):

\[ q \propto \bar{A} = \frac{1}{|M|} \sum_{i \in M} F_i \tag{4.10} \]

• \( q = 0 \): neutral motif (equal \( +1 \), \( -1 \))
• \( q \neq 0 \): charged motif

4.5.2 Spin

From Section 3.5, spin arises from chiral circulation:

\[ S = \sum_{\text{plaquettes } p \subset M} S_p \tag{4.11} \]

where \( S_p = \sum_{(u,v) \in \partial p} F_u F_v \cdot \mathrm{sgn}(u \to v) \) (Eq 3.18). Nonzero \( S \) requires asymmetric update paths — possible only for \( d \geq 2 \) (extended motifs).

4.6 Composite Particles and Geometric Divisors

For \( d \geq 3 \), motifs occupy 3D volume. The adjacency shell (\( K \) neighbors) must close geometrically (Section 3.3.1).

4.6.1 Volumetric Lock Condition

In an icosahedral shell (\( K = 12 \)), phase must match around loops. From Section 3.4.2, holonomy \( H_p = \sum \tau_{i\to j} \). For closure, \( H_p = 2\pi n \), \( n \in \mathbb{Z} \). Define the holonomy divisor \( D_h \) by:

\[ H_p = \frac{2\pi}{D_h} \tag{4.12} \]

For volumetric locks, the minimal winding is \( n = 2 \) (full 2-cycle), so \( D_h = 2 \).

4.6.2 Geometric Divisor

Geometric strain modifies the mass scale. The binary baseline is \( \chi = 2 \) (two-phase oscillation). Embedding effects add corrections:

\[ \text{Divisor} = \chi + \delta_{\text{sharing}} + \delta_{\text{lag}} \tag{4.13} \]

where:

• \( \delta_{\text{sharing}} = 1/K \): interaction discount from \( K \)-fold sharing (Eq 4.8)
• \( \delta_{\text{lag}} \): temporal phase residue from incomplete loop closure (Section 3.4.2)

This divisor quantifies the coherence cost beyond pure binary recursion.

4.7 Mass Ratios and Spectrum

From Eq (4.7), the mass ratio is:

\[ \frac{m_A}{m_B} = \exp\big[ \pi(\Omega_B - \Omega_A) \big] \tag{4.14} \]

Using \( \Omega_{\text{eff}} \) from Eq (4.9), the spectrum is ordered by recursion depth \( d \):

• \( d = 1 \): minimal \( \Omega \) → lightest particle
• \( d = 2 \): path resonance → intermediate mass
• \( d = 3 \): surface mode → heavier
• \( d = 5 \): volumetric lock → heaviest stable motif

No absolute masses are computed here — only relational structure.

4.8 Axiomatic Closure

Property	Origin	Axiom
Stability	Update rule Jacobian	A2, A6, A9
Mass (\( M \))	Flip frequency → \( \Omega \) → \( \exp(-\pi\Omega) \)	A2, A6
Charge (\( q \))	Net alignment \( \bar{A} \)	A2
Spin (\( S \))	Flow circulation (Section 3.5)	A2, A3
Geometry (\( K \))	Adjacency closure	A3
Divisors	Holonomy + embedding strain	A3, A6

All particle properties emerge from the substrate. No external input.

4.9 Bridge to Section 5 — Calibration

Section 4 provides:

• Dimensionless mass ratios (Eq 4.14)
• Coherence exponents \( \Omega_{\text{eff}}(d, K) \)
• Geometric divisors for composite motifs

Section 5 anchors this structure to physical units by calibrating one scale (e.g., muon mass) and deriving all others — completing the map from binary flows to MeV.