Section 4 — Emergent Particles and Mass (TFP, v11.1)

Section 4 — Emergent Particles and Mass (TFP, v11.1)

Stable Motifs of the Discrete Substrate
By John Gavel

4.0 Overview

Particles are stable temporal recursions of the discrete binary-flow substrate. No external entities are postulated: motifs that satisfy the substrate update rules persist; motifs that do not decay.

All intrinsic particle properties derive directly from:

• Binary flows \(F_i \in \{+1,-1\}\) (A2 / Section 1)
• Adjacency graph with \(K = 12\) neighbors (A3 / Section 3)
• Synchronous update tick \(\tau_0\) (A5 / Section 2)
• Tension minimization (A6 / Section 2)
• Finite handshake capacity \(H = 132\) and total operational budget \(H_\text{total} = 140\) (A10 / Section 3.2)

Clarification on \(H_\text{total}\):

• Each node has \(K = 12\) neighbors. Directed pairwise handshakes per tick: \(H = K \times (K - 1) = 12 \times 11 = 132\).
• For 3D volumetric motifs, an additional \(2^3 = 8\) combinatorial states are required for addressing the corners of a local cube.
• Total operational per-tick budget: \(H_\text{total} = H + 8 = 140\).

All particle motifs are constrained by this total budget; exceeding it produces unavoidable latency (“stutter”) that gives rise to mass.

4.1 The Three Pillars of Particle Existence

A motif stabilizes into a particle when it satisfies three mechanical conditions:

• Temporal Oscillation: The motif repeats with frequency \(\omega\) determined by the eigenmodes of the local Jacobian \(J\) (Section 3.1.3).
• Topological Circulation (Spin): To avoid handshake collisions, the motif schedules updates around loops, producing a quantized winding number.
• Relational Stutter (Mass): Maintaining oscillation and circulation consumes the finite handshake budget \(H_\text{total}\), creating a recurring latency above the vacuum baseline \(\delta\) (Section 3.2.4).

4.2 Stable Recursions

A particle is a periodic solution on a finite motif \(M\):

\[ F_i(t + T) = F_i(t), \quad \forall i \in M \]

Stability: Linearized updates \[ \delta F(t + \tau_0) = J \cdot \delta F(t) \] are stable if \(|\lambda_j| < 1\) for all eigenvalues \(\lambda_j\) of \(J\). Perturbations decay according to local tension-minimizing dynamics.

Phase Slip: If the motif fails the stability criterion, synchronization errors exceed the coherence length \(L_c\), causing the motif to unravel into transient waves.

Recursion Depth \(d\):

• \(T = d \cdot \tau_0\)
• \(d = 1\): minimal loop (electron-class)
• \(d = 3\): volumetric closure (baryon-class)

4.3 Mass Derivation

Mass is not substance; it is the bit-consumption rate of a stable eigenmode within the finite handshake budget.

4.3.1 1-Tick Baseline (Electron)

The vacuum baseline stutter \(\delta\) is defined in Section 3.2.4 (Folding Tax).
Electron: minimal sustainable stutter above \(\delta\).
Mechanism: single-site closed loop flipping once per tick (\(d = 1\)).
Preserves identity (A7 / Section 1) and fits within \(H_\text{total} = 140\) (A10 / Section 3).
Operational anchor: 1 unit extra processing latency = 1 electron mass.

4.3.2 Volumetric Closure (Proton)

Proton: 3D volumetric motif coordinating all \(K = 12\) neighbors (A3 / Section 3).
Simplex Scaling: 3D closure projected onto 1D temporal flows (A5 / Section 2) introduces a volumetric synchronization cost.
Effective budget: only \(H_\text{effective} = H_\text{total} \times 11/12\) is usable per tick due to saturation constraints.

Mass ratio (symbolic):

\[ \frac{M_p}{M_e} \approx \frac{H_\text{effective} \cdot 1.25}{\delta} \]

Proton is heavier because volumetric synchronization consumes more of the operational budget.

4.4 Charge and Spin (Operational Definitions)

• Charge \(q\): Net alignment bookkeeping \[ q = \sum_i F_i \] Conservation follows from binary flow preservation (A2).
• Spin \(S\): Update scheduling around loops to avoid collisions (A8 / Section 2). Quantization arises from the discrete 12-neighbor shell (A3).
• Pauli Exclusion: Two motifs cannot occupy the same 12-neighbor shell unless update phases are orthogonal — a pigeonhole constraint on \(H_\text{total}\) (A10).

4.5 Summary of Particle Properties

Property	Operational Basis	Source
Mass	Latency due to finite \(H_\text{total}\)	Section 3.2.4
Charge	Alignment of binary flows \(F_i\)	A2 / Section 1
Spin	Scheduled circulation in adjacency loops	A3, A8 / Section 2–3
Stability	Eigenvalues of Jacobian \(J\)	Section 3.1.3
Recursion Depth	Temporal period \(T = d \cdot \tau_0\)	A5 / Section 2

Key point: Every property is a logical consequence of Section 1–3 axioms, finite lattice topology, and discrete update rules. Nothing is added ad hoc.

4.6 Bridge to Section 5

With particles, mass, charge, and spin fully expressed in dimensionless lattice units, Section 5 anchors these quantities to physical units. The substrate constants \(H\), \(K\), \(\delta\), \(L_c\), and \(\tau_0\) map onto energy, time, and length scales using the proton anchor \(\Psi\).