Particles as Emergent Excitations from Binary Crossing Rules Without Putting Anything In By Hand
Dunstan Low — companion to Binary \(\infty_0\) — ontologia.co.uk
Abstract
The bilateral mesh has exactly two vacuum states: ingress (\(\varphi=-1\)) and egress (\(\varphi=+1\)). A domain wall connecting them — a transition from one vacuum to the other — is a kink soliton, which is the bilateral crossing event \(U_\times\). Kinks are stable, move at constant velocity, and conserve energy. An antikink is the reverse crossing. When a kink and antikink collide, they either annihilate (producing radiation) or form a bound oscillating state (bion). This is particle physics without any particles put in by hand: the binary rule \(\partial^2\varphi = -\lambda\varphi(\varphi^2-1)\) generates them automatically. Three simulations are presented: a single kink (particle stability), a kink-antikink collision (particle interaction), and energy conservation throughout.
\(\partial_t^2\varphi = \partial_x^2\varphi - \lambda\varphi(\varphi^2-1)\) — bilateral field equation
\(\varphi_{\rm kink}(x,t) = \tanh\!\bigl(\gamma(x-vt)\bigr)\) — particle solution
\(E_0 = \tfrac{4\sqrt{\lambda}}{3}\) — rest energy from first principles
I. The Bilateral Vacuum and Domain Walls
The bilateral field \(\varphi\) has a double-well potential \(V(\varphi) = \tfrac{\lambda}{4}(\varphi^2-1)^2\) with minima at \(\varphi = \pm 1\). These are the two bilateral vacua: ingress (0-face, \(\varphi=-1\)) and egress (1-face, \(\varphi=+1\)). The potential parameter \(\lambda\) is fixed by the bilateral crossing amplitude: \(\lambda = A_{\rm bilateral}/2 = 1\) in natural units.
A kink is a smooth transition from \(\varphi=-1\) to \(\varphi=+1\) over a finite region — a domain wall. It satisfies the bilateral field equation and is an exact solution:
The kink has rest energy \(E_0 = 4\sqrt{\lambda}/3 = 4/3\) and rest mass \(m = E_0/c^2\). It is a genuine particle: localised, stable, moves without spreading, obeys relativistic kinematics with the bilateral speed of light \(c = t_1/2\pi\).
II. Simulation A — Single Kink: Stability
A single kink initialised at \(x = -8\) with velocity \(v\). Press Play to watch it propagate stably. Change the velocity with the slider.
Single kink propagation — \(\varphi(x,t)\) field
t = 0.00 kink at x = −8.00
press Play to start
Result A — Particle stability
The kink propagates without spreading or deforming for arbitrarily long times. Its shape \(\tanh(\gamma(x-vt))\) is preserved exactly. Width contracts with velocity (Lorentz contraction: \(\xi_{\rm obs} = \xi/\gamma\)). Energy grows with velocity (\(E = \gamma E_0\)). These are all relativistic particle properties, emerging from a simple scalar field with no particles assumed.
III. Simulation B — Kink-Antikink Collision
A kink (moving right) and antikink (moving left) approach and collide. Choose the outcome: slow collision produces a bion (bound oscillating state); fast collision produces annihilation and radiation.
Kink-antikink collision — \(\varphi(x,t)\) field
speed:
t = 0.00
press Play to start
Result B — Particle interaction without input
At low velocity: the kink and antikink form a bion — an oscillating bound state, like a particle-antiparticle bound state (positronium). At high velocity: they pass through each other with a phase shift, emitting radiation. At critical velocities: complex resonance structure (windows of transmission). None of this was programmed — it emerges from the bilateral field equation applied to two domain walls.
IV. Simulation C — Energy Conservation
The total energy \(E = \int[\tfrac{1}{2}\dot\varphi^2 + \tfrac{1}{2}(\partial_x\varphi)^2 + V(\varphi)]\,dx\) is tracked through the kink-antikink collision. The symplectic (leapfrog) integrator conserves it to better than 0.2%.
Energy \(E(t)/E_0\) during collision — should stay flat
energy tracking — press Play
Result C — Conservation law
Energy is conserved to <0.2% throughout the collision, including the complex bion phase. This is the bilateral analogue of energy-momentum conservation in particle physics. The conserved charge is the topological charge \(Q = \tfrac{1}{2}[\varphi(+\infty)-\varphi(-\infty)] \in \{-1,0,+1\}\), which is exactly the bilateral crossing count modulo 2.
V. Binary Cellular Automaton — Gliders as Particles
A simpler but equally telling demonstration: the bilateral XOR rule on a binary lattice. Each cell at time \(t+1\) is the XOR of its two neighbours at time \(t\). Starting from a single excited bit, stable "gliders" (moving patterns) emerge spontaneously. Hover any cell.
The kink soliton is the 1+1D analogue of the instantons and monopoles that appear in the full 4D bilateral mesh. The topological charge \(Q\) is the bilateral analogue of baryon number. The bion (oscillating bound state) is the analogue of the proton-antiproton bound state. The energy \(E_0 = 4\sqrt{\lambda}/3\) will become, in the full framework, the hadron mass scale — determined by the bilateral coupling \(\lambda\) which is set by the crossing amplitude \(A_n=2\). No free parameters.
Bilateral soliton simulation. φ⁴ field equation integrated by symplectic leapfrog. Energy conservation verified <0.2%. Binary cellular automaton: XOR rule (Rule 90). Particles emerge from binary rules without any particle input. Dunstan Low · ontologia.co.uk