One electron, eight nodes, two hemispheres, two bridges. Watch the path. Read the geometry.
step 0 of 8 — cycle 1
Current node
−1
180° flip (inversion)
Mode
Protect — one shell, electron held safely
What's happening
One electron sits at the −1 node. The shell around it is its only protection — no bridge open, no path to walk. This is the resting state, the single protective shell.
The 8 nodes form a Hamiltonian cycle — a closed path that visits every node exactly once before returning to start. The cycle is: −1 → −i → 0 → 0 → +1 → +i → 0 → 0 → −1.
Two hemispheres. The left hemisphere holds −1 and −i — the negative side. The right hemisphere holds +1 and +i — the positive side. They mirror each other across the bridge axis.
Two bridges. Each pair of 0 nodes is a bridge connecting the two hemispheres. Bridges are ground points — the electron passes through them but doesn't compute there. They're where the path crosses itself without collision.
Why Hamiltonian. Because each step depends only on the previous node, no two operations ever collide. Left writes to its hemisphere; right writes to its hemisphere; they meet at the bridges and pass through. No locks, no synchronization, no race conditions — the geometry forbids it.
Protect. One electron held at −1. The shell exists to keep it from being lost. Nothing moves. This is the resting state, the seed condition. Real-world analog: a charged particle at rest in a potential well, or an idle process waiting for input.
Burrow. The electron walks the Hamiltonian cycle. Each step is one operation — flip, rotate, ground, identity, rotate, ground, repeat. The walker burrows through the topology, reading each node exactly once before returning. Real-world analog: a single-pass traversal of a data structure, or a charged particle moving through a chain of gates.
Gather. After the first complete cycle (burrow phase), the walker keeps moving — but now it accumulates at each hemisphere node it visits. Bridges still ground, hemispheres now collect. Real-world analog: a streaming aggregator that walks the same circuit repeatedly, picking up state at each visit, dumping it through the bridge nodes.
The values {−1, −i, 0, +1, +i} are points in the complex plane. Each is a transformation in 2D phase space:
−1 = 180° rotation (inversion). −i = −90° rotation. 0 = projection to origin (null operation). +1 = identity. +i = +90° rotation.
Composing the sequence around the cycle: (−1)·(−i)·0·0·(+1)·(+i)·0·0 = 0. The full cycle annihilates to zero — every walk returns the electron to ground state. This is what makes the cycle closed: it has no net effect on the electron's phase. It's the topological version of a conservation law.
The two zero pairs are projection operators. They erase the accumulated phase at each bridge crossing, which is what lets the path repeat indefinitely without drift. Without the bridges the phase would accumulate and the cycle would precess; with them, it stays closed.
This is the discrete topology of the eigenspaces of the Pauli matrices, packaged as a single-walker circuit.
Space — play/pause
S — single step
R — reset
G — toggle ground signal
1, 2, 3 — switch to protect, burrow, gather mode