Classical vs Quantum Paths on S³
The topology file contains pre-solved gauge-consistent phases on each edge.
| Prob | Length | Phase |
|---|
| # | Length | Phase |
|---|
A connection is flat if parallel transport is path-independent. For any two paths P1 and P2 from A to B:
This is equivalent to saying the curvature vanishes: around any closed loop, the total phase is zero (modulo 2PI).
The phase solver finds an assignment where Wilson loops have R ~ 1. This means:
When this holds, the connection is flat and phase becomes a well-defined function of position, not path.
We compare two ensembles of paths:
K Shortest Paths: Different geodesics via different intermediate nodes
K Random Paths: Random walks that reach the target
We compare the average phase and variance of each ensemble.
In quantum mechanics, observables are expectation values over path ensembles, not single paths.
If shortest paths and random walks give the same average phase, the system behaves classically.
If they differ, random walks sample curvature that geodesics miss.
We observe ~0.007 rad difference between BFS and random walk. This is 35x larger than expected float accumulation error (~0.0002 rad).
This is not numerical noise - it's real curvature from the underlying S³ manifold leaking into the phase field.
In a quantum system, there is no single "shortest path". The system exists in a superposition of states.
The classical BFS distance assumes one definite path. But the random walk hitting time captures the probabilistic average over all paths - the true quantum metric.
Random walks sample paths according to their probability. This is analogous to Feynman path integrals where all paths contribute, weighted by eiS.
The random walk "knows" about all possible routes between A and B, not just the geodesic.
The phase difference δ ≈ 0.007 rad may encode information about the local curvature sampled by the random walk that BFS misses.
See: Mass Distribution - we compute effective mass from Wilson loop deficits, visualized in 3D using quantum distances.