Axiom 3
Multi-Scale Persistence
Emergence cascades through magnitudes of scale
Formal Definition
d(Gs(x), Gs′(x)) ≤ ε across scale intervals [s, s′]
Real entities persist as recognizable patterns under coarse-graining. A genuine entity must be visible whether you zoom in or out, ruling out scale-dependent artifacts.
In the Hypercube
The Gaussian distribution is visible at every level: individual vertex occupancies, plane-level statistics, and global metrics all show the same bell-curve pattern. The Hamming weight planes provide natural scales of observation.
Micro Scale
Individual agents at specific vertices show local clustering consistent with the global distribution.
Macro Scale
Aggregate plane occupancies form the Gaussian centered at planes 4-5, matching binomial coefficients.