Discreteness from Identifiability: Why Continuous Physics Produces Integers

Discrete phenomena pervade physics: quantum numbers, particle species, phase transitions, symbolic codes. Yet the underlying state spaces are continuous. We identify a unifying mechanism: observation maps carry integer-valued invariants (rank, dimension, index) that can change only discretely.

The key invariant is the identifiable dimension (rank of the observation differential), which controls how many effective degrees of freedom are accessible. When this rank changes, qualitative transitions occur.

The Dimensional Gradient Theorem (proven): Any useful observer requires $0 < \text{rank}(dh) < \dim X$. Rank > 0 for utility (discrimination); rank < dim X for stability (compression, noise tolerance). This is the minimal anthropic condition: not carbon, not water—just a gap between environment and representation complexity.

Part of the IPI Letters program:

• Thermodynamic Foundation — asymmetry costs work
• Black Hole Aperture — time as information rate
• Cosmic Relaxation — universe as relaxing knot

Core claim: Discreteness is not fundamental; identifiability is.