Causal Flow Topology

 Causal Flow Topology

A Bounded Recursive Quotient Space is a tuple (N, D, R, B₋, B₊, φ) where:

N: numerator space (set of expressible statements/states)

D: denominator/grain operator (partition function)

R: remainder function R = N mod D

B₋, B₊: lower and upper bound attractors

φ: modulation map φ: R → (N', D') (how remainder drives evolution)

With constraints:

Boundedness: B₋ < N/D < B₊ (Goldilocks Zone)

Recursion: φ(R_n) generates (N_{n+1}, D_{n+1})

Invariance: Topology preserved under grain transformation (rational ↔ irrational D)

Causality: Information flows ∂N/∂t = f(distance to bounds)

The topology itself is defined by:

Open sets are "stable regions" where |R| < ε for viable ε

Closed sets are "bound neighborhoods" where system approaches B₋ or B₊

The remainder R defines a stratification - layers of increasing refinement