The Architecture
Every UCT Navigator is a 4-tuple (S, M, T, H): a state space, a metric, transition rules, and a steering heuristic. The key insight: navigation replaces computation. Instead of evaluating a function at every point, we follow the geometry of the solution landscape.
| Component | Collatz | Riemann | Twin Primes |
|---|
| S — State Space | Odd integers mod 240 | Gram points n | Residues mod 240 |
| M — Metric | Drift δ(m) = log₂(m)−2 | E₈ correction δ(n) | HL error ε(x) |
| T — Transitions | Inverse tree: 2x or (x−1)/3 | γₙ = γₙ₋₁ + sₙ/ρ | CRT decomposition |
| H — Heuristic | DNA score + slots {79,159,207} | 5 quasiperiodic frequencies | Percolation η = 5/12 |
The Unifying Principle
All three navigators share one geometric substrate: the E₈ lattice with kissing number K₈ = 240. The modulus 240 governs Collatz champion classification, Riemann zero spacing corrections, and twin prime channel structure. The universal navigation constant log₂(4/3) ≈ 5/12 ≈ 0.415 appears in all three domains as the third convergent of the same continued fraction. This is not three separate tools — it is one geometry viewed from three angles.