Navokoj: Physics-Inspired SAT Framework

What It Solves#

Constraint satisfaction as geometric flow — treating constraints as flowing geometry rather than penalties to minimize. Navokoj is the reference implementation of the Arithmetic Manifold theory.

Key Innovation#

Three-Sector Architecture:

Arithmetic Sector: Prime-weighted operators prevent symmetry traps
Geometric Sector: Continuous manifold enables gradient flows
Dynamic Sector: Adiabatic quench implements Hamiltonian dynamics

Core Principle: Discrete problems → Continuous relaxation → Gradient flow → Discrete solution.

Energy Definition#

$E = -\mathbf{w} \cdot \log P(\text{clause satisfied})$

Each clause contributes to the energy based on its prime weight and satisfaction probability.

Adiabatic Quench#

The system is cooled slowly through the phase transition:

$\beta(t) = \beta_0 + r t$

“Adiabatic” means slow enough that the system stays in equilibrium — it always knows where the ground state is.

Results#

Problem	Variables	Clauses	Success Rate
3-SAT (critical density)	50	~215	99.4%
Graph Coloring	50 nodes	—	100%
AI Escargot Sudoku	729	8,850	99.85%

Website#

Live: navokoj.shunyabar.foo | API: api.navokoj.shunyabar.foo

For the production framing, read the Navokoj launch post.

Key Files#

navokoj/README.md — Overview
navokoj/MANIFOLD_IMPLEMENTATION.md — The reference implementation of Arithmetic Manifold theory

Connection to Core Vision#

Navokoj is the pure theory implementation — it’s explicitly described as “the reference implementation of the Arithmetic Manifold theory.” It demonstrates that:

Prime-weighted operators provide unique gradient identity
Geometric flow on the manifold is superior to discrete search
The adiabatic quench successfully navigates phase transitions