Navokoj: The Arithmetic Manifold in Production

Published June 2026

Introducing Navokoj#

Navokoj is a general-purpose engine for finding coherent structure inside astronomically large discrete spaces. It converts complex business planning into automated decisions — scheduling, routing, resource allocation — with a 92.57% satisfaction rate on the SAT 2024 Industrial Track (4,199 problems).

The Problem with Traditional SAT Solvers#

Classical CDCL (Conflict-Driven Clause Learning) solvers hit a wall on certain problem structures:

XOR constraints destroy learned-clause heuristics
Critical density problems ( $\alpha \approx 4.27$ ) cause exponential CNF blow-up
Large-scale industrial problems (1M+ variables) timeout without partial results

Navokoj’s Approach: The Arithmetic Manifold#

Navokoj implements the Arithmetic Manifold theory from ShunyaBar Labs:

Prime-weighted operators provide unique spectral identity to each constraint
Geometric flow on a continuous manifold instead of discrete search
Adiabatic cooling navigates phase transitions smoothly
Casimir-inspired forces pull satisfying assignments together

flowchart LR
    Problem[Discrete constraint problem] --> Encode[Prime-weighted encoding]
    Encode --> Flow[Continuous geometric flow]
    Flow --> Diagnose[DEFEKT diagnostics]
    Flow --> Solve[Navokoj solver engines]
    Diagnose --> Engine{Engine choice}
    Engine --> Nano[Nano]
    Engine --> Mini[Mini]
    Engine --> Pro[Pro]
    Nano --> Result[Best-effort assignment]
    Mini --> Result
    Pro --> Result
    Solve --> Result

Core Innovation: Continuous Relaxation#

Variables are embedded in $[0, 2\pi)^K$ :

x_i = \frac{1}{2}(1 + \cos \theta_i)

This ensures continuous values while maintaining discrete interpretability. The energy landscape is smoothed via heat kernel diffusion:

\text{Tr}(e^{-tL}) \approx Z(\beta)

Real Performance#

Problem Type	Variables	Clauses	Satisfaction	Time
129-SAT (Ultra-High-k)	200	1,000,000	100%	9-10 min
Ramsey R(5,5,5) N=52	2.6M	7.8M	100%	17 min
Random 3-SAT (1M Scale)	1,000,000	4,260,000	92.15%	171s
Supply Chain	435,000	1,300,000	97.18%	67.7s

Engines#

Engine	Best For	Satisfaction	Speed
Nano	Real-time APIs, massive scale	3.24%	Ultra-fast
Mini	Balanced optimization	31.37%	10.64/sec
Pro	Mission-critical verification	92.57%	7.90/sec

DEFEKT: Diagnostic Intelligence#

Before running expensive solvers, DEFEKT gives you an MRI-style scan:

{
  "solvability_score": 84,
  "status": "likely_solvable",
  "recommendation": "Use pro-deepthink on H100 GPU for optimal satisfaction"
}

Getting Started#

import requests

response = requests.post(
    "https://api.navokoj.shunyabar.foo/v1/solve",
    headers={"Authorization": "Bearer YOUR_KEY"},
    json={
        "expression": "(employee_a | employee_b) & (shift_morning -> manager_present)",
        "engine": "mini",
    },
)

result = response.json()
# {"success": true, "satisfaction_rate": 1.0, "assignment": {...}}

Phase Transition Analysis#

The critical density $\alpha = 4.27$ marks the hardest SAT problems:

Clause Density	Median Time	Satisfaction
1-2 (sweet spot)	39ms	97.8%
3-4	684ms	98.9%
4-5 (phase transition)	1.93s	98.2%
10-25 (Very Hard)	14.1s	95.4%
50+ (Monster)	16.2s	92.57%

The Arithmetic Manifold in Production#

Navokoj proves that the Arithmetic Manifold theory scales:

Prime weighting works at million-variable scale
Phase transitions can be navigated, not just detected
Continuous relaxation beats discrete search on structured problems
Real-time constraints (scheduling, routing) are solvable in <1 second

The math is open source at github.com/sethuiyer/navokoj. The production engine pushes satisfaction to 100% whenever feasible.