Sponsor Preview: Fund the Research, Run the Solver
How GitHub Sponsors supports NitroSAT research and unlocks Navokoj compute credits for evaluation.
Read the noteWriting / Field notesIdeas are useful
Ideas are useful
when they leave a trace.
Essays and working notes on arithmetic geometry, constraint intelligence, computational physics, and the systems required to make the theory useful.
Follow via RSS ↗Follow curiosity
down the stack.
Begin with intuition and utility. Continue only as far as the question keeps pulling you.
What is this?
Discover
Enter through a working product, a physical intuition, or an open implementation.
Navokoj — Constraint Intelligence RuntimeThe operational surface: model + deadline → actionable decision.↗Casimir SAT SolverBoolean logic treated as interacting physical states.↗NitroSATThe open-source SAT and MaxSAT implementation.↗BAHA — Branch-Aware OptimizationLambert-W branches as structured escape routes.↗Documentation HubThe navigable spine of the ecosystem.→
How does it become useful?
Understand
Follow the path from solver research to a runtime that applications and organizations can operate.
Navokoj API DocumentationRequests, responses, engines, limits, and integration contracts.↗The Road to Enterprise: From Spectral Phase Transitions to a Production RuntimeThe evolution into a routed, billed, deployable runtime.→Building a Constraint Runtime, Not Just a SolverThe complete constrained-decision lifecycle.→From NitroSAT Research to the Navokoj RuntimeWhere open research ends and the operated runtime begins.→What Happens When the Deadline Arrives?The meaning of solved, feasible, partial, timeout, and error.→Navokoj Pricing: Plans, Workloads, and ComputePlans, workloads, and hardware economics.→
What does it do concretely?
Observe
Follow the experimental and engineering pieces that readers use to evaluate behavior, boundaries, and scale.
Empirical Evaluation of Navokoj Constraint Solver APIAPI behavior across edge cases and scaling regimes.→Probing the Ramsey Limit: Phase Transitions in High-Dimensional LogicSatisfiable, critical, and frustrated regimes.→Kubernetes Placement: 2 Million Constraints, 100% SatisfactionGlobal placement reasoning at multimillion-clause scale.→Navokoj: Computing That Never Fails ClosedThe resilience-computing and best-effort decision thesis.→Beyond SAT: PSPACE-Complete Problems Solved via Continuous OptimizationContinuous optimization beyond ordinary SAT.→Navokoj EvidenceBenchmark campaigns, artifacts, and integrity records.↗
What mechanisms are underneath?
Investigate
Move from application behavior into continuous geometry, multiplicative structure, dynamics, and stopping.
Multiplicative PINN FrameworkMultiplicative constraint enforcement beyond SAT.↗STOP Operator ManuscriptA mathematical theory of when computation should stop.→Multiplicative Archaeology: Recovering Interaction History from Digital AtomsNumbers as compressed histories of interaction.→Navokoj: A Physics Engine for Logic | Constraint Intelligence PlatformDiscrete logic as evolution through an energy landscape.→Training Dynamics for Discrete Constraint SatisfactionConstraint flow through the lens of neural training dynamics.→The Six Research Tribes of ShunyaBarSix research functions composing into one pipeline.→
Why does the architecture recur?
Research
Enter the foundational program: arithmetic identity, geometric motion, and thermodynamic transition.
The Product Is the Surface. The Research Goes Deeper.The institutional thesis and guided entrance.→Unified Research AbstractSixteen frameworks, one program.↗Mathematical MechanismThe self-stabilizing optimizer in compact form.↗The ShunyaBar FunctionalThe arithmetic–geometric action principle.↗Phase Transition and Symmetry BreakingThe quantum-statistical foundation.↗All long-form researchContinue through the complete research corpus.↗
Browse by format.
Every article, grouped by what kind of reading experience it provides.
Product explainers
What Navokoj does, who it serves, and how the runtime changes an operational workflow.
The Road to Enterprise: From Spectral Phase Transitions to a Production Runtime
How Navokoj evolved from a differentiable SAT experiment into a production constraint runtime with SUTRA, GPU routing, verification, usage billing, and customer-controlled deployment.
Read the noteBuilding a Constraint Runtime, Not Just a Solver
The product architecture behind deadline-bounded decisions: models, execution, repair, verification, and evidence.
Read the noteConstraint Intelligence for AI Agents
Agents can propose actions probabilistically; Navokoj helps make the resulting decision admissible, inspectable, and executable.
Read the noteFrom NitroSAT Research to the Navokoj Runtime
Why ShunyaBar Labs keeps an open research release while building a commercial constraint runtime around it.
Read the noteNavokoj Pricing: Plans, Workloads, and Compute
How Navokoj combines monthly workload entitlements with explicit CPU and GPU compute settlement.
Read the noteWhat Happens When the Deadline Arrives?
Navokoj is designed for deadline-bounded decisions: here is what an application receives when a solve cannot finish perfectly.
Read the noteWhy Customers Switch to Navokoj
A practical constraint runtime for decisions that are expensive, slow, opaque, or difficult to integrate.
Read the noteNavokoj: The Arithmetic Manifold in Production
How Navokoj turns discrete constraint problems into continuous geometric flow, with production benchmarks and verifiable results.
Read the noteThe Physics-Informed Control Plane for Agentic Systems
Turn probabilistic agent behavior into reliable, enterprise-grade action with Navokoj's constraint-governed execution layer. Computing that never fails closed.
Read the noteIntroducing NitroSAT: A High-Performance, Physics-Informed MaxSAT Solver Now Available via Navokoj API
ShunyaBar Labs is proud to announce the public release of NitroSAT — a next-generation MaxSAT approximator that achieves exceptional satisfaction rates on massive, real-world constraint problems while maintaining linear scaling in the number of clauses.
Read the noteThe Road to Enterprise (Legacy Report)
The physics behind Navokoj, the barriers already crossed, and the engineering path from public beta to enterprise readiness.
Read the noteEngineering case studies
Concrete workloads, API paths, deployment patterns, and implementation decisions.
Q-State Solves a 9x9 Sudoku Through the Navokoj API
A live Q-State API run solved an 81-variable Sudoku model with 100% satisfaction and zero conflicts.
Read the noteSUTRA on the API: A Small Weighted Constraint Problem, Solved End to End
A concrete WCNF-style API request using engine:nitro, with hard-feasibility, soft preferences, routing, and verification metadata in one response.
Read the noteDeploying Navokoj in Private and Air-Gapped Environments
A deployment model for teams that need constraint execution close to their data, networks, and operational controls.
Read the noteFrom Workforce Data to a Verified Nurse Roster
How a scheduling workflow turns staffing data, policies, and preferences into a time-bounded operational decision.
Read the noteThe Seven Trust Boundaries of a Private Navokoj Deployment
Seven layers of privacy, supply-chain integrity, offline licensing, and binary protection behind the Navokoj deployment model.
Read the noteKubernetes Placement: 2 Million Constraints, 100% Satisfaction
Navokoj, the Fault-Tolerant Constraint Intelligence Engine, delivers a case study in placement safety and outage prevention with 5,000 variables and 2 million constraints.
Read the noteExperimental reports
Measured campaigns, benchmark results, failure modes, and reproducible observations.
Empirical Evaluation of Navokoj Constraint Solver API
Production verification of anytime constraint solving with partial satisfaction semantics. 47 test cases, scaling analysis, and failure mode characterization.
Read the noteBeyond SAT: PSPACE-Complete Problems Solved via Continuous Optimization
Navokoj, the Fault-Tolerant Constraint Intelligence Engine, provides an overview of PSPACE verification with graceful degradation instead of binary failure.
Read the noteProbing the Ramsey Limit: Phase Transitions in High-Dimensional Logic
Navokoj, the Fault-Tolerant Constraint Intelligence Engine, analyzes phase transitions when perfect solutions become impossible.
Read the noteMathematical frameworks
The formal structures and computational mechanisms behind the lab's systems.
The Product Is the Surface. The Research Goes Deeper.
Navokoj is one operational expression of a larger research program. A guide for readers who want to follow the mathematics into ShunyaBar Labs' research site.
Read the noteMultiplicative Archaeology: Recovering Interaction History from Digital Atoms
A framework for treating numbers as structured histories, connecting digit topology, interaction cost, and the archaeology of factorization.
Read the noteThe Six Research Tribes of ShunyaBar
Six research cohorts compose into one pipeline: weight, transform, satisfy, relax, observe, and coordinate.
Read the noteTraining Dynamics for Discrete Constraint Satisfaction
Examining the parallel between gradient-based neural network training and continuous optimization for NP-hard constraint problems.
Read the noteNavokoj: A Physics Engine for Logic | Constraint Intelligence Platform
Revolutionary Constraint Intelligence Platform that treats Boolean logic as continuous dynamical systems. 347ms median latency vs 45s classical solvers. Physics-inspired optimization for NP-complete problems.
Read the noteExploratory hypotheses
Forward-looking connections and research directions that are still being developed.
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