**Category**: Theorem Prover Health Monitoring

**Category:** Phase 3 Core - Distributed Reasoning

Sheaf neural network coordination via graph Laplacians for distributed consensus and harmonic inference. Use when coordinating multi-agent systems, building sheaf-aware GNNs, or implementing distributed consensus protocols.

Čech cohomology for local-to-global consistency verification in code structure and data schemas.

Darwin Gödel Machine patterns for self-improving AI agents with open-ended code evolution. Use for building agents that autonomously improve their own capabilities, modify their codebases, and evolve through interaction.

Segal types for synthetic ∞-categories. Binary composites exist uniquely up to homotopy. Foundation for topological chemputer.

**Category**: Graph Optimization + Remediation

Rezk types (complete Segal spaces). Local univalence: categorical isomorphisms ≃ type-theoretic identities.

Ramanujan graphs and Alon-Boppana spectral optimality for edge growth rules. Optimal expanders with λ₂ ≤ 2√(d-1) bound.

Topological data analysis for stable feature verification across filtrations of code complexity.

Metaskill that fans out on every interaction, using interaction entropy as SplitMixTernary seed for maximum synergistic parallelism

**Category:** Phase 3 Core - Geometric Deep Learning

Operad Composition Skill (PLUS +1)

**Category**: Theorem Dependency Analysis

Kolmogorov complexity as the ultimate intelligence measure. Shortest program that outputs data.

Kan Extensions Skill (ERGODIC 0)

JaxLife open-ended agentic simulator for emergent behavior, tool use, and cultural accumulation. Use for artificial life simulations, emergent agent behavior, and open-ended evolution research.

Ihara zeta function for graphs: non-backtracking walks, prime cycles, and spectral analysis via det(I - uB).

Schmidhuber's Gödel Machine: Self-improving systems that prove their own improvements. Darwin Gödel Machine (DGM) combines evolution with formal verification.

Bengio's GFlowNets: Generative Flow Networks that sample proportionally to reward. Diversity over maximization for causal discovery and molecule design.

Free Monad Generation Skill (PLUS +1)

Hinton's Forward-Forward algorithm for local learning without backpropagation. Use for biologically plausible learning, on-chip training, memory-efficient networks, or parallel layer-wise training.

**Category:** Phase 3 Core - Correctness Guarantees

**Category:** Phase 3 Core - Alternative Learning Paradigms