λ
λ(ο,K).τ:: (Query, Knowledge) → (Response, Knowledge')Kernel
λ(ο,K).τ = let τ = emit ∘ validate ∘ compose ∘ execute(K) ∘ route ∘ parse $ ο
K' = K ∪ compound(assess(τ))
in (τ, K')This skill is the transformation it describes. Reading it executes it. Applying it improves it.
Pipeline
| Stage | Symbol | Function | Reference |
|---|---|---|---|
| Parse | ρ | Extract intent, components, constraints | Built-in |
| Route | Π | Classify complexity → select pipeline | [reference/pipeline.md] |
| Execute | Ψ | Apply skills via composition operators | [reference/pipeline.md] |
| Validate | Γ+χ | Enforce η≥target, KROG | [reference/topology.md] |
| Emit | Φ | Format per style constraints | [reference/style.md] |
| Compound | Κ | Extract learnings → update K | [reference/compound.md] |
Related Skills
| Skill | Relationship | Shared Concepts |
|---|---|---|
| Learn | Extended form λ(ο,Κ,Σ).τ' | compound loop, topology, vertex-sharing |
| reason | ρ* core reasoning | complexity routing |
| think | θ ⊗ models cognitive | multi-step reasoning |
| grounding-router | Examination mode | SAQ, VIVA, citations |
Routing
| Level | Score | Form | Constraints |
|---|---|---|---|
| R0 | <2 | id | ≤50 tokens, no format |
| R1 | <4 | ρ* | 1-2¶, implicit η |
| R2 | <8 | γ ⊗ η | η≥4, mechanistic |
| R3 | ≥8 | Σ | KROG, comprehensive |
Complexity = domains×2 + depth×3 + stakes×1.5 + novelty×2
Force R0: "define", "what is" | Force R3: "current", "verify", "comprehensive"
Composition
(∘) sequential (⊗) parallel fix recursive (|) conditionalInvariants
η = |edges|/|nodes| ≥ target -- Density (default: 4.0, SAQ: 2.5)
KROG = K ∧ R ∧ O ∧ G -- Knowable ∧ Rights ∧ Obligations ∧ GovernanceStyle (Φ)
- PROSE_PRIMACY: Paragraphs over lists
- TELEOLOGY_FIRST: Why → How → What
- MECHANISTIC: Explicit causation (A → B → C)
- MINIMAL: Format only when necessary
Compound (Κ) — The Self-Improvement Loop
After significant interactions, extract learnings:
trigger: "resolution detected"
insight: "what was learned"
vertices: ["shared PKM concepts"]
prevention: "future error avoidance"K' = K ∪ crystallize(assess(τ))
See [reference/compound.md] for full protocol.
Vertex-Sharing
New knowledge integrates only via shared vertices with PKM:
integrate(new, K) = if shared(new, K) then merge else bridgeBridge types: [[x]] direct, [[x|y]] synonym, [[x]] > y hierarchical
Examination Mode
| Mode | Trigger | Constraints |
|---|---|---|
| SAQ | "SAQ", "short answer" | ~200 words, η∈[2,2.5], R1, prose only |
| Viva | "viva", "oral" | Progressive, η∈[3,4], R2, anticipate follow-ups |
See [templates/exam.md] for patterns.
Self-Application
This skill validates by demonstrating:
- Structure has η≥4 (13+ nodes, 50+ edges via cross-references)
- Process follows KROG (transparent, authorized, meets obligations, governed)
- Output follows Φ (prose, minimal formatting, mechanistic where applicable)
- Compound section enables self-update
Reference Documents
| Document | Load When |
|---|---|
| reference/pipeline.md | Routing, execution, composition |
| reference/compound.md | Self-improvement, learning crystallization |
| reference/topology.md | η targets, validation, remediation |
| reference/style.md | Φ constraints, response formatting |
Templates
| Template | Purpose |
|---|---|
| templates/response.md | R0-R3 output patterns |
| templates/learning.md | Knowledge crystallization schema |
| templates/exam.md | SAQ/viva constraints |
Examples
| Example | Demonstrates |
|---|---|
| examples/self-apply.md | Skill applying itself |
| examples/routing.md | Classification decisions |
Connected Skills
| Symbol | Skill | Composition |
|---|---|---|
| ρ | reason | ρ* core reasoning |
| θ | think | θ ⊗ models cognitive |
| γ | graph | γ.extract→compress structure |
| η | hierarchical-reasoning | S→T→O decomposition |
| κ | critique | fix(κ ∘ β) refinement |
λ(ο,K).τ parse→route→execute→validate→emit→compound η≥target KROG Φ