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    (|) conditional

Invariants

η = |edges|/|nodes| ≥ target    -- Density (default: 4.0, SAQ: 2.5)
KROG = K ∧ R ∧ O ∧ G            -- Knowable ∧ Rights ∧ Obligations ∧ Governance

Style (Φ)

1. PROSE_PRIMACY: Paragraphs over lists
2. TELEOLOGY_FIRST: Why → How → What
3. MECHANISTIC: Explicit causation (A → B → C)
4. 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 bridge

Bridge 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

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λ(ο,K).τ    parse→route→execute→validate→emit→compound    η≥target KROG Φ