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)

• 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 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

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