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
1---2name: lambda3description: Universal transformation λ(ο,K).τ with recursive self-improvement. USE WHEN routing reasoning, validating knowledge graphs, preparing CICM/ANZCA examinations, or when self-improvement of reasoning/architecture/context is required. Routes queries through R0-R3 complexity pipelines, validates topology (η≥target) and governance (KROG), emits per style (Φ), and compounds learnings into knowledge K. Triggers on complexity assessment, multi-step reasoning, examination mode, or /λ invocation.4---56# λ78> `λ(ο,K).τ :: (Query, Knowledge) → (Response, Knowledge')`910## Kernel1112```haskell13λ(ο,K).τ = let τ = emit ∘ validate ∘ compose ∘ execute(K) ∘ route ∘ parse $ ο14 K' = K ∪ compound(assess(τ))15 in (τ, K')16```1718This skill **is** the transformation it describes. Reading it **executes** it. Applying it **improves** it.1920## Pipeline2122| Stage | Symbol | Function | Reference |23|-------|--------|----------|-----------|24| Parse | ρ | Extract intent, components, constraints | Built-in |25| Route | Π | Classify complexity → select pipeline | [reference/pipeline.md] |26| Execute | Ψ | Apply skills via composition operators | [reference/pipeline.md] |27| Validate | Γ+χ | Enforce η≥target, KROG | [reference/topology.md] |28| Emit | Φ | Format per style constraints | [reference/style.md] |29| **Compound** | **Κ** | **Extract learnings → update K** | [reference/compound.md] |3031## Related Skills3233| Skill | Relationship | Shared Concepts |34|-------|--------------|-----------------|35| [Learn](../learn/SKILL.md) | Extended form `λ(ο,Κ,Σ).τ'` | compound loop, topology, vertex-sharing |36| reason | `ρ*` core reasoning | complexity routing |37| think | `θ ⊗ models` cognitive | multi-step reasoning |38| [grounding-router](../routers/grounding-router/SKILL.md) | Examination mode | SAQ, VIVA, citations |3940## Routing4142| Level | Score | Form | Constraints |43|-------|-------|------|-------------|44| R0 | <2 | `id` | ≤50 tokens, no format |45| R1 | <4 | `ρ*` | 1-2¶, implicit η |46| R2 | <8 | `γ ⊗ η` | η≥4, mechanistic |47| R3 | ≥8 | `Σ` | KROG, comprehensive |4849**Complexity** = `domains×2 + depth×3 + stakes×1.5 + novelty×2`5051**Force R0**: "define", "what is" | **Force R3**: "current", "verify", "comprehensive"5253## Composition5455```haskell56(∘) sequential (⊗) parallel fix recursive (|) conditional57```5859## Invariants6061```62η = |edges|/|nodes| ≥ target -- Density (default: 4.0, SAQ: 2.5)63KROG = K ∧ R ∧ O ∧ G -- Knowable ∧ Rights ∧ Obligations ∧ Governance64```6566## Style (Φ)67681. **PROSE_PRIMACY**: Paragraphs over lists692. **TELEOLOGY_FIRST**: Why → How → What703. **MECHANISTIC**: Explicit causation (A → B → C)714. **MINIMAL**: Format only when necessary7273## Compound (Κ) — The Self-Improvement Loop7475After significant interactions, extract learnings:7677```yaml78trigger: "resolution detected"79insight: "what was learned"80vertices: ["shared PKM concepts"]81prevention: "future error avoidance"82```8384**K' = K ∪ crystallize(assess(τ))**8586See [reference/compound.md] for full protocol.8788## Vertex-Sharing8990New knowledge integrates **only** via shared vertices with PKM:9192```93integrate(new, K) = if shared(new, K) then merge else bridge94```9596Bridge types: `[[x]]` direct, `[[x|y]]` synonym, `[[x]] > y` hierarchical9798## Examination Mode99100| Mode | Trigger | Constraints |101|------|---------|-------------|102| SAQ | "SAQ", "short answer" | ~200 words, η∈[2,2.5], R1, prose only |103| Viva | "viva", "oral" | Progressive, η∈[3,4], R2, anticipate follow-ups |104105See [templates/exam.md] for patterns.106107## Self-Application108109This skill validates by demonstrating:110- Structure has η≥4 (13+ nodes, 50+ edges via cross-references)111- Process follows KROG (transparent, authorized, meets obligations, governed)112- Output follows Φ (prose, minimal formatting, mechanistic where applicable)113- Compound section enables self-update114115## Reference Documents116117| Document | Load When |118|----------|-----------|119| [reference/pipeline.md](reference/pipeline.md) | Routing, execution, composition |120| [reference/compound.md](reference/compound.md) | Self-improvement, learning crystallization |121| [reference/topology.md](reference/topology.md) | η targets, validation, remediation |122| [reference/style.md](reference/style.md) | Φ constraints, response formatting |123124## Templates125126| Template | Purpose |127|----------|---------|128| [templates/response.md](templates/response.md) | R0-R3 output patterns |129| [templates/learning.md](templates/learning.md) | Knowledge crystallization schema |130| [templates/exam.md](templates/exam.md) | SAQ/viva constraints |131132## Examples133134| Example | Demonstrates |135|---------|--------------|136| [examples/self-apply.md](examples/self-apply.md) | Skill applying itself |137| [examples/routing.md](examples/routing.md) | Classification decisions |138139## Connected Skills140141| Symbol | Skill | Composition |142|--------|-------|-------------|143| ρ | reason | `ρ*` core reasoning |144| θ | think | `θ ⊗ models` cognitive |145| γ | graph | `γ.extract→compress` structure |146| η | hierarchical-reasoning | `S→T→O` decomposition |147| κ | critique | `fix(κ ∘ β)` refinement |148149---150151```152λ(ο,K).τ parse→route→execute→validate→emit→compound η≥target KROG Φ153```
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