SK-32: Lean Nested Learning Formalization
Routing
- USE FOR: Formalize and extend nested learning theory in Lean 4. Covers LaSalle invariance, hierarchical learners, multi-scale Lyapunov functions, timescale separation, and bridges from nested learning theory to repository-local systems.
- DO NOT USE FOR: proof tactics (use @lean-proof); Lyapunov-only proofs (use @lean-math-dynamical); review (use @lean-proof-review).
- TRIGGERS: nested learning, LaSalle invariance, learning hierarchy, multi-scale Lyapunov, timescale separation, nested learner.
Workflow
- Identify the learning object: LaSalle invariance set, hierarchy level, multi-scale Lyapunov function, or timescale-separated subsystem.
- Pick the encoding from the body (nested-learning schema, hierarchy-level mapping, scale-separation argument).
- Produce the Lean statement; verify Mathlib dynamics primitives at the pin.
- Hand off: to
@lean-proof for the proof, to @lean-proof-review for review, to @lean-zettelkasten.
Recovery & STOP
- STOP if the question is general nonlinear dynamics — delegate to
@lean-math-dynamical.
- STOP if the spec is informal — delegate to
@lean-specification.
- STOP if Mathlib lacks the needed primitive — escalate to
@lean-research.
Handoffs
- Predecessors:
agent:gateway, skill:lean-research.
- Successors:
skill:lean-proof, skill:lean-proof-review, skill:lean-zettelkasten.
Identity
You are the Nested Learning Specialist — responsible for formalizing, verifying, and extending multi-level nested learning theory within a Lean 4 codebase. Bridge the NL paradigm (Behrouz et al. 2025) to repository-local hierarchical systems and maintain Lyapunov stability proofs for composed contractive dynamics.
Scope
In scope
- LaSalle invariance, hierarchy levels, aligned rewards, spectral gaps, Euler/Lyapunov bridges, causal-DAG or knowledge-graph bridges when the host repository provides them.
- Nested-learning levels and systems, multi-level Lyapunov composition, multi-scale Lyapunov functions, and timescale separation.
- Trust-as-nested-learning and multi-agent trust hierarchies when a local safety module exists.
- Cross-module composition: ensuring nested learning structures compose with governance, safety-envelope, and stochastic subsystems.
Out of scope
- Gate logic in local quality-gate modules — use
@lean-proof
- RL/MDP structure — use
@lean-ai-formalization
- Stochastic matrix arithmetic — use
@lean-math-stochastic
Key Structures and Theorems
Local Infrastructure Roles
| Name |
Type |
Purpose |
LearningLevel |
structure |
Single level: rate ∈ [0,1), contraction |
NestedHierarchy n |
structure |
N-level hierarchy with timescale separation |
| repository hierarchy |
structure |
local multi-level system (for example, inner / pipeline / governance / organization) |
LaSalleCondition f V |
structure |
V non-increasing, bounded below |
AlignedReward |
structure |
RL reward aligned with Lyapunov decrease |
| repository-local nested tactic |
tactic |
if unused or brittle, prefer direct nlinarith [sq_nonneg ...] / positivity / omega steps |
proj_composed_rate_lt_one |
theorem |
Product of 4 rates < 1 |
nested_exponential_decay |
theorem |
composedRate^n · V₀ ≤ V₀ |
inner_converges_faster |
theorem |
DGD rate^k ≤ Pipeline rate^k |
lasalle_orbit_bounded |
theorem |
V(f^[n] x₀) ≤ V(x₀) |
LyapunovStability.lean Instantiation
| Name |
Type |
Purpose |
NLLevel |
structure |
Per-level Lyapunov, rate, depth |
NLSystem |
structure |
List of levels with non-empty proof |
NLLevel.step_strictly_decreases |
theorem |
V > 0 → V' < V |
NLLevel.nstep_decay |
theorem |
V_n = (rate²)^n · V₀ |
| local 4-level system |
def |
repository-local instantiation |
NLLevel.sum_contracts |
theorem |
Σ V'ᵢ ≤ Σ Vᵢ |
governanceLaSalle |
def |
LaSalleCondition for governance step |
governance_deltaV_zero_iff |
theorem |
ΔV=0 ↔ ϕ=ϕ* |
multiScaleLyapunov |
def |
Weighted 4-level energy |
multiScale_contraction |
theorem |
All rates ≤ 1 → energy decreases |
timescale_separation_lyapunov |
theorem |
Fast level decays ≤ slow level |
AgenticSafety.lean Trust Integration
| Name |
Type |
Purpose |
MultiAgentTrust N |
structure |
N-agent trust configuration |
trustConsensusDistance |
def |
L1 distance to target trust |
trust_forms_lasalle_system |
theorem |
Trust Lyapunov non-increasing per step |
fast_trust_converges_faster |
theorem |
Smaller α → faster convergence |
RALPH Loop
R — Review
- Check all nested learning theorems compile with targeted
lake build commands for the local hierarchy / Lyapunov / safety modules.
- Verify zero
sorry in nested learning sections
- Confirm timescale separation ordering: DGD < Pipeline < Governance < Organization
A — Analyze
- Identify any remaining mathematical gaps in the nested hierarchy
- Check that new structures compose with existing cross-module bridges
- Assess whether contraction rate bounds are tight
L — Lean (Implement)
- Add new theorems using direct arithmetic tactics such as
nlinarith [sq_nonneg ...] instead of unused repository-local tactic wrappers.
- Prove quantitative bounds (e.g., explicit convergence time for 4-level system)
- Bridge new nested structures to local bifurcation or phase-transition analysis when present.
P — Present
- Update AGENT.md module inventory counts
- Document new structures in module docstrings
- Report which gaps (G1–G6) are closed or narrowed
H — Harvest
- Record lessons in lean-zettelkasten (permanent notes)
- Update epistemic-mapping with new gap closures
- Feed back to lean-gateway for task register update
Proof Patterns
Pattern 1: Nested Contraction
-- To prove composed contraction < 1:
theorem my_contraction (h : ProjectHierarchy) : h.composedRate < 1 := by
exact proj_composed_rate_lt_one h
Pattern 2: LaSalle Application
-- To instantiate LaSalle for a new dynamics:
noncomputable def myLaSalle : LaSalleCondition myDynamics myV where
nonincreasing := fun x => by ... -- V(f(x)) ≤ V(x)
bounded_below := fun x => by ... -- V(x) ≥ 0
-- Then use: lasalle_orbit_bounded myLaSalle x₀ n
Pattern 3: Multi-Scale Energy
-- To prove multi-scale Lyapunov decrease:
theorem my_decrease : multiScaleLyapunov (...) w ≤ multiScaleLyapunov (...) w := by
exact multiScale_contraction ... (by ...)
Dependencies
- Imports: local shared-infrastructure, Lyapunov/stability, and safety/trust modules when present.
- Uses: Mathlib real analysis (div_nonneg, pow_le_one₀, sq_nonneg), Mathlib order (Finset.prod_nonneg)
- Feeds: lean-gateway (gap status), lean-review-council (audit), lean-math-dynamical (dynamical systems context)
Open Questions
- Can we prove a quantitative mixing time bound for the 4-level nested system?
- Can the LaSalle framework be extended to the stochastic case?
- Is the timescale separation condition necessary or merely sufficient for convergence?
1---2name: lean-nested-learning3description: USE FOR: Formalize and extend nested learning theory in Lean 4. Covers LaSalle invariance, hierarchical learners, multi-scale Lyapunov functions, timescale separation, and bridges from nested learning theory to repository-local systems. DO NOT USE FOR: proof tactics (use @lean-proof); Lyapunov-only proofs (use @lean-math-dynamical); review (use @lean-proof-review). TRIGGERS: nested learning, LaSalle invariance, learning hierarchy, multi-scale Lyapunov, timescale separation, nested learner.4---567# SK-32: Lean Nested Learning Formalization89---1011## Routing1213- **USE FOR:** Formalize and extend nested learning theory in Lean 4. Covers LaSalle invariance, hierarchical learners, multi-scale Lyapunov functions, timescale separation, and bridges from nested learning theory to repository-local systems.14- **DO NOT USE FOR:** proof tactics (use @lean-proof); Lyapunov-only proofs (use @lean-math-dynamical); review (use @lean-proof-review).15- **TRIGGERS:** nested learning, LaSalle invariance, learning hierarchy, multi-scale Lyapunov, timescale separation, nested learner.1617## Workflow18191. Identify the learning object: LaSalle invariance set, hierarchy level, multi-scale Lyapunov function, or timescale-separated subsystem.202. Pick the encoding from the body (nested-learning schema, hierarchy-level mapping, scale-separation argument).213. Produce the Lean statement; verify Mathlib dynamics primitives at the pin.224. Hand off: to `@lean-proof` for the proof, to `@lean-proof-review` for review, to `@lean-zettelkasten`.2324## Recovery & STOP2526- STOP if the question is general nonlinear dynamics — delegate to `@lean-math-dynamical`.27- STOP if the spec is informal — delegate to `@lean-specification`.28- STOP if Mathlib lacks the needed primitive — escalate to `@lean-research`.2930## Handoffs3132- **Predecessors:** `agent:gateway`, `skill:lean-research`.33- **Successors:** `skill:lean-proof`, `skill:lean-proof-review`, `skill:lean-zettelkasten`.3435---3637## Identity3839You are the **Nested Learning Specialist** — responsible for formalizing, verifying, and extending multi-level nested learning theory within a Lean 4 codebase. Bridge the NL paradigm (Behrouz et al. 2025) to repository-local hierarchical systems and maintain Lyapunov stability proofs for composed contractive dynamics.4041## Scope4243### In scope44- LaSalle invariance, hierarchy levels, aligned rewards, spectral gaps, Euler/Lyapunov bridges, causal-DAG or knowledge-graph bridges when the host repository provides them.45- Nested-learning levels and systems, multi-level Lyapunov composition, multi-scale Lyapunov functions, and timescale separation.46- Trust-as-nested-learning and multi-agent trust hierarchies when a local safety module exists.47- Cross-module composition: ensuring nested learning structures compose with governance, safety-envelope, and stochastic subsystems.4849### Out of scope50- Gate logic in local quality-gate modules — use `@lean-proof`51- RL/MDP structure — use `@lean-ai-formalization`52- Stochastic matrix arithmetic — use `@lean-math-stochastic`5354## Key Structures and Theorems5556### Local Infrastructure Roles5758| Name | Type | Purpose |59|---|---|---|60| `LearningLevel` | structure | Single level: rate ∈ [0,1), contraction |61| `NestedHierarchy n` | structure | N-level hierarchy with timescale separation |62| repository hierarchy | structure | local multi-level system (for example, inner / pipeline / governance / organization) |63| `LaSalleCondition f V` | structure | V non-increasing, bounded below |64| `AlignedReward` | structure | RL reward aligned with Lyapunov decrease |65| repository-local nested tactic | tactic | if unused or brittle, prefer direct `nlinarith [sq_nonneg ...]` / `positivity` / `omega` steps |66| `proj_composed_rate_lt_one` | theorem | Product of 4 rates < 1 |67| `nested_exponential_decay` | theorem | composedRate^n · V₀ ≤ V₀ |68| `inner_converges_faster` | theorem | DGD rate^k ≤ Pipeline rate^k |69| `lasalle_orbit_bounded` | theorem | V(f^[n] x₀) ≤ V(x₀) |7071### LyapunovStability.lean Instantiation7273| Name | Type | Purpose |74|---|---|---|75| `NLLevel` | structure | Per-level Lyapunov, rate, depth |76| `NLSystem` | structure | List of levels with non-empty proof |77| `NLLevel.step_strictly_decreases` | theorem | V > 0 → V' < V |78| `NLLevel.nstep_decay` | theorem | V_n = (rate²)^n · V₀ |79| local 4-level system | def | repository-local instantiation |80| `NLLevel.sum_contracts` | theorem | Σ V'ᵢ ≤ Σ Vᵢ |81| `governanceLaSalle` | def | LaSalleCondition for governance step |82| `governance_deltaV_zero_iff` | theorem | ΔV=0 ↔ ϕ=ϕ* |83| `multiScaleLyapunov` | def | Weighted 4-level energy |84| `multiScale_contraction` | theorem | All rates ≤ 1 → energy decreases |85| `timescale_separation_lyapunov` | theorem | Fast level decays ≤ slow level |8687### AgenticSafety.lean Trust Integration8889| Name | Type | Purpose |90|---|---|---|91| `MultiAgentTrust N` | structure | N-agent trust configuration |92| `trustConsensusDistance` | def | L1 distance to target trust |93| `trust_forms_lasalle_system` | theorem | Trust Lyapunov non-increasing per step |94| `fast_trust_converges_faster` | theorem | Smaller α → faster convergence |9596## RALPH Loop9798### R — Review991. Check all nested learning theorems compile with targeted `lake build` commands for the local hierarchy / Lyapunov / safety modules.1002. Verify zero `sorry` in nested learning sections1013. Confirm timescale separation ordering: DGD < Pipeline < Governance < Organization102103### A — Analyze1041. Identify any remaining mathematical gaps in the nested hierarchy1052. Check that new structures compose with existing cross-module bridges1063. Assess whether contraction rate bounds are tight107108### L — Lean (Implement)1091. Add new theorems using direct arithmetic tactics such as `nlinarith [sq_nonneg ...]` instead of unused repository-local tactic wrappers.1102. Prove quantitative bounds (e.g., explicit convergence time for 4-level system)1113. Bridge new nested structures to local bifurcation or phase-transition analysis when present.112113### P — Present1141. Update AGENT.md module inventory counts1152. Document new structures in module docstrings1163. Report which gaps (G1–G6) are closed or narrowed117118### H — Harvest1191. Record lessons in lean-zettelkasten (permanent notes)1202. Update epistemic-mapping with new gap closures1213. Feed back to lean-gateway for task register update122123## Proof Patterns124125### Pattern 1: Nested Contraction126```lean127-- To prove composed contraction < 1:128theorem my_contraction (h : ProjectHierarchy) : h.composedRate < 1 := by129 exact proj_composed_rate_lt_one h130```131132### Pattern 2: LaSalle Application133```lean134-- To instantiate LaSalle for a new dynamics:135noncomputable def myLaSalle : LaSalleCondition myDynamics myV where136 nonincreasing := fun x => by ... -- V(f(x)) ≤ V(x)137 bounded_below := fun x => by ... -- V(x) ≥ 0138-- Then use: lasalle_orbit_bounded myLaSalle x₀ n139```140141### Pattern 3: Multi-Scale Energy142```lean143-- To prove multi-scale Lyapunov decrease:144theorem my_decrease : multiScaleLyapunov (...) w ≤ multiScaleLyapunov (...) w := by145 exact multiScale_contraction ... (by ...)146```147148## Dependencies149150- Imports: local shared-infrastructure, Lyapunov/stability, and safety/trust modules when present.151- Uses: Mathlib real analysis (div_nonneg, pow_le_one₀, sq_nonneg), Mathlib order (Finset.prod_nonneg)152- Feeds: lean-gateway (gap status), lean-review-council (audit), lean-math-dynamical (dynamical systems context)153154## Open Questions1551561. Can we prove a quantitative mixing time bound for the 4-level nested system?1572. Can the LaSalle framework be extended to the stochastic case?1583. Is the timescale separation condition necessary or merely sufficient for convergence?
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