obstruction-learning

# Obstruction Learning Skill

Learn topological ASI via random walk obstruction detection and Čech H⁰ cohomology.

## Metadata

| Property | Value |
|----------|-------|
| **Name** | obstruction-learning |
| **Trit** | -1 (VALIDATOR) |
| **Category** | Topological Verification |
| **Dependencies** | sheaf-cohomology, ramanujan-expander, gay-mcp |

## Core Concept

**Obstructions are H⁰ generators** - irreducible elements that block global consistency from local patches.

```
Čech Cohomology: H⁰(U, F) = ker(d₀: F(U) → ∏ᵢⱼ F(Uᵢ ∩ Uⱼ))

Obstruction detected when:
  - GF(3) conservation violated (sum ≢ 0 mod 3)
  - Voice triads don't harmonize
  - Skill compositions conflict
  - Local patches fail to glue globally
```

## Random Walk Reconstruction

### The 69-Skill Walk

Sample 69 skills from the 181-skill manifold:

```bash
# Execute random walk
just random-walk-69

# Verify GF(3) conservation
just verify-gf3

# Track cumulative obstructions
just random-walk-obstruction 69
```

### Obstruction Detection

```sql
-- Find unbalanced cells in 23³ grid
SELECT cell_id, skill_count, trit_sum, gf3_status
FROM cell_density 
WHERE gf3_status = 'UNBALANCED';

-- H⁰ generators by trit class
SELECT trit, COUNT(*) as generators
FROM skills 
GROUP BY trit;
```

## Mathematical Foundations

### Čech Cohomology

For a covering U = {Uᵢ} of skill space:

```
H⁰(U, F) = { s ∈ F(U) | d₀(s) = 0 }

where d₀: F(U) → ∏ F(Uᵢ ∩ Uⱼ)
maps global sections to intersection restrictions
```

**Obstruction** = element of H⁰ that prevents gluing.

### GF(3) as Cohomology

The GF(3) conservation law is a discrete cohomology:

```
Trit assignment: skill → {-1, 0, +1}
Coboundary: d(triad) = sum of trits mod 3

H⁰ = { triads | d(triad) = 0 } = balanced triads
Obstruction = triad with d ≠ 0
```

### Ramanujan Mixing

Random walks on Ramanujan expanders mix optimally:

```
λ₂ ≤ 2√(d-1)     [Alon-Boppana bound]
gap = d - λ₂      [Spectral gap]
τ_mix = O(log n / gap)  [Mixing time]
```

## Workflow

### 1. Pre-Interaction Sync

```bash
just pre-interaction
# Syncs plurigrid/asi arena + hdresearch/duck
# Loads GF(3) skill triad
# Computes spectral awareness
```

### 2. Random Walk Sampling

```bash
# Sample without replacement (maximal coverage)
just random-walk-69

# Sample with replacement (GF(3) conservation)
just random-walk 23
```

### 3. Obstruction Detection

```bash
# Find H⁰ generators
just obstruction-h0

# Detect unbalanced cells
just obstruction-detect

# Balance with complementary skill
just obstruction-balance -1  # Find validators to add
just obstruction-balance +1  # Find generators to add
```

### 4. Audio Generation

Convert obstruction traces to sound:

```bash
just audio-from-trace
```

## Integration Patterns

### With Voice Enforcement

```toml
# voice-enforcement.toml
[triads.obstruction]
validator = "Milena (Enhanced)"   # -1: detects obstruction
coordinator = "Petra (Premium)"   # 0: mediates resolution
generator = "Federica (Premium)"  # +1: proposes fix
sum = 0
```

### With Dune Orthogonalization

The 23×23×23 grid maps skills to:

| Axis | Dimensions |
|------|------------|
| DATA | chain_indexing → real_time_streaming |
| INTERFACE | sql_query_engine → ai_copilot |
| INFRASTRUCTURE | kubernetes → multi_tenant_isolation |

### With World Extractable Value

```
WEV = PoA - 1 = extractable coordination loss

Obstruction → WEV > 0
Resolution → WEV → 0
Global consistency → Optimal coordination
```

## Commands

```bash
# Full ASI learning loop
just asi-learn

# Spectral bounds
just spectral-bounds

# World Extractable Value
just wev-compute

# Voice obstruction analysis
just voice-obstructions
```

## Skill Triad

This skill belongs to the **topological verification** triad:

| Role | Skill | Trit |
|------|-------|------|
| VALIDATOR | **obstruction-learning** | -1 |
| COORDINATOR | sheaf-cohomology | 0 |
| GENERATOR | persistent-homology | +1 |

**Sum = 0** ✓ GF(3) conserved

## References

- Bott & Tu, *Differential Forms in Algebraic Topology*
- Lurie, *Higher Topos Theory*
- Riehl-Shulman, *Synthetic ∞-categories*
- QRI, *Symmetry Theory of Valence*



## Scientific Skill Interleaving

This skill connects to the K-Dense-AI/claude-scientific-skills ecosystem:

### Graph Theory
- **networkx** [○] via bicomodule
  - Universal graph hub

### Bibliography References

- `general`: 734 citations in bib.duckdb

## Cat# Integration

This skill maps to **Cat# = Comod(P)** as a bicomodule in the equipment structure:

```
Trit: 0 (ERGODIC)
Home: Prof
Poly Op: ⊗
Kan Role: Adj
Color: #26D826
```

### GF(3) Naturality

The skill participates in triads satisfying:
```
(-1) + (0) + (+1) ≡ 0 (mod 3)
```

This ensures compositional coherence in the Cat# equipment structure.