# Padic Ultrametric Embedding

> P-adic ultrametric distance for UMAP/itUMAP/HNSW with Snowflake Arctic 1024-bit embeddings on Apple Silicon via MLX. Full SPI trace down to Metal ops.

- Skill: `plurigrid/padic-ultrametric-embedding` (Agent Skill, multi-file: 4 files)
- Install (CLI): `npx skillmds@latest add plurigrid/padic-ultrametric-embedding`
- Raw SKILL.md: https://api.skillmd.com/api/skills/plurigrid/padic-ultrametric-embedding/raw
- Safety review: pending
- Works with: Claude Code, Claude.ai, OpenAI Codex
- Category: AI & ML
- Author: plurigrid (https://skillmd.com/u/plurigrid)
- Updated: 2026-09-17
- Page: https://skillmd.com/skills/plurigrid/padic-ultrametric-embedding

---

# P-adic Ultrametric Embedding Skill

Non-Archimedean geometry for skill embeddings. Introduces p-adic distance as foundation for understanding UMAP, itUMAP, and HNSW search structures.

## Why P-adic?

The **ultrametric inequality** (strong triangle inequality):

```
d(x, z) ≤ max(d(x, y), d(y, z))
```

This gives:
- **Hierarchical clustering** - all triangles are isoceles
- **Natural tree structure** - skills cluster into clopen balls
- **No "in-between"** - either close (same prefix) or far (different subtree)

## Full Stack

```
┌─────────────────────────────────────────────────────────────────┐
│                    P-ADIC EMBEDDING STACK                       │
├─────────────────────────────────────────────────────────────────┤
│                                                                 │
│  Layer 7: UMAP/itUMAP                                          │
│           ↓ geodesic-preserving projection                     │
│                                                                 │
│  Layer 6: HNSW Index                                           │
│           ↓ navigable small-world graph                        │
│                                                                 │
│  Layer 5: Snowflake Arctic 1024-bit                            │
│           ↓ semantic embedding space                           │
│                                                                 │
│  Layer 4: MLX Operations                                       │
│           ↓ matmul, attention, pooling                         │
│                                                                 │
│  Layer 3: Metal Kernels                                        │
│           ↓ GPU shaders on Apple Silicon                       │
│                                                                 │
│  Layer 2: P-adic Valuation                                     │
│           ↓ v_p(x) = largest k where p^k | x                   │
│                                                                 │
│  Layer 1: SPI (Strong Parallelism Invariance)                  │
│           ↓ deterministic parallel execution                   │
│                                                                 │
│  Layer 0: Content ID + Normal Form                             │
│           cq | jq diffed with narya.el semantics               │
│                                                                 │
└─────────────────────────────────────────────────────────────────┘
```

## GF(3) Triad

```
padic-ultrametric-embedding (0) ⊗ skill-embedding-vss (-1) ⊗ gay-mcp (+1) = 0 ✓
```

## Core Mathematics

### P-adic Valuation

For prime p (default p=2):
```python
def p_adic_valuation(n: int, p: int = 2) -> int:
    """v_p(n) = largest k such that p^k divides n."""
    if n == 0: return float('inf')
    k = 0
    while n % p == 0:
        n //= p
        k += 1
    return k
```

### P-adic Norm

```python
def p_adic_norm(n: int, p: int = 2) -> float:
    """|n|_p = p^(-v_p(n)) — smaller for "larger" divisibility."""
    v = p_adic_valuation(n, p)
    return 0.0 if v == float('inf') else p ** (-v)
```

### Ultrametric Distance on Embeddings

```python
def padic_ultrametric_distance(emb_a: np.ndarray, emb_b: np.ndarray, p: int = 2) -> float:
    """
    d_p(a, b) = max_i |a_i - b_i|_p
    
    Satisfies strong triangle: d(x,z) ≤ max(d(x,y), d(y,z))
    """
    diff = emb_a - emb_b
    scale = 2 ** 32  # Fixed-point conversion
    diff_int = (diff * scale).astype(np.int64)
    
    norms = [p_adic_norm(abs(int(d)), p) if d != 0 else 0.0 for d in diff_int]
    return max(norms) if norms else 0.0
```

## Content ID + Normal Form

Every skill with an `id:` field gets:

```python
@dataclass
class ContentID:
    id: str           # The skill's unique identifier
    content: str      # Raw content
    normal_form: str  # Canonicalized (cq/jq normalized)
    hash: str         # SHA-256 of normal form
    source: str       # 'cq' | 'jq' | 'narya'
```

### cq Normalization (S-expression style)

```python
def cq_normalize(content: str) -> str:
    """Normalize as EDN/S-expr: balance parens, collapse whitespace."""
    content = re.sub(r'\s+', ' ', content)
    return content.strip()
```

### jq Normalization (JSON style)

```python
def jq_normalize(content: str) -> str:
    """Normalize as JSON: sort keys, compact."""
    data = json.loads(content)
    return json.dumps(data, sort_keys=True, separators=(',', ':'))
```

### Narya Diff Semantics

```python
@dataclass
class NaryaDiff:
    before: str
    after: str
    delta: Dict[str, Any]  # {added: N, removed: M, changed: N+M}
    birth: List[str]       # New content lines
    death: List[str]       # Removed content lines
    
    def to_narya_witness(self) -> Dict:
        """Format as proof witness for narya.el."""
        return {
            'before': sha256(self.before)[:16],
            'after': sha256(self.after)[:16],
            'delta': self.delta,
            'birth': len(self.birth),
            'death': len(self.death),
            'impact': self.delta['changed'] > 0
        }
```

## MLX Operation Trace

Every embedding generation is traced:

```python
@dataclass
class MLXTrace:
    operation: str           # 'tokenize', 'embedding_lookup', 'attention', etc.
    input_shapes: List[Tuple[int, ...]]
    output_shape: Tuple[int, ...]
    metal_kernel: Optional[str]  # 'gather', 'matmul_4bit', 'softmax', etc.
    flops: int
    memory_bytes: int
```

### Traced Operations

| Operation | Metal Kernel | FLOPS Formula |
|-----------|--------------|---------------|
| embedding_lookup | gather | seq_len × dim |
| attention | matmul_4bit, softmax | 2 × seq² × dim |
| ffn | matmul_4bit | 4 × seq × dim² |
| layer_norm | layer_norm | 2 × seq × dim |
| pooling | mean | seq × dim |

## SPI (Strong Parallelism Invariance)

```python
class SPIVerifier:
    """Verifies determinism across parallel executions."""
    
    def __init__(self, seed: int):
        self.seed = seed
        self.traces: List[MLXTrace] = []
        self.checksums: List[str] = []
    
    def log_trace(self, trace: MLXTrace):
        self.traces.append(trace)
        checksum = hashlib.sha256(
            f"{trace.operation}:{trace.output_shape}:{self.checksums[-1] if self.checksums else '0'}"
            .encode()
        ).hexdigest()[:16]
        self.checksums.append(checksum)
    
    def verify_chain(self) -> bool:
        """Verify checksum chain integrity."""
        return len(self.checksums) == len(self.traces)
```

## Usage

### Index Skills with P-adic Distance

```python
from padic_ultrametric import PAdicSkillIndex

index = PAdicSkillIndex('/path/to/skills', seed=1069, prime=2)
content_ids = index.index_skills_with_ids()

# Find nearest by p-adic (not Euclidean!)
neighbors = index.padic_nearest('bisimulation-game', k=5)
for name, eucl, padic in neighbors:
    print(f"{name}: eucl={eucl:.4f}, p-adic={padic:.6f}")
```

### Find Skills with Content IDs

```python
# Skills with id: field and normal form
for name, cid in content_ids.items():
    print(f"{name}: {cid.id} (hash: {cid.hash})")
```

### Ultrametric Clustering

```python
clusters = index.find_ultrametric_clusters(threshold=0.3)
for cluster in clusters[:5]:
    print(f"Cluster: {cluster}")
```

### Narya-style Diff

```python
diff = index.diff_skills('skill-a', 'skill-b')
witness = diff.to_narya_witness()
print(f"Added: {witness['birth']}, Removed: {witness['death']}")
```

### SPI Report

```python
report = index.spi_report()
print(f"Seed: {report['seed']}, Chain valid: {report['chain_valid']}")
print(f"Total FLOPS: {report['total_flops']:,}")
```

## UMAP Connection

UMAP uses nearest-neighbor graphs; p-adic ultrametric gives **hierarchical** graphs:

| Property | Euclidean NN | P-adic Ultrametric |
|----------|--------------|---------------------|
| Triangle | Weak: d(x,z) ≤ d(x,y) + d(y,z) | Strong: d(x,z) ≤ max(d(x,y), d(y,z)) |
| Clusters | Spherical | Clopen balls (tree) |
| Interpolation | Smooth gradients | Discrete jumps |
| Graph structure | k-NN graph | Bruhat-Tits tree |

### itUMAP Enhancement

Iterative UMAP with p-adic warm-start:
1. Compute p-adic tree structure
2. Initialize UMAP with tree distances
3. Refine with geodesic optimization

## Dependencies

```bash
uv pip install mlx-embeddings duckdb numpy
```

## Files

- `padic_ultrametric.py` - Core implementation
- `skill_embedding_vss.py` - Basic VSS without p-adic
- `trifurcate_walk.py` - GF(3) random walks through embedding space

## Invariants

```yaml
invariants:
  - name: ultrametric_property
    predicate: "∀x,y,z: d(x,z) ≤ max(d(x,y), d(y,z))"
    scope: per_triple
    
  - name: content_id_uniqueness
    predicate: "distinct id: → distinct hash"
    scope: per_skill
    
  - name: spi_determinism
    predicate: "same seed → same checksum chain"
    scope: per_index
```

---

## End-of-Skill Interface

## References

- [P-adic Analysis](https://en.wikipedia.org/wiki/P-adic_number) - Non-Archimedean metric
- [Bruhat-Tits Trees](https://en.wikipedia.org/wiki/Bruhat%E2%80%93Tits_building) - P-adic symmetric spaces
- [UMAP](https://arxiv.org/abs/1802.03426) - McInnes et al.
- [HNSW](https://arxiv.org/abs/1603.09320) - Malkov & Yashunin
- [Snowflake Arctic](https://huggingface.co/Snowflake/snowflake-arctic-embed-l-v2.0)
- [MLX](https://github.com/ml-explore/mlx) - Apple's ML framework


---

## Autopoietic Marginalia

> **The interaction IS the skill improving itself.**

Every use of this skill is an opportunity for worlding:
- **MEMORY** (-1): Record what was learned
- **REMEMBERING** (0): Connect patterns to other skills  
- **WORLDING** (+1): Evolve the skill based on use



*Add Interaction Exemplars here as the skill is used.*