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):
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
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
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:
@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)
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)
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
@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:
@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)
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
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
# Skills with id: field and normal form
for name, cid in content_ids.items():
print(f"{name}: {cid.id} (hash: {cid.hash})")
Ultrametric Clustering
clusters = index.find_ultrametric_clusters(threshold=0.3)
for cluster in clusters[:5]:
print(f"Cluster: {cluster}")
Narya-style Diff
diff = index.diff_skills('skill-a', 'skill-b')
witness = diff.to_narya_witness()
print(f"Added: {witness['birth']}, Removed: {witness['death']}")
SPI Report
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:
- Compute p-adic tree structure
- Initialize UMAP with tree distances
- Refine with geodesic optimization
Dependencies
uv pip install mlx-embeddings duckdb numpy
Files
padic_ultrametric.py- Core implementationskill_embedding_vss.py- Basic VSS without p-adictrifurcate_walk.py- GF(3) random walks through embedding space
Invariants
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 - Non-Archimedean metric
- Bruhat-Tits Trees - P-adic symmetric spaces
- UMAP - McInnes et al.
- HNSW - Malkov & Yashunin
- Snowflake Arctic
- 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.