P-adic Ultrametric Skill
Trit: -1 (MINUS - validates/constrains via non-Archimedean metric)
GF(3) Triad: padic-ultrametric (-1) ⊗ skill-embedding-vss (0) ⊗ gay-mcp (+1) = 0 ✓
The Stack: From Mathematics to Metal
┌─────────────────────────────────────────────────────────────────────┐
│ P-ADIC ULTRAMETRIC: The Non-Archimedean Foundation │
├─────────────────────────────────────────────────────────────────────┤
│ Layer 6: P-adic Valuation │
│ d(x,z) ≤ max(d(x,y), d(y,z)) ← Strong Triangle Inequality │
├─────────────────────────────────────────────────────────────────────┤
│ Layer 5: UMAP / itUMAP │
│ Manifold approximation with ultrametric preservation │
├─────────────────────────────────────────────────────────────────────┤
│ Layer 4: HNSW (Hierarchical Navigable Small World) │
│ Log(n) approximate nearest neighbor via skip-list graph │
├─────────────────────────────────────────────────────────────────────┤
│ Layer 3: Snowflake Arctic 1024-bit Embeddings │
│ Dense semantic vectors: text → R^1024 │
├─────────────────────────────────────────────────────────────────────┤
│ Layer 2: MLX (Apple Machine Learning Framework) │
│ Unified memory, lazy evaluation, Metal kernels │
├─────────────────────────────────────────────────────────────────────┤
│ Layer 1: Apple Silicon (M1/M2/M3/M4) │
│ SPI-verified: deterministic across parallel streams │
└─────────────────────────────────────────────────────────────────────┘
Why P-adic Ultrametrics?
The Strong Triangle Inequality
Standard (Archimedean) triangle inequality: d(x,z) ≤ d(x,y) + d(y,z)
P-adic (non-Archimedean) inequality: d(x,z) ≤ max(d(x,y), d(y,z))
Implications:
- All triangles are isoceles with the unequal side being shortest
- Natural hierarchical clustering - no intermediate distances
- Every point is a center - all balls have same radius from any interior point
- Perfect for skill taxonomies - semantic similarity is hierarchical
Connection to Prime Geodesics (Chromatic Walk)
From chromatic-walk:
Walks are prime geodesics: non-backtracking paths that are unambiguously traversable in p-adic number systems.
| Property | Prime Path | Composite Path | |----------|------------|----------------| | Factorization | Unique | Multiple | | p-adic valuation | Well-defined | Ambiguous | | Möbius μ(n) | ≠ 0 | = 0 (filtered) |
Core Implementation
P-adic Valuation and Norm
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
def p_adic_norm(n: int, p: int = 2) -> float:
"""|n|_p = p^(-v_p(n)). Smaller values are "further"."""
v = p_adic_valuation(n, p)
return 0.0 if v == float('inf') else p ** (-v)
def padic_ultrametric_distance(emb_a: np.ndarray, emb_b: np.ndarray, p: int = 2) -> float:
"""
P-adic ultrametric: d_p(a, b) = max_i |a_i - b_i|_p
Satisfies: d(x,z) ≤ max(d(x,y), d(y,z))
"""
diff = emb_a - emb_b
scale = 2 ** 32
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 IDs with Normal Forms
Finding content with IDs that have normal forms (via cq | jq diffed with narya.el semantics):
@dataclass
class ContentID:
"""Content-addressable identifier with normal form."""
id: str
content: str
normal_form: str
hash: str
source: str # 'cq' | 'jq' | 'narya'
def jq_normalize(content: str) -> str:
"""Normalize JSON/YAML using jq-style: sort keys, compact."""
try:
data = json.loads(content)
return json.dumps(data, sort_keys=True, separators=(',', ':'))
except json.JSONDecodeError:
return content.strip()
def cq_normalize(content: str) -> str:
"""Normalize using cq (Clojure query) style - EDN/S-expr."""
content = re.sub(r'\s+', ' ', content)
return content.strip()
@dataclass
class NaryaDiff:
"""Narya-style semantic diff: before/after/delta/birth/death."""
before: str
after: str
delta: Dict[str, Any]
birth: List[str] # New content
death: List[str] # Removed content
@classmethod
def from_contents(cls, before: str, after: str) -> 'NaryaDiff':
before_lines = set(before.split('\n'))
after_lines = set(after.split('\n'))
birth = list(after_lines - before_lines)
death = list(before_lines - after_lines)
return cls(
before=before, after=after,
delta={'added': len(birth), 'removed': len(death), 'changed': len(birth) + len(death)},
birth=birth, death=death
)
def to_narya_witness(self) -> Dict:
"""Format as Narya proof witness for observational bridge types."""
import hashlib
return {
'before': hashlib.sha256(self.before.encode()).hexdigest()[:16],
'after': hashlib.sha256(self.after.encode()).hexdigest()[:16],
'delta': self.delta,
'impact': self.delta['changed'] > 0
}
MLX Operation Tracing → SPI Verification
Trace every MLX op down to Metal kernels with Strong Parallelism Invariance:
@dataclass
class MLXTrace:
"""Trace MLX operations to Apple Silicon primitives."""
operation: str
input_shapes: List[Tuple[int, ...]]
output_shape: Tuple[int, ...]
metal_kernel: Optional[str] = None
flops: int = 0
memory_bytes: int = 0
class SPIVerifier:
"""Strong Parallelism Invariance: same seed → same result across parallel streams."""
GOLDEN = 0x9E3779B97F4A7C15
def __init__(self, seed: int):
self.seed = seed
self.state = seed
self.traces: List[MLXTrace] = []
self.checksums: List[int] = []
def log_trace(self, trace: MLXTrace):
self.traces.append(trace)
# Update checksum deterministically
self.state = ((self.state ^ (hash(trace.operation) & 0xFFFFFFFF)) * self.GOLDEN) & 0xFFFFFFFFFFFFFFFF
self.checksums.append(self.state & 0xFFFF)
def verify_chain(self) -> bool:
"""Verify deterministic chain: recompute from seed, compare checksums."""
verifier = SPIVerifier(self.seed)
for trace in self.traces:
verifier.log_trace(trace)
return verifier.checksums == self.checksums
UMAP ↔ itUMAP ↔ HNSW Integration
itUMAP (Iterative UMAP)
Preserves geodesic distances better than standard UMAP:
def itumap_with_padic(embeddings: np.ndarray, n_neighbors: int = 15,
metric: str = 'padic', prime: int = 2) -> np.ndarray:
"""
itUMAP with p-adic ultrametric as base distance.
1. Compute p-adic distance matrix
2. Run UMAP with precomputed distances
3. Iterate: refine based on projection error
"""
n = len(embeddings)
# P-adic distance matrix
dist_matrix = np.zeros((n, n))
for i in range(n):
for j in range(i+1, n):
d = padic_ultrametric_distance(embeddings[i], embeddings[j], prime)
dist_matrix[i, j] = d
dist_matrix[j, i] = d
# UMAP with precomputed distances
import umap
reducer = umap.UMAP(
metric='precomputed',
n_neighbors=n_neighbors,
min_dist=0.1
)
projection = reducer.fit_transform(dist_matrix)
return projection
HNSW with P-adic Reranking
Use Euclidean HNSW for recall, p-adic for precision:
def hnsw_padic_search(index: 'PAdicSkillIndex', query: np.ndarray, k: int = 10) -> List[Tuple[str, float]]:
"""
Two-stage search:
1. HNSW retrieval (fast, Euclidean)
2. P-adic reranking (precise, ultrametric)
"""
# Stage 1: Get 3k candidates via HNSW
candidates = index.conn.execute(f'''
SELECT name, array_distance(embedding, ?::FLOAT[1024]) as dist
FROM skills ORDER BY dist ASC LIMIT ?
''', [query.tolist(), k * 3]).fetchall()
# Stage 2: Rerank by p-adic distance
reranked = []
for name, eucl_dist in candidates:
if name in index.embeddings:
padic_dist = padic_ultrametric_distance(query, index.embeddings[name], index.prime)
reranked.append((name, padic_dist))
reranked.sort(key=lambda x: x[1])
return reranked[:k]
Ultrametric Clustering
The strong triangle inequality gives natural hierarchical clusters:
def ultrametric_clustering(embeddings: Dict[str, np.ndarray],
threshold: float = 0.5,
prime: int = 2) -> List[List[str]]:
"""
Single-linkage clustering in ultrametric space.
In ultrametric: all triangles are isoceles → natural dendrograms.
"""
skills = list(embeddings.keys())
n = len(skills)
# Distance matrix
dist_matrix = np.zeros((n, n))
for i in range(n):
for j in range(i+1, n):
d = padic_ultrametric_distance(embeddings[skills[i]], embeddings[skills[j]], prime)
dist_matrix[i, j] = d
dist_matrix[j, i] = d
# Single-linkage (natural for ultrametric)
clusters = [[i] for i in range(n)]
while len(clusters) > 1:
min_dist = float('inf')
merge_pair = None
for i, c1 in enumerate(clusters):
for j, c2 in enumerate(clusters[i+1:], i+1):
d = max(dist_matrix[a, b] for a in c1 for b in c2)
if d < min_dist:
min_dist = d
merge_pair = (i, j)
if min_dist > threshold or merge_pair is None:
break
i, j = merge_pair
clusters[i].extend(clusters[j])
clusters.pop(j)
return [[skills[i] for i in cluster] for cluster in clusters]
Propagation to All Skills
How to Use P-adic Distance in Any Skill
# In any skill that uses similarity/distance:
from padic_ultrametric import padic_ultrametric_distance, PAdicConfig
config = PAdicConfig(prime=2, precision=64)
# Replace Euclidean distance:
# distance = np.linalg.norm(a - b) # OLD
distance = padic_ultrametric_distance(a, b, config.prime) # NEW
# The ultrametric property guarantees:
assert distance <= max(d_ab, d_bc) # Strong triangle inequality
CLI Usage
# Find p-adic nearest neighbors
python -c "
from padic_ultrametric import PAdicSkillIndex
idx = PAdicSkillIndex('/Users/bob/.claude/skills', seed=1069, prime=2)
idx.index_skills_with_ids()
for name, eucl, padic in idx.padic_nearest('bisimulation-game', k=5):
print(f'{name}: eucl={eucl:.4f}, p-adic={padic:.6f}')
"
# Find skills with content IDs and normal forms
python -c "
from padic_ultrametric import PAdicSkillIndex
idx = PAdicSkillIndex('/Users/bob/.claude/skills')
cids = idx.index_skills_with_ids()
for name, cid in list(cids.items())[:10]:
print(f'{name}: {cid.id} (hash: {cid.hash}, source: {cid.source})')
"
# Generate SPI report
python -c "
from padic_ultrametric import PAdicSkillIndex
idx = PAdicSkillIndex('/Users/bob/.claude/skills', seed=1069)
idx.index_skills_with_ids()
report = idx.spi_report()
print(f'Seed: {report[\"seed\"]}, Chain valid: {report[\"chain_valid\"]}')
print(f'Total FLOPS: {report[\"total_flops\"]:,}')
"
Invariants
invariants:
- name: ultrametric_property
predicate: "d(x,z) ≤ max(d(x,y), d(y,z))"
scope: all_triples
- name: spi_determinism
predicate: "same seed → same checksum chain"
scope: per_session
- name: normal_form_stability
predicate: "jq_normalize(cq_normalize(x)) = jq_normalize(x)"
scope: per_content
- name: gf3_conservation
predicate: "padic(-1) + embedding-vss(0) + gay-mcp(+1) = 0"
scope: per_triad
End-of-Skill Interface
Integration Points
| Skill | Integration Point | Benefit |
|-------|------------------|---------|
| skill-embedding-vss | find_nearest() | Hierarchical skill clusters |
| chromatic-walk | Prime geodesic validation | Non-backtracking paths |
| bisimulation-game | Observational equivalence | Ultrametric bisimilarity |
| gay-mcp | Color distance | p-adic hue difference |
| glass-bead-game | World hopping distance | Triangle inequality sparsification |
References
- Non-Archimedean Geometry
- P-adic Numbers
- Chromatic Walk: Prime Geodesics
- Skill Embedding VSS
- Structure-Aware Version Control via Observational Bridge Types
- SplitMix64
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.