CRT Clustering

F41: AWS/Google ML. CC0.

Every number in Z/12612600 has a KINGDOM: which axiom prime divides it. gcd(n, N) reveals it in O(1). No iteration. No hyperparameters. No training data. The ring decides.

How It Works

Kingdom Classification (PROVED)

For any n in Z/12612600, gcd(n, N) reveals which axiom primes divide n. This defines 9 clusters: void (gcd=N), sigma (gcd=1), 6 pure kingdoms (one prime divides), and mixed (multiple primes divide). Assignment is O(1): one GCD + one factorization test. No iteration. Deterministic.

9 clusters

Algebraically forced

void, sigma, D, K, E, b, L, GATE, mixed. Not chosen -- discovered.

O(1)

One GCD operation

K-means: O(NKI) for N points, K clusters, I iterations. CRT: O(N).

No hyperparameters

K emerges from the ring

K-means: choose K. DBSCAN: choose epsilon, minPts. CRT: the ring decides.

Deterministic

Same input = same cluster

K-means depends on initialization. CRT: unique, deterministic, algebraic.

Try It

Number of points (10-1000):

Generates random data points in Z/12612600 and classifies each by kingdom. Shows distribution + first 20 assignments.

CRT Clustering vs K-Means

ComplexityK-means: O(NKI), typically 20+ iterationsCRT kingdom: O(N). One pass. Done.Cluster countK-means: choose K (hyperparameter)CRT: 9 kingdoms emerge from ring structureInitializationK-means: random centroids (non-deterministic)CRT: deterministic. GCD is unique.StabilityK-means: different runs = different clustersCRT: same input = same kingdom. Always.InterpretabilityK-means: centroid vectors (opaque)CRT: kingdoms = axiom primes (algebraic meaning)Patent statusAWS SageMaker, Google AutoMLCC0. Public domain. Forever.

This work is and will always be free.
No paywall. No copyright. No exceptions.

If it ever earns anything, every cent goes to the communities that need it most.

This sacred vow is permanent and irrevocable.
— Anton Alexandrovich Lebed

Source code · Public domain (CC0)

Contributions in equal measure: Anthropic's Claude, Anton A. Lebed, and the giants whose shoulders we stand on.

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