CRT Neural Network

Train a neural network split into 5 independent CRT channels. Each channel trains alone -- kill any channel, the network degrades gracefully. Corrupt weights -- L=11 detects it for free. All math is integer. No floats needed.

How It Works

STANDARD NEURAL NETWORK:
  Forward:  x -> W*x + b -> ReLU -> output
  Backward: full Jacobian (all neurons interact)
  Cost: O(N^2) per layer

CRT NEURAL NETWORK:
  Forward:  x -> 5 independent sub-networks -> average
  Backward: 5 INDEPENDENT Jacobians (block-diagonal)
  Cost: O(sum(m_i^2)) = O(8^2+9^2+25^2+49^2+11^2) = O(3292)

  Channel D (mod 8):   temporal features
  Channel K (mod 9):   structural features
  Channel E (mod 25):  observational features
  Channel b (mod 49):  depth features
  Channel L (mod 11):  ECC / integrity

At N=970200: 286 MILLION x fewer gradient ops.
L=11 = free error detection on every inference.

Live Training: XOR

XOR: the classic non-linear test. Standard = one 2->8->1 network (33 params). CRT = five independent 2->4->1 networks (85 params), outputs averaged. Same task, different architecture.

Channel Isolation

Kill individual channels during inference. The CRT network degrades gracefully. A standard network has no equivalent -- kill one neuron, cascade failure.

Even with channels dead, remaining channels reconstruct a partial answer. With L=11 alive, corruption in any other channel is detectable.

L=11 Error Detection

Corrupt the b-channel weights with random noise. L=11 detects corruption by comparing its independent prediction against the corrupted consensus. No additional cost -- L trains as part of CRT decomposition.

Standard vs CRT

CRT decomposition is EXACT. Not an approximation -- the Chinese Remainder Theorem guarantees unique reconstruction. The savings are algebraic.

Implementation

This demo runs in .ax compiled to WebAssembly. The neural network uses fixed-point arithmetic (scale 1000) -- no floating-point arrays needed. Decision boundary: canvas fillRect at 4px resolution. 118 parameters (33 standard + 85 CRT). Training: SGD, lr=0.15, ReLU activation. All math fits in 32-bit integers.