tiny-mod4-verified

Formally verified MOD-4 circuit. Single-layer threshold network computing modulo-4 arithmetic with 100% accuracy.

Architecture

Component	Value
Inputs	8
Outputs	1 (per residue class)
Neurons	4 (one per residue 0-3)
Parameters	36 (4 × 9)
Weights	[1, 1, 1, -3, 1, 1, 1, -3]
Bias	0
Activation	Heaviside step

Key Properties

• 100% accuracy (256/256 inputs correct)
• Coq-proven correctness
• Algebraic weight pattern: (1, 1, 1, 1-m) repeating
• Computes Hamming weight mod 4
• Compatible with neuromorphic hardware

Algebraic Pattern

MOD-4 uses the repeating pattern [1, 1, 1, -3]:

• Positions 1-3: weight = 1
• Position 4: weight = 1-4 = -3
• Positions 5-7: weight = 1
• Position 8: weight = 1-4 = -3

This creates a cumulative sum that cycles mod 4.

Usage

import torch
from safetensors.torch import load_file

weights = load_file('mod4.safetensors')

def mod4_circuit(bits):
    # bits: list of 8 binary values
    inputs = torch.tensor([float(b) for b in bits])
    weighted_sum = (inputs * weights['weight']).sum() + weights['bias']
    # Output represents cumulative sum mod 4
    return weighted_sum.item()

# Test
print(mod4_circuit([1,0,0,0,0,0,0,0]))  # 1 mod 4 = 1
print(mod4_circuit([1,1,1,1,0,0,0,0]))  # 4 mod 4 = 0
print(mod4_circuit([1,1,1,1,1,0,0,0]))  # 5 mod 4 = 1

Verification

Coq Theorem:

Theorem mod4_correct_residue_0 : forall x0 x1 x2 x3 x4 x5 x6 x7,
  mod4_is_zero [x0; x1; x2; x3; x4; x5; x6; x7] =
  Z.eqb ((Z.of_nat (hamming_weight [x0; x1; x2; x3; x4; x5; x6; x7])) mod 4) 0.

Proven axiom-free using:

1. Algebraic correctness: Weight pattern proven to maintain mod-4 invariant
2. Universal quantification: Verified for all 8-bit inputs
3. Parametric instantiation: Instantiates mod_m_weights_8 with m=4

Full proof: coq-circuits/Modular/Mod4.v

Residue Distribution

For 8-bit inputs (256 total):

• Residue 0: 72 inputs
• Residue 1: 64 inputs
• Residue 2: 56 inputs
• Residue 3: 64 inputs

Citation

@software{tiny_mod4_verified_2025,
  title={tiny-mod4-verified: Formally Verified MOD-4 Circuit},
  author={Norton, Charles},
  url={https://huggingface.co/phanerozoic/tiny-mod4-verified},
  year={2025}
}