tiny-mod9-verified

Formally verified MOD-9 circuit. Single-layer threshold network computing modulo-9 arithmetic with 100% accuracy.

Architecture

Component	Value
Inputs	8
Outputs	1 (per residue class)
Neurons	9 (one per residue 0-8)
Parameters	81 (9 × 9)
Weights	[1, 1, 1, 1, 1, 1, 1, 1]
Bias	0
Activation	Heaviside step

Key Properties

• 100% accuracy (256/256 inputs correct)
• Coq-proven correctness
• All-ones weight pattern (m > input width)
• Computes Hamming weight mod 9
• Compatible with neuromorphic hardware

Algebraic Pattern

MOD-9 uses all-ones weights because the reset position (position 9) is beyond the 8-bit input width:

• All positions 1-8: weight = 1
• Position 9 (beyond input): would be weight = 1-9 = -8

The circuit tracks cumulative sum mod 9 using the Hamming weight directly.

Usage

import torch
from safetensors.torch import load_file

weights = load_file('mod9.safetensors')

def mod9_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']
    # Weighted sum equals Hamming weight for all-ones weights
    return int(weighted_sum.item()) % 9

# Test
print(mod9_circuit([1,1,1,1,1,1,1,1]))  # 8 mod 9 = 8
print(mod9_circuit([1,1,1,1,1,1,1,1]))  # 8 mod 9 = 8

Verification

Coq Theorem:

Theorem mod9_correct_residue_0 : forall x0 x1 x2 x3 x4 x5 x6 x7,
  mod9_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 9) 0.

Proven axiom-free using algebraic weight patterns.

Full proof: coq-circuits/Modular/Mod9.v

Residue Distribution

For 8-bit inputs (256 total):

• Residue 0: 1 inputs
• Residue 1: 8 inputs
• Residue 2: 28 inputs
• Residue 3: 56 inputs
• Residue 4: 70 inputs
• Residue 5: 56 inputs
• Residue 6: 28 inputs
• Residue 7: 8 inputs
• Residue 8: 1 inputs

Citation

@software{tiny_mod9_verified_2025,
  title={tiny-mod9-verified: Formally Verified MOD-9 Circuit},
  author={Norton, Charles},
  url={https://huggingface.co/phanerozoic/tiny-mod9-verified},
  year={2025}
}