tiny-mod11-verified

Formally verified MOD-11 circuit. Single-layer threshold network computing modulo-11 arithmetic with 100% accuracy.

Architecture

Component	Value
Inputs	8
Outputs	1 (per residue class)
Neurons	11 (one per residue 0-10)
Parameters	99 (11 × 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 11
• Compatible with neuromorphic hardware

Algebraic Pattern

MOD-11 uses all-ones weights because the reset position (position 11) is beyond the 8-bit input width:

• All positions 1-8: weight = 1
• Position 11 (beyond input): would be weight = 1-11 = -10

The circuit tracks cumulative sum mod 11 using the Hamming weight directly.

Usage

import torch
from safetensors.torch import load_file

weights = load_file('mod11.safetensors')

def mod11_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']
    return int(weighted_sum.item()) % 11

# Test
print(mod11_circuit([1,1,1,1,1,1,1,1]))  # 8 mod 11 = 8
print(mod11_circuit([1,1,1,0,0,0,0,0]))  # 3 mod 11 = 3

Verification

Coq Theorem:

Theorem mod11_correct_residue_0 : forall x0 x1 x2 x3 x4 x5 x6 x7,
  mod11_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 11) 0.

Proven axiom-free using algebraic weight patterns.

Full proof: coq-circuits/Modular/Mod11.v

Residue Distribution

For 8-bit inputs (256 total), limited to residues 0-8:

• 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
• Residues 9-10: 0 inputs (unreachable with 8-bit input)

Citation

@software{tiny_mod11_verified_2025,
  title={tiny-mod11-verified: Formally Verified MOD-11 Circuit},
  author={Norton, Charles},
  url={https://huggingface.co/phanerozoic/tiny-mod11-verified},
  year={2025}
}