tiny-Majority-verified

Formally verified majority gate for 8-bit inputs. Single threshold neuron computing majority function with 100% accuracy.

Architecture

Component	Value
Inputs	8
Outputs	1
Neurons	1
Parameters	9
Weights	[1, 1, 1, 1, 1, 1, 1, 1]
Bias	-5
Activation	Heaviside step

Key Properties

• 100% accuracy (256/256 inputs correct)
• Coq-proven correctness
• Single threshold neuron
• Integer weights
• Fires when ≥5 of 8 inputs are true
• Equivalent to 5-out-of-8 threshold

Usage

import torch
from safetensors.torch import load_file

weights = load_file('majority.safetensors')

def majority_gate(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 >= 0)

# Test
print(majority_gate([0,0,0,0,0,0,0,0]))  # 0 (no majority)
print(majority_gate([1,1,1,1,0,0,0,0]))  # 0 (4/8, not majority)
print(majority_gate([1,1,1,1,1,0,0,0]))  # 1 (5/8, majority!)
print(majority_gate([1,1,1,1,1,1,1,1]))  # 1 (8/8, majority)

Verification

Coq Theorem:

Theorem majority_correct : forall x0 x1 x2 x3 x4 x5 x6 x7,
  majority_circuit [x0; x1; x2; x3; x4; x5; x6; x7] =
  majority_spec [x0; x1; x2; x3; x4; x5; x6; x7].

Proven axiom-free via three methods:

1. Exhaustive: Verified on all 256 inputs
2. Universal: Quantified proof over all boolean combinations
3. Algebraic: Characterized via hamming weight ≥ 5

Algebraic characterization:

Theorem majority_hamming_weight (xs : list bool) :
  length xs = 8 ->
  majority_circuit xs = true <-> hamming_weight xs >= 5.

Full proof: coq-circuits/Threshold/Majority.v

Circuit Operation

Input with k true bits produces weighted sum: k*1 - 5 = k - 5

• k < 5: weighted_sum < 0 → output 0 (no majority)
• k ≥ 5: weighted_sum ≥ 0 → output 1 (majority)

Applications

• Voting systems
• Fault-tolerant computing
• Consensus protocols
• Error correction (majority voting)

Citation

@software{tiny_majority_prover_2025,
  title={tiny-Majority-verified: Formally Verified Majority Gate},
  author={Norton, Charles},
  url={https://huggingface.co/phanerozoic/tiny-Majority-verified},
  year={2025}
}