README.md

---
license: mit
tags:
- formal-verification
- coq
- threshold-logic
- neuromorphic
---

# tiny-4OutOf8-verified

Formally verified 4-out-of-8 threshold gate. Single threshold neuron with 100% accuracy.

## Architecture

| Component | Value |
|-----------|-------|
| Inputs | 8 |
| Outputs | 1 |
| Neurons | 1 |
| Parameters | 9 |
| Weights | [1, 1, 1, 1, 1, 1, 1, 1] |
| Bias | -4 |
| Activation | Heaviside step |

## Key Properties

- 100% accuracy (256/256 inputs correct)
- Coq-proven correctness
- Single threshold neuron
- Integer weights
- Fires when at least 4 of 8 inputs are true


## Usage

```python
import torch
from safetensors.torch import load_file

weights = load_file('fouroutof8.safetensors')

def fouroutof8_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({func}_gate([0,0,0,0,0,0,0,0]))  # 0 (0/4, below threshold)
print({func}_gate([1,1,1,0,0,0,0,0]))  # 0 (3/8, below threshold)
print({func}_gate([1,1,1,1,0,0,0,0]))  # 1 (4/8, at threshold)
print(fouroutof8_gate([1,1,1,1,1,1,1,1]))  # 1 (8/8, above threshold)
```

## Verification

**Coq Theorem**:
```coq
Theorem fourout_eight_correct : forall x0 x1 x2 x3 x4 x5 x6 x7,
  fourout_eight_circuit [x0; x1; x2; x3; x4; x5; x6; x7] =
  fourout_eight_spec [x0; x1; x2; x3; x4; x5; x6; x7].
```

Proven axiom-free via:
1. **Exhaustive**: All 256 inputs verified
2. **Universal**: Quantified proof over boolean combinations
3. **Algebraic**: Hamming weight ≥ 4

Full proof: [coq-circuits/Threshold/FourOutOfEight.v](https://github.com/CharlesCNorton/coq-circuits/blob/main/coq/Threshold/FourOutOfEight.v)

## Circuit Operation

Input with h true bits (Hamming weight h):
- Weighted sum: h - 4
- Output: 1 if h ≥ 4, else 0

## Citation

```bibtex
@software{tiny_4outof8_prover_2025,
  title={tiny-4OutOf8-verified: Formally Verified 4-out-of-8 threshold gate},
  author={Norton, Charles},
  url={https://huggingface.co/phanerozoic/tiny-4OutOf8-verified},
  year={2025}
}
```