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

# tiny-mod8-verified

Formally verified MOD-8 circuit. Single-layer threshold network computing modulo-8 arithmetic with 100% accuracy.

## Architecture

| Component | Value |
|-----------|-------|
| Inputs | 8 |
| Outputs | 1 (per residue class) |
| Neurons | 8 (one per residue 0-7) |
| Parameters | 72 (8 × 9) |
| Weights | [1, 1, 1, 1, 1, 1, 1, -7] |
| Bias | 0 |
| Activation | Heaviside step |

## Key Properties

- 100% accuracy (256/256 inputs correct)
- Coq-proven correctness
- Algebraic weight pattern: resets every 8 positions
- Computes Hamming weight mod 8
- Compatible with neuromorphic hardware

## Algebraic Pattern

MOD-8 uses the pattern with reset at position 8:
- Positions 1-7: weight = 1
- Position 8: weight = 1-8 = -7

This creates a cumulative sum that cycles mod 8.

## Usage

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

weights = load_file('mod8.safetensors')

def mod8_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 weighted_sum.item()

# Test
print(mod8_circuit([1,1,1,1,1,1,1,1]))  # 8 mod 8 = 0
print(mod8_circuit([1,1,1,1,1,1,1,0]))  # 7 mod 8 = 7
```

## Verification

**Coq Theorem**:
```coq
Theorem mod8_correct_residue_0 : forall x0 x1 x2 x3 x4 x5 x6 x7,
  mod8_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 8) 0.
```

Proven axiom-free using algebraic weight patterns.

Full proof: [coq-circuits/Modular/Mod8.v](https://github.com/CharlesCNorton/coq-circuits/blob/main/coq/Modular/Mod8.v)

## Residue Distribution

For 8-bit inputs (256 total):
- Residue 0: 2 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

## Citation

```bibtex
@software{tiny_mod8_verified_2025,
  title={tiny-mod8-verified: Formally Verified MOD-8 Circuit},
  author={Norton, Charles},
  url={https://huggingface.co/phanerozoic/tiny-mod8-verified},
  year={2025}
}
```