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

# 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

```python
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**:
```coq
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](https://github.com/CharlesCNorton/coq-circuits/blob/main/coq/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

```bibtex
@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}
}
```