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---
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}
}
```
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