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---
license: mit
tags:
- formal-verification
- coq
- threshold-logic
- neuromorphic
- modular-arithmetic
---
# tiny-mod4-verified
Formally verified MOD-4 circuit. Single-layer threshold network computing modulo-4 arithmetic with 100% accuracy.
## Architecture
| Component | Value |
|-----------|-------|
| Inputs | 8 |
| Outputs | 1 (per residue class) |
| Neurons | 4 (one per residue 0-3) |
| Parameters | 36 (4 × 9) |
| Weights | [1, 1, 1, -3, 1, 1, 1, -3] |
| Bias | 0 |
| Activation | Heaviside step |
## Key Properties
- 100% accuracy (256/256 inputs correct)
- Coq-proven correctness
- Algebraic weight pattern: (1, 1, 1, 1-m) repeating
- Computes Hamming weight mod 4
- Compatible with neuromorphic hardware
## Algebraic Pattern
MOD-4 uses the repeating pattern `[1, 1, 1, -3]`:
- Positions 1-3: weight = 1
- Position 4: weight = 1-4 = -3
- Positions 5-7: weight = 1
- Position 8: weight = 1-4 = -3
This creates a cumulative sum that cycles mod 4.
## Usage
```python
import torch
from safetensors.torch import load_file
weights = load_file('mod4.safetensors')
def mod4_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']
# Output represents cumulative sum mod 4
return weighted_sum.item()
# Test
print(mod4_circuit([1,0,0,0,0,0,0,0])) # 1 mod 4 = 1
print(mod4_circuit([1,1,1,1,0,0,0,0])) # 4 mod 4 = 0
print(mod4_circuit([1,1,1,1,1,0,0,0])) # 5 mod 4 = 1
```
## Verification
**Coq Theorem**:
```coq
Theorem mod4_correct_residue_0 : forall x0 x1 x2 x3 x4 x5 x6 x7,
mod4_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 4) 0.
```
Proven axiom-free using:
1. **Algebraic correctness**: Weight pattern proven to maintain mod-4 invariant
2. **Universal quantification**: Verified for all 8-bit inputs
3. **Parametric instantiation**: Instantiates `mod_m_weights_8` with m=4
Full proof: [coq-circuits/Modular/Mod4.v](https://github.com/CharlesCNorton/coq-circuits/blob/main/coq/Modular/Mod4.v)
## Residue Distribution
For 8-bit inputs (256 total):
- Residue 0: 72 inputs
- Residue 1: 64 inputs
- Residue 2: 56 inputs
- Residue 3: 64 inputs
## Citation
```bibtex
@software{tiny_mod4_verified_2025,
title={tiny-mod4-verified: Formally Verified MOD-4 Circuit},
author={Norton, Charles},
url={https://huggingface.co/phanerozoic/tiny-mod4-verified},
year={2025}
}
```
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