---
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}
}
```