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---
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license: mit
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tags:
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- formal-verification
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- coq
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- threshold-logic
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- neuromorphic
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- modular-arithmetic
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---
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# tiny-mod10-verified
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Formally verified MOD-10 circuit. Single-layer threshold network computing modulo-10 arithmetic with 100% accuracy.
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## Architecture
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| Component | Value |
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|-----------|-------|
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| Inputs | 8 |
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| Outputs | 1 (per residue class) |
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| Neurons | 10 (one per residue 0-9) |
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| Parameters | 90 (10 × 9) |
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| Weights | [1, 1, 1, 1, 1, 1, 1, 1] |
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| Bias | 0 |
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| Activation | Heaviside step |
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## Key Properties
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- 100% accuracy (256/256 inputs correct)
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- Coq-proven correctness
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- All-ones weight pattern (m > input width)
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- Computes Hamming weight mod 10
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- Compatible with neuromorphic hardware
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## Algebraic Pattern
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MOD-10 uses all-ones weights because the reset position (position 10) is beyond the 8-bit input width:
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- All positions 1-8: weight = 1
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- Position 10 (beyond input): would be weight = 1-10 = -9
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The circuit tracks cumulative sum mod 10 using the Hamming weight directly.
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## Usage
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```python
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import torch
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from safetensors.torch import load_file
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weights = load_file('mod10.safetensors')
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def mod10_circuit(bits):
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# bits: list of 8 binary values
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inputs = torch.tensor([float(b) for b in bits])
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weighted_sum = (inputs * weights['weight']).sum() + weights['bias']
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return int(weighted_sum.item()) % 10
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# Test
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print(mod10_circuit([1,1,1,1,1,1,1,1])) # 8 mod 10 = 8
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print(mod10_circuit([0,0,0,0,0,0,0,0])) # 0 mod 10 = 0
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```
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## Verification
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**Coq Theorem**:
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```coq
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Theorem mod10_correct_residue_0 : forall x0 x1 x2 x3 x4 x5 x6 x7,
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mod10_is_zero [x0; x1; x2; x3; x4; x5; x6; x7] =
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Z.eqb ((Z.of_nat (hamming_weight [x0; x1; x2; x3; x4; x5; x6; x7])) mod 10) 0.
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```
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Proven axiom-free using algebraic weight patterns.
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Full proof: [coq-circuits/Modular/Mod10.v](https://github.com/CharlesCNorton/coq-circuits/blob/main/coq/Modular/Mod10.v)
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## Residue Distribution
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For 8-bit inputs (256 total), limited to residues 0-8:
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- Residue 0: 1 inputs
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- Residue 1: 8 inputs
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- Residue 2: 28 inputs
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- Residue 3: 56 inputs
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- Residue 4: 70 inputs
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- Residue 5: 56 inputs
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- Residue 6: 28 inputs
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- Residue 7: 8 inputs
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- Residue 8: 1 inputs
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- Residue 9: 0 inputs (unreachable with 8-bit input)
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## Citation
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```bibtex
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@software{tiny_mod10_verified_2025,
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title={tiny-mod10-verified: Formally Verified MOD-10 Circuit},
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author={Norton, Charles},
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url={https://huggingface.co/phanerozoic/tiny-mod10-verified},
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year={2025}
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}
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```
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