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README.md

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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)
+
+## 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}
+}
+```