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README.md

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+---
+license: mit
+tags:
+- formal-verification
+- coq
+- threshold-logic
+- neuromorphic
+- minority
+---
+
+# tiny-Minority-prover
+
+Formally verified minority gate for 8-bit inputs. Single threshold neuron computing minority function with 100% accuracy.
+
+## Architecture
+
+| Component | Value |
+|-----------|-------|
+| Inputs | 8 |
+| Outputs | 1 |
+| Neurons | 1 |
+| Parameters | 9 |
+| Weights | [-1, -1, -1, -1, -1, -1, -1, -1] |
+| Bias | 3 |
+| Activation | Heaviside step |
+
+## Key Properties
+
+- 100% accuracy (256/256 inputs correct)
+- Coq-proven correctness
+- Single threshold neuron
+- Integer weights
+- Fires when ≤3 of 8 inputs are true
+- Complement of majority (inverted weights)
+
+## Usage
+
+```python
+import torch
+from safetensors.torch import load_file
+
+weights = load_file('minority.safetensors')
+
+def minority_gate(bits):
+    # bits: list of 8 binary values
+    inputs = torch.tensor([float(b) for b in bits])
+    weighted_sum = (inputs * weights['weight']).sum() + weights['bias']
+    return int(weighted_sum >= 0)
+
+# Test
+print(minority_gate([0,0,0,0,0,0,0,0]))  # 1 (minority)
+print(minority_gate([1,1,1,0,0,0,0,0]))  # 1 (3/8, minority)
+print(minority_gate([1,1,1,1,0,0,0,0]))  # 0 (4/8, not minority)
+print(minority_gate([1,1,1,1,1,1,1,1]))  # 0 (no minority)
+```
+
+## Verification
+
+**Coq Theorem**:
+```coq
+Theorem minority_correct : forall x0 x1 x2 x3 x4 x5 x6 x7,
+  minority_circuit [x0; x1; x2; x3; x4; x5; x6; x7] =
+  minority_spec [x0; x1; x2; x3; x4; x5; x6; x7].
+```
+
+Proven axiom-free via three methods:
+1. **Exhaustive**: Verified on all 256 inputs
+2. **Universal**: Quantified proof over all boolean combinations
+3. **Algebraic**: Characterized via hamming weight ≤ 3
+
+**Algebraic characterization**:
+```coq
+Theorem minority_hamming_weight (xs : list bool) :
+  length xs = 8 ->
+  minority_circuit xs = true <-> hamming_weight xs <= 3.
+```
+
+Full proof: [coq-circuits/Threshold/Minority.v](https://github.com/CharlesCNorton/coq-circuits)
+
+## Circuit Operation
+
+Input with k true bits produces weighted sum: -k + 3
+
+- k ≤ 3: weighted_sum ≥ 0 → output 1 (minority)
+- k > 3: weighted_sum < 0 → output 0 (not minority)
+
+## Applications
+
+- Inverted voting systems
+- Fault detection (low activation)
+- Sparse pattern detection
+- Neuromorphic hardware
+
+## Citation
+
+```bibtex
+@software{tiny_minority_prover_2025,
+  title={tiny-Minority-prover: Formally Verified Minority Gate},
+  author={Norton, Charles},
+  url={https://huggingface.co/phanerozoic/tiny-Minority-prover},
+  year={2025}
+}
+```