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README.md

@@ -1,97 +1,97 @@
----
-license: mit
-tags:
-- formal-verification
-- coq
-- threshold-logic
-- neuromorphic
-- functionally-complete
----
-
-# tiny-NAND-verified
-
-Formally verified NAND gate. Single threshold neuron computing negated conjunction with 100% accuracy.
-
-## Architecture
-
-| Component | Value |
-|-----------|-------|
-| Inputs | 2 |
-| Outputs | 1 |
-| Neurons | 1 |
-| Parameters | 3 |
-| Weights | [-1, -1] |
-| Bias | 1 |
-| Activation | Heaviside step |
-
-## Key Properties
-
-- 100% accuracy (4/4 inputs correct)
-- Coq-proven correctness
-- Single threshold neuron
-- Integer weights
-- Commutative: NAND(x,y) = NAND(y,x)
-- Functionally complete (can build any Boolean function)
-- Self-dual: NAND(x,x) = NOT(x)
-
-## Usage
-
-```python
-import torch
-from safetensors.torch import load_file
-
-weights = load_file('nand.safetensors')
-
-def nand_gate(x, y):
-    # Heaviside: weighted_sum + bias >= 0
-    inputs = torch.tensor([float(x), float(y)])
-    weighted_sum = (inputs * weights['weight']).sum() + weights['bias']
-    return int(weighted_sum >= 0)
-
-# Test
-print(nand_gate(0, 0))  # 1
-print(nand_gate(0, 1))  # 1
-print(nand_gate(1, 0))  # 1
-print(nand_gate(1, 1))  # 0
-```
-
-## Verification
-
-**Coq Theorem**:
-```coq
-Theorem nand_correct : forall x y, nand_circuit x y = negb (andb x y).
-```
-
-Proven axiom-free with properties:
-- Commutativity
-- Self-duality (NAND(x,x) = NOT(x))
-- Functional completeness
-- Identity with true gives NOT
-- Absorption with false gives true
-
-Full proof: [coq-circuits/Boolean/NAND.v](https://github.com/CharlesCNorton/coq-circuits)
-
-## Circuit Operation
-
-Input combination produces weighted sum:
-- (0,0): 0*(-1) + 0*(-1) + 1 = 1 >= 0 → 1
-- (0,1): 0*(-1) + 1*(-1) + 1 = 0 >= 0 → 1
-- (1,0): 1*(-1) + 0*(-1) + 1 = 0 >= 0 → 1
-- (1,1): 1*(-1) + 1*(-1) + 1 = -1 < 0 → 0
-
-Fails to fire only when both inputs are true.
-
-## Functional Completeness
-
-NAND is functionally complete - any Boolean function can be built from NAND gates alone. This makes it particularly important for circuit composition.
-
-## Citation
-
-```bibtex
-@software{tiny_nand_prover_2025,
-  title={tiny-NAND-verified: Formally Verified NAND Gate},
-  author={Norton, Charles},
-  url={https://huggingface.co/phanerozoic/tiny-NAND-verified},
-  year={2025}
-}
-```
+---
+license: mit
+tags:
+- formal-verification
+- coq
+- threshold-logic
+- neuromorphic
+- functionally-complete
+---
+
+# tiny-NAND-verified
+
+Formally verified NAND gate. Single threshold neuron computing negated conjunction with 100% accuracy.
+
+## Architecture
+
+| Component | Value |
+|-----------|-------|
+| Inputs | 2 |
+| Outputs | 1 |
+| Neurons | 1 |
+| Parameters | 3 |
+| Weights | [-1, -1] |
+| Bias | 1 |
+| Activation | Heaviside step |
+
+## Key Properties
+
+- 100% accuracy (4/4 inputs correct)
+- Coq-proven correctness
+- Single threshold neuron
+- Integer weights
+- Commutative: NAND(x,y) = NAND(y,x)
+- Functionally complete (can build any Boolean function)
+- Self-dual: NAND(x,x) = NOT(x)
+
+## Usage
+
+```python
+import torch
+from safetensors.torch import load_file
+
+weights = load_file('nand.safetensors')
+
+def nand_gate(x, y):
+    # Heaviside: weighted_sum + bias >= 0
+    inputs = torch.tensor([float(x), float(y)])
+    weighted_sum = (inputs * weights['weight']).sum() + weights['bias']
+    return int(weighted_sum >= 0)
+
+# Test
+print(nand_gate(0, 0))  # 1
+print(nand_gate(0, 1))  # 1
+print(nand_gate(1, 0))  # 1
+print(nand_gate(1, 1))  # 0
+```
+
+## Verification
+
+**Coq Theorem**:
+```coq
+Theorem nand_correct : forall x y, nand_circuit x y = negb (andb x y).
+```
+
+Proven axiom-free with properties:
+- Commutativity
+- Self-duality (NAND(x,x) = NOT(x))
+- Functional completeness
+- Identity with true gives NOT
+- Absorption with false gives true
+
+Full proof: [coq-circuits/Boolean/NAND.v](https://github.com/CharlesCNorton/coq-circuits/blob/main/coq/Boolean/NAND.v)
+
+## Circuit Operation
+
+Input combination produces weighted sum:
+- (0,0): 0*(-1) + 0*(-1) + 1 = 1 >= 0 → 1
+- (0,1): 0*(-1) + 1*(-1) + 1 = 0 >= 0 → 1
+- (1,0): 1*(-1) + 0*(-1) + 1 = 0 >= 0 → 1
+- (1,1): 1*(-1) + 1*(-1) + 1 = -1 < 0 → 0
+
+Fails to fire only when both inputs are true.
+
+## Functional Completeness
+
+NAND is functionally complete - any Boolean function can be built from NAND gates alone. This makes it particularly important for circuit composition.
+
+## Citation
+
+```bibtex
+@software{tiny_nand_prover_2025,
+  title={tiny-NAND-verified: Formally Verified NAND Gate},
+  author={Norton, Charles},
+  url={https://huggingface.co/phanerozoic/tiny-NAND-verified},
+  year={2025}
+}
+```