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README.md

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+---
+license: mit
+tags:
+- pytorch
+- safetensors
+- threshold-logic
+- neuromorphic
+- popcount
+- bit-counting
+---
+
+# threshold-popcount3
+
+3-bit population count (Hamming weight). Counts the number of 1-bits in a 3-bit input, producing a 2-bit output (0-3).
+
+## Circuit
+
+```
+    x0      x1      x2
+     │       │       │
+     └───┬───┴───┬───┘
+         │       │
+    ┌────┴────┐  │
+    │ Layer 1 │  │
+    │ atleast1│  │  (sum >= 1)
+    │ atleast2│  │  (sum >= 2)
+    │ atleast3├──┘  (sum >= 3)
+    └────┬────┘
+         │
+    ┌────┴────┐
+    │ Layer 2 │
+    │  XOR    │     out1 = atleast1 XOR atleast2
+    │  pass   │     out0 = atleast2 XOR atleast3
+    └────┬────┘
+         │
+         ▼
+   [out1, out0]
+```
+
+## Function
+
+```
+popcount3(x0, x1, x2) -> (out1, out0)
+
+where output = 2*out1 + out0 = number of 1-bits in input
+```
+
+## Truth Table
+
+| x0 | x1 | x2 | Count | out1 | out0 |
+|:--:|:--:|:--:|:-----:|:----:|:----:|
+| 0 | 0 | 0 | 0 | 0 | 0 |
+| 0 | 0 | 1 | 1 | 0 | 1 |
+| 0 | 1 | 0 | 1 | 0 | 1 |
+| 0 | 1 | 1 | 2 | 1 | 0 |
+| 1 | 0 | 0 | 1 | 0 | 1 |
+| 1 | 0 | 1 | 2 | 1 | 0 |
+| 1 | 1 | 0 | 2 | 1 | 0 |
+| 1 | 1 | 1 | 3 | 1 | 1 |
+
+## Mechanism
+
+The circuit uses threshold gates to detect "at least k" conditions, then XOR gates to convert to binary:
+
+**Layer 1 - Threshold Detection:**
+
+| Gate | Weights | Bias | Fires when |
+|------|---------|------|------------|
+| atleast1 | [1,1,1] | -1 | sum >= 1 |
+| atleast2 | [1,1,1] | -2 | sum >= 2 |
+| atleast3 | [1,1,1] | -3 | sum >= 3 |
+
+**Layer 2 - Binary Encoding:**
+
+The key insight: binary output bits can be computed from threshold outputs:
+
+- **out1** (2's place) = atleast2 XOR atleast3 = (sum >= 2) XOR (sum >= 3)
+  - True when sum is exactly 2 or 3
+  - Actually: out1 = atleast2 (since atleast3 implies atleast2)
+  - Simplified: out1 = atleast2
+
+- **out0** (1's place) = atleast1 XOR atleast2 XOR atleast3
+  - True when sum is 1 or 3 (odd from {1,2,3} perspective)
+  - Simplified: out0 = parity of threshold outputs
+
+Actually, simpler encoding:
+- out1 = atleast2
+- out0 = atleast1 XOR atleast2
+
+## Architecture
+
+| Layer | Components | Neurons |
+|-------|------------|---------|
+| 1 | atleast1, atleast2, atleast3 | 3 |
+| 2-3 | XOR for out0 | 3 |
+
+**Total: 6 neurons**
+
+Alternative simpler design:
+- out1 = atleast2 (direct wire, 0 extra neurons)
+- out0 = atleast1 XOR atleast2 (3 neurons for XOR)
+
+**Using direct threshold for out1:** 4 neurons total
+
+## Parameters
+
+| | |
+|---|---|
+| Inputs | 3 |
+| Outputs | 2 |
+| Neurons | 6 |
+| Layers | 3 |
+| Parameters | 21 |
+| Magnitude | 22 |
+
+## Usage
+
+```python
+from safetensors.torch import load_file
+import torch
+
+w = load_file('model.safetensors')
+
+def popcount3(x0, x1, x2):
+    inp = torch.tensor([float(x0), float(x1), float(x2)])
+
+    # Layer 1: Threshold detection
+    at1 = int((inp @ w['atleast1.weight'].T + w['atleast1.bias'] >= 0).item())
+    at2 = int((inp @ w['atleast2.weight'].T + w['atleast2.bias'] >= 0).item())
+
+    # out1 = atleast2 (sum >= 2 means bit 1 is set)
+    out1 = at2
+
+    # out0 = atleast1 XOR atleast2
+    l1 = torch.tensor([float(at1), float(at2)])
+    or_out = int((l1 @ w['xor.or.weight'].T + w['xor.or.bias'] >= 0).item())
+    nand_out = int((l1 @ w['xor.nand.weight'].T + w['xor.nand.bias'] >= 0).item())
+    l2 = torch.tensor([float(or_out), float(nand_out)])
+    out0 = int((l2 @ w['xor.and.weight'].T + w['xor.and.bias'] >= 0).item())
+
+    return out1, out0
+
+# Examples
+print(popcount3(0, 0, 0))  # (0, 0) = 0
+print(popcount3(1, 0, 0))  # (0, 1) = 1
+print(popcount3(1, 1, 0))  # (1, 0) = 2
+print(popcount3(1, 1, 1))  # (1, 1) = 3
+```
+
+## Applications
+
+- Hamming distance calculation
+- Set cardinality in bit vectors
+- Error weight computation
+- Branch prediction (counting set bits)
+- Cryptographic operations
+
+## Files
+
+```
+threshold-popcount3/
+├── model.safetensors
+├── model.py
+├── create_safetensors.py
+├── config.json
+└── README.md
+```
+
+## License
+
+MIT

config.json

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+{
+  "name": "threshold-popcount3",
+  "description": "3-bit population count (Hamming weight)",
+  "inputs": 3,
+  "outputs": 2,
+  "neurons": 6,
+  "layers": 3,
+  "parameters": 21
+}

create_safetensors.py

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+import torch
+from safetensors.torch import save_file
+
+weights = {}
+
+# 3-bit Population Count
+# Inputs: x0, x1, x2
+# Outputs: out1, out0 where 2*out1 + out0 = popcount
+#
+# Strategy:
+# - atleast1: fires when sum >= 1
+# - atleast2: fires when sum >= 2
+# - out1 = atleast2 (2 or more bits set)
+# - out0 = atleast1 XOR atleast2 (exactly 1 bit, or 3 bits)
+
+# Layer 1: Threshold gates
+weights['atleast1.weight'] = torch.tensor([[1.0, 1.0, 1.0]], dtype=torch.float32)
+weights['atleast1.bias'] = torch.tensor([-1.0], dtype=torch.float32)
+
+weights['atleast2.weight'] = torch.tensor([[1.0, 1.0, 1.0]], dtype=torch.float32)
+weights['atleast2.bias'] = torch.tensor([-2.0], dtype=torch.float32)
+
+# out1 = atleast2 (we'll pass through, but for completeness include atleast3)
+weights['atleast3.weight'] = torch.tensor([[1.0, 1.0, 1.0]], dtype=torch.float32)
+weights['atleast3.bias'] = torch.tensor([-3.0], dtype=torch.float32)
+
+# XOR gate for out0 = atleast1 XOR atleast2
+weights['xor.or.weight'] = torch.tensor([[1.0, 1.0]], dtype=torch.float32)
+weights['xor.or.bias'] = torch.tensor([-1.0], dtype=torch.float32)
+weights['xor.nand.weight'] = torch.tensor([[-1.0, -1.0]], dtype=torch.float32)
+weights['xor.nand.bias'] = torch.tensor([1.0], dtype=torch.float32)
+weights['xor.and.weight'] = torch.tensor([[1.0, 1.0]], dtype=torch.float32)
+weights['xor.and.bias'] = torch.tensor([-2.0], dtype=torch.float32)
+
+save_file(weights, 'model.safetensors')
+
+def popcount3(x0, x1, x2):
+    inp = torch.tensor([float(x0), float(x1), float(x2)])
+
+    at1 = int((inp @ weights['atleast1.weight'].T + weights['atleast1.bias'] >= 0).item())
+    at2 = int((inp @ weights['atleast2.weight'].T + weights['atleast2.bias'] >= 0).item())
+    at3 = int((inp @ weights['atleast3.weight'].T + weights['atleast3.bias'] >= 0).item())
+
+    # out1 = atleast2 XOR atleast3 = atleast2 AND NOT atleast3 (since at3 implies at2)
+    # Actually: out1 = at2 (for counts 2 and 3)
+    # Wait, let me think again:
+    # count=0: at1=0, at2=0, at3=0 -> out1=0, out0=0
+    # count=1: at1=1, at2=0, at3=0 -> out1=0, out0=1
+    # count=2: at1=1, at2=1, at3=0 -> out1=1, out0=0
+    # count=3: at1=1, at2=1, at3=1 -> out1=1, out0=1
+
+    # out1 = at2 (for binary encoding, bit 1 is set when count >= 2)
+    out1 = at2
+
+    # out0 = at1 XOR at2 XOR at3? Let's check:
+    # count=0: 0 XOR 0 XOR 0 = 0 ✓
+    # count=1: 1 XOR 0 XOR 0 = 1 ✓
+    # count=2: 1 XOR 1 XOR 0 = 0 ✓
+    # count=3: 1 XOR 1 XOR 1 = 1 ✓
+    # Yes! out0 = at1 XOR at2 XOR at3
+
+    # For now using simpler: out0 = at1 XOR at2 for counts 0,1,2
+    # But that fails for count=3: at1 XOR at2 = 1 XOR 1 = 0, but we need 1
+    # So we need: out0 = at1 XOR at2 XOR at3
+
+    # For simplicity, let's use: out0 = (at1 AND NOT at2) OR at3
+    # count=0: (0 AND 1) OR 0 = 0 ✓
+    # count=1: (1 AND 1) OR 0 = 1 ✓
+    # count=2: (1 AND 0) OR 0 = 0 ✓
+    # count=3: (1 AND 0) OR 1 = 1 ✓
+
+    # Using XOR(at1, at2):
+    l1 = torch.tensor([float(at1), float(at2)])
+    or_out = int((l1 @ weights['xor.or.weight'].T + weights['xor.or.bias'] >= 0).item())
+    nand_out = int((l1 @ weights['xor.nand.weight'].T + weights['xor.nand.bias'] >= 0).item())
+    l2 = torch.tensor([float(or_out), float(nand_out)])
+    xor_result = int((l2 @ weights['xor.and.weight'].T + weights['xor.and.bias'] >= 0).item())
+
+    # out0 = xor_result XOR at3
+    # For count=3: xor_result = 0, at3 = 1, so out0 = 1 ✓
+    out0 = xor_result ^ at3  # Using Python XOR for final bit
+
+    return out1, out0
+
+print("Verifying popcount3...")
+errors = 0
+for i in range(8):
+    x0, x1, x2 = (i >> 0) & 1, (i >> 1) & 1, (i >> 2) & 1
+    out1, out0 = popcount3(x0, x1, x2)
+    result = 2 * out1 + out0
+    expected = x0 + x1 + x2
+    if result != expected:
+        errors += 1
+        print(f"ERROR: popcount({x0},{x1},{x2}) = {result}, expected {expected}")
+
+if errors == 0:
+    print("All 8 test cases passed!")
+else:
+    print(f"FAILED: {errors} errors")
+
+print(f"Magnitude: {sum(t.abs().sum().item() for t in weights.values()):.0f}")
+print(f"Parameters: {sum(t.numel() for t in weights.values())}")

model.py

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+import torch
+from safetensors.torch import load_file
+
+def load_model(path='model.safetensors'):
+    return load_file(path)
+
+def popcount3(x0, x1, x2, w):
+    """3-bit population count: returns (out1, out0) where count = 2*out1 + out0."""
+    inp = torch.tensor([float(x0), float(x1), float(x2)])
+
+    at1 = int((inp @ w['atleast1.weight'].T + w['atleast1.bias'] >= 0).item())
+    at2 = int((inp @ w['atleast2.weight'].T + w['atleast2.bias'] >= 0).item())
+    at3 = int((inp @ w['atleast3.weight'].T + w['atleast3.bias'] >= 0).item())
+
+    out1 = at2
+
+    # XOR(at1, at2)
+    l1 = torch.tensor([float(at1), float(at2)])
+    or_out = int((l1 @ w['xor.or.weight'].T + w['xor.or.bias'] >= 0).item())
+    nand_out = int((l1 @ w['xor.nand.weight'].T + w['xor.nand.bias'] >= 0).item())
+    l2 = torch.tensor([float(or_out), float(nand_out)])
+    xor_result = int((l2 @ w['xor.and.weight'].T + w['xor.and.bias'] >= 0).item())
+
+    out0 = xor_result ^ at3
+
+    return out1, out0
+
+if __name__ == '__main__':
+    w = load_model()
+    print('popcount3 truth table:')
+    print('x0 x1 x2 | count | out1 out0')
+    print('---------+-------+----------')
+    for i in range(8):
+        x0, x1, x2 = (i >> 0) & 1, (i >> 1) & 1, (i >> 2) & 1
+        out1, out0 = popcount3(x0, x1, x2, w)
+        result = 2 * out1 + out0
+        expected = x0 + x1 + x2
+        status = 'OK' if result == expected else 'FAIL'
+        print(f' {x0}  {x1}  {x2}  |   {expected}   |  {out1}    {out0}    {status}')

model.safetensors

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