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README.md

@@ -9,7 +9,7 @@ tags:
 
 # threshold-exactly4outof8
 
-Exactly-4-out-of-8 detector. Fires when exactly four inputs are active - the balanced case.
+Exactly-4-out-of-8 detector. Fires when precisely half the inputs are active. The tie detector.
 
 ## Circuit
 
@@ -21,9 +21,8 @@ Exactly-4-out-of-8 detector. Fires when exactly four inputs are active - the bal
        ┌───────┴───────┐
        ▼               ▼
   ┌─────────┐     ┌─────────┐
-  │ AtLeast4│     │ AtMost4 │
-  │ w: +1×8 │     │ w: -1×8 │
-  │ b:  -4  │     │ b:  +4  │
+  │  ≥ 4    │     │  ≤ 4    │
+  │ b = -4  │     │ b = +4  │
   └─────────┘     └─────────┘
        │               │
        └───────┬───────┘
@@ -33,24 +32,104 @@ Exactly-4-out-of-8 detector. Fires when exactly four inputs are active - the bal
           └─────────┘
                │
                ▼
-          (HW = 4?)
+            tie?
 ```
 
-## Significance
+## The Center of the Distribution
 
-This is the "tie" detector for 8 inputs - fires when exactly half are active. Neither majority nor minority.
+HW = 4 is special:
 
-## Truth Table
+- **Maximum entropy**: C(8,4) = 70 is the largest binomial coefficient for n=8
+- **Perfect balance**: 4 ones, 4 zeros
+- **Self-complementary**: Flipping all bits maps HW=4 to itself (though not necessarily the same pattern)
 
-| HW | Result |
-|----|--------|
-| 0-3 | 0 |
-| 4 | **1** |
-| 5-8 | 0 |
+This circuit fires on 70 of 256 inputs - the mode of the distribution.
 
-## Architecture
+## Neither Majority Nor Minority
 
-**3 neurons, 21 parameters, 2 layers**
+| Circuit | Condition | Fires at HW=4? |
+|---------|-----------|----------------|
+| Majority | HW ≥ 5 | No |
+| Minority | HW ≤ 3 | No |
+| **Exactly4** | HW = 4 | **Yes** |
+
+Exactly4 catches what both Majority and Minority miss. It's the gap between them - the deadlock.
+
+## Threshold Arithmetic
+
+**AtLeast4**: Sum all inputs with weight +1, subtract 4.
+```
+x₀ + x₁ + x₂ + x₃ + x₄ + x₅ + x₆ + x₇ - 4 ≥ 0
+```
+Fires when 4 or more inputs are active.
+
+**AtMost4**: Sum all inputs with weight -1, add 4.
+```
+-x₀ - x₁ - x₂ - x₃ - x₄ - x₅ - x₆ - x₇ + 4 ≥ 0
+```
+Fires when 4 or fewer inputs are active.
+
+Their intersection is exactly 4.
+
+## The Binomial Peak
+
+| HW | C(8,k) | Exactly4? |
+|----|--------|-----------|
+| 0 | 1 | - |
+| 1 | 8 | - |
+| 2 | 28 | - |
+| 3 | 56 | - |
+| **4** | **70** | **YES** |
+| 5 | 56 | - |
+| 6 | 28 | - |
+| 7 | 8 | - |
+| 8 | 1 | - |
+
+The distribution is symmetric around 4. This circuit sits at the apex.
+
+## Parameters
+
+| Component | Weights | Bias |
+|-----------|---------|------|
+| AtLeast4 | all +1 | -4 |
+| AtMost4 | all -1 | +4 |
+| AND | [+1, +1] | -2 |
+
+**Total: 3 neurons, 21 parameters, 2 layers**
+
+## Usage
+
+```python
+from safetensors.torch import load_file
+import torch
+
+w = load_file('model.safetensors')
+
+def exactly4(bits):
+    inp = torch.tensor([float(b) for b in bits])
+    atleast = int((inp * w['atleast.weight']).sum() + w['atleast.bias'] >= 0)
+    atmost = int((inp * w['atmost.weight']).sum() + w['atmost.bias'] >= 0)
+    comb = torch.tensor([float(atleast), float(atmost)])
+    return int((comb * w['and.weight']).sum() + w['and.bias'] >= 0)
+
+# Balanced: alternating pattern
+bits = [1, 0, 1, 0, 1, 0, 1, 0]
+print(exactly4(bits))  # 1
+
+# Majority (5) - not a tie
+bits = [1, 1, 1, 0, 1, 0, 1, 0]
+print(exactly4(bits))  # 0
+```
+
+## Files
+
+```
+threshold-exactly4outof8/
+├── model.safetensors
+├── model.py
+├── config.json
+└── README.md
+```
 
 ## License