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README.md

@@ -9,7 +9,7 @@ tags:
 
 # threshold-exactly7outof8
 
-Exactly-7-out-of-8 detector. Fires when exactly seven inputs are active.
+Exactly-7-out-of-8 detector. Fires when precisely seven inputs are active. The lone-dissenter detector.
 
 ## Circuit
 
@@ -21,9 +21,8 @@ Exactly-7-out-of-8 detector. Fires when exactly seven inputs are active.
        ┌───────┴───────┐
        ▼               ▼
   ┌─────────┐     ┌─────────┐
-  │ AtLeast7│     │ AtMost7 │
-  │ w: +1×8 │     │ w: -1×8 │
-  │ b:  -7  │     │ b:  +7  │
+  │  ≥ 7    │     │  ≤ 7    │
+  │ b = -7  │     │ b = +7  │
   └─────────┘     └─────────┘
        │               │
        └───────┬───────┘
@@ -33,24 +32,107 @@ Exactly-7-out-of-8 detector. Fires when exactly seven inputs are active.
           └─────────┘
                │
                ▼
-          (HW = 7?)
+      one dissenter?
 ```
 
-## Significance
+## The Singleton Complement
 
-Near-unanimity detector. Exactly one dissenter among eight. The single-error tolerance case.
+This circuit detects the complement of Exactly1: instead of one voice in the wilderness, it's one holdout against consensus.
 
-## Truth Table
+| Circuit | Pattern | Interpretation |
+|---------|---------|----------------|
+| Exactly1 | 00000001 | Pioneer, outlier |
+| **Exactly7** | 11111110 | Holdout, veto |
 
-| HW | Result |
-|----|--------|
-| 0-6 | 0 |
-| 7 | **1** |
-| 8 | 0 |
+Both fire on exactly 8 input patterns (one for each position).
 
-## Architecture
+## Near-Unanimity
 
-**3 neurons, 21 parameters, 2 layers**
+7/8 = 87.5% - overwhelming but not complete agreement:
+
+- **Hung jury**: 11-1 in a 12-person jury (scaled to 8)
+- **Single point of failure**: All systems go except one
+- **Error correction**: A single bit flip from all-ones
+
+The circuit identifies the liminal state between consensus and unanimity.
+
+## Dual of Exactly1
+
+| Circuit | HW | Count | Interpretation |
+|---------|-----|-------|----------------|
+| Exactly1 | 1 | 8 | One active |
+| **Exactly7** | 7 | 8 | One inactive |
+
+They are perfect duals. Bitwise NOT maps each Exactly1 input to a unique Exactly7 input.
+
+## The Holdout Problem
+
+In consensus systems:
+
+| HW | Status | Exactly7 |
+|----|--------|----------|
+| 6 | Multiple dissenters | 0 |
+| **7** | **Single holdout** | **1** |
+| 8 | Complete consensus | 0 |
+
+Exactly7 identifies the unique scenario: overwhelming support with precisely one objector. This is useful for:
+
+- Identifying the blocking vote
+- Detecting near-consensus states
+- Finding the single bit error in a codeword
+
+## Parameters
+
+| Component | Weights | Bias |
+|-----------|---------|------|
+| AtLeast7 | all +1 | -7 |
+| AtMost7 | all -1 | +7 |
+| 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 exactly7(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)
+
+# One holdout at position 3
+bits = [1, 1, 1, 0, 1, 1, 1, 1]
+print(exactly7(bits))  # 1
+
+# Complete unanimity - no holdout
+bits = [1, 1, 1, 1, 1, 1, 1, 1]
+print(exactly7(bits))  # 0
+```
+
+## Contrast with 7-out-of-8
+
+| Circuit | Condition | Fires on |
+|---------|-----------|----------|
+| 7-out-of-8 | HW ≥ 7 | HW = 7, 8 (9 patterns) |
+| **Exactly7** | HW = 7 | HW = 7 only (8 patterns) |
+
+The difference: 7-out-of-8 includes unanimity. Exactly7 excludes it.
+
+## Files
+
+```
+threshold-exactly7outof8/
+├── model.safetensors
+├── model.py
+├── config.json
+└── README.md
+```
 
 ## License