threshold-exactly3outof8

Exactly-3-out-of-8 detector. Fires when precisely three inputs are active. The triple detector.

Circuit

  x₀ x₁ x₂ x₃ x₄ x₅ x₆ x₇
   │  │  │  │  │  │  │  │
   └──┴──┴──┴──┼──┴──┴──┴──┘
               │
       ┌───────┴───────┐
       ▼               ▼
  ┌─────────┐     ┌─────────┐
  │  ≥ 3    │     │  ≤ 3    │
  │ b = -3  │     │ b = +3  │
  └─────────┘     └─────────┘
       │               │
       └───────┬───────┘
               ▼
          ┌─────────┐
          │   AND   │
          └─────────┘
               │
               ▼
         exactly 3?

Why Three Matters

Three is the smallest odd number greater than one. In voting:

• Byzantine fault tolerance: 3f+1 nodes tolerate f failures
• Quorum systems: Often require ⌈n/2⌉ + 1 = 3 for n=4
• Graph theory: A triangle is the minimal clique

There are C(8,3) = 56 input patterns that fire this circuit - exactly double the pairs.

The Threshold Sandwich

The circuit squeezes Hamming weight between two bounds:

Lower bound (AtLeast3):     Upper bound (AtMost3):
  +x₀ + x₁ + ... + x₇         -x₀ - x₁ - ... - x₇
  ─────────────────────       ─────────────────────
           -3                          +3

  fires if HW ≥ 3             fires if HW ≤ 3

The AND gate demands both conditions simultaneously. Only HW = 3 satisfies both.

Input Distribution

HW	Count	Exactly3
0	1	0
1	8	0
2	28	0
3	56	1
4	70	0
5	56	0
6	28	0
7	8	0
8	1	0

The binomial distribution peaks at HW=4, but HW=3 and HW=5 are tied at 56 patterns each.

Parameters

Component	Weights	Bias
AtLeast3	all +1	-3
AtMost3	all -1	+3
AND	[+1, +1]	-2

Total: 3 neurons, 21 parameters, 2 layers

Duality with Exactly5

Exactly3 and Exactly5 are complementary around the center:

Circuit	Fires on	Count	Complement
Exactly3	HW = 3	56	HW = 5
Exactly5	HW = 5	56	HW = 3

If you flip all 8 bits, an Exactly3 input becomes Exactly5, and vice versa.

Usage

from safetensors.torch import load_file
import torch

w = load_file('model.safetensors')

def exactly3(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)

# Triangle: positions 0, 3, 7 active
bits = [1, 0, 0, 1, 0, 0, 0, 1]
print(exactly3(bits))  # 1

Files

threshold-exactly3outof8/
├── model.safetensors
├── model.py
├── config.json
└── README.md

License

MIT