threshold-exactly6outof8

Exactly-6-out-of-8 detector. Fires when precisely six inputs are active. The supermajority detector.

Circuit

  x₀ x₁ x₂ x₃ x₄ x₅ x₆ x₇
   │  │  │  │  │  │  │  │
   └──┴──┴──┴──┼──┴──┴──┴──┘
               │
       ┌───────┴───────┐
       ▼               ▼
  ┌─────────┐     ┌─────────┐
  │  ≥ 6    │     │  ≤ 6    │
  │ b = -6  │     │ b = +6  │
  └─────────┘     └─────────┘
       │               │
       └───────┬───────┘
               ▼
          ┌─────────┐
          │   AND   │
          └─────────┘
               │
               ▼
       supermajority?

The Three-Quarters Threshold

6/8 = 75% - a common supermajority requirement:

• Constitutional amendments: Many require 2/3 or 3/4
• Corporate bylaws: Often need 75% for major decisions
• Consensus protocols: BFT systems often use ⌈2n/3⌉

With 8 inputs, 6 is the first integer ≥ 75%.

Tolerance Analysis

Active	Inactive	Tolerance	This circuit
5	3	Can lose 0	0
6	2	Can lose 1	1
7	1	Can lose 2	0
8	0	Can lose 3	0

Exactly6 detects the state with precisely one-defection tolerance. Lose any vote and you drop below supermajority. Gain any vote and you have margin to spare.

Dual of Exactly2

Circuit	HW	Count	Bit-flip image
Exactly2	2	28	Exactly6
Exactly6	6	28	Exactly2

Flipping all bits maps pairs to sextets. Both fire on exactly 28 inputs.

The Veto Position

In a system requiring 6/8 supermajority:

• 2 dissenters can block any action (they need 3 to pass minority position)
• 6 supporters are exactly enough (no defection allowed)

Exactly6 catches this critical configuration - the minimum winning coalition.

Parameters

Component	Weights	Bias
AtLeast6	all +1	-6
AtMost6	all -1	+6
AND	[+1, +1]	-2

Total: 3 neurons, 21 parameters, 2 layers

Usage

from safetensors.torch import load_file
import torch

w = load_file('model.safetensors')

def exactly6(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)

# Supermajority: exactly 6
bits = [1, 1, 1, 1, 1, 1, 0, 0]
print(exactly6(bits))  # 1

# Near-unanimity: 7 is too many
bits = [1, 1, 1, 1, 1, 1, 1, 0]
print(exactly6(bits))  # 0

Files

threshold-exactly6outof8/
├── model.safetensors
├── model.py
├── config.json
└── README.md

License

MIT