threshold-atmost1outof8

At-most-1-out-of-8 detector. Fires when zero or one inputs are active. The sparsity detector.

Circuit

  x₀ x₁ x₂ x₃ x₄ x₅ x₆ x₇
   │  │  │  │  │  │  │  │
   └──┴──┴──┴──┼──┴──┴──┴──┘
               ▼
          ┌─────────┐
          │ w: -1×8 │
          │ b:  +1  │
          └─────────┘
               │
               ▼
           HW ≤ 1?

A single threshold neuron. The negative weights mean each active input subtracts from the sum.

The Sparsity Bound

This circuit enforces extreme sparsity:

• HW = 0: Complete silence (allowed)
• HW = 1: Single signal (allowed)
• HW ≥ 2: Multiple signals (rejected)

It's the upper bound for "at most one thing happening."

Mechanism

sum = -x₀ - x₁ - x₂ - x₃ - x₄ - x₅ - x₆ - x₇ + 1
    = 1 - HW

HW	Sum	Output
0	+1	1
1	0	1
2	-1	0
3+	< -1	0

The bias of +1 grants a budget of one active input. Any more exceeds the budget.

Dual of AtLeast7

Circuit	Condition	Fires when
AtLeast7	HW ≥ 7	Dense (7-8 active)
AtMost1	HW ≤ 1	Sparse (0-1 active)

Under bitwise NOT, AtMost1 inputs map to AtLeast7 inputs. They detect opposite extremes of the distribution.

Applications

• Mutex verification: At most one process holds the lock
• One-hot validation: Valid encoding has exactly 0 or 1 bit set
• Conflict detection: Multiple requests trigger an error

The AtMost-k Family

Circuit	Weights	Bias	Fires when
AtMost1	all -1	+1	HW ≤ 1
AtMost2	all -1	+2	HW ≤ 2
AtMost3	all -1	+3	HW ≤ 3
...	all -1	+k	HW ≤ k

All single-layer circuits. The bias determines the threshold.

Parameters

Component	Value
Weights	[-1, -1, -1, -1, -1, -1, -1, -1]
Bias	+1
Total	9 parameters

Single neuron, single layer.

Usage

from safetensors.torch import load_file
import torch

w = load_file('model.safetensors')

def atmost1(bits):
    inp = torch.tensor([float(b) for b in bits])
    return int((inp * w['weight']).sum() + w['bias'] >= 0)

# Empty: allowed
print(atmost1([0,0,0,0,0,0,0,0]))  # 1

# Singleton: allowed
print(atmost1([0,0,0,1,0,0,0,0]))  # 1

# Pair: rejected
print(atmost1([0,1,0,1,0,0,0,0]))  # 0

Files

threshold-atmost1outof8/
├── model.safetensors
├── model.py
├── config.json
└── README.md

License

MIT