threshold-lessthan

8-bit less-than comparator. Returns 1 if a < b, 0 otherwise.

Circuit

a0 b0    a1 b1    a2 b2    a3 b3    a4 b4    a5 b5    a6 b6    a7 b7
 │ │ 0    │ │      │ │      │ │      │ │      │ │      │ │      │ │
 └┬┘│     └┬┘      └┬┘      └┬┘      └┬┘      └┬┘      └┬┘      └┬┘
  ▼ │      ▼        ▼        ▼        ▼        ▼        ▼        ▼
┌───┐│  ┌───┐    ┌───┐    ┌───┐    ┌───┐    ┌───┐    ┌───┐    ┌───┐
│FS0│┴──│FS1│────│FS2│────│FS3│────│FS4│────│FS5│────│FS6│────│FS7│──►(a<b)
└───┘   └───┘    └───┘    └───┘    └───┘    └───┘    └───┘    └───┘
  │       │        │        │        │        │        │        │
  ▼       ▼        ▼        ▼        ▼        ▼        ▼        ▼
 d0      d1       d2       d3       d4       d5       d6       d7
      (difference bits unused - only final borrow matters)

Uses 8 cascaded full subtractors. The final borrow output indicates a < b.

Mechanism

The circuit computes a - b using subtraction:

• If a ≥ b: no borrow propagates, result is non-negative
• If a < b: borrow propagates out, indicating underflow

The difference bits (d0-d7) are computed but unused. Only the final borrow matters.

Truth Table (sample)

a	b	a < b
0	0	0
0	1	1
1	0	0
127	128	1
255	0	0
0	255	1
100	100	0
99	100	1

Full Subtractor Structure

        a       b
        │       │
        └───┬───┘
            ▼
       ┌─────────┐
       │   HS1   │
       └─────────┘
        │       │
       d1       b1
        │        \
        │   bin   \
        └──┬──┘    \
           ▼        \
      ┌─────────┐    \
      │   HS2   │     │
      └─────────┘     │
       │       │      │
       d       b2     │
               │      │
               └──┬───┘
                  ▼
              ┌──────┐
              │  OR  │
              └──────┘
                  │
                  ▼
                bout

Architecture

Component	Per FS	Total (8 FSs)
Neurons	9	72
Parameters	27	216

Layers: 32 (8 FSs × 4 layers each)

Comparison Family

Circuit	Condition	Implementation
LessThan	a < b	borrow_out of (a - b)
Equal	a = b	NOR of all XOR bits
GreaterThan	a > b	LessThan(b, a)

Usage

from safetensors.torch import load_file

w = load_file('model.safetensors')

def less_than(a, b):
    """a, b: 8-bit lists [a0..a7] (LSB first)
    Returns: 1 if a < b, 0 otherwise"""
    # See model.py for full implementation
    pass

# Example: 99 < 100?
a = [(99 >> i) & 1 for i in range(8)]
b = [(100 >> i) & 1 for i in range(8)]
result = less_than(a, b)  # Returns 1

Files

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

License

MIT