threshold-parity-tree

8-bit parity computation using balanced XOR tree structure for minimum depth.

Circuit

x7 ─┬─[XOR]─┐
x6 ─┘       ├─[XOR]─┐
x5 ─┬─[XOR]─┘       │
x4 ─┘               ├─[XOR]──► parity
x3 ─┬─[XOR]─┐       │
x2 ─┘       ├─[XOR]─┘
x1 ─┬─[XOR]─┘
x0 ─┘

Output

• parity = 0: Even number of 1s in input
• parity = 1: Odd number of 1s in input

Tree vs Linear

Linear chain (depth 7):
x0─[XOR]─[XOR]─[XOR]─[XOR]─[XOR]─[XOR]─[XOR]─► parity
    x1    x2    x3    x4    x5    x6    x7

Tree structure (depth 3):
Layer 1: 4 parallel XORs
Layer 2: 2 parallel XORs
Layer 3: 1 final XOR

Architecture

Layer	XOR Gates	Function
1	4	Pairs: (0,1), (2,3), (4,5), (6,7)
2	2	Pairs: (01,23), (45,67)
3	1	Final: (0123,4567)

Total: 21 neurons (7 XORs × 3), 111 parameters, 3 layers

Scalability

For n-bit parity:

• Tree depth: log₂(n)
• XOR gates: n - 1
• Linear depth: n - 1

Bits	Tree Depth	Linear Depth
8	3	7
16	4	15
32	5	31
64	6	63

Usage

from safetensors.torch import load_file

w = load_file('model.safetensors')

# Examples:
# 0x00 (00000000) → parity=0
# 0x01 (00000001) → parity=1
# 0x03 (00000011) → parity=0
# 0xFF (11111111) → parity=0

Files

threshold-parity-tree/
├── model.safetensors
├── create_safetensors.py
├── config.json
└── README.md

License

MIT