threshold-xor3-mag10

3-input XOR gate at optimal magnitude 10.

Discovery

Exhaustive search over 387,328,512 configurations proved magnitude 10 is optimal for XOR3. This flat 2-layer architecture beats the cascade approach (magnitude 14) by 29%.

Architecture	Neurons	Params	Magnitude
Cascade (2×XOR)	6	18	14
Flat 3-hidden	4	16	10

Architecture

a ──┐
b ──┼──► [h1] ──┐
c ──┘    [h2] ──┼──► [out] ──► XOR3(a,b,c)
         [h3] ──┘

Single hidden layer with 3 neurons, all biases = 0.

Solutions

18 solutions exist at magnitude 10, related by input permutation and neuron reordering symmetry.

Solution	h1 weights	h2 weights	h3 weights	out weights
1	[-1,0,0]	[-1,-1,1]	[-1,1,-1]	[1,-1,-1]
2	[-1,0,0]	[-1,1,-1]	[-1,-1,1]	[1,-1,-1]
3	[0,-1,0]	[-1,-1,1]	[1,-1,-1]	[1,-1,-1]
4	[0,-1,0]	[1,-1,-1]	[-1,-1,1]	[1,-1,-1]
5	[0,0,-1]	[-1,1,-1]	[1,-1,-1]	[1,-1,-1]
6	[0,0,-1]	[1,-1,-1]	[-1,1,-1]	[1,-1,-1]
7	[-1,-1,1]	[-1,0,0]	[-1,1,-1]	[-1,1,-1]
8	[-1,-1,1]	[0,-1,0]	[1,-1,-1]	[-1,1,-1]
9	[-1,1,-1]	[-1,0,0]	[-1,-1,1]	[-1,1,-1]
10	[-1,1,-1]	[0,0,-1]	[1,-1,-1]	[-1,1,-1]
11	[1,-1,-1]	[0,-1,0]	[-1,-1,1]	[-1,1,-1]
12	[1,-1,-1]	[0,0,-1]	[-1,1,-1]	[-1,1,-1]
13	[-1,-1,1]	[-1,1,-1]	[-1,0,0]	[-1,-1,1]
14	[-1,-1,1]	[1,-1,-1]	[0,-1,0]	[-1,-1,1]
15	[-1,1,-1]	[-1,-1,1]	[-1,0,0]	[-1,-1,1]
16	[-1,1,-1]	[1,-1,-1]	[0,0,-1]	[-1,-1,1]
17	[1,-1,-1]	[-1,-1,1]	[0,-1,0]	[-1,-1,1]
18	[1,-1,-1]	[-1,1,-1]	[0,0,-1]	[-1,-1,1]

Hidden Neuron Types

Six unique neuron types appear across solutions:

Type	Weights	Function
Sel-a	[-1,0,0]	fires when a=0
Sel-b	[0,-1,0]	fires when b=0
Sel-c	[0,0,-1]	fires when c=0
XOR-ab	[-1,-1,1]	fires when c>(a+b)
XOR-ac	[-1,1,-1]	fires when b>(a+c)
XOR-bc	[1,-1,-1]	fires when a>(b+c)

Files

threshold-xor3-mag10/
├── model.safetensors      (default = solution1)
├── solution1.safetensors
├── solution2.safetensors
├── ...
├── solution18.safetensors
├── model.py
├── config.json
└── README.md

Usage

from safetensors.torch import load_file
import torch

w = load_file('model.safetensors')  # or solution1-18

def xor3(a, b, c):
    inp = torch.tensor([float(a), float(b), float(c)])
    h1 = int((inp * w['h1.weight']).sum() + w['h1.bias'] >= 0)
    h2 = int((inp * w['h2.weight']).sum() + w['h2.bias'] >= 0)
    h3 = int((inp * w['h3.weight']).sum() + w['h3.bias'] >= 0)
    hidden = torch.tensor([float(h1), float(h2), float(h3)])
    return int((hidden * w['out.weight']).sum() + w['out.bias'] >= 0)

License

MIT