Build the first working two-layer XOR network

Now implement the shortest complete path for the artifact. The working mechanism is: compute two ReLU hidden activations, combine them into an output score, and trace how intermediate representations separate the cases. Keep every intermediate value available for inspection; hiding it behind a framework would make this lesson harder to reason about. The output should be deterministic for this fixture. Only after this path works should you generalize the data source or user interface.

The artifact's user-facing goal is specific: show why a hidden layer can solve XOR while a single straight decision boundary cannot. Its accepted input is the four binary XOR examples, represented as two floats and one binary target. Those statements are intentionally narrower than “build an AI system.” Narrow scope lets us inspect every input and expected result, and it prevents a toy result from being presented as a production claim. This is the chapter's first end-to-end implementation. Run it twice and verify identical output.

Save this as lesson.py and run python3 lesson.py. It uses only the language standard library, so the example is reproducible offline.

def xor_net(x):
 h1=max(0,x[0]-x[1]); h2=max(0,x[1]-x[0])
 return int(h1+h2>=.5)
print([xor_net(x) for x in [(0,0),(0,1),(1,0),(1,1)]])

Expected output: [0, 1, 1, 0]. Exact floating-point formatting may vary slightly, but the asserted behavior must not. Read the output as evidence about this stage, not merely proof that the interpreter started.

Print both hidden activations for all four XOR rows. A hidden unit that is zero for every row is dead and contributes nothing; revisit its weights before changing the output threshold. When a dot product fails, print both dimensions and repair the architecture, not the input by truncation. Keep the linear baseline visible: XOR's value is demonstrating representation, not merely obtaining four correct labels.

At the implementation stage, save the smallest failing fixture beside the expected result. Change one cause at a time and rerun the exact command printed above; that makes the repair reviewable and keeps this chapter's progressive artifact reproducible.

Assert the exact XOR truth table, then calculate hidden-unit activation coverage and prediction margin. Compare with the best single linear boundary, which cannot classify all four corners. Perturb each input slightly only if the declared domain permits non-binary values, and state that this fixed-weight demonstration explains a forward pass rather than proving a trained deep model will generalize.

For this implementation page, preserve the fixture and result as evidence for the next page. Label observations separately from conclusions: a passing assertion establishes the behavior it names, while broader usefulness requires the chapter's full evaluation set and stated operating limits.