Build the first working single-neuron logic-gate trainer

Now implement the shortest complete path for the artifact. The working mechanism is: form z=w1x1+w2x2+b, pass it through a sigmoid, calculate binary cross-entropy, and update all parameters with gradients. 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: learn weights and a bias that reproduce the OR truth table instead of hand-tuning until one example happens to pass. Its accepted input is four binary input pairs with binary OR targets. 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.

import math
samples=[((0,0),0),((0,1),1),((1,0),1),((1,1),1)]; w=[0.,0.]; b=0.
for _ in range(2000):
 for x,y in samples:
  p=1/(1+math.exp(-(w[0]*x[0]+w[1]*x[1]+b))); e=p-y
  w=[w[j]-.2*e*x[j] for j in range(2)]; b-=.2*e
print([round(v,2) for v in w],round(b,2))

Expected output: two positive weights and a negative bias. 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.

Trace z, sigmoid probability, error, and each parameter update for one OR example. If loss stays near 0.693, confirm that the bias changes and that labels are numeric zero or one. Clamp only the exponential input for numerical stability; do not clamp the learned probability so aggressively that gradients disappear. When rounded accuracy looks perfect, inspect probability margins to catch a brittle decision near 0.5.

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.

Evaluate all four truth-table rows every time—there is no reason to sample a four-row domain. Show cross-entropy, exact accuracy, minimum distance from 0.5, and learned weights. Compare against constant-zero and constant-one baselines. The neuron has mastered OR only when every row passes and the probabilities are comfortably on the correct side of the threshold.

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.