How every line is checked

Most answer keys ask you to trust them. This one shows its work about its work.

Each line carries a label

✓ ProvedThe computer algebra system SymPy showed this line equals the one before it: it reduced the difference to zero.

✓ Checked independentlyA separate program proved it another way, usually one written for that line and saved so anyone can rerun it.

≈ Checked numericallyThe two sides agree at every point sampled. That is strong evidence, but it is not a proof, and it is never called one.

ReviewedA sentence (setting up a model, choosing a method) that no computer can check. A reviewer read it and found it sound.

Not checkedNothing has looked at it yet.

The answer is checked a second way

Separately from the solution, every answer is confirmed by a route the solution doesn't take: a derivative is recomputed directly, an antiderivative is differentiated back, an area is found by numerical integration, a differential equation's solution is substituted back into the equation, a maximum is found by sampling thousands of points.

The explanation is judged separately

A solution can be correct line by line and still explain a step badly. A panel of AI models reads every solution and objects to anything unclear or misleading. When they disagree, a stronger model reads it, rules on each objection, and writes a small program to settle any line the algebra system couldn't. Every verdict stays on the problem's receipt, dated, including the ones that were overruled.

Nihil obstat

“Nothing stands in the way.” A problem carries it when every line is machine-checked (or, for a sentence, read and found sound), the answer is confirmed a second way, and no objection to the explanation stands.

Where the problems come from

The problems are original, built by program so that their answers are known before a solution is written, and they are organized the way a three-semester calculus sequence is taught. New problems are added every night.

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