#Axiom

Formal mathematics, verifiably checked — Axiom returns proofs a computer can verify, step by step, instead of answers that merely sound right.

#Overview

Axiom is the Lab's institution for formal mathematics. You pose a problem, and Axiom reasons through it step by step, then produces a result that has been machine-checked rather than merely asserted. Where a quick answer might sound confident and still be wrong, Axiom is built for correctness — every step is verified by a formal proof system, so its proofs are checked by a computer, not guessed.

This matters whenever the answer has to be right, not just plausible: proofs, formal reasoning, and problems where a single wrong step invalidates the whole result. Axiom gives you a solution you can actually rely on and verify for yourself — and because it shares the Lab's memory, it works in your style and helps you invent new solutions, not just confirm old ones.

#What you can do

  • State a problem in plain language — an equation, an integral, a proof — and get a rigorous, worked solution step by step.
  • Work across 10 math domains with clean LaTeX rendering, so the result is readable as well as correct.
  • Hand it an image of an equation and have Axiom recognize it, so you do not have to retype the math.
  • Get machine-checked rigor. Every proof is verified by a formal proof system, so the answer is one a computer can confirm — not an approximation that merely looks right.
  • Invent new solutions, not just check known ones — Axiom reasons toward results and helps you find a way through.
  • Keep your solutions as artifacts you can reopen, reuse, and build on later.

#How to use it

  1. Open Axiom from the Lab.
  2. State your problem clearly. Give the full setup — the definitions, assumptions, and exactly what you want shown or solved — or hand it an image of the equation. The more precise the statement, the cleaner the result.
  3. Let Axiom work it through. It reasons rigorously toward a verifiable solution rather than a fast approximation.
  4. Read the result. Follow the steps from your premises to the conclusion in clean LaTeX, and check any point you want to confirm.
  5. Refine if needed. If the problem was not quite stated as you meant, restate it more precisely and run it again.
Tip

A clear, complete problem statement is the single biggest lever on a clean result. Spell out your assumptions rather than leaving them implied.

#Examples

What you ask:

text
Prove that the sum of two even integers is always even.

What you see: A worked, machine-checked proof that proceeds from the definition of an even integer to the conclusion, each step verified so you can follow and confirm the reasoning, rendered in clean LaTeX.

What you ask:

text
Show whether the square root of 2 is irrational, with the full argument.

What you see: A rigorous solution presented as a structured result, leading step by step to a conclusion a computer can verify — a proof, not an approximation.

#Tips

  • State the problem precisely. Include your definitions and assumptions; ambiguity in the question becomes ambiguity in the answer.
  • Ask for the full argument when you want every step shown, not just the final answer.
  • Use the image-to-equation recognition to skip retyping complex math — hand Axiom a picture of the problem.
  • Reach for Axiom on correctness-critical work — proofs and formal reasoning — and keep quick numeric questions in Chat overview.