theorem
/ THEE-uh-rem /
A theorem is a statement that has been proven true — not just tested, not just believed, but locked down by an unbroken chain of logic. You start from a handful of ground rules everyone agrees on (the axioms), and step by airtight step you show the statement must follow. Once that chain is complete, the theorem is true forever, the way 2 + 2 = 4 was true yesterday and will be true a billion years from now. It's the difference between "this bridge has held so far" and "this bridge cannot fall."
That permanence is what makes theorems the bedrock of mathematics. A scientist's law can be overturned by tomorrow's experiment, but a proven theorem never expires — the Pythagorean theorem about right triangles has stood, unbeaten, for over two thousand years. Build a new result on top of a theorem and you can trust the foundation completely.
The crucial contrast is with a conjecture: a statement that looks true, that mountains of examples seem to support, but that nobody has yet proven. A conjecture is a promising hunch; a theorem is a settled fact. The famous Fermat's Last Theorem was, for 358 years, really only a conjecture — it earned the name "theorem" in full only in 1994, when Andrew Wiles finally supplied the proof.
The Pythagorean theorem: in any right triangle, the square of the longest side equals the sum of the squares of the other two — proven, not merely observed.
"Theorem" comes from the Greek theōrēma, "a thing looked at" or "contemplated" — from the same root as "theater," something you behold. The biggest result in a paper is the "theorem"; a small stepping-stone toward it is a lemma, and a quick free consequence is a corollary.