How Do Upper and Lower Bounds Work?

Suppose a problem asks for the smallest possible value of some quantity. A lower bound proves the answer cannot be smaller than a number. An upper bound proves it cannot be larger. Together they create an interval of possibility.

Two different kinds of evidence

Lower bounds usually come from a construction: here is an object that achieves value 43, so the optimum is at least 43. Upper bounds usually come from a converse or impossibility argument: every valid object is limited to value 46, so the optimum is at most 46. They are not interchangeable, even when they happen to use similar mathematics.

The gap is information

If both bounds agree, the answer is exact. If they do not, the gap is not a failure of the record; it is the record. Saying 43 ≤ X ≤ 46 rules out every value outside that interval and makes the next useful contribution obvious: improve one side, or find a proof that closes the interval.

Why wording matters

Every bound depends on scope. Change the dimension, allowed operations, error tolerance, or asymptotic regime, and you may be asking a different question. That is why the Registry stores Claims against a versioned specification instead of treating a number as context-free.

Read the frontier

On a canonical Limit page, look for the direction symbol, the value, the scope, and the evidence behind it. A construction can show what is achievable. Only a matching upper-bound argument can show that nothing better is possible.