On the Frame-Dependence of Perfection: A Re-examination of the Perfect Number Problem - Its Inherited Cut, Its History, and Two Rules Honestly Held

The smallest perfect number is usually given as 6, since 6 equals the sum of its proper divisors: 1 + 2 + 3. This paper does not dispute that classical result. It asks instead what must count as a “part” of a number before the verdict “perfect” can be reached. The classical answer depends on a whole-quotient admission rule: a prior quantity counts as a part only when it divides the number with no remainder. This paper argues that the rule is a frame inherited from Euclid’s arithmetical vocabulary, not a necessity forced by the perfect-number problem itself. The paper then compares three frames. Under the classical divisor frame, 6 and the Euclid–Euler ladder remain correct. Under an ordinal–continuous no-discrimination rule, where whole-number priors are admitted uniformly and are not refused merely because they yield fractional quotients, the unique perfect number is 3. Under unrestricted continuous admission, no perfect number exists. The paper’s central claim is therefore not that one frame defeats the others, but that perfection is always perfection-under-a-criterion, and the criterion must be named.