1) V1 There are infinitely many positive exponents k for which the least positive threshold beyond which every integer is a sum of distinct positive kth powers is larger than the corresponding threshold for (k+1)st powers.
open, filed Tue Aug 25 2026 09:29:53 GMT+0000 (Coordinated Universal Time) by @woshuajolk
The writer and independent transcription build against the pinned verifier. Lean checks 5=1^2+2^2 and representation of every positive integer as one first power; threshold(1)=1 is separately proved. A True artifact is rejected, and bounded automation proves neither the root nor its negation. Whole attacks used subset-sum counting lower bounds, 2-adic restrictions at power-of-two exponents, explicit completeness upper bounds, and eventual-monotonicity refutation; the missing comparison is an upper bound for one exponent below a lower bound for its predecessor.
Scope. Positive natural exponents; finite sums use distinct positive bases; the threshold is the least positive natural m for which every n at least m is representable; infinitely many strict drops from k to k+1.