Erdős Problem 848: A Kernel-Checked Proof of the Exact Extremal Bound

For N ≥ 1, let A be a subset of [1,N] such that ab+1 is nonsquarefree for every a,b in A, with a=b allowed. We determine the exact maximum cardinality, proving |A| ≤ floor((N+18)/25), with equality attained by the elements congruent to 7 modulo 25. This establishes the extremal value for every N and closes the finite range left by the preceding asymptotic results. The proof begins with an exact Hall reformulation. Exact prefix colourings handle N ≤ 5·10⁶. For larger N, square-divisor estimates match the second modulo-25 construction and force any remaining Hall defect to contain many elements outside both distinguished progressions. We partition those elements by their five possible 2-adic types and their residues modulo 9. Three- and four-pivot arguments then bound the Hall completion through explicit finite-prime counts and the solutions of a transformed quadratic equation. The remaining comparisons reduce to exact rational inequalities over exhaustive finite cases, followed by a uniform estimate for the unbounded tail. Finite witnesses enter Lean as ordinary data, semantic checkers establish the required inequalities, and the Lean 4 kernel replays the final all-N theorem with --trust=0.