AN EXACT RATIONAL COUNTEREXAMPLE TO ERDŐS'S SQUARE-PACKING CONJECTURE f(k²+1)=k (PROBLEM #106)

Abstract

This preprint gives an explicit exact rational counterexample to Erdős's square-packing conjecture f(k²+1)=k, catalogued as Erdős Problem #106. The primary-source formulation of Erdős and Graham (1975) uses total circumference instead: if F_EG(n) is the maximum total circumference, then F_EG(n)=4f(n), so their conjecture F_EG(k²+1)=4k is exactly equivalent to the modern side-length statement. For k=5, the preprint constructs 26 squares inside a unit square with pairwise disjoint interiors and exact total side length 4471148/894225 = 5 + 23/894225 > 5; equivalently, their exact total circumference is 17884592/894225 = 20 + 92/894225 > 20. Thus the same witness contradicts both normalizations of the same conjecture. The witness is specified entirely by rational data. Containment and all 325 pairwise non-overlap conditions are checked by two independently implemented exact-arithmetic methods: a Separating Axis Theorem verifier and a convex-polygon half-plane clipping verifier. The preprint does not determine the optimal value f(26), nor does it settle the individual k=3 or k=4 cases. External reproduction and peer review remain pending.