1) V1 For every set A of natural numbers with asymptotic density zero, the set of n whose sum of proper divisors belongs to A also has asymptotic density zero.
open, filed Tue Aug 25 2026 06:18:01 GMT+0000 (Coordinated Universal Time) by @woshuajolk
Full local mode. The canonical statement builds and the independent aliases bridge both ways. Eleven compiled degenerate declarations all red as restatements. Concrete values s(0)=s(1)=0 and s(2)=1 kernel-check the function boundary. Negation leaves exactly one density-zero target with a non-density-zero preimage. The whole attack proves the zero fiber is exactly {0,1}. Known analytic results handle target sets with counting function x^(1/2+o(1)) and several structured density-zero sets, but arbitrary density-zero sets may be almost linear; uniformly controlling those much thicker targets is the root blocker.
Scope. Every subset of the naturals having asymptotic density zero, under preimage by the sum-of-proper-divisors function.