GCD Witness Lemmas

A streamlined working draft version of the full proof in Section 2 of Document 31, compressing the algebraic core of the gcd witness into three quick-reference lemmas. Generic witness: for an odd prime $p$ not in the multiplicative witness set, there exists a valid witness $\ell$ such that $p\nmid n_\ell$ if and only if $p$ does not divide the $\gcd_\ell n_\ell$ of the entire set—therefore, the only possible failures are the finite odd prime factors of this gcd. Fixed multiplicative prime: if $p$ itself is a multiplicative prime, the witness must be a distinct $\ell\ne p$; this is a finite leave-one-out check. Nonsplit FW witness: the same gcd lemma holds when restricted to the subset of nonsplit multiplicative primes, and its odd gcd factors are exactly the only primes that could cause the standard FW-H3 witness to fail. The document concludes the algebraic essence of the entire mechanism in one sentence: this is the exact algebraic reason why the all-prime witness problem can be finite-ized.