PFP II: Signed Certificates, Resumable Search, and a Restricted Path-Pairing Obstruction

When a search fails to find a sign transformation, three explanations remain: the transformation does not exist, the search has not finished, or the allowed transformations were too restrictive. This paper gives a small implementation within the Proposal Fidelity Protocol (PFP) that keeps these possibilities distinct. A finite signed graph admits either a vertex-sign assignment or an odd closed walk, and each outcome comes with a witness that a host checks independently; this is Harary's balance criterion presented as a checking protocol. For an exact real symmetric matrix in a fixed basis, the same check decides whether diagonal sign changes can make every nonzero off-diagonal entry negative, and a finite stoquastic matrix has an entrywise nonnegative thermal exponential. A deterministic bounded search checks candidate assignments in fixed batches, records the unexamined suffix as UNKNOWN, and resumes only after replaying every earlier segment; a skipped segment or a changed presentation is refused. An exact Fock-space example on a three-site triangle with hopping 1, on-site repulsion 2, and inverse temperature 1/4 enumerates all 192 rooted three-hop sequences, 96 positive and 96 negative. One negative sequence has integrated weight −exp(−1)/384; enumeration at length three and rational bounds for every other length show that no positive sequence has the same magnitude, so no sign-reversing pairing of individual sequences with equal weights covers it. This rules out that pairing operation on that carrier, not other representations, regroupings, or estimators. Exact Python checkers, an optional native Rust graph checker, tests, a companion notebook, and demonstrations accompany the manuscript. No new graph theorem, sampling-efficiency result, or general sign-problem claim is made.