NP-completeness of the initial odd-ear prefix

The variable-threshold decision problem beta(G)>=t for the maximum initial consecutive odd prefix of an open ear decomposition is NP-complete, even on finite simple 2-vertex-connected graphs with a universal vertex whose deletion leaves a connected bipartite graph. The initial circuit and one-edge ears are counted; the first even ear ends the prefix. This gives a full complexity answer to Sebo's OWR 2014 Question 4 and Benchetrit-Sebo's 2015 Question 3.3. It does not assert P!=NP, fixed-threshold hardness, closed-ear hardness or a theorem about the total number of odd ears. The six-page note combines the classical factor-critical formulation with an explicit cone identity and a polynomial Set Cover reduction. Complete proofs include all preprocessing branches within the cone subclass, private-element perfect matchings, arbitrary partial blocks and omitted distinguished vertices, the exact loss identity d_i-k(a_i-b_i)=b_i(2k-a_i)>=0, connectivity forcing coverage, explicit polynomial encoding and NP certificates. The key estimate is k(r-s)<=T