Positivity structure of the flavour-determinant vertex expansion at odd generation number (SQG Companion)

Every determinantal sign-free criterion is closed at odd generation number for the 16 of Spin(10). A resolution must therefore come from outside the determinantal class. This paper develops the structure of the remaining candidate — the flavour-determinant sextic V_6 = \det_{\sigma\tau} F + \mathrm{h.c.} — in its continuous-time (world-line) representation and settles the pointwise form of the positivity question on every geometry class. Proved results include: vanishing of all odd orders by a particle–hole cancellation theorem; exact adjointness of the two bond orientations; non-negativity of the second-order weight on any geometry; and, after a sector-walk reformulation, all-order positivity on a single bond (the interaction is simultaneously diagonal in an eleven-member commuting family and the vertex is entrywise non-negative after an explicit diagonal sign gauge). Beyond one bond the pointwise question is closed in the negative in two independent places: winding×antiwinding sequences on loops carry negative weight, and at a branch point (four-site star) explicit order-six words of strictly negative weight exist. Because the offending words are leg-balanced, their weights are invariant under every diagonal sign gauge and every per-leg rescaling; the obstruction recurs at every even order with magnitude growing as (n_f!)^k. The earlier tree conjecture is thereby refuted. What survives is the grouped statement: the weight summed over vertex orientations at fixed times is positive in every test on every geometry and is precisely the object evaluated by a fermion-bag sampler. It is stated as the single remaining conjecture.