The flavour-determinant sextic at odd generation number: pointwise obstruction and the surviving grouped conjecture (SQG Companion)

This note reformulates the world-line positivity of the flavour-determinant (sextic) vertex in exact algebraic terms and closes it on a single bond. The quartic Hamiltonian is a positive scalar on every joint (E, N_A) sector. Consequently every continuous-time weight becomes a positive linear combination of closed sector-walk traces of the vertex blocks. In a joint eigenbasis of an eleven-member commuting family (gauge Casimir and weight, per-flavour charges, per-site-per-flavour charges, and the per-flavour hopping spins) the interaction is simultaneously diagonal for all couplings and the vertex is, after an explicit diagonal sign gauge, entrywise nonnegative. Every single-bond weight is therefore nonnegative at every order, every time configuration and every coupling: the single-bond conjecture is a theorem (deterministic certificates, su(2) proxy at n_f=3). Off trees, pointwise positivity fails while grouping restores it. On the three-site ring the Dyson integrand is nonnegative at every order after exact summation over vertex assignments at fixed times; odd grouped weights vanish identically on bipartite lattices by a staggered-phase argument; and W_2≥0 on any geometry. At the first branch point (four-site star) explicit order-six words with negative weight exist, while the corresponding grouped weights dominate them by four to five orders of magnitude. The pointwise conjecture is thereby settled on every geometry class — proved on the bond, positive on chains, false at branchings — and the grouped (fermion-bag) statement remains as the single surviving, algorithmically relevant form of the conjecture.