The classical Hessian is not its contraction

For a formal bosonic multiplet ${\phi }^A$, a quadratic classical observable may be written schematically as

If a covariance and all pairings have been supplied, a prospective contraction has the form

Equation (eq:prospective) is only a displayed target until the product, pullback, or pairing between its distributions is defined. It is not evaluated by knowing $K_{AB}$ classically.

Fermionic fields require the corresponding graded signs. A common bosonic-looking formula is not a generator-level sign check for a BRST or BV multiplet.