  In Dirac-Bergmann constrained dynamics, a first-class constraint typically
does not _alone_ generate a gauge transformation. Each first-class constraint
in Maxwell's theory generates a change in the electric field E by an arbitrary
gradient, spoiling Gauss's law. The secondary constraint p^i,_i=0 still holds,
but being a function of derivatives of momenta (mere auxiliary fields), it is
not directly about the observable electric field (a function of derivatives of
A), which couples to charge. Only a special combination of the two first-class
constraints, the Anderson-Bergmann-Castellani gauge generator G, leaves E
unchanged. Likewise only that combination leaves the canonical action
invariant---an argument independent of observables. If one uses a first-class
constraint to generate instead a canonical transformation, one partly strips
the canonical coordinates of physical meaning as electromagnetic potentials,
vindicating the Anderson-Bergmann Lagrangian orientation of interesting
canonical transformations. The need to keep gauge-invariant q,t-dH/dp=-E-p=0
supports using the gauge generator and primary Hamiltonian rather than the
separate first-class constraints and the extended Hamiltonian.
  Partly paralleling Pons's criticism, it is shown that Dirac's proof that a
first-class primary constraint generates a gauge transformation, by comparing
evolutions from _identical_ initial data, cancels out and hence fails to detect
the alterations made to the initial state. It also neglects the arbitrary
coordinates multiplying the secondary constraints _inside_ the canonical
Hamiltonian. Thus the gauge-generating property has been ascribed to the
primaries alone.
  Hence the Dirac conjecture about secondary first-class constraints as
generating gauge transformations rests upon a false presupposition about
primary first-class constraints. Clarity about electromagnetism help with GR.
