= Solution
\b[Matching <spacetime symmetries> to <dynamical symmetries> is a powerful demand for a coherent interpretation, but an unconditional equality theorem does not follow from logic or from <general covariance>.] A <spacetime symmetry> preserves the structure declared physically significant: for <Minkowski spacetime>, the <metric tensor> and relevant orientations. A <dynamical symmetry> takes solutions to solutions with the laws' fixed backgrounds held fixed. The appropriate comparison concerns external transformations on the same physical quantities; internal <gauge symmetries> have no reason to be isometries of <spacetime>.
One direction becomes a theorem after important premises are supplied. Suppose all fixed background structure is geometrical and the matter equations are constructed covariantly from it. An automorphism of that full structure then takes solutions of those equations to solutions. For example, the <Klein-Gordon equation> $(\eta^{ab}\partial_a\partial_b+m^2)\phi=0$, in units $c=\hbar=1$, has the <Poincaré symmetry> of the fixed <Minkowski metric>. This is an explanation of symmetry inheritance conditional on a coupling prescription; it does not derive that prescription from the bare existence of a <metric tensor>.
A counterexample exposes the missing premise. On the same <Minkowski spacetime>, add a fixed timelike <vector field> $n^a$ and a matter term
$$
\mathcal L=-\frac12\eta^{ab}\partial_a\phi\partial_b\phi
-\frac{\alpha}{2}(n^a\partial_a\phi)^2.
$$
For generic nonzero $\alpha$, a boost which preserves $\eta$ but changes $n$ is not a symmetry with that background held fixed. The full background has fewer symmetries than the <metric tensor> alone. Writing the equations in arbitrary coordinates and transforming $n$ as a <vector field> gives <general covariance>, not <Lorentz invariance>. Conversely, redundant unobservable geometrical structure could make the declared <spacetime symmetry> group artificially too small. Accidental symmetries of a restricted matter sector could make its <dynamical symmetry> group too large.
There are deeper motivations for avoiding such mismatches. If all measuring devices obey <Lorentz-invariant dynamics>, their behaviour supports a common <Minkowski metric> interpretation; adding an undetectable preferred rest structure sacrifices explanatory economy. If a preferred structure affects some device, then the original metric-only geometry was incomplete as a description of physical background structure. These observations support either revising the geometry or explaining the universal matter symmetry, but they are methodological and interpretive arguments, not an automatic metaphysical identity.<Minkowski space-time: a glorious non-entity by Harvey R. Brown and Oliver Pooley>[§§3–5, rods, clocks and the relation between geometry and matter dynamics]{ref}
Two qualifications matter. <Boundary conditions> and <initial conditions> can select a rest frame in an individual solution while the laws remain boost symmetric. Also, a symmetry of the complete theory need not be detectable by transforming only a subsystem relative to its environment. The best principle therefore matches the physically relevant geometry to the external symmetries of the complete laws, after all backgrounds and couplings are included. Calling it merely a bookkeeping preference understates its explanatory value; calling it an unavoidable consequence of covariance overstates its logical force.
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