= Solution
For translations represented by $U(a)=e^{ia_\mu P^\mu}$, the energy-momentum generators have a joint spectral measure. The <spectrum condition> is
$$
\boxed{\operatorname{Sp}(P)\subseteq\overline V_+=\{p:p^0\ge0,\ p_\mu p^\mu\ge0\},}
$$
using signature $(+,-,-,-)$. It excludes negative-energy and spacelike <momentum> states in the vacuum representation, apart from the allowed zero <momentum> of the vacuum. It is stronger than bounding the Hamiltonian below in just one chosen frame: energy is nonnegative in every future-directed inertial frame. Thermal infinite-system representations need not satisfy the same vacuum <spectrum condition>, so the representation matters.
The condition supports stability and controls analyticity. For a translation-invariant vacuum, insertion of the spectral measure into a two-point function gives a form
$$
W(x)=\langle\Omega|\phi(x)\phi(0)|\Omega\rangle=\int_{\overline V_+}e^{-ip\cdot x}\,d\mu(p).
$$
Replacing $x$ by $x-iy$, with $y$ in the future timelike cone, inserts the decaying factor $e^{-p\cdot y}$ and gives analytic continuation into the appropriate tube, subject to the usual distributional bounds. This is a consequence of spectral support, not an assertion of compact spatial support of $W$.
Positive energy is not a commutator condition or a guarantee of finite-speed particle-position propagation. It supplies none of the region-algebra inclusions used in <primitive causality> by itself. Its tension with sharp localization is precisely why the particle-localization discussion needs separate hypotheses. \b[Spectrum, spacelike commutation and operational <no-signalling> constrain different structures and cannot simply substitute for one another.]
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