= Solution
<Primitive causality> is an initial-region completeness requirement. A state's full restriction to a suitable region $O$ determines expectations of <observables> in its future <domain of dependence>. The <domain of dependence> is not the whole causal future: it contains points whose past-inextendible causal curves all meet the specified initial data region. Merely being reachable by one causal curve from $O$ is not enough.
A standard local <time-slice axiom> makes this precise when $O$ contains a neighbourhood of a <Cauchy slice> of its causal development:
$$
\boxed{\mathcal A(O)=\mathcal A(D(O)),\qquad\mathcal A(D^+(O))\subseteq\mathcal A(O),}
$$
with the regions and algebra identifications understood in the specified local net. Definitions applying <primitive causality> to all bounded open regions can be stronger than a time-slice property stated only for suitable causally convex Cauchy neighbourhoods; these formulations should not be conflated without extra hypotheses.
For a local hyperbolic Klein–Gordon equation, a later field is determined by the field and its normal derivative, or <canonical momentum>, on a <Cauchy slice>. Its value can be written using the causal fundamental solution, with integration supported where the past <light cone> meets the initial slice. If that support lies in the initial patch, the later smeared field is generated by the patch's initial field and <momentum> operators. Their algebra therefore generates the <observables> of the patch's <domain of dependence>. This is the local differential-equation mechanism behind the time-slice property.
The required information is the entire restricted <quantum state>: it assigns expectations to every <observable> and product in the initial algebra. Mean values of the initial field and <momentum> alone do not determine their fluctuations or higher correlations and thus are insufficient. Knowledge of the restricted state fixes later <probability distributions> and correlations; it does not fix which individual outcome a future measurement will produce.
Finally, <primitive causality> concerns propagation of information already encoded in initial data, while <microcausality> controls compatibility of spacelike measurements and <quantum no-signalling> controls possible interventions. The <spectrum condition> adds a separate positivity restriction. \b[A causal initial-value evolution, positive energy and <no-signalling> are complementary requirements, not equivalent definitions of quantum locality.]
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