= Solution
<R-parity> is the multiplicative $\mathbb Z_2$ quantum number
$$
\boxed{R_p=(-1)^{3(B-L)+2s},}
$$
where $B,L,s$ are <baryon number>, <lepton number> and spin. Ordinary <Standard Model> particles have $R_p=+1$ and their superpartners have $R_p=-1$. Gauge invariance alone allows <MSSM> <superpotential> terms $LLE^c$, $LQD^c$, $U^cD^cD^c$ and $LH_u$. Simultaneous baryon- and lepton-number violation can induce rapid <proton decay>. Exact <R-parity> forbids these renormalizable terms, since the associated <matter parity> is odd; it does not forbid every higher-dimensional proton-decay operator, for example $QQQL$.
Two consequences are \b[superpartner production in pairs from ordinary-particle initial states] and \b[stability of the lightest supersymmetric particle]: a single parity-odd particle cannot decay only into parity-even particles. Cascades therefore end in the <lightest supersymmetric particle>; if it is neutral and weakly interacting, it produces missing-energy signatures.
A continuous <R-symmetry> rotates $\theta$ and the <supercharges>. With $R(\theta)=1$, the <superpotential> has <R-charge> two and the fermion in a <chiral superfield> has one less charge than its scalar. <R-parity> is only a discrete restriction, represented by combining <matter parity> with $\theta\mapsto-\theta$, and need not preserve a full continuous <R-symmetry>. For example, a <gaugino> <Majorana mass term> violates an unbroken continuous <R-symmetry> but preserves <R-parity>. Thus proton protection by parity and continuous R-charge selection rules are distinct requirements.
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