= Solution
The <hierarchy problem> is visible in the one-loop top-quark contribution to the <Higgs boson> mass parameter,
$$
\delta m_H^2\sim-\frac{3|y_t|^2}{8\pi^2}\Lambda^2.
$$
In a supersymmetric theory the corresponding <top squark>[stop] loop has the opposite quadratic term. If supersymmetry is softly broken, the remaining correction is roughly
$$
|\delta m_H^2|\sim\frac{3|y_t|^2}{8\pi^2}m_{\widetilde t}^2
\log\frac{\Lambda}{m_{\widetilde t}}.
$$
The two relevant one-loop <Feynman diagram>[Feynman diagrams] are a Higgs two-point function with a closed top-quark loop and the analogous stop loop. Requiring the residual correction not to exceed the electroweak scale by many orders of magnitude gives the <naturalness> expectation $m_{\widetilde t}$, and broadly other <sparticle> masses, at a few TeV or below. This is an order-of-magnitude argument rather than a mass bound.
<R-parity> is
$$
R_p=(-1)^{3(B-L)+2s},
$$
where $B$ is <baryon number>, $L$ is <lepton number> and $s$ is <spin angular momentum>. Standard Model particles have $R_p=+1$, while their superpartners have $R_p=-1$. The renormalizable <R-parity violation>[R-parity-violating] superpotential of the <Minimal supersymmetric Standard Model> is
$$
W_{\rm RPV}=\frac12\lambda_{ijk}L_iL_jE_k^c
+\lambda'_{ijk}L_iQ_jD_k^c
+\frac12\lambda''_{ijk}U_i^cD_j^cD_k^c
+\kappa_iL_iH_u,
$$
with $\lambda_{ijk}=-\lambda_{jik}$ and $\lambda''_{ijk}=-\lambda''_{ikj}$.
Combining $\lambda'_{i1k}L_iQ_1D_k^c$ with $\lambda''_{11k}U_1^cD_1^cD_k^c$ produces <proton decay> through exchange of a virtual right-handed down-type <squark>. The tree diagram has $u$ and $d$ entering the baryon-number-violating vertex, an internal $\widetilde d_{Rk}$ line, and the lepton-number-violating vertex emitting $\bar\nu_i$ and a light quark. Hadronization gives
$$
p\longrightarrow\pi^++\bar\nu_i.
$$
The invisible particle is an <antineutrino> of any family $i=1,2,3$, and electric-charge conservation requires a positively charged <pion>. The antisymmetry of $\lambda''_{11k}$ requires $k=2$ or $3$.
Integrating out a squark of mass $m_{\widetilde q}$ gives a dimension-six interaction with coefficient $C\sim\lambda'\lambda''/m_{\widetilde q}^2$. Ignoring order-one hadronic and phase-space factors,
$$
\Gamma_p\sim\frac{|\lambda'\lambda''|^2}{8\pi}
\frac{m_p^5}{m_{\widetilde q}^4}.
$$
For $m_p\sim1\ \mathrm{GeV}$ and $m_{\widetilde q}\sim10^4\ \mathrm{GeV}$, the factor before the couplings is about $10^{-18}$--$10^{-17}\ \mathrm{GeV}$. A lifetime above $10^{34}$ years is about $10^{65}\ \mathrm{GeV}^{-1}$, so $\Gamma_p\lesssim10^{-65}\ \mathrm{GeV}$ and
$$
\boxed{|\lambda'\lambda''|\lesssim10^{-24}}
$$
at order-of-magnitude accuracy. Hadronic matrix elements and flavour choices alter the numerical coefficient, not the conclusion that simultaneous baryon- and lepton-number violation is extraordinarily constrained.
The matter-parity factor $(-1)^{3(B-L)}$ is odd for every quark or lepton superfield and even for either Higgs superfield. Each term in $W_{\rm RPV}$ contains an odd number of matter superfields, so <R-parity> forbids all four types of term. Standard MSSM Yukawa interactions contain two matter superfields and remain allowed.
Finally, the requested four-point interaction contains a squark, an antisquark, a $W$ boson and a gluon. It comes from the squark <gauge-covariant kinetic term>
$$
(D_\mu\widetilde Q)^\dagger D^\mu\widetilde Q,
\qquad
D_\mu=\partial_\mu+ig_sG_\mu^aT^a+ig_2W_\mu^It^I+\cdots.
$$
The cross term is
$$
2g_sg_2\,\widetilde Q^\dagger T^at^I\widetilde Q\,G_\mu^aW^{I\mu},
$$
because the colour and weak generators commute. With all fields incoming, its <Feynman rule> is
$$
2ig_sg_2g^{\mu\nu}(T^a)_{ji}(t^I)_{BA},
$$
with the conventional $1/\sqrt2$ conversion when $W^I$ is replaced by $W^\pm$.
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