= Solution
At each primary hit $p$, send a <shadow ray> toward each point light. An especially clear parameterization is
$$
s_j(t)=p+t(\ell_j-p),\qquad 0<t<1.
$$
If an opaque scene surface intersects this open segment, the source is hidden. The finite interval is essential: an object beyond the light must not cast a shadow at $p$. Exclude the launch surface with a small geometrically appropriate origin offset or positive minimum parameter, so <floating-point arithmetic> does not create a false self-intersection.
Introduce a visibility factor $V_j(p)$ equal to zero when blocked and one otherwise. The modified equation is
$$
\boxed{I(p)=k_aI_a+\sum_j V_j(p)I_j(p)\left[k_d\max(0,\hat N\cdot\hat L_j)+k_s\chi_j\max(0,\hat R_j\cdot\hat V)^n\right].}
$$
A blocked light loses both its diffuse and specular contributions. The simple ambient approximation remains. Opaque point lights produce hard shadow boundaries. Use a consistent sidedness policy for occluders: an opaque closed solid blocks from either crossing side, so a two-sided blocker test is appropriate even when primary visibility uses <back-face culling>. An any-hit query can stop as soon as it finds one blocker before the light.
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