= Solution
The instantaneous-decay approximation keeps the <scale factor> and physical volume fixed across the decay. Energy conservation and rapid thermalization give
$$
\rho_{r,+}=\rho_{r,-}+\rho_{\phi,-}\simeq\frac{3M_{\rm Pl}^2}{t_d^2},
\qquad
s_+=c_1g_\star\left(\frac{3M_{\rm Pl}^2}{c_2g_\star t_d^2}\right)^{3/4}.
$$
Meanwhile the abundance relation gives $s_-=\rho_{\phi,-}/(Ym)\simeq3M_{\rm Pl}^2/(Ym\,t_d^2)$. Dividing yields the \b[entropy-injection factor]
$$
\boxed{\frac{s_+}{s_-}\simeq
\frac{c_1}{3}\left(\frac3{c_2}\right)^{3/4}g_\star^{1/4}
\frac{Ym\sqrt{t_d}}{\sqrt{M_{\rm Pl}}}.}
$$
There is a numerical error in the printed coefficient: part (c) omits the factor $1/3$. The denominator $s_-=3M_{\rm Pl}^2/(Ym\,t_d^2)$ supplies that factor, while the radiation energy contributes $3^{3/4}$ to the numerator. Thus the consistent coefficient is $c_1 3^{-1/4}c_2^{-3/4}$, rather than $c_1 3^{3/4}c_2^{-3/4}$. Both cannot follow from the stated part (b) using the same decay-time approximation.
Its increase is especially transparent without dropping the original radiation. Let $R=\rho_{\phi,-}/\rho_{r,-}$. At unchanged $g_\star$, $T_+^4/T_-^4=1+R$, so
$$
\boxed{\frac{s_+}{s_-}=\left(\frac{T_+}{T_-}\right)^3
=(1+R)^{3/4}>1.}
$$
Relic domination means $R\gg1$, so the leading estimate is much larger than one. This is <entropy production by decay of a dominant relic>.
There is no conflict with part (a). The equilibrium gas before the decay was a separately conserved, reversibly expanding system. During decay it receives energy from the nonthermal relic, and the decay products thermalize irreversibly. Its <cosmological perfect-fluid continuity equation> now has an energy-injection source, so $d(\rho_rV)+P_r\,dV$ does not vanish. The assumptions behind <cosmological entropy conservation> therefore fail during the event. Moreover the zero-chemical-potential equilibrium expression in part (a) cannot be imposed on the decoupled relic as though it were already part of that same equilibrium radiation bath. The <Second law of thermodynamics> permits, and here requires, the resulting entropy increase.
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