= Solution
A <sufficient statistic> has a conditional law of $X$ given $T$ independent of the parameter. An ordinary <ancillary statistic> has a distribution independent of the whole parameter. A <partial ancillary statistic> for $\psi$ has a law free of $\psi$, though it may depend on the <nuisance parameter> $\lambda$. The <conditionality principle> uses the conditional experiment given the observed value of $S$; removing $\lambda$ additionally requires the conditional law of $C$ given $S$ to be free of it.
The factorization requested in the PDF expresses this stronger <ancillary conditioning that eliminates a nuisance parameter>. There is also a normalization typo: the conditional factor must satisfy $\sum_c\phi_C(c,s;\psi)=1$ \b[for each fixed $s$], rather than the printed double sum over $s$ and $c$.
Under that nuisance-eliminating interpretation, take $\phi_0(x)$ to be the parameter-free conditional probability of $X=x$ given $(C,S)$, $\phi_C(c,s;\psi)$ to be the conditional probability of $C=c$ given $S=s$, and $\phi_S(s;\lambda)$ the marginal probability of $S=s$. The conditional probability rule gives $f=\phi_0\phi_C\phi_S$, with the three separate normalizations. Conversely summing over the fibers first gives the joint law $\phi_C\phi_S$, then summing over $c$ gives $\phi_S$ and dividing gives the required conditional law $\phi_C$. The factorization also proves sufficiency of $(C,S)$.
This equivalence is not valid for ordinary ancillarity alone: a constant $S$ is always ancillary, and $T=X$ is sufficient, but conditioning on a constant generally leaves the nuisance parameter present. Thus the distinction matters to the conclusion.
For the independent <gamma distributed> sample, the joint <log-likelihood> is
$$
\ell(a,b)=na\log b-n\log\Gamma(a)+(a-1)\sum_j\log x_j-b\sum_jx_j.
$$
The proposed factorization would write this as a data-only term plus a function of $a$ and data plus a function of $b$ and data. It would therefore force its mixed derivative to vanish. In fact
$$
\boxed{\frac{\partial^2\ell}{\partial a\,\partial b}=\frac nb\ne0.}
$$
The <mixed log-likelihood derivative obstruction to parameter separation> rules out this exact nuisance-eliminating factorization. It does not rule out ordinary ancillary statistics: a constant statistic still qualifies, and $(\sum\log X_j,\sum X_j)$ is sufficient for the full two-parameter family.
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