Theorem 5.16.18B2A67FDA22D2

Qualification and proof. For the general germ statement, the precise formulation is the local one: divisors range over modifications of a sufficiently small neighborhood on which $f$ is represented. The proved uniform valuative criterion gives exactly \[f\in\mathcal I(\varphi)_x\quad\Longleftrightarrow\quad\exists\varepsilon>0:\ \operatorname{ord}_Ef+A_X(E)\geq(1+\varepsilon)\nu(\varphi,E)\] for all such divisors whose centers contain $x$. The same $\varepsilon$ must work for every divisor.

In the analytic-singularity clause, one must test all components of the total pole divisor on the log resolution whose centers contain $x$, including nonexceptional components. Restricting the test to exceptional divisors is false: on $\mathbb C$ take $\varphi=\log|z|^2$ and the identity resolution. There are no exceptional divisors, but $1\notin\mathcal I(\varphi)_0$ since $|z|^{-2}$ is not locally integrable.

Here is the finite test with this correction. On a log resolution write $\varphi\circ\pi=\sum_i a_i\log|s_i|^2+O(1)$ and let $b_i$ be the order of the holomorphic Jacobian along $D_i=\{s_i=0\}$. If $c_i=\operatorname{ord}_{D_i}(f\circ\pi)$, change of variables makes the integrand, up to a bounded factor and a holomorphic residual factor, a product of $|s_i|^{2(c_i+b_i-a_i)}$. At a generic point of each divisor this residual factor is nonzero. The one-variable integral $\int_0^\delta r^{2(c_i+b_i-a_i)+1}\,\mathrm dr$ is finite exactly when $c_i+b_i+1>a_i$. Since the divisor is simple normal crossings, Fubini proves sufficiency for their simultaneous inequalities and the generic-point calculation proves necessity. Thus the test is $\operatorname{ord}_{D_i}f+A_X(D_i)>\nu(\varphi,D_i)$ for all the indicated total pole components. This is also the log-resolution integrability criterion. Resolution itself remains the permitted Hironaka input.