= Solution
\b[The printed proximity integral is missing a factor $1/(2\pi)$.] Denote that literal integral by $m_{\mathrm{print}}$ and write $\overline m=m_{\mathrm{print}}/(2\pi)$. The correct <Annular spherical Jensen formula>, with the printed definitions of $N$ and $T$, is
$$
\boxed{T(r)=I(a)\log(1/r)+N(r;a)+\overline m(r;a).}
$$
The following derivation proves this identity and also isolates the normalization issue.
For the <chordal metric> put $u_a(z)=-\log k(f(z),a)$ and let $M_a(s)=(2\pi)^{-1}\int_0^{2\pi}u_a(se^{i\theta})\,d\theta$. In a finite target coordinate,
$$
u_a=\tfrac12\log(1+|f|^2)-\log|f-a|+\tfrac12\log(1+|a|^2).
$$
For $a=\infty$ use $u_\infty=\tfrac12\log(1+|f|^2)$. Differentiating away from the preimages of $a$ gives half the squared <spherical derivative of a meromorphic function>. At an $a$-point of local degree $d$, $u_a=-d\log|z-z_0|+O(1)$; the second <distributional derivatives> give the Laplacian of $\log|z-z_0|$ is $2\pi\delta_{z_0}$. Thus, including <poles> by changing spherical coordinates,
$$
\Delta u_a=\frac12\left(\frac{2|f'|}{1+|f|^2}\right)^2-2\pi\sum_{f(z)=a}\deg_z(f)\,\delta_z.
$$
Because $a$ is off the image of $|z|=1$, $u_a$ is smooth near that circle and $I(a)=M_a'(1)$ is finite and independent of $r$.
Apply <Green's second identity> to the harmonic weight $w(z)=\log(|z|/r)$ on $r<|z|<1$. On the outer boundary $w=\log(1/r)$ and $\partial_nw=1$; on the inner boundary $w=0$ and $\partial_nw=-1/r$. Hence
$$
\int_{r<|z|<1}w\Delta u_a\,dA=2\pi\left[M_a'(1)\log(1/r)+M_a(r)-M_a(1)\right].
$$
The distributional formula makes the left side $2\pi T(r)-2\pi N(r;a)$, and $M_a(r)-M_a(1)=\overline m(r;a)$, proving the corrected <Annular spherical Jensen formula>. Initially take an inner circle avoiding $a$-points. Logarithmic singularities have continuous angular means, and a point on the inner circle has zero logarithmic counting weight, so continuity extends the identity to every $0<r\le1$.
There is no way to absorb the missing factor into a different constant $I(a)$. For the <holomorphic function> $f(z)=z$ and $a=\infty$, direct integration gives
$$
N(r;\infty)=0,\qquad m_{\mathrm{print}}(r;\infty)=\pi\log\frac{1+r^2}{2},\qquad T(r)=\frac12\log\frac1r+\frac12\log\frac{1+r^2}{2}.
$$
The value of $(T-m_{\mathrm{print}})/\log(1/r)$ tends to $1/2$ as $r\downarrow0$ and to $\pi$ as $r\uparrow1$. Therefore \b[the literal printed identity has no constant $I(\infty)$], whereas the corrected identity has $I(\infty)=1/2$.
For the singularity criterion, define the nondecreasing normalized spherical area
$$
s(t)=\frac1{4\pi}\int_{e^{-t}<|z|<1}\left(\frac{2|f'|}{1+|f|^2}\right)^2dA,\qquad t\ge0.
$$
<Tonelli's theorem> applied to the logarithmic weight gives $T(e^{-l})=\int_0^l s(t)\,dt$. Averages of a nondecreasing function converge to its supremum: for fixed $t_0<l$, $(1-t_0/l)s(t_0)\le T(e^{-l})/l\le s(l)$. Thus the limit, not just the lower limit, exists in $[0,\infty]$ and equals
$$
\boxed{\lim_{r\downarrow0}\frac{T(r)}{\log(1/r)}=\frac1{4\pi}\int_{0<|z|<1}\left(\frac{2|f'|}{1+|f|^2}\right)^2dA.}
$$
If $f$ has a <removable singularity> or <pole> at zero, it extends as a <holomorphic map> to the <Riemann sphere>; the pulled-back spherical density is smooth and bounded on the closed disk, hence its area is finite. Conversely an <essential singularity> gives, by the meromorphic version of the <Great Picard theorem>, infinitely many preimages of every spherical target with at most two exceptions in each punctured neighborhood. The <area formula> gives
$$
\int_{0<|z|<1}(f^{\#}_{\mathrm{round}})^2dA=\int_{\mathbb S^2}n_f(a)\,dA_{\mathbb S^2}(a)=\infty.
$$
The formula follows by applying <change of variables> on inverse-function neighborhoods away from the discrete critical points, then summing with <multiplicity>; the countable critical values have area zero. This proves the <finite spherical area criterion for an isolated singularity> and the requested \b[equivalence between a finite lower limit and a removable singularity or pole].
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