= Central contribution to kurtosis
{title2=$C_h=\mathbb E[Z^4\mathbf1_{\{|Z|\leq h\}}]$}
For a <standard score> $Z$ and $h>0$, split <kurtosis> into the contribution $C_h$ from within $h$ <standard deviations> of the <mean> and the contribution $T_h$ from outside:
$$
\kappa=C_h+T_h,\qquad C_h=\mathbb E[Z^4\mathbf1_{\{|Z|\leq h\}}],\qquad 0\leq C_h\leq h^4.
$$
The <indicator function> selects the central outcomes. The bound follows directly from $Z^4\leq h^4$ there. Thus $C_h/\kappa\leq h^4/\kappa$: when <kurtosis> is large, any fixed central range contributes a small fraction. This is a precise reason to emphasize extreme <standard scores> rather than peak height.<Kurtosis as Peakedness, 1905 – 2014. R.I.P. by Peter H. Westfall>[§5, Why Tails Mostly Determine Kurtosis, inequality (1) and following fixed-h argument]{ref}
For the <rare-outlier three-point distribution> with $q=1/100$, $C_1=0$ and $T_1=100$. At the other endpoint $q=1$, all <probability mass> lies at $Z=\pm1$, so $C_1=\kappa=1$: central dominance is possible when <kurtosis> is small. More generally, the <Cauchy-Schwarz inequality> gives $\kappa=\mathbb E[Z^4]\geq(\mathbb E[Z^2])^2=1$; the equally weighted outcomes $\pm1$ attain this minimum.
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