For approval-based committee elections, the sizewise least-core value $Γ(P)$ minimizes the maximum excess support of a deviation over size-$k$ committees. Deviations are nonempty and contain at most $k$ candidates, with support measured relative to their Hare quota. A negative value certifies core nonemptiness, while $-Γ(P)$ is the largest worst-case quota shortfall secured by any committee. We quantify this margin on candidate-disjoint approval chains, a domain whose qualitative core nonemptiness is already known. We first factor each committee's deviation score over disjoint components. For chains, a closed local table yields the exact value and an attaining committee in polynomial time for rational weights. For nonnegative normalized real weights, let $q_{\mathrm{act}}\ge1$ count the connected components of the positive-support incidence graph. Then \[ Γ(P)\le-\frac{1}{k\min\{q_{\mathrm{act}},k+1\}}. \] Explicit extremal profiles show that this dependence on the active-component count is sharp. The margin is $1$-Lipschitz in total variation on the full ballot space. Each negative margin therefore certifies a neighborhood with nonempty core, even when perturbations leave the chain domain.

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