We argue, with systematic empirical evidence, that a large language model's political ideology is not a fixed point, but a conditional distribution $\mathbb{P}($position$\mid$context$)$ over a real political space. We evaluate nine current LLMs using a unified measurement framework anchored by VAA-CHES projection models, which map responses onto three validated dimensions (lrgen, lrecon, galtan) across six contextual axes. Our findings reveal high sensitivity to context: persuasive framing and under-represented languages displace coordinates by up to 0.57 and 0.52 units, respectively, while chain-of-thought reasoning often amplifies rather than dampens paraphrase instability. Despite this local plasticity, the model cohort occupies a remarkably narrow Overton envelope overall, occupying roughly one-third the spread of major European parties. Supported by a multi-trait multi-method (MTMM) analysis, we conclude that a single point cannot summarize LLM political behavior; it must be characterized as a shape. Our code and data are publicly available at https://github.com/sakhadib/LLM-Ideoplasticity.

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