Existing continual-learning methods protect parameters, replayed examples, or Euclidean feature subspaces. When applied to hyperbolic multimodal models, they do not explicitly preserve the Lorentz geometry that jointly encodes within-modality similarity, cross-modal correspondence, and semantic hierarchy; sequential updates can therefore retain task scores while still distorting previously learned relations. We address this gap with Hyperbolic Multimodal Continual Learning (HMCL). We show that preserving the old multimodal geometry amounts to restricting all modalities to one shared hyperbolic isometry, which induces a family of admissible first-order parameter changes. We formulate a joint closest-admissible (CA) correction that retains the shared rotation best matching the candidate modal updates; its minimal-rotation (MR) special case fixes this rotation to zero. Both variants correct the displacement realized by AdamW, and task anchoring bounds within-task accumulation while preserving learning freedom. Across a unified 16-task classification-retrieval stream with three hyperbolic backbones, HMCL improves final performance and backward transfer over sequential fine-tuning and four continual-learning baselines; HMCL-CA gives the highest Overall score on every backbone. A modality-extended stream confirms the retrieval gains. Representation analyses find 81.2 to 95.5 percent less radial, angular, cross-modal, and paired-distance drift; ImageNet-WordNet results show better semantic ancestry and radial hierarchy.

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