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Direct adaptation to streaming scenarios often leads to catastrophic inference performance degradation due to the severe mismatch between training and streaming inference. To bridge this gap, we present the first autoregressive (AR) models tailored for streaming TSE. Our approach introduces a Chunk-wise Interleaved Splicing Paradigm that ensures highly efficient and stable streaming inference. To ensure the coherence between the extracted speech segments, we design a historical context refinement mechanism that mitigates boundary discontinuities by leveraging historical information. 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