Feeding the Void: Co-evolution of the SIDM-seeded Black Hole and the dark halo after core-collapse

While the gravothermal collapse of self-interacting dark matter (SIDM) halos provides a compelling mechanism for seeding supermassive black holes (BHs), the post-collapse macroscopic co-evolution of the BH and its host halo remains largely unexplored. In this paper, we introduce a novel N-body framework to dynamically model the dark matter accretion onto the central BH. By bracketing the accretion rate with two complementary phenomenological schemes---a `multi-round core-collapse' scenario and a `continuous accretion' scenario---we establish bounds on the BH mass growth trajectory. We find that the BH mass reaches 5%-12% of the total halo mass within 2 Gyr post-collapse. Concurrently, this sustained mass depletion drives a `gravothermal supply chain' that forces outer dark matter inward, severely depleting the inner halo density (2r s). This unique structural imprint qualitatively matches the recently detected anomaly in the strong lens JVAS B1938+666 (detection V), where the best-fit lens model consists of an unresolved central point mass and a shallow, uniform-surface-density extended disk, offering a distinct observational signature for SIDM-seeded BHs.