A causal video generator produces two key--value (KV) states for every segment: the KV it uses while denoising that segment, and the KV it writes into the cache for later segments to read. In a pretrained generator the two already differ substantially, later segments attend to them differently, and replacing the written KV with the generation KV degrades the continuation. We propose dKV Forcing (dKV denotes differentiable KV) to learn a dedicated memory representation from its future use: a writer-specific low-rank residual on the shared backbone separates memory writing from generation and is trained through the future reads of what it writes. Learning a write from its future use needs gradients through the cache, which the mutable cache in common use cannot provide. dKV Forcing replaces the training-time cache by persistent differentiable pages with local rematerialization and, in one temporal traversal, computes exactly the gradient that a two-pass reconstruction of the same objective computes. One traversal computes the full memory gradient at $7\%$ generator-update overhead over detached-write training, with $28\%$ less resident memory and $20.8\%$ shorter training cycles than the two-pass reconstruction. Over Self Gradient Forcing (SGF), the same objective trained with the same traversal, writer adaptation raises VBench-Long aesthetic quality by $6.6$ points at 60\,s and $8.5$ points at 240\,s with the highest subject and background consistency, and rollout dropout of the residual keeps most of this gain while restoring the motion of training without write gradients.

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