Beyond Endpoint Sufficiency: Inner-Time Trajectory as a Representation Channel in Iterative Systems

Many iterative models treat the final converged state as the representation. We study endpoint insufficiency: convergence can erase task distinctions, and the trajectory can carry reliability information even when the endpoint remains the answer. We formalize representations as $\mathrm{Rep}(x)=(z_T(x),r(z_{0:T}(x)))$, give a contractive construction with coincident endpoints and a label-informative signed-area functional, and introduce audits with strong endpoint, residualization, permutation, and orientation controls. Exploratory KITTI reliability audits use frozen RAFT-Things and SEA-RAFT checkpoints not trained on KITTI. At $80\%$ retained coverage, passive history reduces retained flow error by $8.1\%$ beyond a learned endpoint-plus-convergence baseline on RAFT-Things; simple convergence reduces it by $4.8\%$ beyond endpoint plus native uncertainty on SEA-RAFT. Richer geometry adds no resolved utility beyond convergence profiles, and the KITTI-finetuned RAFT checkpoint supplies a negative boundary. Controlled dynamics, looped Transformers, diffusion features, and a residual-null visual case expose distinct mechanisms and limits. These results support endpoint-sufficiency auditing, not universal trajectory access: retain history when it improves the declared target beyond strong inexpensive controls.