Iterative algorithms are integral to many statistical procedures, whereas sampling uncertainty along their iteration is generally not quantified. We develop simultaneous inference for discrete trajectories generated from high-dimensional estimating equations, with iteration-specific population estimands that preserve algorithmic regularization. Our proposed fixed-Jacobian bootstrap propagates perturbations along the observed trajectory without repeatedly running the original nonlinear algorithm. We establish uniform first-order expansions and bootstrap validity over possibly infinite sets of contrasts and iterations under stability, entropy and moment conditions. Simulations in sparse linear regression, paired Fisher discrimination, and temporal-difference learning demonstrate accurate simultaneous coverage. A micro-randomized trial application illustrates inference for a treatment trajectory with participant-level clustering.

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