When to Intervene: Predictive Memory in a Langton Highway

How much of a self-written environment must a predictive state retain when a simple observation is supplemented by a local intervention? We answer this question for one exact standard-highway trace of the classical Langton ant. Every source has finite black support and the same complete uncontrolled future. Nevertheless, predicting the first T turns for every possible time of a single current-cell toggle requires Θ(T²) source-dependent bits in the worst case. An explicit matrix of protected readout lanes gives the lower bound: the chosen intervention time selects one row, which is read autonomously before permanent standard-highway recovery. The construction retains the entire field, has word-independent mass and signed row/column charges, and admits exact finite-horizon response counts. More generally a menu of at most q ≤ T possible trigger times requires Θ(qT) bits, uniformly for large T. Arbitrary further toggles do not increase the quadratic order; their exact quotient is determined by the free colors in an alternating-axis causal region. A binary-entropy argument preserves the quadratic lower bound under a specified nonzero prediction error. The result distinguishes predictive storage from execution time and passive regularity from sufficiency under future interaction.