Uniform Decay, Heterogeneous Causes: Long-Horizon Failure in Long-Context Agents

Context windows now reach a million tokens, but the horizon an agent can sustain has not grown in proportion, and a single accuracy number cannot say why. We measure the question with a task whose horizon is set analytically and whose every step is individually adjudicable: the two-agent $\log(N)$-Questions game of \citet{potash2019}, ported to frontier models. A questioner sees $N$ Wikipedia lead paragraphs, up to 126{,}000 tokens, and must identify a secretly chosen target in exactly $\log_2 N$ yes/no questions put to an answerer who sees only the target. Running both roles on one model removes any capability gap, and the context is an aggregation rather than a retrieval probe: a well-balanced question requires estimating what \emph{fraction} of all $N$ documents carries a property, so no sparse subset of the input suffices. Across 408 games at $N{=}4$ to 1024, win rate declines with horizon length at $r=-0.973$ and is captured by a single per-round reliability parameter, $\mathrm{win}=p^{\log_2 N}$ with $p=0.928$. That form presumes each round fails independently, which we test rather than assume: the per-round failure rate is flat across the horizon ($\chi^2=10.5$, $\mathrm{df}=9$, $p=0.31$), and a failure at one round does not raise the chance of one at the next. The causes beneath that uniform decay are not uniform. Adjudicating every game against three judges (2{,}931 judgements each) and attributing each loss to its first failure splits them between answer errors and discrimination failures in roughly equal measure, 47--55\% and 42--49\% depending on the judge and stable across horizon lengths. Neither cause is explained by capacity or deliberation: the partition deficit is as large at $N{=}8$ as at $N{=}1024$, and the reasoning trace grows from 5.7k to 11.4k tokens per call between the first round and the ninth while the candidate set shrinks 500-fold, buying no measurable reliability. Two per-step failures compound identically, and it is the compounding, not either mechanism, that bounds the horizon.