Universality Profiles: How Much Task Coverage Can a Fixed Model Sustain?

We study *sustainable breadth*: for a *fixed* algorithm and a finite budget, how broad is the family of tasks on which it can succeed while remaining fast? For a frozen foundation model, an in-context learner, or a multi-task procedure, this is the question of how much coverage a single fixed model can sustain at a target error, a quantity observed to saturate in practice but left unquantified by existing theory. Rather than fixing a task class and optimizing over algorithms as in minimax theory, we fix the algorithm and quantify the complexity of the subset it can cover. We formalize this through the *universality profile*, the packing complexity of the algorithm's success set at a target accuracy or regret level. Our main converse shows that success on $\exp (B)$ distinguishable tasks requires the transcript to reveal at least $\Omega (B)$ information about the underlying task; once this information is upper bounded, one obtains a quantitative breadth–convergence frontier. In online prediction with expert advice, the classical minimax law $\Theta (\sqrt{T\log N})$ becomes an explicit universality law with budget $\log N$, and varying only the prior support of Hedge yields different algorithmic breadth on the same ambient class. In a packed $K$-armed bandit identification family with $\exp (\Theta (K))$ environments, the framework yields a sharp lower bound $T\gtrsim K/{\Delta }^2$ together with a matching, up to logarithmic factors, uniform-sampling upper bound. A local linear-bandit analysis turns the same proof skeleton into a two-sided structured frontier, with a lower bound that holds for every algorithm and a least-squares achievability guarantee at the scale set by full-parameter estimation. On a fixed expert class, we moreover characterize the success set of a given algorithm explicitly, showing that the profile ranges over a whole spectrum of values as the algorithm varies while the ambient minimax hardness stays put. A finite-feature kernel witness shows that capacity bounds breadth in a nonparametric setting, and real-data checks on digit classification make the same breadth–budget law visible outside simulation: saturation at the class packing, a phase-transition knee, and coverage bounded by the capacity of a frozen representation. The point is not to replace minimax hardness, but to add an algorithm-centric axis that measures the task coverage achieved by a particular procedure.