Rank Envelopes and Finite-Sample Geometry in Split Conformal Prediction

Split conformal prediction (SCP) provides finite-sample, distribution-free marginal coverage through discrete calibration ranks. In small-calibration, high-coverage regimes, however, the rank grid may be too coarse to represent the desired coverage level, causing SCP to become conservative or vacuous. We develop rank-indexed quantile envelopes that characterize this finite-sample resolution through order-statistic geometry. The envelopes partition the rank space into forced-in, ambiguous, and forced-out regions and yield explicit finite-sample coverage bounds for prediction rules constrained by this geometry. The resulting construction separates rank decisions that are fixed by finite-sample calibration from those that remain unresolved and can therefore be handled by different efficiency mechanisms. When randomization is acceptable, envelope mixing uses the unresolved region to attain exact marginal target coverage whenever the target lies within the attainable coverage range. When historical, prior, or model-based information about the score distribution is available, reference-guided rules use this information only within the unresolved region while retaining explicit distribution-free coverage control. The same construction extends naturally to exchangeable strata, providing stratum-wise finite-sample guarantees when effective calibration sizes are small because coverage is required separately across classes or subgroups. The framework also clarifies the limits of deterministic improvement under finite rank resolution and recovers standard SCP as the ambiguity region vanishes. Experiments in regression and classification illustrate how randomization, auxiliary information, and stratification can improve informativeness in small-calibration regimes while making their corresponding coverage guarantees explicit.