Why it is a question
THEORY-248, narrowed on 2026-10-19, holds that the expected generalisation error is at most √(2σ²R_E(ε)/n) + ε, with R_E(ε) the least information about the sample carried by any surrogate output within ε in expected generalisation error. For an L-Lipschitz loss, R_E(ε) is at most the rate-distortion function of the output’s law at distortion ε/(2L), which can be finite where I(S;W) is infinite. Until that date, THEORY-248’s promote_when asked for the bound to be evaluated for such an algorithm and found non-trivial. At the owner’s direction, from the promotion audit’s batch 019 (ADR-047), the THEORY was narrowed and promoted on the proved bound, and the condition moved here unchanged.
It is open because the bound depends on the unknown data distribution, through P_W, and LIT-236 evaluates it only for a Gaussian mean, with a units error (NOTE-780). Finite is not the same as non-vacuous.
What would answer it
The condition formerly on THEORY-248, unchanged. A bound of this form, R_E(ε) or the rate-distortion function of the output’s law, is evaluated for an algorithm whose I(S;W) is infinite, such as deterministic training of a network with continuous weights, and gives a non-trivial value. The distortion must be measured in the units the Lipschitz condition uses. Correlations between estimated compressibility and test error across training settings cannot settle it, since they do not show that the bound holds or is tight.