The Cost of Recovering Information: Sharp Exactification and Optimal Retention of Gaussian Experiments

Approximate statistical sufficiency bounds decision loss, but exact reconstruction may require additional observations. This paper determines sharp recovery and retention laws for finite labeled binary Gaussian experiments under sensitivity-one scalar Gaussian acquisition, a deterministic finite query cap, and a precision budget holding on every execution. With fixed positive mass at the greatest source precision, the worst exact-recovery cost from reverse deficiency eta has order log⁻²(1/eta), uniformly in mixture size. Optimal retention with at most M Gaussian components has cost exp[-Theta(sqrt(M))]. The paper also distinguishes finite-bit from component retention, gives growing-memory laws, and establishes a sharp common-record alphabet law for two fixed future acquisition budgets. Theorems pair constructions with converse bounds under their stated assumptions. The observation model permits ideal real Gaussian samples and measurable decoders; practical efficiency is a separate question. The accompanying supplement contains proof details, reproducible rational decoder certificates, and Lean sources with an explicit coverage table. The manuscript is partially Lean-verified; some printed constants and extensions remain outside the formalized statements. AI assistance is disclosed in the manuscript. This is a preprint, not a claim of external peer review.