The Price of a Blind Jump: How Rounds and Distributions Create Learning Paths

How do adaptive rounds and input distributions change the cost of learning hierarchical structure? We answer this question through the blind-jump width, the number of unknown coordinates a learner must locate together. For fixed-link product juntas with square-integrable likelihood ratios and full detectability, we identify the exact ambient exponent of $R$-batch full statistical-query (SQ) cost $q/{\tau }^2$ as the $R$-wave carrier-closure width ${\mathrm{BLeap}}_R$. Equal nonadaptive and sequential endpoints can nevertheless yield reversed difficulty rankings at intermediate rounds. A hidden-rung block staircase exhibits three regimes for the same support-recovery task. Under Rademacher inputs, depth $D$ and block width $b$ give exponent $b\lceil D/R\rceil$, with explicit binomial bounds for growing structural parameters. Bounded amplitude perturbations at dispersion $u=\Theta (1/(bD))$ create unary response-energy witnesses: $n$ nonadaptive full SQs recover the support, with an IID implementation using $O_D(b\log n)$ rows and $O(nN)$ arithmetic at fixed signal-to-noise ratio. Every fixed symmetric standardized nonconstant-modulus law instead requires exponentially many passive samples in $b$, at fixed depth and signal-to-noise ratio. These results quantify how rounds build useful carriers and how distribution design supplies them immediately. Lean 4 formalizations accompany the carrier calculus, lower-bound success estimates, and selected analytic identities.