When Do Rankings Beat Signs? Gradient Direction Recovery under Even Nuisance

Can ranking a batch of antipodal query pairs recover a gradient direction with fewer evaluations than using within-pair signs alone? We study fixed-center recovery in $d$ dimensions for a linear signal plus an unknown differentiable even nuisance, counting function evaluations and feedback rounds separately. Shared pair offsets create an information hierarchy: Mirror retains within-pair signs, Reverse couples reversed cross-pair orders to cancel offsets, and Full enforces all ranking inequalities. Relative shell variation is the nuisance range over the query sphere normalized by the linear-signal scale. When it is at most $c\epsilon /\sqrt{d}$, Reverse localizes every consistent direction to angular accuracy $\epsilon$ using $\tilde{O}(d+\sqrt{d/\epsilon })$ evaluations in one round at fixed confidence, without a variation bound supplied to the decoder. This matches a lower bound up to logarithmic factors, versus the $\Theta (d/\epsilon )$ one-round sign-only minimax cost on the same class. Over well-conditioned positive-definite quadratics with no common relative-curvature bound, the $R$-round minimax cost is instead $\Theta (dR({\epsilon }^{-1/R}-1))$, even with cumulative rankings. We extend validity beyond exact evenness under supplied error bounds. Matched-endpoint experiments certify information beyond signs and short cycles. On two ridge tasks under joint evaluation and round caps, short-cycle recovery attains lower terminal mean gaps than the tested batch-capable controls. An additional optimization benefit from fitting the complete ranking remains unestablished.