The Price of Reliable Rotary Retrieval: Task Geometry, Shared Spectra, and Minimal Adaptation

Distinct rotary encodings need not permit reliable retrieval at a reasonable query–key norm. We study this gap in a supplied-address, position-only soft-attention channel, measuring reliability by total off-target mass. Our central result determines the exact price of reusing a representation: relative to an $m$-block spectrum redesigned for each task, a spectrum frozen across all lengths and addresses needs exactly $m+1$ two-dimensional rotation blocks for constant-factor amplitude competitiveness. The obstruction is geometric. Task-specific boundary retrieval can exploit a linear slope, but a frozen spectrum efficient on interior tasks eventually surrounds the target in phase space and cancels that advantage. Stability imposes a separate constraint. For maximum displacement $L$, the maximal critical frequency-error radius is $1/(2L)$, with feasibility strictly below the threshold. No finite frozen spectrum retains this tolerance order at every length; one length-adaptive frequency restores it while preserving the optimal amplitude order. A finite convex calibration with a dimension-independent approximation factor turns these guarantees into checkable budget certificates. Standard-frequency comparisons and reported checkpoint interventions distinguish representation cost from phase and allocation failures. Together, the results characterize reliable rotary addressing under an explicit resource contract.