The Cost of Eigenspace Restriction in Shared Attention Control

Soft prompts and prefixes control a frozen Transformer through learned records: a record's key decides when attention retrieves it, and its value decides what it returns. Confining records to a low-dimensional subspace cuts their parameters, and the value map's highest-gain eigenspace is the natural spectral choice. We show that this choice can require exponentially more records. In a two-head scalar attention interface with fixed projections and readout, we count the equal-weight records needed to approximate a growing-amplitude target m tanh q, equivalently a fixed target under growing attention gain. At every fixed relative tolerance we determine the optimal exponential count rate exactly, both for unrestricted records and for the eigenspace restriction, and the restricted rate is strictly larger below an explicit tolerance. The converse is an entropy-area bound: equal-weight records pay for how far attention moves from uniform. Because the restriction ties every value to its key, attention must keep moving across the query interval, whereas unrestricted records can saturate retrieval early and let a palette of values supply the rest; counted constructions attain both rates. The loss persists for thin full-dimensional strips around the eigenspace, nearby fixed-key value maps and, under a disk-margin condition, an RMSNorm realization, and computer-certified explicit libraries exhibit it at moderate amplitude in four parameter settings. In frozen RoBERTa-base, one unrestricted shared key/value record gives lower SST-2 held-out loss than up to 256 subspace-confined records, and the strongest confined arms stay above it with a fivefold training budget; on AG News, at the same block, the differences are at most 0.021 nats. In the scalar interface, finite optimization favors unrestricted records at every tested count under uniform queries, with reversals under some query distributions.