Prompt learning adapts CLIP to downstream recognition by replacing hand-written templates with learned continuous context vectors, which in Context Optimization (CoOp) form a dense prompt matrix
P∈Rm×d trained from only a few examples per class. We study whether this matrix is over-parameterized by factorizing it as
P=BA, which cuts the trainable prompt parameters from
md to
r(m+d), and to
rd once the token-side factor
B is fixed. Across seven few-shot benchmarks and two CLIP backbones, low-rank prompts match or improve dense CoOp at far fewer parameters, with the clearest gains on low-shot base-to-new generalization. We then find that the token-side factor need not be learned at all: fixing
B to a Gaussian, orthogonal, SVD-derived, or even random basis and training only the embedding-side factor
A stays on par with the fully trainable factorization, and a source-trained
B offers no advantage over a random one. A prompt-factor asymmetry and a local update-space dimension gap show why fixing
B is far less restrictive than fixing
A, and a smoothness-only guarantee certifies that optimizing
A over a fixed
B converges. In the CLIP prompt setting, the embedding-side coefficients carry the adaptation while the token basis can simply be fixed.