On Two-Knob Projection for Supervisory Control of Large Models (Toward a Theory of Alignment Ops)

We study whether diverse supervisory controls for large models admit a low-dimensional representation. We formalize a two-coordinate projection (a, τ) capturing (i) alignment style relative to a reference family and (ii) randomness versus idempotency, and define the Control Sufficiency Index (CSI) to quantify fidelity. We give simple structural results (two-dimensional sufficiency under regularity, a onedimensional impossibility for mixed greedy-sampling regimes, Lipschitz stability of service indicators, and preservation under non-expansive meta-supervision) and test the projection on decoding controls and best-of-N. On small models, temperature and top-p vary τ monotonically with limited cross-sensitivity in a, while best-of-N collapses τ. We view these as first steps toward a compact control surface; we do not claim universal sufficiency across tasks or models. We also propose theory of Alignment Ops in Appendix.