Toward a Two-Knob Projection for Supervisory Control of Language 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 $(a,\tau)$ 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 one-dimensional 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 $\tau$ monotonically with limited cross-sensitivity in $a$, while best-of-$N$ collapses $\tau$. 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.