The Degrees-of-Freedom Knob: A Control-Theoretic, Data-Anchored Proxy Model of Operational Capability Under Constraint Regimes (1656–2026)

Fundamental physical laws fix the underlying dynamical degrees of freedom of Nature, butsocieties operate within a time-dependent operational subset: the set of actions that can bereliably measured, prepared, controlled, scaled, and legally/physically executed. We formalizea “degrees-of-freedom knob” as a time-varying admissible-control constraint U (t) acting on asocio-technical dynamical system, thereby shrinking (or expanding) the reachable set in statespace without changing the plant physics. We focus on hazard-class operational reachability(high-externality actuators) and show that a measurable constraint surrogate can reduce real-ized operational degrees of freedom even while the general scientific capability frontier growsexponentially. We propose (i) a rigorous operational-DOF definition via reachability volume andGramian surrogates; (ii) a data-anchored frontier capability proxy C(t) using three observablechannels: time/frequency metrology, compute density, and societal energy throughput; and (iii)a measurable constraint surrogate S(t) defined as a normalized Restricted-Actuator Index (RAI)derived from in-force multilateral instruments that ban or gate specific experimental modali-ties and weaponization lifecycle primitives. Finally, we specify an AI/AGI Capability-GatingProtocol (ACGP) that implements the knob in software through capability-conditioned disclo-sure and capability-conditioned tool access, structured as policy decision/enforcement pointswith auditable, multi-party-controlled overrides. Importantly, the protocol does not attempt tosuppress derivable science; it gates execution accelerants and enforces anti-capture safeguards(auditability, multi-party control, periodic recalibration). The resulting framework is falsifiable:it predicts that increases in metrology and automation must be matched by either (a) payoff-reduction hardening (e.g., cryptographic migration) or (b) tighter actuator gating to maintaina stable viability envelope.