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, but
societies operate within a time-dependent operational subset: the set of actions that can be
reliably measured, prepared, controlled, scaled, and legally/physically executed. We formalize
a “degrees-of-freedom knob” as a time-varying admissible-control constraint U (t) acting on a
socio-technical dynamical system, thereby shrinking (or expanding) the reachable set in state
space 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 grows
exponentially. We propose (i) a rigorous operational-DOF definition via reachability volume and
Gramian surrogates; (ii) a data-anchored frontier capability proxy C(t) using three observable
channels: 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-Gating
Protocol (ACGP) that implements the knob in software through capability-conditioned disclo-
sure and capability-conditioned tool access, structured as policy decision/enforcement points
with auditable, multi-party-controlled overrides. Importantly, the protocol does not attempt to
suppress 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 maintain
a stable viability envelope.