The Closure Criterion: When Does the Induced Moduli Evolution Become an Intrinsic Dynamical System?

P77 produced the induced moduli trajectory t ↦ [ρt] and its velocity decomposition d[ρt]/dt
= DΠ ∘ DH ∘ DS (dκt/dt) — an extrinsic velocity, pushed forward from the P12 coupling
dynamics. The natural next question is whether this motion is governed by an intrinsic law
on the constitution space, a vector field d[ρt]/dt = X([ρt]) depending on position alone — the
object every classical moduli-dynamics programme assumes. We do not assume it.
Following the method of P71 (analyticity derived, not postulated), P73 (the principal bundle
reached, not assumed) and P77 (the flow induced, not postulated), this paper
characterises the condition under which an intrinsic vector field exists, rather than
constructing one. The central result is the Closure Criterion (Theorem 1): the induced
evolution can be written as an autonomous intrinsic vector field X([ρ]) if and only if the
coupling admits a closure map — a rule dκt/dt = g([ρt]) expressing the coupling velocity
as a function of the present constitution alone. The criterion is sharp and two-sided.
Positive side: under closure, KA acquires a genuine dynamical system on X(Sig, G), and
the language of attractors, basins, invariant sets, and Lyapunov functions becomes
available (the prerequisite for the inevitable-collapse basin of P76–P77). Negative side:
without closure — the generic case, where the coupling is driven externally (an adversary,
an open environment) — no intrinsic vector field exists on the constitution space; there is
only the externally driven trajectory of P77. This is the honest separation many frameworks
skip: they silently write X([ρ]) and proceed. P78 shows that in the generic KA setting such an
X need not exist at all. The criterion sets the agenda for KA 11.0: P79 asks which system
classes produce closure; P80 builds the dynamics where they do. KA 11.0 thereby acquires a
clean kinematics / dynamics divide: P77 gave the kinematics; intrinsic dynamics exist
exactly under the closure condition this paper identifies.