= Solution
\b[Specify what “onset” means.] I would distinguish the first local, perhaps short-lived <subduction> event from persistent, self-sustaining recycling in a connected global network of <tectonic plates>. Modern <plate tectonics> combines relatively rigid moving <lithosphere>, localized boundary deformation, <seafloor spreading> and sustained convergence. A sinking drip or a single imposed descending sheet is not sufficient evidence of that global regime. Different processes may become established at different times, so a transition interval may be a better hypothesis than an instantaneous worldwide switch.
\b[Construct competing physical models.] Solve conservation of <mass>, <momentum> and <energy> for evolving <mantle convection> and <lithosphere>, including <temperature>- and stress-dependent <rheology>, brittle yielding, <plastic deformation>, melting, compositional <buoyancy> and relevant phase changes. Use <geodynamic modelling> to compare a <stagnant lid>, episodic overturn, drip-dominated recycling and a sustained <mobile lid>. In a simplified incompressible creeping-flow framework,
$$
\nabla\cdot\mathbf u=0,\qquad
-\nabla p+\nabla\cdot(2\eta\dot{\boldsymbol\epsilon})+\rho\mathbf g=0,
$$
$$
\rho c_p\left(\frac{\partial T}{\partial t}+\mathbf u\cdot\nabla T\right)
=\nabla\cdot(k\nabla T)+\rho H+\Phi.
$$
Here $\mathbf u$ is <velocity>, $p$ <pressure>, $\eta$ effective <viscosity>, $\dot{\boldsymbol\epsilon}$ the <strain-rate tensor>, $\rho$ <density>, $\mathbf g$ <gravitational acceleration>, $c_p$ <specific heat capacity>, $k$ <thermal conductivity>, $H$ radiogenic power per unit <mass>, and $\Phi$ dissipative heating per unit <volume>. These equations are a framework: compressibility, latent heat and chemical transport may require additional terms. A simple <yield stress> law $\tau_y=C+\mu(\rho gz-P_f)$ represents cohesion $C$, friction coefficient $\mu$ and <pore pressure> $P_f$, illustrating why hydration and interface weakening can change the outcome.
\b[Vary the early-planet controls rather than assuming the answer.] Explore plausible initial mantle <potential temperature>, <radiogenic heating>, <crust> thickness and composition, <water> inventory, grain-size evolution, surface <temperature>, and boundary strength. Hotter <mantle> lowers <viscosity> but can produce thicker buoyant <basaltic crust>; it does not automatically make sustained <subduction> easier. A cold strong plate can transmit <slab pull> yet resist bending and initiation. <Water weakening of rocks>, melt-induced weakening, inherited faults and mechanically weak <sediments> can change this balance. Test proposed triggers such as <mantle plume> impingement or gravitational instability without presupposing an established global plate-force system.
Useful diagnostics include the <Rayleigh number>,
$$
\mathrm{Ra}=\frac{\rho g\alpha\Delta T D^3}{\eta\kappa},
$$
with <thermal expansion coefficient> $\alpha$, layer thickness $D$, <thermal diffusivity> $\kappa$ and <temperature> contrast $\Delta T$, together with surface mobility, the localization of deformation, plate longevity, melt production and material recycling. A large <Rayleigh number> establishes vigorous <convection> under suitable conditions, not necessarily a <mobile lid>. A convincing model should sustain one-sided <subduction>, connected boundary evolution and <seafloor spreading> after any artificial trigger is removed. Compare initiation thresholds with continuation thresholds: <hysteresis> can allow an established regime to persist under conditions in which it would not spontaneously start.
As an example of the type of experiment, three-dimensional models have tested <mantle plume>-induced <subduction initiation>, finding sensitivity to plate <buoyancy>, magmatic weakening and hydration of the interface. The scientific use of such results here is to identify a feasible mechanism and its predicted signatures, not to adopt their success as a direct measurement of an ancient onset date.<Plate tectonics on the Earth triggered by plume-induced subduction initiation by Taras V. Gerya and others>[2015, modeled initiation conditions and alternative dripping behavior]{ref}
\b[Connect a model regime to a date through independent evidence.] Couple the experiments to candidate planetary cooling histories, then predict observable <pressure-temperature-time paths>, magmatic compositions, deformation patterns and recycling rates for each regime. If independently dated rocks constrain a thermal or compositional state, ask which regimes can reproduce that state and the observed geological associations. Repeat over uncertain initial conditions and constitutive parameters, check numerical resolution, and compare two- and three-dimensional results where geometry matters. Models that prescribe a descending <slab>, fix plate <velocity>, or insert a weak boundary may be excellent forward models of a working <subduction zone> but do not demonstrate spontaneous initiation.
The result is a conditional statement: persistent <plate tectonics> is physically feasible under a range of conditions, and dated geological observations may favor that range at a particular time. Uncertain cooling histories, several feasible regimes and uncertain <rheology> prevent numerical feasibility alone from dating onset. The strongest constraint comes when the same parameter range explains multiple independently dated observations and alternative regimes fail those tests.
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