Definition. Agentic post-training run [ftip-0074]
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    An agentic post-training run fixes a finite round count \(N_{\rm ag}\geq 0\), an initial state \(\Sigma _0\), a fixed accounting dimension \(m\geq 1\), a componentwise budget \(b_{\rm ag}\in \mathbb R_+^m\), and the round specifications of Definition [ftip-0073]. Put \(\Omega _{\rm ag}=\prod _{n=0}^{N_{\rm ag}-1}\Omega _n\), using a singleton for the empty product. Once those specifications and their exact versions \(v_{\rm ag}\) are fixed, the run is the map

    \[ \mathsf {Run}_{\Sigma _0,v_{\rm ag}}:\Omega _{\rm ag}\longrightarrow \mathcal M_{\rm exec}\times \mathcal R_{\rm ag}^{N_{\rm ag}} \times \mathbb R_+^m. \]

    Write a realized round-seed tuple as

    \[ \omega _{\rm ag}=(\omega _0,\ldots ,\omega _{N_{\rm ag}-1}). \]

    The run recursively constructs \(\Sigma _1,\ldots ,\Sigma _{N_{\rm ag}}\); the seed tuple and state sequence are empty beyond \(\Sigma _0\) when \(N_{\rm ag}=0\). Its cumulative realized cost is \[ c_{\rm ag}=\sum _{n=0}^{N_{\rm ag}-1}c_n. \] The empty sum is \(0\in \mathbb R_+^m\) when \(N_{\rm ag}=0\). The map returns \((M_{N_{\rm ag}},\mathcal L_{N_{\rm ag}},c_{\rm ag})\). A run claiming the hard budget must declare a pre-admission or stopping rule that guarantees \(c_{\rm ag}\preceq b_{\rm ag}\).

    The run declaration includes the task sampler, environments, serializers, rollout inference settings, feedback and credit maps, update maps, stopping rule, seed laws, implementation versions, and resource account. The round records retain every realized seed, version, decision, and cost. Leaving one of these fields implicit defines a family of runs rather than one reproducible intervention.