Definition. One agentic post-training round [ftip-0073]
AGENTDRAFTED
At round \(n\), fix a task \(\mathsf T_n\), its law
\(\mu _{\mathsf T_n}\), and a versioned interactive realization
\(\mathsf {Env}^{\upsilon _n}_{\mathsf T_n}\). Let \(\mathcal I_{\rm art}\) and
\(\mathcal I_{\rm inf}\) be declared identifier spaces. Let
\(a_n\in \mathcal I_{\rm art}\) be an immutable content identifier resolving the
artifact \(M_n\), and let \(\iota _n\in \mathcal I_{\rm inf}\) record all serializer,
decoding, tool, and stopping settings. The policy--inference stamp
\(\nu _n=(a_n,\iota _n)\in \mathcal I_{\rm art}\times \mathcal I_{\rm inf}\)
resolves a non-anticipating policy \(\pi _n\). Let \(\Omega _n^{\rm task}\),
\(\Omega _n^{\rm roll}\), \(\Omega _n^{\rm feed}\),
\(\Omega _n^{\rm credit}\), and \(\Omega _n^{\rm update}\) be declared seed
spaces. The rollout seed contains both policy-decoding and environment
randomness. Write
\(\Omega _n=\Omega _n^{\rm task}\times \Omega _n^{\rm roll}\times
\Omega _n^{\rm feed}\times \Omega _n^{\rm credit}\times
\Omega _n^{\rm update}\), and let
\(\omega _n=(\omega _n^{\rm task},\omega _n^{\rm roll},
\omega _n^{\rm feed},\omega _n^{\rm credit},\omega _n^{\rm update})\) be sampled
from a declared joint law on \(\Omega _n\). Write \(\mathcal Z_{\mathsf T_n}\)
for the stopped trajectories admitted by the versioned realization.
For declared feedback and credit spaces \(\mathcal F_n\) and
\(\mathcal G_n\), decision space
\(\mathcal D_n=\{\mathsf {accept},\mathsf {reject}\}\), accounting dimension
\(m\geq 1\), and exact implementation versions \(v_n^{\rm S}\), \(v_n^{\rm R}\),
\(v_n^{\rm F}\), \(v_n^{\rm C}\), \(v_n^{\rm U}\), and \(v_n^{\rm A}\), the round
specifies maps
\[
\begin {aligned}
\mathsf S_n^{v_n^{\rm S}}&:
\Omega _n^{\rm task}\longrightarrow \mathcal X_{\mathsf T_n},\\
\mathsf R_n^{v_n^{\rm R}}&:
\mathcal X_{\mathsf T_n}\times \Omega _n^{\rm roll}
\longrightarrow \mathcal Z_{\mathsf T_n},\\
\mathsf F_n^{v_n^{\rm F}}&:
\mathcal X_{\mathsf T_n}\times \mathcal Z_{\mathsf T_n}
\times \Omega _n^{\rm feed}\longrightarrow \mathcal F_n,\\
\mathsf C_n^{v_n^{\rm C}}&:
\mathcal X_{\mathsf T_n}\times \mathcal Z_{\mathsf T_n}
\times \mathcal F_n\times \Omega _n^{\rm credit}
\longrightarrow \mathcal G_n,\\
\mathsf U_n^{v_n^{\rm U}}&:
\mathcal M_{\rm exec}\times \mathcal Z_{\rm opt}\times \mathcal G_n
\times \Omega _n^{\rm update}
\longrightarrow
\mathcal D_n\times
\mathcal M_{\rm exec}\times \mathcal Z_{\rm opt},\\
\mathsf A_n^{v_n^{\rm A}}&:
\mathcal M_{\rm exec}\times \mathcal Z_{\rm opt}
\times \mathcal X_{\mathsf T_n}\times \mathcal Z_{\mathsf T_n}\\
&\quad {}\times \mathcal F_n\times \mathcal G_n\times \mathcal D_n\\
&\quad {}\times \mathcal M_{\rm exec}\times \mathcal Z_{\rm opt}
\times \Omega _n\longrightarrow \mathbb R_+^m.
\end {aligned}
\]
The accounting map receives the starting state, proposed state, decision,
and realized seeds, so update work and rejected proposals remain observable in
the cost record.
They produce
\[
\begin {aligned}
X_n&=\mathsf S_n^{v_n^{\rm S}}(\omega _n^{\rm task}),\\
Z_n&=\mathsf R_n^{v_n^{\rm R}}(X_n,\omega _n^{\rm roll}),\\
F_n&=\mathsf F_n^{v_n^{\rm F}}(X_n,Z_n,\omega _n^{\rm feed}),\\
\Gamma _n&=\mathsf C_n^{v_n^{\rm C}}
(X_n,Z_n,F_n,\omega _n^{\rm credit}),\\
(d_n,\widetilde M_{n+1},\widetilde s_{n+1})
&=\mathsf U_n^{v_n^{\rm U}}
(M_n,s_n,\Gamma _n,\omega _n^{\rm update}),\\
c_n&=\mathsf A_n^{v_n^{\rm A}}(
M_n,s_n,X_n,Z_n,F_n,\Gamma _n,d_n,
\widetilde M_{n+1},\widetilde s_{n+1},\omega _n).
\end {aligned}
\]
The task sampler pushes its seed law forward to \(\mu _{\mathsf T_n}\).
Conditional on \(X_n\), the rollout sampler pushes its seed law forward to the
interaction law of Remark [ftip-0028] under
\(\mathsf {Env}^{\upsilon _n}_{\mathsf T_n}\) and \(\pi _n\).
If \(d_n=\mathsf {accept}\), set
\((M_{n+1},s_{n+1})=(\widetilde M_{n+1},\widetilde s_{n+1})\); if
\(d_n=\mathsf {reject}\), set \((M_{n+1},s_{n+1})=(M_n,s_n)\). The realized
cost \(c_n\) includes proposal work even after rejection. With
\(v_n^{\rm ops}=(v_n^{\rm S},v_n^{\rm R},v_n^{\rm F},v_n^{\rm C},
v_n^{\rm U},v_n^{\rm A})\), append
\[
\begin {aligned}
r_n={}&(\mathsf T_n,X_n,\nu _n,\upsilon _n,Z_n,F_n,\Gamma _n,d_n,
\widetilde M_{n+1},\widetilde s_{n+1},M_{n+1},s_{n+1},
\omega _n,v_n^{\rm ops},c_n),\\
\mathcal L_{n+1}={}&\operatorname {append}(\mathcal L_n,r_n).
\end {aligned}
\]
The record belongs to \(\mathcal R_{\rm ag}\) from Definition [ftip-004U] and
determines \(\Sigma _{n+1}\) without rewriting an earlier record.
