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\title{\bf Governance Under Lag: A Geometric Framework for Recoverability}
\author{[Authors anonymized for submission]}

\newcommand{\x}{\mathbf{x}}
\newcommand{\xt}{\tilde{\mathbf{x}}}
\newcommand{\A}{\mathcal{A}}
\newcommand{\Rset}{\mathcal{R}}
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\newcommand{\Cset}{\mathcal{C}}
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\newcommand{\Lam}{\Lambda}
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\begin{document}
\maketitle
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\begin{abstract}
We extend the admissibility and recoverability stack to systems operating under temporal lag. Observation delay, decision delay, and actuation delay together create a window during which the system evolves without corrective intervention. We formalize lag-reachable sets, define robust and existential recoverability under delay, prove monotonic degradation with lag, and introduce a lag-adjusted Rigel number. The framework shows how delay transforms recoverability from a pointwise property into a set-based condition.
\end{abstract}
\section{Lag Decomposition}
Let total lag be
\[
\tau=\tau_{obs}+\tau_{dec}+\tau_{act}.
\]
\section{Lag-Reachable Sets}
Let the uncontrolled lag dynamics be
\[
\dot{\x}=f_0(\x,d),
\]
with bounded disturbance $d\in\mathcal{D}$.
The lag-reachable set from an initial state $\x_0$ is
\[
\mathcal{L}(\x_0,\tau)=\{\x(\tau): \dot{\x}=f_0(\x,d),\ \x(0)=\x_0,\ d\in\mathcal{D}\}.
\]
The safe lag-reachable set is
\[
\mathcal{L}_{safe}(\x_0,\tau)=\{\x(\tau)\in\mathcal{L}(\x_0,\tau): \x(t)\notin\Cset,\ \forall t\in[0,\tau]\}.
\]
\section{Time-Conditioned Recoverability}
Define robust lag recoverability by
\[
\x_0\in\Rrob(\tau)\iff \mathcal{L}_{safe}(\x_0,\tau)\subseteq \Rset(0).
\]
Define existential lag recoverability by
\[
\x_0\in\mathcal{R}_{ex}(\tau)\iff \mathcal{L}_{safe}(\x_0,\tau)\cap \Rset(0)\neq\emptyset.
\]
\section{Monotonic Degradation}
If $\tau_1<\tau_2$, then robust and existential recoverable regions shrink monotonically:
\[
\Rrob(\tau_2)\subseteq \Rrob(\tau_1),\qquad
\mathcal{R}_{ex}(\tau_2)\subseteq \mathcal{R}_{ex}(\tau_1).
\]
\section{Critical Lag}
For each state $\x_0$, define the critical lag
\[
\tau_{crit}(\x_0)=\sup\{\tau:\x_0\in\Rrob(\tau)\}.
\]
\section{Lag-Adjusted Rigel Number}
Define
\[
\Ri(\x_0;\tau)=
\begin{cases}
\inf_{\x_\tau\in\mathcal{L}_{safe}(\x_0,\tau)} \Ri(\xt_\tau), & \text{if }\mathcal{L}_{safe}(\x_0,\tau)\subseteq \Rset(0),\\
0, & \text{otherwise.}
\end{cases}
\]
This score is nonincreasing in $\tau$ and vanishes once lag exceeds the critical threshold.
\section{Illustrative Regimes}
\begin{table}[h]
\centering
\caption{Critical Lag (Illustrative)}
\begin{tabular}{lcccc}
\hline
System & $\tau_{typical}$ & $\tau_{crit}$ & Sensitivity & Regime \\
\hline
Drone & 0.8s & $\sim$0.6s & High & Sub-second critical \\
Human & 48hr & $\sim$72hr & Moderate & Multi-day ceiling \\
Consensus & 5min & $\sim$15min & Low & Wide tolerance \\
\hline
\end{tabular}
\end{table}
\section{Conclusion}
Lag transforms recoverability from a pointwise property into a set-based condition over reachable states. This paper provides the temporal bridge needed before commit-time enforcement can be stated meaningfully.
\end{document}
