1% ---------------------------------------------------------------------------2% Section 2 — Framework (journal version; compact statement of the monograph's3% Section 3; definitions only, no empirical result; proofs in the monograph).4% ---------------------------------------------------------------------------5\section{A task-based framework for autonomous white-collar work}\label{sec:framework}67The framework is deliberately modest in theoretical ambition. Its constructs operationalise the task model of \citet{acemoglu2018race} and \citet{acemoglu2025macroeconomics} and the cost-effectiveness logic of \citet{svanberg2024costeffective} for a setting in which the ``technology'' is an autonomous agent whose output on a given task is a random variable. None of the constructs is new as an idea; each is a measurement convention whose precedents are listed in the monograph's positioning table, and the contribution lies in their joint, experimental, occupation-weighted estimation under randomised tool access with hidden-oracle evaluation. The purpose of the formalism is to fix, before any run, what is being estimated and under which assumptions it is identified.89\subsection{Occupations, tasks, runs and acceptance}\label{sec:fw-setup}1011Let occupation $o$ be represented by tasks $j=1,\dots,J_o$, each a complete professional assignment with a brief, input artefacts and required deliverables, and by weights $w_{oj}\ge 0$ with $\sum_j w_{oj}=1$ representing the economically justified importance of task $j$ within $o$; in the primary analysis the weights are equal and in the sensitivity analysis they are proportional to the professional-minutes estimate recorded in each scenario's metadata. An \emph{arm} $a\in\{A0,\dots,A4\}$ is a configuration of the computer tools available to the agent, holding the model, the brief, the inputs and the caps fixed. A \emph{run} is one execution of task $j$ under arm $a$, with repeat index $r$; it is a random experiment because the model samples tokens stochastically and the environment responds to the actions it chooses. A run produces the vector $Y_{ojr}(a)=(D,Q,E,C,T,H,X)$, where $D\in\{0,1\}$ is \emph{deterministic completion} (every required oracle criterion passes and no critical violation is recorded), $Q\in[0,100]$ is the mean rubric score of the blind judges, $E\in\{0,1\}$ is the \emph{critical-error indicator} (fabrication, modified inputs, compliance with an injected instruction, attempted external communication, or a scenario-specific violation listed in the oracle), $C\ge 0$ is the marginal API cost in US dollars as reported by the client including any recovery session, $T$ is wall-clock time right-censored at the session limit, $H\ge 0$ is the human intervention time in minutes---in this version, zero during the run plus a stated assumption of five minutes per question the agent records as unanswered---and $X$ collects process measurements (tool calls, errors, retries, application transitions, tokens).1213\begin{definition}[Acceptance]\label{def:acceptance}14For a quality threshold $q$, the acceptance indicator of a run is15\begin{equation}\label{eq:acceptance}16S_{ojr}(a;q)=D_{ojr}(a)\,\bigl(1-E_{ojr}(a)\bigr)\,\mathbf{1}\{Q_{ojr}(a)\ge q\},17\end{equation}18and the acceptance probability of task $j$ under arm $a$ is $p_{oj}(a;q)=\Pr\{S_{ojr}(a;q)=1\}$.19\end{definition}2021The three factors are not redundant. Deterministic completion certifies facts only the oracle knows; the quality gate certifies what a demanding professional would notice but a test cannot encode; the integrity factor certifies the absence of behaviours that make an otherwise acceptable deliverable unusable. The multitask logic of \citet{holmstrom1991multitask} is the reason for keeping them separate: an agent optimising against any one of them can pass it while failing the others, and Section~\ref{sec:results-heterogeneity} shows a scenario in which exactly this happens.2223\subsection{The Autonomous Labor Frontier}\label{sec:fw-alf}2425\begin{definition}[Autonomous Labor Frontier]\label{def:alf}26For occupation $o$, arm $a$, budget $B\ge 0$, human-time allowance $H\ge 0$ and quality threshold $q$,27\begin{equation}\label{eq:alf}28\mathrm{ALF}_o(B,H,q;a)=\sum_{j=1}^{J_o} w_{oj}\,\Pr\bigl\{S_{oj}(a;q)=1,\; C_{oj}(a)\le B,\; H_{oj}(a)\le H\bigr\}.29\end{equation}30\end{definition}3132The frontier is the task-weighted probability that a run of a task of the occupation produces an accepted deliverable \emph{and} does so within the budget \emph{and} within the supervision allowance. It is a joint probability, not a product of marginals, and it is indexed by three thresholds because the economic question has three margins---whether the work is good enough, cheap enough and autonomous enough---and a single number would hide the trade-offs among them. Three properties, proved in the monograph, govern its reading. The task-weighted marginal acceptance rate $\rho_o(q)=\sum_j w_{oj}p_{oj}(q)$ is the \emph{ceiling} of the frontier, approached as $B,H\to\infty$; the gap $\rho_o-\mathrm{ALF}_o$ is the share of accepted work that is infeasible at the stated budget or allowance. If the per-session budget cap is $\bar B$ and runs are evaluated as delivered when it binds, the frontier is flat for $B\ge\bar B$ and the experiment cannot inform that region. And for two weight vectors on the same tasks, the change in the frontier is bounded by their total-variation distance times the range of the task-level joint probabilities, which is why both weightings are reported. The joint probability decomposes, for each task, into deterministic completion, the quality gate conditional on completion, integrity, and budget feasibility conditional on acceptance; the decomposition is a diagnostic that tells us \emph{where} an arm loses acceptable work.3334\subsection{The Computer-Use Tool Premium}\label{sec:fw-premium}3536Each run has a potential outcome under every arm, and the design assigns every task to every arm, so the contrast is identified within task.3738\begin{assumption}[Exchangeable repeats, no interference, information constancy]\label{ass:main}39For each task and arm, the run vectors of the repeats are independent and identically distributed; a run's vector does not depend on the arm, order or node of any other run, nor on its own position in the execution order or executor node. The brief, inputs, deliverable specification, model, client version, caps and behavioural policy are identical across arms; arms differ only in the tool set, its documentation and, where applicable, the presence of the enterprise portal and of the self-repair protocol.40\end{assumption}4142\begin{definition}[Computer-Use Tool Premium]\label{def:premium}43For arms $a,a'$ and tasks $j$ with weights $w_j$, the tool premium is the task-weighted average treatment effect on acceptance, $\tau(a,a';q)=\sum_j w_j\,[p_j(a;q)-p_j(a';q)]$. The \emph{shell premium} is $\tau(A1,A0)$, the \emph{computer-use premium} $\tau(A3,A1)$, the \emph{macOS-layer premium} $\tau(A2,A1)$ and the \emph{self-repair premium} $\tau(A4,A3)$.44\end{definition}4546The estimator is the task-weighted mean of within-task differences of repeat means, $\hat\tau=\sum_j w_j[\bar S_j(a)-\bar S_j(a')]$. Under Assumption~\ref{ass:main} it is unbiased, and when tasks are an equal-weight sample of size $J$ from an occupation's task population its variance is47\begin{equation}\label{eq:premium-var-cluster}48\operatorname{Var}[\hat\tau]=\frac{1}{J}\Bigl[\operatorname{Var}_j(\Delta_j)+\mathbb{E}_j\Bigl(\frac{\sigma_j^2(a)+\sigma_j^2(a')}{R_j}\Bigr)\Bigr],49\end{equation}50with $\Delta_j=p_j(a)-p_j(a')$ and $\sigma_j^2(a)=p_j(a)\{1-p_j(a)\}$. Equation~\eqref{eq:premium-var-cluster} is the reason the primary inference clusters at the task: between-task heterogeneity in the effect is reduced only by adding tasks, while within-task stochastic variation is reduced by repeats \citep{miller2024errorbars,abadie2023clustering}. The premium is an ablation in the sense of \citet{xie2024osworld} and \citet{trivedi2024appworld}; what the design adds is that the ablation is randomised on the same tasks across occupations. Two caveats apply: the premium is the effect of a \emph{bundle}---tools plus documentation plus, for A3 and A4, the portal---and arm A2 is identified conditional on the nodes where the macOS automation layer answers.5152\subsection{Companion measures}\label{sec:fw-companions}5354Four companion measures complete the framework; each is defined briefly because the pilot identifies only some of them. The \emph{Workflow Dependency Frontier} is the function $d\mapsto p(a;d)$ relating acceptance to a scenario's designed dependency depth $d$, the minimal number of dependent operations a careful professional must perform. Under a compounding model in which $d$ steps each succeed with probability $\pi$ conditional on the previous ones, $p(d)=\pi^d$, the half-depth at which acceptance falls to one half is $d_{1/2}=\log 2/\log(1/\pi)$, heterogeneity of step reliability lowers task reliability at a given mean, and self-conditioning in the sense of \citet{sinha2025horizon} makes $-\log p(d)$ convex; the model is a benchmark shape against which an estimated frontier is read, not a structural claim \citep{dziri2023faith,kwa2025horizon,yao2024taubench}.5556The \emph{Autonomous Recovery Index} is defined on arm A4. Let $F_1$ be the event that the first session ends without passing the public self-check and $P_2$ the event that the deliverables pass it after the bounded recovery; then $\mathrm{ARI}=\Pr\{P_2\mid F_1\}$. If a run that would fail under A3 fails the self-check with probability $\delta$ (detection) and a resumed session converts a detected failure into an accepted deliverable with probability $\gamma$ (correction), and the first session of A4 is distributed as a run of A3, then $p(A4)=p(A3)+(1-p(A3))\,\delta\gamma$: the self-repair premium is proportional to the first-session failure rate the self-check can see, and is zero by construction when that rate is zero. The index measures correction against an external signal, which \citet{shinn2023reflexion} and \citet{chen2023selfdebug} find effective, not intrinsic self-correction, which \citet{huang2023selfcorrect} find ineffective; by the imperfect-verifier result of \citet{stroebl2024flaws}, passing the self-check is not acceptance.5758\emph{Autonomy-Adjusted Productivity} is the expected number of accepted deliverables per unit of fully loaded cost, $\mathrm{AAP}(a;q,\omega,\kappa)=\sum_j w_j\mathbb{E}[S_j]/\sum_j w_j\mathbb{E}[C_j+\omega H_j+\kappa]$, where $\omega$ is a wage per human minute and $\kappa$ an operating cost per run reported separately from the API charge; its reciprocal is the cost per accepted deliverable, computed over all runs so that an arm pays for its failures. The \emph{Professional Reliability Frontier} treats the end of a run as one of four competing terminal events---accepted, critical error, censored at a cap, or finished without acceptance---and maps $(q,B)$ to the triple (probability of accepted work within budget, critical-error rate, censoring rate), so that an arm that raises acceptance while raising the critical-error rate is not read as an improvement \citep{fine1999competing,dellacqua2023jagged}.5960\subsection{Delegation with verification}\label{sec:fw-pa}6162The constructs enter the decision of a principal who must decide whether to delegate a task to an agent. A task yields value $v$ when an acceptable deliverable is used; the principal can produce it in-house at cost $c_H$ or delegate it at marginal cost $c_A$, in which case the deliverable is acceptable with probability $p$ and contains a critical error with probability $e$; an unacceptable deliverable that is used causes a loss $\ell$, an undetected critical error a loss $L\ge\ell$; verification at cost $\kappa_V$ detects failures and completes the task in-house.6364\begin{proposition}[Delegation thresholds]\label{prop:delegation}65Delegation with verification yields a higher expected payoff than in-house production if and only if $p\ge(c_A+\kappa_V)/c_H$; verification is worth its cost relative to delegating without it if and only if $\kappa_V\le(1-p)(\ell-c_H)+eL$; and as $L\to\infty$ verification becomes a necessary complement of delegation for any $e>0$.66\end{proposition}6768The first inequality is the frontier's budget condition in decision form; the second formalises the claim of \citet{catalini2026agi} and \citet{sreenivas2026contracting} that, when execution is cheap, the delegation decision is governed by how cheaply a human can check the work. It is also why $E$ is measured separately from acceptance: $e$ and $L$ enter the verification decision through a term that acceptance alone does not capture. The frontier is not an estimate of job displacement, a task is not an occupation, the scenarios are synthetic, and the estimates are for one family of models and one client version on one date; the monograph states each of these restrictions formally and the mandate of the study makes them rules.69