Minimal Oversight: Uncertainty-Aware Governance for Delegated AI Systems

# Minimal Oversight: Uncertainty-Aware Governance for Delegated AI Systems
Source: [https://arxiv.org/html/2606.15563](https://arxiv.org/html/2606.15563)
###### Abstract

AI systems increasingly delegate decisions to specialized models, evaluators, tools, and supervisory controllers\. The central AI problem is no longer only model accuracy, but uncertainty\-aware governance: how much autonomy to grant, which evidence should calibrate trust, what performance ceiling a delegated AI system can sustain, and when human intervention becomes necessary\. We propose the Minimum Sufficient Oversight Principle \(MSO\), a variational principle for principled autonomy delegation: minimize governance burden on the Fisher information manifold subject to a delivery constraint\. The resulting Euler–Lagrange solution yields a water\-filling allocation of governed delegation across the task space\. Building on a revealed\-action governed delegation channel model, we prove a capacity theorem for stationary symbolwise review policies, derive a local first\-order approximation relating workflow complexity to quality degradation, and give a drift\-dominated autonomy\-time scaling law linking intervention timing to effective capacity, complexity, and drift\. Within this framework, masking appears as a structural AI\-governance pathology: corrected performance can hide the competence signal needed to calibrate trust\. Synthetic simulations and a semi\-real reconstructed workflow support design prescriptions including upstream\-first correction, sensitivity\-based intervention, and explicit feasibility checks before autonomy is expanded\. The result is a computable framework for uncertainty, planning, and oversight in delegated AI systems\. A companion Python package is available at[https://github\.com/crbazevedo/delegation\-lab](https://github.com/crbazevedo/delegation-lab)\.

Keywords:artificial intelligence; uncertainty in AI; autonomous agents; expert systems; oversight; trust calibration; planning under uncertainty\.

Working symbols used throughout\.State:σraw\\sigma\_\{\\mathrm\{raw\}\}\(raw competence\),σcorr\\sigma\_\{\\mathrm\{corr\}\}\(delivered quality\),H​\(W\)H\(W\)\(workflow complexity\)\.Control:α​\(x\)\\alpha\(x\)\(governed\-delegation allocation\),d​\(x\)d\(x\)\(delegated\-scope indicator\),KKandK/NK/N\(discrete review capacity and coverage\),BB\(average review budget\)\.Limits:CopC\_\{\\mathrm\{op\}\}\(operational quality ceiling\),Cdel​\(B\)C\_\{\\mathrm\{del\}\}\(B\)\(governed\-channel capacity\),BeffB\_\{\\mathrm\{eff\}\}\(effective autonomy buffer\),Tauto∗T^\{\*\}\_\{\\mathrm\{auto\}\}\(autonomy time\)\.Diagnostics:M∗M^\{\*\}\(masking index\),DC​\(v\)\\mathrm\{DC\}\(v\)\(delegation centrality\)\.

Notation discipline\.We useKKfor*discrete*corrector capacity in synchronous or queueing\-style models, soK/NK/Nis the fraction of items that can be reviewed in a cycle\. We useBBfor the*average review\-cost budget*in the governed\-channel model of Definition 1\. When a governance policy is written asπ=\(K,ρ,ϕ\)\\pi=\(K,\\rho,\\phi\), itsKKcomponent denotes the same discrete review\-capacity resource as inK/NK/N, specialized to the policy level\. We reserveCopC\_\{\\mathrm\{op\}\}for the operational quality ceiling used in design and experiments, andCdel​\(B\)C\_\{\\mathrm\{del\}\}\(B\)for the information\-theoretic capacity of the governed delegation channel\.

Modern AI systems increasingly rely on*delegated decision structures*: one model proposes an answer, another evaluates it, tools supply evidence, and a supervising policy decides what may proceed\. The central AI design problem is no longer only how accurate an individual model is\. It is how uncertainty should be represented, how trust should be calibrated from evidence, how much autonomy to grant, where to place oversight, and when human intervention becomes necessary as the system evolves\.

This paper develops a theory ofprincipled autonomy delegationfor such systems\. The aim is to make delegation computable rather than heuristic\. Given an agent’s demonstrated competence, the difficulty structure of the task space, and a limited governance budget, the theory should tell us four things: how oversight should be allocated, what performance ceiling the delegated system can actually achieve, how workflow complexity degrades that ceiling, and how long the system can operate before intervention is required\.

We formalize this with theMinimum Sufficient Oversight Principle\(MSO\): minimize total governance burden on the Fisher information manifold subject to a delivery constraint\. The resulting allocation is a water\-filling solution over the task space\. From this principle, and from an explicit channel model of governed delegation, we derive a family of results: an optimal governed\-delegation allocation, a stationary delegation\-capacity theorem, a local process\-complexity sensitivity law, and a drift\-dominated autonomy\-time scaling law from first\-passage theory\. These quantities support a standard AI workflow: represent uncertainty, plan feasible autonomy, monitor drift, and intervene when the evidence no longer supports delegation\. Graph topology appears only because it determines how uncertainty, correction, and error propagation affect trust calibration\.

Within this broader framework, one structural failure mode appears immediately\. In delegated systems, the process that preserves output quality can also destroy the information needed to calibrate trust\. When corrected performance is used as the basis for authority, oversight can hide deterioration in the producing agent\. We call thismasking\. It is not the main object of the theory; it is the first pathology the theory exposes and resolves by separating raw competence from corrected output quality\.

The paper’s thesis is therefore broader than measurement correction\. It is that autonomy delegation can be governed by a principled variational and information\-theoretic framework, rather than by ad hoc routing rules, intuition, or static safety margins\. Masking is one consequence of that framework\. The larger contribution is a calculable design logic for delegated systems: how topology, workflow complexity, review capacity, and measured competence jointly determine what can be delegated, under what oversight, for how long\.

A concrete example helps fix ideas\. Consider an AI\-assisted software\-delivery workflow: a generator proposes code, an evaluator inspects it, testing and security tools provide evidence, and a supervisory gate decides what ships\. The designer must decide which evidence calibrates trust, how much autonomy each stage receives, whether the system can support the required quality target at all, and how often humans must intervene\. The MSO turns those questions into explicit quantities rather than heuristics\.

Category scope\.The paper is positioned as a contribution to artificial intelligence, especially uncertainty in AI, expert\-system governance, and planning under limited evidence\. It uses delegated workflows and graph motifs as a modeling substrate, but it does not study coordination protocols or distributed\-agent negotiation as its primary object\.

A note on theoretical status\.The results in this paper span a range of rigor\. We mark each as:Theorem\(proved under stated assumptions\),Proposition\(derived under approximations\), orEmpirical Law\(observed in simulations\)\. This demarcation appears at first use of each result and is summarized in Section 4\.

Outline\.Section 1 develops the theory: the MSO, the Fisher metric, the Return Operator, delegation graphs, the Euler–Lagrange water\-filling solution, the delegation capacity, process entropy, and the autonomy time\. Section 2 connects to existing frameworks\. Section 3 presents numerical validation across eight conditions\. Section 4 discusses implementation, limitations, and conclusions\.

## 1The Principle

We begin with the operational setup: a delegation, its measurements, and the principle that governs it\. The theory rests on three primitives \(distinction, record, agency\); here we proceed directly from their operational consequences\. The term distinction is used in the operational sense of drawing a boundary between observable states, not as a dependence on Spencer\-Brown’s calculus of indications\(Spencer\-Brown,[1969](https://arxiv.org/html/2606.15563#bib.bib24)\)\.

### The delegation setup

Adelegationconsists of a principalAA, an agentBB, a scopeSS\(the set of tasksBBis authorized to perform\), and a correctorCCthat observesBB’s outputs and applies corrections\. At each pointx∈Sx\\in S, the agent produces outcomes; the corrector reviews a subset\. Both raw and corrected outcomes enter the record stateR​\(t\)R\(t\)\. From these records we compute two quantities that will drive the entire theory:

- •Evidential supportσraw​\(x,t\)\\sigma\_\{\\mathrm\{raw\}\}\(x,t\): the empirical success rate ofBB’s*uncorrected*outcomes atxx, computed as the running fraction of correct outputs in the accumulated recordsR​\(t\)R\(t\)\. This is a Bernoulli parameter in\[0,1\]\[0,1\]that measuresBB’s demonstrated competence\. \(It is related to the mutual information byI​\(R;outcome\)=1−H​\(σraw\)I\(R;\\,\\mathrm\{outcome\}\)=1\-H\(\\sigma\_\{\\mathrm\{raw\}\}\), whereHHis the binary entropy function\.\)
- •Corrected supportσcorr​\(x,t\)\\sigma\_\{\\mathrm\{corr\}\}\(x,t\): the empirical success rate computed from outcomes that include the corrector’s interventions\. This measures the system’s output quality\.

Together, these two quantities define the minimal informational state needed for principled autonomy delegation: one signal for agent competence and one for delivered system quality\. Their separation also resolves the masking failure mode introduced above\. Authorization must be based onσraw\\sigma\_\{\\mathrm\{raw\}\}—not onσcorr\\sigma\_\{\\mathrm\{corr\}\}, which conflates the agent’s competence with the corrector’s diligence\.

### The Minimum Sufficient Oversight Principle

Why minimize oversight cost? The intuition is that oversight is*expensive*: every unit of review, monitoring, or corrective capacity spent on one scope point is a unit unavailable elsewhere\. This is an informational and organizational cost, not a thermodynamic lower bound in Landauer’s sense\(Landauer,[1961](https://arxiv.org/html/2606.15563#bib.bib16)\)\. A code reviewer spending time re\-checking a reliable module has less capacity for the fragile one\. Minimizing oversight cost means spending the review budget*efficiently*—concentrating attention where it produces the most quality improvement, not where the agent is already competent\. This is not a philosophical preference—it is the variational structure of the problem, analogous to Shannon’s water\-filling \(power is allocated to channels proportionally to their capacity for improvement, not uniformly\)\.

The central control variable is thegoverned\-delegation intensityα​\(x,t\)∈\[0,1\]\\alpha\(x,t\)\\in\[0,1\]\. It is the fraction or weight of task classxxallowed to be handled by the delegated system during intervalttunder the available governance regime\. Thusα\\alphais not the same object as the discrete review fractionK/NK/Nor the average review budgetBB:KKandBBdescribe the scarce review resource, whileα\\alphadescribes how much delegated workload is placed under that resource\. A baseline trust policy may map evidential support to a maximum allowed intensity,

α​\(x,t\)≤αmax​\(x,t\)=G​\(σraw​\(x,t\)\),\\alpha\(x,t\)\\leq\\alpha\_\{\\max\}\(x,t\)=G\\bigl\(\\sigma\_\{\\mathrm\{raw\}\}\(x,t\)\\bigr\),\(1\)whereGGis monotone non\-decreasing\. The MSO then optimizes the actual allocationα\\alphabeneath this ceiling\. This distinction avoids conflating two different questions: how much authority a task class is allowed to receive and how scarce oversight effort should be distributed within that allowed region\.

The Minimum Sufficient Oversight Principle \(MSO\)The delegation allocates the minimum sufficient governance burden over scope and time, subject to a delivery constraint:minα​∫0T∫Sα​\(x,t\)2​detg​\(x,t\)​dx​dtsubject to∫Sα​σraw​dx≥pmin​\|S\|\.\\min\_\{\\alpha\}\\;\\int\_\{0\}^\{T\}\\\!\\\!\\int\_\{S\}\\alpha\(x,t\)^\{2\}\\,\\sqrt\{\\det g\(x,t\)\}\\;\\mathrm\{d\}x\\,\\mathrm\{d\}t\\quad\\text\{subject to\}\\quad\\int\_\{S\}\\alpha\\,\\sigma\_\{\\mathrm\{raw\}\}\\;\\mathrm\{d\}x\\geq p\_\{\\min\}\\,\|S\|\.\(2\)*In words:*allocate only the oversight burden—measured in the Fisher information geometry—sufficient to meet the quality target\. The quadratic costα2\\alpha^\{2\}reflects diminishing returns: concentrating all governed workload at one point is more expensive than spreading it, just as concentrating all power on one channel is inefficient in Shannon’s water\-filling\.

*Why quadratic cost?*A linear cost∫α​g​dx\\int\\alpha\\sqrt\{g\}\\,\\mathrm\{d\}xwould yield a knapsack\-like solution that selects points by the ratiog/σ\\sqrt\{g\}/\\sigmarather than allocating proportionally\. The quadratic formα2\\alpha^\{2\}models the realistic property that the marginal cost of oversight is*increasing*: doubling review effort at a point costs four times as much\. This is the standard assumption in resource allocation theory and produces the water\-filling solution that is the hallmark of optimal allocation in information theory\(Cover and Thomas,[2006](https://arxiv.org/html/2606.15563#bib.bib7)\)\.

*Scope and geometry\.*The MSO above treats the delegated scopeSSas fixed\. If scope itself is a control variable, introduced​\(x\)∈\{0,1\}d\(x\)\\in\\\{0,1\\\}and optimize

minα,d​∫Sd​\(x\)​α​\(x\)2​detg​\(x\)​𝑑xs\.t\.∫Sd​\(x\)​α​\(x\)​σ​\(x\)​𝑑x≥pmin​∫Sd​\(x\)​𝑑x\.\\min\_\{\\alpha,d\}\\int\_\{S\}d\(x\)\\alpha\(x\)^\{2\}\\sqrt\{\\det g\(x\)\}\\,dx\\quad\\text\{s\.t\.\}\\quad\\int\_\{S\}d\(x\)\\alpha\(x\)\\sigma\(x\)\\,dx\\geq p\_\{\\min\}\\\!\\int\_\{S\}d\(x\)\\,dx\.This endogenous\-scope extension is useful, but it also exposes a cherry\-picking issue: without a coverage or task\-value constraint such as∫Sd​\(x\)​w​\(x\)​𝑑x≥Wmin\\int\_\{S\}d\(x\)w\(x\)\\,dx\\geq W\_\{\\min\}, the optimum may delegate only a small high\-competence subset\. The short version of the paper therefore focuses on the fixed\-scope problem and treats scope selection as an outer design constraint\.

The volume elementdetg\\sqrt\{\\det g\}is the Fisher information metric\(Čencov,[1982](https://arxiv.org/html/2606.15563#bib.bib6);Amari and Nagaoka,[2000](https://arxiv.org/html/2606.15563#bib.bib3)\)\. For Bernoulli outcomes with success probabilityσ\\sigma,

g​\(σ\)=1σ​\(1−σ\)\.g\(\\sigma\)=\\frac\{1\}\{\\sigma\(1\-\\sigma\)\}\.\(3\)This is the metric in the local coordinateσ\\sigma\. If the task coordinate is an external coordinatexxwith success curveσ​\(x\)\\sigma\(x\), the pullback metric isgx=\(∂xσ\)2/\[σ​\(1−σ\)\]g\_\{x\}=\(\\partial\_\{x\}\\sigma\)^\{2\}/\[\\sigma\(1\-\\sigma\)\]\. The discrete models below index each cell by its empirical competence value, so equation \([3](https://arxiv.org/html/2606.15563#S1.E3)\) is the intended local metric\.

### The Return Operator

The delegation evolves through a cyclic process we call theReturn OperatorRR:BBoperates→\\toCCcorrects→\\torecords accumulate→\\toσ\\sigmaupdates→\\toGGupdatesα\\alpha→\\toscope adjusts\. We derive the dynamics ofσ\\sigmafrom first principles\.

At each pointxx, in each time intervald​t\\mathrm\{d\}t, the agent produces an outcome with probabilityη​\(x,t\)​d​t\\eta\(x,t\)\\,\\mathrm\{d\}tof being observed, whereη\\etais the observation rate\. Each observation updatesσ\\sigmatoward the agent’s true competenceσskill​\(x\)\\sigma\_\{\\mathrm\{skill\}\}\(x\), because the Bayesian posterior on competence, given a new outcome, moves in the direction of the data\. Simultaneously, old records lose relevance at rateδ\\delta—the environment shifts, skills change, and stale evidence no longer applies\. When stale evidence is removed, the estimate should not necessarily decay to zero; it should decay toward a prior support levelσ0​\(x\)\\sigma\_\{0\}\(x\)\(for example,1/21/2for an uninformative Bernoulli prior, or0for a conservative “no demonstrated support” score\)\. Combining these two effects:

∂σraw∂t=η​\(x,t\)​\[σskill​\(x\)−σraw​\(x,t\)\]⏟learning:σmoves toward truth−δ​\[σraw​\(x,t\)−σ0​\(x\)\]⏟forgetting:σrelaxes to prior support\.\\frac\{\\partial\\sigma\_\{\\mathrm\{raw\}\}\}\{\\partial t\}=\\underbrace\{\\eta\(x,t\)\\bigl\[\\sigma\_\{\\mathrm\{skill\}\}\(x\)\-\\sigma\_\{\\mathrm\{raw\}\}\(x,t\)\\bigr\]\}\_\{\\text\{learning: $\\sigma$ moves toward truth\}\}\\;\-\\;\\underbrace\{\\delta\\bigl\[\\sigma\_\{\\mathrm\{raw\}\}\(x,t\)\-\\sigma\_\{0\}\(x\)\\bigr\]\}\_\{\\text\{forgetting: $\\sigma$ relaxes to prior support\}\}\.\(4\)This is a linear relaxation equation, structurally identical to a leaky integrator in neuroscience or a first\-order low\-pass filter in engineering\. The agent “charges up” towardσskill\\sigma\_\{\\mathrm\{skill\}\}through observations and relaxes toward the prior through decay\. The balance determines the equilibrium\.

Fixed point\.*Assumptions: binary outcomes \(Bernoulli\), stationaryσskill\\sigma\_\{\\mathrm\{skill\}\}, constantη\\eta,δ\\delta, andσ0\\sigma\_\{0\}\.*Setting∂σ/∂t=0\\partial\\sigma/\\partial t=0and solving:

σraw∗=η​σskill\+δ​σ0η\+δ\.\\sigma\_\{\\mathrm\{raw\}\}^\{\*\}=\\frac\{\\eta\\;\\sigma\_\{\\mathrm\{skill\}\}\+\\delta\\,\\sigma\_\{0\}\}\{\\eta\+\\delta\}\.\(5\)The intuition is immediate:σ∗\\sigma^\{\*\}is a weighted average of the truth \(σskill\\sigma\_\{\\mathrm\{skill\}\}\) and the prior supportσ0\\sigma\_\{0\}, with weightsη\\eta\(observation\) andδ\\delta\(decay\)\. When observations are frequent relative to decay \(η≫δ\\eta\\gg\\delta\),σ∗→σskill\\sigma^\{\*\}\\to\\sigma\_\{\\mathrm\{skill\}\}—the system learns the truth\. When decay dominates \(δ≫η\\delta\\gg\\eta\),σ∗→σ0\\sigma^\{\*\}\\to\\sigma\_\{0\}—the system returns to its prior\. The numerical examples below use the conservative support conventionσ0=0\\sigma\_\{0\}=0, under which equation \([5](https://arxiv.org/html/2606.15563#S1.E5)\) reduces to the simpler expressionσraw∗=η​σskill/\(η\+δ\)\\sigma\_\{\\mathrm\{raw\}\}^\{\*\}=\\eta\\sigma\_\{\\mathrm\{skill\}\}/\(\\eta\+\\delta\)\. Forη=10\\eta=10,δ=2\\delta=2, andσskill=0\.80\\sigma\_\{\\mathrm\{skill\}\}=0\.80, this givesσ∗=0\.667\\sigma^\{\*\}=0\.667\. With an uninformative Bernoulli priorσ0=0\.5\\sigma\_\{0\}=0\.5, the same parameters would giveσ∗=0\.750\\sigma^\{\*\}=0\.750\.

Convergence\.The linearized dynamics around the fixed point decay at rateη\+δ\\eta\+\\delta—the spectral gap ofRR\. The distance to equilibrium decreases as\|σ​\(t\)−σ∗\|∝e−\(η\+δ\)​t\|\\sigma\(t\)\-\\sigma^\{\*\}\|\\propto e^\{\-\(\\eta\+\\delta\)t\}\. Convergence is faster when the observation rate is high and the evidence memory is short\. The time constant itself does not scale with the number of scope pointsNN; what improves with largerNNis the statistical uncertainty of an average over independent cells, whose variance shrinks asO​\(1/N\)O\(1/N\)\.

The corrector’s effect onσ\\sigma\.The correctorCCcatches a fractionccof the agent’s errors\. The*corrected*fixed point is:

σcorr∗=σraw∗\+\(1−σraw∗\)×c\.\\sigma\_\{\\mathrm\{corr\}\}^\{\*\}=\\sigma\_\{\\mathrm\{raw\}\}^\{\*\}\+\(1\-\\sigma\_\{\\mathrm\{raw\}\}^\{\*\}\)\\times c\.\(6\)The masking index follows directly:M∗=σcorr∗/σraw∗=1\+\(1−σraw∗\)​c/σraw∗M^\{\*\}=\\sigma\_\{\\mathrm\{corr\}\}^\{\*\}/\\sigma\_\{\\mathrm\{raw\}\}^\{\*\}=1\+\(1\-\\sigma\_\{\\mathrm\{raw\}\}^\{\*\}\)c/\\sigma\_\{\\mathrm\{raw\}\}^\{\*\}\. For our worked example \(σraw∗=0\.667\\sigma\_\{\\mathrm\{raw\}\}^\{\*\}=0\.667,c=0\.70c=0\.70\):M∗=1\+0\.333×0\.70/0\.667=1\.35M^\{\*\}=1\+0\.333\\times 0\.70/0\.667=1\.35\. The corrector makes the system appear 35% more competent than it actually is\. This gap is the masking problem in numerical form\.

### Delegation graphs for uncertainty propagation

Delegated AI workflows are often directed acyclic graphs \(DAGs\): the code generator’s output feeds an evaluator, whose corrected output fans out to unit tests, static analysis, and security scanning, all of which fan in to a merge gate \(Figure[1](https://arxiv.org/html/2606.15563#S1.F1)A\)\. An agent’s effective skill depends on the quality of its inputs:

σskill,eff​\(v\)=σskill​\(v\)×AGG​\(σcorr​\(u1\),…,σcorr​\(uk\)\),\\sigma\_\{\\mathrm\{skill,eff\}\}\(v\)=\\sigma\_\{\\mathrm\{skill\}\}\(v\)\\;\\times\\;\\mathrm\{AGG\}\\bigl\(\\sigma\_\{\\mathrm\{corr\}\}\(u\_\{1\}\),\\ldots,\\sigma\_\{\\mathrm\{corr\}\}\(u\_\{k\}\)\\bigr\),\(7\)whereu1,…,uku\_\{1\},\\ldots,u\_\{k\}are the parents ofvvin the DAG andAGG\\mathrm\{AGG\}is an aggregation function\. The choice ofAGG\\mathrm\{AGG\}is not arbitrary—it is determined by the task structure at the merge node:

- •AGG=∏\\mathrm\{AGG\}=\\prod\(product\): each input contributes an independent dimension\. A merge gate in a code pipeline requires both correct logic*and*correct style—errors in either degrade the merge\. This is the most common case and produces the highest masking, because errors from all parents compound multiplicatively\.
- •AGG=min\\mathrm\{AGG\}=\\min\(weakest link\): all inputs must be correct for the merge to succeed\. A safety\-critical system that gates on “all checks pass” is limited by its weakest component\. This produces the most fragile fan\-in but the most “honest”—masking at the merge is dominated by a single parent\.
- •AGG=weighted mean\\mathrm\{AGG\}=\\text\{weighted mean\}: inputs have different importance\. A recommendation system that blends product relevance \(weight 0\.6\) and user sentiment \(weight 0\.4\) averages quality across inputs\. This dilutes both errors and masking\.

Infan\-out, one node’s failure cascades to all children simultaneously\. Infan\-in, errors from parents sharing an upstream source are correlated \(*diamond pattern*\), creating conditional fragility at the merge node \(quality drops29%29\\%when the shared source fails\)\. The masking index compounds super\-multiplicatively with depth \(Figure[1](https://arxiv.org/html/2606.15563#S1.F1)B\): a five\-layer pipeline where the first layer hasM∗=1\.35M^\{\*\}=1\.35and the index*increases*with depth \(to∼1\.5\{\\sim\}1\.5,1\.71\.7,2\.02\.0,2\.32\.3at subsequent layers\) hasMtotal∗=∏Mi∗=14\.3M^\{\*\}\_\{\\mathrm\{total\}\}=\\prod M^\{\*\}\_\{i\}=14\.3—exceeding\(1\.35\)5=4\.5\(1\.35\)^\{5\}=4\.5, the prediction from naively assuming uniform masking\.

### The Euler–Lagrange Solution

The MSO \([2](https://arxiv.org/html/2606.15563#S1.E2)\) is a constrained optimization over the governed\-delegation fieldα​\(x,t\)\\alpha\(x,t\)\. We solve it using the calculus of variations\. Introducing a Lagrange multiplierλ\\lambdafor the delivery constraint, the Lagrangian is:

The Lagrangianℒ=∫0T∫Sα2​detg​dx​dt−λ​\(∫Sα​σraw​dx−pmin​\|S\|\)\.\\mathcal\{L\}=\\int\_\{0\}^\{T\}\\\!\\\!\\int\_\{S\}\\alpha^\{2\}\\,\\sqrt\{\\det g\}\\;\\mathrm\{d\}x\\,\\mathrm\{d\}t\\;\-\\;\\lambda\\Bigl\(\\int\_\{S\}\\alpha\\,\\sigma\_\{\\mathrm\{raw\}\}\\,\\mathrm\{d\}x\-p\_\{\\min\}\\,\|S\|\\Bigr\)\.

Taking the functional derivative and setting it to zero:

δ​ℒδ​α=2​α​\(x\)​g​\(x\)−λ​σraw​\(x\)=0⟹α∗​\(x\)=λ​σraw​\(x\)2​g​\(x\)\.\\frac\{\\delta\\mathcal\{L\}\}\{\\delta\\alpha\}=2\\,\\alpha\(x\)\\,\\sqrt\{g\(x\)\}\-\\lambda\\,\\sigma\_\{\\mathrm\{raw\}\}\(x\)=0\\quad\\Longrightarrow\\quad\\alpha^\{\*\}\(x\)=\\frac\{\\lambda\\,\\sigma\_\{\\mathrm\{raw\}\}\(x\)\}\{2\\,\\sqrt\{g\(x\)\}\}\.Sinceg​\(σ\)=1/σ​\(1−σ\)g\(\\sigma\)=1/\\sigma\(1\-\\sigma\)for Bernoulli outcomes \(equation[3](https://arxiv.org/html/2606.15563#S1.E3)\),g=1/σ​\(1−σ\)\\sqrt\{g\}=1/\\sqrt\{\\sigma\(1\-\\sigma\)\}and:

The Euler–Lagrange Solution \(Water\-Filling\)α∗​\(x,t\)=min⁡\(αmax​\(x,t\),λ2​σraw​\(x\)​σraw​\(x\)​\(1−σraw​\(x\)\)\)\.\\alpha^\{\*\}\(x,t\)=\\min\\\!\\Bigl\(\\alpha\_\{\\max\}\(x,t\),\\;\\;\\frac\{\\lambda\}\{2\}\\,\\sigma\_\{\\mathrm\{raw\}\}\(x\)\\,\\sqrt\{\\sigma\_\{\\mathrm\{raw\}\}\(x\)\\,\(1\-\\sigma\_\{\\mathrm\{raw\}\}\(x\)\)\}\\Bigr\)\.\(8\)*In words:*governed delegation is allocated proportionally to the productσraw⋅σ​\(1−σ\)\\sigma\_\{\\mathrm\{raw\}\}\\cdot\\sqrt\{\\sigma\(1\-\\sigma\)\}, which peaks at intermediate competence \(σ≈0\.75\\sigma\\approx 0\.75\)\. Points where the agent is moderately competent receive the most delegated workload under governance \(highest marginal return of review\-supported operation\)\. Very weak points \(σ≪0\.5\\sigma\\ll 0\.5\) receive less \(review has diminishing returns when most outputs need rework\)\. Very strong points \(σ→1\\sigma\\to 1\) also receive less \(review rarely finds errors\)\. See Box 1 for a worked example\.

The multiplierλ\\lambdais determined by the delivery constraint:∫Sα∗​σraw​dx=pmin​\|S\|\\int\_\{S\}\\alpha^\{\*\}\\,\\sigma\_\{\\mathrm\{raw\}\}\\,\\mathrm\{d\}x=p\_\{\\min\}\\,\|S\|\. The solution distributes governed workload according to a water\-filling rule on the Fisher manifold, where the “water level”λ/2\\lambda/2is set by the delivery target\. The allocation peaks at intermediate competence \(σ≈0\.75\\sigma\\approx 0\.75\) and tapers at both extremes, paralleling Shannon’s water\-filling for power allocation across parallel channels\(Cover and Thomas,[2006](https://arxiv.org/html/2606.15563#bib.bib7)\)\.

Three closed\-form quantities follow directly from the solution:

- •Masking index:M∗=σcorr∗/σraw∗M^\{\*\}=\\sigma\_\{\\mathrm\{corr\}\}^\{\*\}/\\sigma\_\{\\mathrm\{raw\}\}^\{\*\}\. WhenM∗\>1M^\{\*\}\>1, the corrector is hiding agent errors from the authorization mechanism\. In our worked example,M∗=1\.35M^\{\*\}=1\.35\. In the experiments of Section 3,M∗≈1\.8M^\{\*\}\\approx 1\.8in the standard single\-layer delegation—the corrector makes the agent appear nearly twice as competent as it is\.
- •Corrector capacity threshold:K/N≥max⁡\(0,\(pmin−σraw∗\)/\[\(1−σraw∗\)​c\]\)K/N\\geq\\max\\\!\\left\(0,\\,\(p\_\{\\min\}\-\\sigma\_\{\\mathrm\{raw\}\}^\{\*\}\)/\[\(1\-\\sigma\_\{\\mathrm\{raw\}\}^\{\*\}\)\\,c\]\\right\), whereKKis the number of outputs the corrector reviews per cycle,NNis the scope size, andccis the corrector’s catch rate\. Forpmin=0\.80p\_\{\\min\}=0\.80,σraw∗=0\.55\\sigma\_\{\\mathrm\{raw\}\}^\{\*\}=0\.55,c=0\.65c=0\.65: the threshold isK/N\>0\.25/0\.2925=0\.85K/N\>0\.25/0\.2925=0\.85\. The corrector must review at least 85% of outputs\. Below this ratio, the delegation cannot maintain quality, regardless of how the governed workload is allocated\.
- •Convergence time:Tcal≈1/\(η\+δ\)T\_\{\\mathrm\{cal\}\}\\approx 1/\(\\eta\+\\delta\)per cell\. Averages over larger scopes have lower sampling variance, but the dynamical time constant remains1/\(η\+δ\)1/\(\\eta\+\\delta\)under the independent\-cell model\.

Together, these quantities answer the question posed in Section 1\. The masking index diagnoses when the oversight process is hiding agent weakness\. The capacity threshold determines when delegation can function at all\. The convergence time predicts how long calibration will take\. And the Euler–Lagrange solution prescribes exactly how much authority to grant at each point—no more, no less\.

When the MSO has no solution\.The variational problem \([2](https://arxiv.org/html/2606.15563#S1.E2)\) is infeasible when the required quality target exceeds the operational ceiling available under the current raw support, corrector catch rate, and review capacity\. In this case, the MSO does not prescribe “more authority”—it prescribes*no delegation*under the current design\. The task must be performed by a more capable agent, assigned more effective correction, or decomposed into subtasks that fall within the current system’s competence\. Theα∗\\alpha^\{\*\}solution is also degenerate whenσraw\\sigma\_\{\\mathrm\{raw\}\}is uniform across the scope \(no information heterogeneity\): the Fisher volume element is constant,α∗​\(x\)\\alpha^\{\*\}\(x\)is the same at every point, and the theory adds no value beyond a uniform policy\.

### Asynchronous delegation dynamics

The synchronous Return Operator is a mean\-field approximation\. Real pipelines are asynchronous: agents process at different rates, correctors queue, events arrive stochastically, and batch refreshes occur on schedules\. A Generalized Stochastic Petri Net \(GSPN\)\(Marsan et al\.,[1984](https://arxiv.org/html/2606.15563#bib.bib17)\)gives the corresponding concurrency model\. In that setting the local support dynamics become

∂σraw​\(v\)∂t=λobs​\(v,𝐦\)​\[σskill,eff​\(v\)−σraw​\(v\)\]−λforget​\(v\)​\[σraw​\(v\)−σ0​\(v\)\],\\frac\{\\partial\\sigma\_\{\\mathrm\{raw\}\}\(v\)\}\{\\partial t\}=\\lambda\_\{\\mathrm\{obs\}\}\(v,\\mathbf\{m\}\)\[\\sigma\_\{\\mathrm\{skill,eff\}\}\(v\)\-\\sigma\_\{\\mathrm\{raw\}\}\(v\)\]\-\\lambda\_\{\\mathrm\{forget\}\}\(v\)\[\\sigma\_\{\\mathrm\{raw\}\}\(v\)\-\\sigma\_\{0\}\(v\)\],\(9\)where𝐦\\mathbf\{m\}is the marking state of the net\. When the corrector pool is uncongested,λobs≈η\\lambda\_\{\\mathrm\{obs\}\}\\approx\\etaand equation \([9](https://arxiv.org/html/2606.15563#S1.E9)\) reduces to the synchronous model\. Under congestion, calibration slows and throughput, utilization, and queue lengths become governance observables\. The GSPN is therefore the operational model; the ODE above is its tractable mean\-field projection\.

### Topology as a governance object

The delegation DAG is not just a wiring diagram; it changes where governance effort has leverage\. Different motifs generate different failure surfaces and therefore different design prescriptions\. Table[1](https://arxiv.org/html/2606.15563#S1.T1)summarizes the four recurring cases used throughout the paper\.

Table 1:Delegation motifs as governance objects\.In this sense, topology enters the theory twice: it determines how errors propagate, and it determines where marginal governance investment produces the largest increase in sustainable autonomy\.

### Delegation capacity

The MSO determines the optimal governed\-delegation allocation for a given pipeline and task distribution\. But what is the*best possible*quality the pipeline can achieve, optimized over all task distributions? This ceiling is thedelegation channel capacity\.

We define it in two related ways, connected through a monotone bridge under the binary symmetric model\. The*primary*definition is operational: it is the supremum of achievable shipped output quality:

Delegation Channel CapacityCop​\(G,K,B\)=supp​\(task\)q∗​\(output\)\.C\_\{\\mathrm\{op\}\}\(G,K,B\)=\\sup\_\{p\(\\mathrm\{task\}\)\}\\;q^\{\*\}\(\\mathrm\{output\}\)\.\(10\)*In words:*the best possible output quality the pipeline can achieve\. No operational policy can exceedCopC\_\{\\mathrm\{op\}\}\.

Hereq∗​\(output\)q^\{\*\}\(\\mathrm\{output\}\)denotes the quality of the output that is actually shipped at the sink\. In an uncorrected output stage,q∗=σraw∗q^\{\*\}=\\sigma^\{\*\}\_\{\\mathrm\{raw\}\}; in a reviewed output stage,q∗=σcorr∗q^\{\*\}=\\sigma^\{\*\}\_\{\\mathrm\{corr\}\}\. The raw signal remains the preferred authorization signal because it is less affected by masking\.

The*information\-theoretic*definition models each node as a Binary Symmetric Channel \(BSC\) with effective error rateεeff​\(v\)=\(1−σskill​\(v\)\)​\(1−c​\(v\)\)\\varepsilon\_\{\\mathrm\{eff\}\}\(v\)=\(1\-\\sigma\_\{\\mathrm\{skill\}\}\(v\)\)\(1\-c\(v\)\), giving node capacityCvBSC=1−Hb​\(εeff​\(v\)\)C\_\{v\}^\{\\mathrm\{BSC\}\}=1\-H\_\{b\}\(\\varepsilon\_\{\\mathrm\{eff\}\}\(v\)\)bits\. The two definitions are related by a monotone bridge: sinceq∗=1−εeffq^\{\*\}=1\-\\varepsilon\_\{\\mathrm\{eff\}\}andHbH\_\{b\}is monotone on\[0,0\.5\]\[0,0\.5\], maximizingq∗q^\{\*\}is equivalent to maximizingCBSCC^\{\\mathrm\{BSC\}\}\. We use the operational definition \(CopC\_\{\\mathrm\{op\}\}as a quality ceiling\) throughout the design and numerical sections, and invoke the BSC formulation only in the proofs of Theorem 1 and Proposition 2 where information\-theoretic tools are needed\.

For a single uncorrected node under the conservativeσ0=0\\sigma\_\{0\}=0convention, the raw\-support ceiling isη/\(η\+δ\)\\eta/\(\\eta\+\\delta\), achieved whenσskill=1\\sigma\_\{\\mathrm\{skill\}\}=1; with output correction, the shipped\-quality ceiling is the correspondingσcorr∗\\sigma^\{\*\}\_\{\\mathrm\{corr\}\}\. For a linear chain of depthDD, each layer receives the corrected output of the previous layer\. Define the*recursive chain quality*by:

σcorr∗​\(i\)=R​\(σskill×σcorr∗​\(i−1\)\),σcorr∗​\(0\)=1,\\sigma^\{\*\}\_\{\\mathrm\{corr\}\}\(i\)=R\\bigl\(\\sigma\_\{\\mathrm\{skill\}\}\\times\\sigma^\{\*\}\_\{\\mathrm\{corr\}\}\(i\{\-\}1\)\\bigr\),\\qquad\\sigma^\{\*\}\_\{\\mathrm\{corr\}\}\(0\)=1,\(11\)whereR​\(⋅\)R\(\\cdot\)is the Return Operator \(equation[5](https://arxiv.org/html/2606.15563#S1.E5)\)\. The operational quality ceiling at depthDDisCop​\(D\)=σcorr∗​\(D\)C\_\{\\mathrm\{op\}\}\(D\)=\\sigma^\{\*\}\_\{\\mathrm\{corr\}\}\(D\)\.

This recursive formula accounts for the fact that each corrector stabilizes the signal before passing it downstream\. The simpler product formulaCprod​\(D\)=\[σcorr∗​\(1\)\]DC\_\{\\mathrm\{prod\}\}\(D\)=\[\\sigma^\{\*\}\_\{\\mathrm\{corr\}\}\(1\)\]^\{D\}treats each layer as if it had the same input quality, producing a conservative lower bound \(see Experiment 6\)\. We use the recursive formula in the numerical experiments \(gap<0\.002<0\.002vs\. simulation\) and the information\-theoretic boundCchainBSC=mini⁡\[1−Hb​\(εeff​\(i\)\)\]C\_\{\\mathrm\{chain\}\}^\{\\mathrm\{BSC\}\}=\\min\_\{i\}\[1\-H\_\{b\}\(\\varepsilon\_\{\\mathrm\{eff\}\}\(i\)\)\]in the proofs \(where the DPI is needed\)\.

The capacity has a striking practical consequence\. If the quality requirementpminp\_\{\\min\}exceedsCopC\_\{\\mathrm\{op\}\},*no governance policy within the existing pipeline can help*\. The designer must change the pipeline: better agents, more correctors, or a different topology\.

The dual quantity is thegovernance overhead: the minimum cost of achieving a given quality target\.

R​\(pmin\)=minπ:σraw∗≥pmin⁡Cost​\(π\),R\(p\_\{\\min\}\)=\\min\_\{\\pi:\\,\\sigma^\{\*\}\_\{\\mathrm\{raw\}\}\\geq p\_\{\\min\}\}\\mathrm\{Cost\}\(\\pi\),whereπ\\piranges over all governance policies andCost​\(π\)\\mathrm\{Cost\}\(\\pi\)is the total corrector budget plus routing overhead\. The dualityR​\(pmin\)≤Budget⇔pmin≤Cop​\(Budget\)R\(p\_\{\\min\}\)\\leq\\mathrm\{Budget\}\\Leftrightarrow p\_\{\\min\}\\leq C\_\{\\mathrm\{op\}\}\(\\mathrm\{Budget\}\)connects the two quantities\.

The governed delegation channel\.

To prove a capacity theorem, we need a precise channel model where governance is not an analogy to coding but a*causal action policy*over a controlled discrete memoryless channel \(DMC\)\.

Definition 1: Governed Delegation ChannelAgoverned delegation channelat nodevvis a cost\-constrained action\-dependent DMC\(𝒳,𝒜,𝒴,PY\|X,A,c,B\)\(\\mathcal\{X\},\\mathcal\{A\},\\mathcal\{Y\},P\_\{Y\|X,A\},c,B\):*Task alphabet:*𝒳=\{0,1\}\\mathcal\{X\}=\\\{0,1\\\}\(correct/incorrect task specification\)\.*Action alphabet:*𝒜=\{0,1\}\\mathcal\{A\}=\\\{0,1\\\}, whereAt=0A\_\{t\}=0\(no review\) andAt=1A\_\{t\}=1\(review\)\. Each action incurs costc​\(At\)c\(A\_\{t\}\), withc​\(0\)=0c\(0\)=0andc​\(1\)=1c\(1\)=1, subject to an average budget:1n​∑t=1n𝔼​\[c​\(At\)\]≤B\\frac\{1\}\{n\}\\sum\_\{t=1\}^\{n\}\\mathbb\{E\}\[c\(A\_\{t\}\)\]\\leq B\.*Output alphabet:*𝒴=\{0,1\}\\mathcal\{Y\}=\\\{0,1\\\}\(correct/incorrect output\)\.*Channel law:*Conditional on actionAt=aA\_\{t\}=a, the node operates as a BSC with crossover probabilityεa\\varepsilon\_\{a\}:PY\|X,A=0=BSC​\(ε0\)P\_\{Y\|X,A=0\}=\\mathrm\{BSC\}\(\\varepsilon\_\{0\}\)\(unreviewed\),PY\|X,A=1=BSC​\(ε1\)P\_\{Y\|X,A=1\}=\\mathrm\{BSC\}\(\\varepsilon\_\{1\}\)\(reviewed\), withε1<ε0\\varepsilon\_\{1\}<\\varepsilon\_\{0\}\. For the delegation setting:ε0=1−σskill​\(v\)\\varepsilon\_\{0\}=1\-\\sigma\_\{\\mathrm\{skill\}\}\(v\)andε1=\(1−σskill​\(v\)\)​\(1−cv\)\\varepsilon\_\{1\}=\(1\-\\sigma\_\{\\mathrm\{skill\}\}\(v\)\)\(1\-c\_\{v\}\)\.*Memoryless property:*P​\(Yn\|Xn,An\)=∏t=1nP​\(Yt\|Xt,At\)P\(Y^\{n\}\|X^\{n\},A^\{n\}\)=\\prod\_\{t=1\}^\{n\}P\(Y\_\{t\}\|X\_\{t\},A\_\{t\}\)\.

*Key distinction:*Governance is not a code applied*after*the channel—it is a causal action that selects*which channel law*is applied at each symbol, under a cost constraint\. The corrector’s review is the action; the review budget is the cost constraint\.

Delegation capacity\.In full generality, the capacity of a governed delegation channel is the supremum of achievable rates under admissible causal governance policies:

Cdel​\(B\)=supπ∈Π​\(B\)lim infn→∞1n​I​\(Xn;Yn\),C\_\{\\mathrm\{del\}\}\(B\)=\\sup\_\{\\pi\\in\\Pi\(B\)\}\\;\\liminf\_\{n\\to\\infty\}\\frac\{1\}\{n\}\\,I\(X^\{n\};Y^\{n\}\),\(12\)whereΠ​\(B\)\\Pi\(B\)is the set of causal governance policies satisfying the average review\-cost budgetBB\. The fully adaptive feedback case is a controlled\-channel problem and is not solved in closed form here\. The theorem below proves the single\-letter capacity for the stationary symbolwise case used in the simulations and design calculations\.

Single\-letter form \(stationary symbolwise policies\)\.When the governance policy is restricted to stationary symbolwise review—i\.e\., each task is independently reviewed with probabilityqq, without dependence on past observations—and the review action is recorded in the governance log, the effective channel output is the pair\(Y,A\)\(Y,A\)\. The action is a Bernoulli random variableA∼Bern​\(q\)A\\sim\\mathrm\{Bern\}\(q\)withq≤Bq\\leq B\(review fraction\), independent ofXX\. Each action induces a BSC, soI​\(X;Y,A\)=I​\(X;Y∣A\)I\(X;Y,A\)=I\(X;Y\\mid A\)andI​\(X;Y∣A=a\)=1−Hb​\(εa\)I\(X;Y\\mid A=a\)=1\-H\_\{b\}\(\\varepsilon\_\{a\}\)forX∼Bern​\(1/2\)X\\sim\\mathrm\{Bern\}\(1/2\)\. Optimizing over admissibleqqgives the revealed\-action capacity:

Cdel​\(B\)=maxq∈\[0,B\]⁡\(1−q\)​\[1−Hb​\(ε0\)\]\+q​\[1−Hb​\(ε1\)\]\.C\_\{\\mathrm\{del\}\}\(B\)=\\max\_\{q\\in\[0,B\]\}\\;\(1\-q\)\\bigl\[1\-H\_\{b\}\(\\varepsilon\_\{0\}\)\\bigr\]\+q\\bigl\[1\-H\_\{b\}\(\\varepsilon\_\{1\}\)\\bigr\]\.\(13\)SinceHb​\(ε1\)<Hb​\(ε0\)H\_\{b\}\(\\varepsilon\_\{1\}\)<H\_\{b\}\(\\varepsilon\_\{0\}\)forε1<ε0<1/2\\varepsilon\_\{1\}<\\varepsilon\_\{0\}<1/2, the optimum isq∗=Bq^\{\*\}=B\(review as much as the budget allows\), giving:

Cdel​\(B\)=\(1−B\)​\[1−Hb​\(ε0\)\]\+B​\[1−Hb​\(ε1\)\]\.C\_\{\\mathrm\{del\}\}\(B\)=\(1\-B\)\\bigl\[1\-H\_\{b\}\(\\varepsilon\_\{0\}\)\\bigr\]\+B\\bigl\[1\-H\_\{b\}\(\\varepsilon\_\{1\}\)\\bigr\]\.If the action log is not available to the decoder or evaluator, the stationary randomized policy induces the averaged BSC with crossoverεavg​\(q\)=\(1−q\)​ε0\+q​ε1\\varepsilon\_\{\\mathrm\{avg\}\}\(q\)=\(1\-q\)\\varepsilon\_\{0\}\+q\\varepsilon\_\{1\}and capacity1−Hb​\(εavg​\(q\)\)1\-H\_\{b\}\(\\varepsilon\_\{\\mathrm\{avg\}\}\(q\)\), which is generally different\. The paper uses the revealed\-action form because governance actions are logged by construction\.

Theorem 1\[Theorem\]: Delegation Capacity*Assumptions:*Each nodevvoperates as a governed delegation channel \(Definition 1\)\. Review actions are stationary, symbolwise, independent ofXX, and recorded; the decoder/evaluator observes the action log\. Nodes process independently conditional on their input\. Tasks are processed in epochs ofnnsymbols, with0≤ε1<ε0<1/20\\leq\\varepsilon\_\{1\}<\\varepsilon\_\{0\}<1/2\.*Achievability:*For any source with entropy rateH​\(X\)<Cdel​\(B\)H\(X\)<C\_\{\\mathrm\{del\}\}\(B\), there exists an encoder, a governance policy satisfying the stationary symbolwise budget, and a decoder such that the block error probabilityPe\(n\)→0P\_\{e\}^\{\(n\)\}\\to 0asn→∞n\\to\\infty\.*Converse:*IfH​\(X\)\>Cdel​\(B\)H\(X\)\>C\_\{\\mathrm\{del\}\}\(B\), then for every encoder, stationary symbolwise admissible governance policy, and decoder:lim infn→∞Pe\(n\)\>0\\liminf\_\{n\\to\\infty\}P\_\{e\}^\{\(n\)\}\>0\.For a cascade ofDDgoverned delegation channels:Cdelchain​\(B1,…,BD\)≤minv⁡Cv​\(Bv\)C\_\{\\mathrm\{del\}\}^\{\\mathrm\{chain\}\}\(B\_\{1\},\\ldots,B\_\{D\}\)\\leq\\min\_\{v\}C\_\{v\}\(B\_\{v\}\), with equality in the binary memoryless cascade under end\-to\-end coding across the composed channel\.

*Proof of achievability\.*

*Step A \(fix a stationary governance law\)\.*Choose a stationary action lawPA=Bern​\(q\)P\_\{A\}=\\mathrm\{Bern\}\(q\)withq≤Bq\\leq B\. Because the action is recorded, the channel output is\(Y,A\)\(Y,A\)and the channel law factorizes as:

PY,A\|X\(q\)​\(y,a\|x\)=PA​\(a\)​PY\|X,A=a​\(y\|x\)\.P\_\{Y,A\|X\}^\{\(q\)\}\(y,a\|x\)=P\_\{A\}\(a\)\\,P\_\{Y\|X,A=a\}\(y\|x\)\.The mutual information isI​\(X;Y,A\)=I​\(X;Y∣A\)I\(X;Y,A\)=I\(X;Y\\mid A\)becauseAAis independent ofXX\.

*Step B \(standard random coding\)\.*For any rateR<I​\(X;Y,A\)PX,PA,PY\|X,AR<I\(X;Y,A\)\_\{P\_\{X\},P\_\{A\},P\_\{Y\|X,A\}\}, standard random coding on the induced DMCPY,A\|X\(q\)P\_\{Y,A\|X\}^\{\(q\)\}produces a sequence of\(2n​R,n\)\(2^\{nR\},n\)block codes withPe\(n\)≤2−n​E​\(R\)P\_\{e\}^\{\(n\)\}\\leq 2^\{\-nE\(R\)\}, whereE​\(R\)\>0E\(R\)\>0is the reliability function of the induced DMC\(Cover and Thomas,[2006](https://arxiv.org/html/2606.15563#bib.bib7)\), Theorem 7\.7\.1\.

*Step C \(optimize over governance\)\.*Take the supremum over admissibleq∈\[0,B\]q\\in\[0,B\]\. The maximum is attained atq∗=Bq^\{\*\}=B, yieldingCdel​\(B\)C\_\{\\mathrm\{del\}\}\(B\)as in equation \([13](https://arxiv.org/html/2606.15563#S1.E13)\)\.

*Proof of converse\.*LetX^n\\hat\{X\}^\{n\}be the decoded task sequence after the full pipeline\. IfPe\(n\)→0P\_\{e\}^\{\(n\)\}\\to 0, Fano’s inequality\(Cover and Thomas,[2006](https://arxiv.org/html/2606.15563#bib.bib7)\), Theorem 2\.10\.1, gives:

n​R=H​\(Xn\)≤I​\(Xn;X^n\)\+n​ϵn,ϵn→0\.nR=H\(X^\{n\}\)\\leq I\(X^\{n\};\\hat\{X\}^\{n\}\)\+n\\epsilon\_\{n\},\\quad\\epsilon\_\{n\}\\to 0\.For the pipeline Markov chain with recorded governance actionsXn→\(Y1n,A1n\)→\(Y2n,A2n\)→⋯→\(YDn,ADn\)→X^nX^\{n\}\\to\(Y\_\{1\}^\{n\},A\_\{1\}^\{n\}\)\\to\(Y\_\{2\}^\{n\},A\_\{2\}^\{n\}\)\\to\\cdots\\to\(Y\_\{D\}^\{n\},A\_\{D\}^\{n\}\)\\to\\hat\{X\}^\{n\}, the DPI gives:

I​\(Xn;X^n\)≤minv⁡I​\(Xn;Yvn,Avn\)\.I\(X^\{n\};\\hat\{X\}^\{n\}\)\\leq\\min\_\{v\}I\(X^\{n\};Y\_\{v\}^\{n\},A\_\{v\}^\{n\}\)\.For each nodevv, under admissible action budgetBvB\_\{v\}:1n​I​\(Xn;Yvn,Avn\)≤Cv​\(Bv\)\\frac\{1\}\{n\}I\(X^\{n\};Y\_\{v\}^\{n\},A\_\{v\}^\{n\}\)\\leq C\_\{v\}\(B\_\{v\}\)\. ThereforeR≤minv⁡Cv​\(Bv\)R\\leq\\min\_\{v\}C\_\{v\}\(B\_\{v\}\)\.□\\square

Corollary \(quality target\)\.The theorem is stated in bits\. To translate to the operational quality targetpminp\_\{\\min\}: for a uniform binary source, mutual information isI​\(X;Y\)=1−Hb​\(1−p\)I\(X;Y\)=1\-H\_\{b\}\(1\-p\)whereppis the symbol correctness probability\. Therefore, a quality targetpminp\_\{\\min\}is achievable if and only if:

1−Hb​\(1−pmin\)<Cdel​\(B\)\.1\-H\_\{b\}\(1\-p\_\{\\min\}\)<C\_\{\\mathrm\{del\}\}\(B\)\.Equivalently:pmin<1−Hb−1​\(1−Cdel​\(B\)\)p\_\{\\min\}<1\-H\_\{b\}^\{\-1\}\(1\-C\_\{\\mathrm\{del\}\}\(B\)\)\. This is the bridge between the information\-theoretic capacity \(in bits\) and the operational quality ceiling \(inσ\\sigma\) used throughout the paper\.

*Remark \(DAGs\)\.*For a general DAG, the chain bottleneck generalizes to a cut\-set bound:

R≤min𝒞∈cuts​\(G\)​∑v∈𝒞Cv​\(Bv\),R\\;\\leq\\;\\min\_\{\\mathcal\{C\}\\in\\mathrm\{cuts\}\(G\)\}\\sum\_\{v\\in\\mathcal\{C\}\}C\_\{v\}\(B\_\{v\}\),under the conditional independence assumptions needed for network factorization\(Cover and Thomas,[2006](https://arxiv.org/html/2606.15563#bib.bib7)\), the network\-information\-theory cut\-set viewpoint ofCover and Thomas\([2006](https://arxiv.org/html/2606.15563#bib.bib7), Section 15\.10\)\. When upstream errors are correlated \(diamond pattern\), the effective capacity is lower in theory; in the mean\-field simulator, the average\-quality penalty is negligible \(<1%<1\\%\), but conditional analysis reveals a1\.4×1\.4\\timesfragility ratio: quality atDDdrops29%29\\%when the shared sourceAAfails \(Section 3\)\.

*Remark \(Fisher prioritization\)\.*The Fisher\-priority review rule \(ϕπ​\(v\)=g​\(σraw​\(v\)\)\\phi\_\{\\pi\}\(v\)=g\(\\sigma\_\{\\mathrm\{raw\}\}\(v\)\)\) is not part of the capacity theorem\. It is a separate result: under a quadratic local loss, Fisher\-prioritized review maximizes the first\-order marginal gain in mutual information per unit review cost\. The practical advantage of Fisher priority over uniform review depends on the heterogeneity ofσ\\sigmaacross scope points; in settings with homogeneous competence, the improvement is negligible\. This belongs as an operational prescription \(Section 4\), not as the constructive policy in the achievability proof\.

### Process entropy and process capacity

The delegation capacity bounds quality for a given pipeline\. But real workflows vary in*complexity*—the number of decisions agents make, the stochasticity of routing, the asynchrony of events\. A deterministic pipeline \(every item follows the same path\) is easier to govern than a stochastic one \(agents choose tools and routes on the fly\)\.

We define theprocess entropyof a workflowWWas the total Shannon entropy of the execution decisions:

H​\(W\)=H​\(routing\)\+H​\(tool calls\)\+H​\(timing\),H\(W\)=H\(\\text\{routing\}\)\+H\(\\text\{tool calls\}\)\+H\(\\text\{timing\}\),\(14\)where each term is defined precisely:H​\(routing\)=∑vH​\(Pv\)H\(\\text\{routing\}\)=\\sum\_\{v\}H\(P\_\{v\}\), wherePvP\_\{v\}is the probability distribution over outgoing edges at nodevv\(deterministic routing hasH​\(Pv\)=0H\(P\_\{v\}\)=0; an agent that chooses amongkkroutes uniformly hasH​\(Pv\)=log2⁡kH\(P\_\{v\}\)=\\log\_\{2\}k\);H​\(tool calls\)=∑vH​\(Qv\)H\(\\text\{tool calls\}\)=\\sum\_\{v\}H\(Q\_\{v\}\), whereQvQ\_\{v\}is the distribution over tool\-call sequences at nodevv; andH​\(timing\)=∑vH​\(Tv\)H\(\\text\{timing\}\)=\\sum\_\{v\}H\(T\_\{v\}\), whereTvT\_\{v\}is the distribution over inter\-event arrival times at nodevv\(synchronous steps haveH​\(Tv\)=0H\(T\_\{v\}\)=0\)\. The additive decomposition assumes that routing, tool, and timing decisions at different nodes are*conditionally independent given the input*—a natural assumption when agents do not share hidden state\.

The following result connects capacity to workflow complexity\. Unlike Theorem 1, which is proved for the revealed\-action stationary governed delegation channel, Proposition 2 relies on a Taylor approximation and an entropy\-variance bound\. It should be read as a*local first\-order sensitivity law*: near a reference workflow regime, added process complexity acts as an approximately linear tax on achievable quality\. It captures the empirically observed linear degradation \(Section 3\), but is not intended as a globally tight bound:

Proposition 2\[Proposition\]: Local Process\-Complexity Sensitivity LawFor a pipeline operating near a reference workflow regime, with channel capacityCopC\_\{\\mathrm\{op\}\}and process entropyH​\(W\)H\(W\)measured relative to that regime:σraw∗​\(output∣W\)≥Cop−λ​H​\(W\),\\sigma^\{\*\}\_\{\\mathrm\{raw\}\}\(\\mathrm\{output\}\\mid W\)\\;\\geq\\;C\_\{\\mathrm\{op\}\}\-\\lambda\\,H\(W\),\(15\)whereλ\\lambdais thegovernance gap coefficient\.*In words:*near a fixed operating regime, each additional bit of workflow complexity reduces achievable output quality by approximatelyλ\\lambda\. This is a local sensitivity, not a universal law: it says how fast quality erodes around the current regime, not how every workflow behaves globally\.

*Proof\.*Model the routing state as a random variableRRwith entropyH​\(R\)=H​\(W\)H\(R\)=H\(W\), so the delegation channel’s transition matrixp​\(Y\|X\)p\(Y\|X\)depends onRR\. Consider a local perturbation around a reference routing regimer0r\_\{0\}where the conditional capacity is twice differentiable\. Using the standard chain\-rule identityI​\(X;Y,R\)=I​\(X;R\)\+I​\(X;Y∣R\)=I​\(X;Y\)\+I​\(X;R∣Y\)I\(X;Y,R\)=I\(X;R\)\+I\(X;Y\\mid R\)=I\(X;Y\)\+I\(X;R\\mid Y\), we obtain

I​\(X;Y\)=I​\(X;Y∣R\)\+I​\(X;R\)−I​\(X;R∣Y\)\.I\(X;Y\)=I\(X;Y\\mid R\)\+I\(X;R\)\-I\(X;R\\mid Y\)\.For exogenous workflow randomness,RRis independent of the sourceXX, soI​\(X;R\)=0I\(X;R\)=0and thereforeI​\(X;Y\)≤I​\(X;Y∣R\)I\(X;Y\)\\leq I\(X;Y\\mid R\)\. Averaging over routing realizations gives

I​\(X;Y\)≤𝔼R​\[I​\(X;Y∣R\)\]≤Cop,I\(X;Y\)\\leq\\mathbb\{E\}\_\{R\}\[I\(X;Y\\mid R\)\]\\leq C\_\{\\mathrm\{op\}\},where the second inequality uses Fano’s inequality with side information\(Cover and Thomas,[2006](https://arxiv.org/html/2606.15563#bib.bib7)\): for each realizationrr, the achievable quality is bounded by the channel capacity,I​\(X;Y∣R=r\)≤CopI\(X;Y\\mid R=r\)\\leq C\_\{\\mathrm\{op\}\}\. The gap betweenI​\(X;Y\)I\(X;Y\)andCopC\_\{\\mathrm\{op\}\}arises because the receiver \(corrector\) must allocate review effort across all possible routing realizations rather than the optimal allocation for the actual realization\. By a second\-order Taylor expansion of the conditional capacity around the reference routing regime, the first\-order term vanishes at the local optimum and the residual gap is bounded by:

Cop−I​\(X;Y\)≤12​maxr⁡\|C′′​\(r\)\|​Var​\(R\)≤λ​H​\(R\),C\_\{\\mathrm\{op\}\}\-I\(X;Y\)\\leq\\frac\{1\}\{2\}\\,\\max\_\{r\}\\\!\\bigl\|C^\{\\prime\\prime\}\(r\)\\bigr\|\\;\\mathrm\{Var\}\(R\)\\leq\\lambda\\,H\(R\),where the last inequality should be read as a local normalized entropy–dispersion bound: for the finite routing alphabets used here, the routing coordinate is scaled to bounded support and the proportionality constant between dispersion and entropy is absorbed intoλ\\lambda\. Thusλ\\lambdais a local property of the channel and routing parametrization, not a universal constant\. Sinceσraw∗\\sigma^\{\*\}\_\{\\mathrm\{raw\}\}is a monotone function ofI​\(X;Y\)I\(X;Y\), the bound transfers:σraw∗≥Cop−λ​H​\(W\)\\sigma^\{\*\}\_\{\\mathrm\{raw\}\}\\geq C\_\{\\mathrm\{op\}\}\-\\lambda H\(W\)\.

*Worked example:*A pipeline withC=0\.80C=0\.80,pmin=0\.50p\_\{\\min\}=0\.50,λ=0\.02\\lambda=0\.02/bit can handle workflows with up toHmax=\(0\.80−0\.50\)/0\.02=15H\_\{\\max\}=\(0\.80\-0\.50\)/0\.02=15bits of process entropy—roughly 15 independent binary routing decisions, or2152^\{15\}equally likely routing traces—before quality drops below threshold\.

*Remark\.*The governance gapλ\\lambdais now a*derived*quantity—the curvature of the conditional capacity as a function of routing state—though in practice it is easier to measure empirically \(Experiment 7 reportsλ≈0\.02\\lambda\\approx 0\.02/bit\)\. Better governance \(Fisher\-prioritized routing\) reducesλ\\lambdaby concentrating review on the most uncertain outcomes, partially compensating for routing uncertainty\.

The maximum process complexity at qualitypminp\_\{\\min\}is therefore:

Hmax​\(pmin\)=Cop−pminλ\.H\_\{\\max\}\(p\_\{\\min\}\)=\\frac\{C\_\{\\mathrm\{op\}\}\-p\_\{\\min\}\}\{\\lambda\}\.This is thedelegation process capacity: the maximum entropy of a workflow that the pipeline can handle while maintaining quality\. Better governance \(higherK/NK/N, Fisher\-prioritized routing\) reducesλ\\lambda, allowing more complex workflows at the same quality\.

It is useful to summarize these constraints by a singleeffective autonomy buffer:

Beff=Cop−pmin−λ​H​\(W\)\.B\_\{\\mathrm\{eff\}\}=C\_\{\\mathrm\{op\}\}\-p\_\{\\min\}\-\\lambda H\(W\)\.\(16\)This buffer is the central geometric quantity of the theory\. WhenBeff\>0B\_\{\\mathrm\{eff\}\}\>0, delegated autonomy is feasible under the model\. WhenBeff=0B\_\{\\mathrm\{eff\}\}=0, the pipeline is exactly at its autonomy cliff\. WhenBeff<0B\_\{\\mathrm\{eff\}\}<0, no governance policy can sustain the requested quality target\. The autonomy time in the next subsection is the temporal survivability of this same buffer under drift and noise\. Table[2](https://arxiv.org/html/2606.15563#S1.T2)summarizes the structural correspondence between Shannon’s information theory and the delegation framework\. Note: this is a*structural*correspondence, not a formal equivalence\. The formal theorem \(Theorem 1\) is stated for the revealed\-action stationary governed delegation channel, not by analogy\.

Table 2:Structural correspondence between information theory and delegation theory \(see Theorem 1 for the formal result\)\.
### Autonomy time

The most operationally relevant quantity is theautonomy timeTauto∗T^\{\*\}\_\{\\mathrm\{auto\}\}: the expected duration a pipeline can operate without human intervention before quality drops belowpminp\_\{\\min\}\.

In the GSPN framework, the system’s state drifts due to distribution shift \(agents’ effective skill degrades at rateμ\\mu\) and fluctuates due to finite sampling \(noise varianceν2\\nu^\{2\}\)\. We model the output quality as a stochastic process:

d​σ​\(t\)=−μeff​d​t\+νeff​d​W​\(t\),\\mathrm\{d\}\\sigma\(t\)=\-\\mu\_\{\\mathrm\{eff\}\}\\,\\mathrm\{d\}t\+\\nu\_\{\\mathrm\{eff\}\}\\,\\mathrm\{d\}W\(t\),whereW​\(t\)W\(t\)is a standard Wiener process\. The autonomy time is the*first\-passage time*ofσ​\(output,t\)\\sigma\(\\text\{output\},t\)belowpminp\_\{\\min\}, starting from the effective operating pointσ0\\sigma\_\{0\}, so that the initial distance to the intervention threshold is precisely the effective autonomy bufferBeffB\_\{\\mathrm\{eff\}\}from equation[16](https://arxiv.org/html/2606.15563#S1.E16)\.

For this drifted Brownian motion, in the ideal constant\-drift model withμeff\>0\\mu\_\{\\mathrm\{eff\}\}\>0, the mean first\-passage time fromσ0\\sigma\_\{0\}to the absorbing barrier atpminp\_\{\\min\}is \(see e\.g\.Redner[2001](https://arxiv.org/html/2606.15563#bib.bib21)\):

Proposition 3\[Proposition\]: Autonomy TimeTauto∗=Cop−pmin−λ​H​\(W\)μeff\.T^\{\*\}\_\{\\mathrm\{auto\}\}=\\frac\{C\_\{\\mathrm\{op\}\}\-p\_\{\\min\}\-\\lambda\\,H\(W\)\}\{\\mu\_\{\\mathrm\{eff\}\}\}\.\(17\)*In words:*the maximum expected time the pipeline can operate without human intervention\. The numerator is the effective autonomy buffer from equation[16](https://arxiv.org/html/2606.15563#S1.E16); the denominator is the drift rate\. Autonomy is proportional to the buffer and inversely proportional to the drift\.

In the ideal Brownian model with constant negative drift, the mean isBeff/μeffB\_\{\\mathrm\{eff\}\}/\\mu\_\{\\mathrm\{eff\}\}and the diffusion term controls the dispersion of hitting times rather than the mean\. When drift is negligible and noise dominates, the mean first\-passage time of an unbounded Brownian motion to a one\-sided barrier is not finite; nevertheless, typical times such as medians and fixed quantiles scale as\(σ0−pmin\)2/νeff2\(\\sigma\_\{0\}\-p\_\{\\min\}\)^\{2\}/\\nu^\{2\}\_\{\\mathrm\{eff\}\}\. The experiments \(Section 3\) are therefore interpreted as testing the drift\-dominated scalingTauto∗∝1/μT^\{\*\}\_\{\\mathrm\{auto\}\}\\propto 1/\\mu, not as a full distributional first\-passage theorem\.

Five factors of autonomy\.Tauto∗T^\{\*\}\_\{\\mathrm\{auto\}\}increases when:

1. 1\.CopC\_\{\\mathrm\{op\}\}is high \(better agents, correctors, topology\)\.
2. 2\.pminp\_\{\\min\}is low \(relaxed quality requirements\)\.
3. 3\.H​\(W\)H\(W\)is low \(simpler, more deterministic workflows\)\.
4. 4\.λ\\lambdais small \(better governance compresses the gap\)\.
5. 5\.μeff\\mu\_\{\\mathrm\{eff\}\}andνeff2\\nu^\{2\}\_\{\\mathrm\{eff\}\}are small \(stable models, large scope, frequent observations\)\.

The minimum human intervention frequency at each node isf​\(v\)=1/Tauto∗​\(v\)f\(v\)=1/T^\{\*\}\_\{\\mathrm\{auto\}\}\(v\), and the optimal intervention schedule minimizes total human review cost:

min​∑vf​\(v\)×cost​\(v\)subject tof​\(v\)≥1/Tauto∗​\(v\)​∀v\.\\min\\sum\_\{v\}f\(v\)\\times\\mathrm\{cost\}\(v\)\\quad\\text\{subject to\}\\quad f\(v\)\\geq 1/T^\{\*\}\_\{\\mathrm\{auto\}\}\(v\)\\;\\;\\forall\\,v\.This is a linear program, solvable in polynomial time\. Nodes with shortTauto∗T^\{\*\}\_\{\\mathrm\{auto\}\}\(high drift, high process entropy, low capacity\) receive more frequent human review; nodes with longTauto∗T^\{\*\}\_\{\\mathrm\{auto\}\}can operate autonomously for extended periods\.

The convergence time for the pipeline to reach anϵ\\epsilon\-neighborhood of operational capacity after deployment \(or after a modification\) is:

Tcal=ln⁡\(1/ϵ\)λ1,T\_\{\\mathrm\{cal\}\}=\\frac\{\\ln\(1/\\epsilon\)\}\{\\lambda\_\{1\}\},whereλ1\\lambda\_\{1\}is the spectral gap of the coupled GSPN generator matrix\. The expected number of observations collected over that interval is approximatelyN​η¯​TcalN\\bar\{\\eta\}T\_\{\\mathrm\{cal\}\}, whereη¯\\bar\{\\eta\}is the average observation rate per scope point\. Fan\-out accelerates convergence \(parallel observations\); resource contention decelerates it \(queuing delays\)\.

![Refer to caption](https://arxiv.org/html/2606.15563v1/figure1_dag_v3.png)Figure 1:The SDLC delegation graph \(σskill=0\.55\\sigma\_\{\\mathrm\{skill\}\}=0\.55,c=0\.65c=0\.65, product aggregation at merges\)\.\(A\)Agent nodes with correctors; blue arrows show output→\\toinput linkage\. The masking indexM∗M^\{\*\}increases with depth, reaching2\.772\.77at the merge gate \(Mtotal∗=14\.3M^\{\*\}\_\{\\mathrm\{total\}\}=14\.3\)\.\(B\)Total masking compounds super\-multiplicatively with depth: linked chains \(solid\) grow faster than the independent prediction \(dashed\)\.\(C\)Removing the reviewer’s corrector cascadesΔ​σraw\\Delta\\sigma\_\{\\mathrm\{raw\}\}to three downstream branches simultaneously\. All values are specific to the stated parameters\.

## 2Connections to Existing Frameworks

The MSO connects to established frameworks in economics and security\. Table[3](https://arxiv.org/html/2606.15563#S2.T3)summarizes the correspondences; we develop the two strongest below\.

Table 3:Correspondences between established frameworks and MSO constructs\.### Principal–agent theory

In the standard principal–agent model\(Jensen and Meckling,[1976](https://arxiv.org/html/2606.15563#bib.bib14);Holmström,[1979](https://arxiv.org/html/2606.15563#bib.bib13)\), the principal designs a contractw​\(y\)w\(y\)mapping observable output to payment, under moral hazard: the agent’s effort is unobservable\. The MSO produces a correspondence: the contractw​\(y\)w\(y\)maps to the governance functionalG​\(σ\)G\(\\sigma\); the outputyymaps toσraw\\sigma\_\{\\mathrm\{raw\}\}; moral hazard maps to masking \(M∗\>1M^\{\*\}\>1\)\. The mechanism collapses to a single monotone functionGGbecause the agent’s “type” is revealed by outcomes the corrector observes—no message space or revelation principle is needed\(Myerson,[1981](https://arxiv.org/html/2606.15563#bib.bib19)\)\.

The MSO also identifies a form of moral hazard absent from classical theory: the information asymmetry created by the*corrector*, not the agent\. The dualσ\\sigmaprescription resolves this by changing how information is*recorded*, not how agents are*rewarded*\. In a complementary direction,Fudenberg and Liang\([2025](https://arxiv.org/html/2606.15563#bib.bib11)\)characterize optimal information disclosure to a single delegate whose alignment is uncertain; the MSO addresses the orthogonal problem of allocating oversight across a*pipeline*of delegates\.

### The principle of least privilege

Saltzer and Schroeder’s\([1975](https://arxiv.org/html/2606.15563#bib.bib22)\)principle—“operate using the least set of privileges necessary”—has remained a heuristic for fifty years\. The MSO provides its variational formulation: equation \([2](https://arxiv.org/html/2606.15563#S1.E2)\) minimizes total oversight cost subject to delivery, and the water\-filling solution \([8](https://arxiv.org/html/2606.15563#S1.E8)\) gives the computable policy:α∗​\(x\)=min⁡\(αmax​\(x\),λ​σ​\(x\)/\(2​g​\(x\)\)\)\\alpha^\{\*\}\(x\)=\\min\(\\alpha\_\{\\max\}\(x\),\\lambda\\sigma\(x\)/\(2\\sqrt\{g\(x\)\}\)\), determined from the system’s own logs and the quality requirementpminp\_\{\\min\}\.

*Scope of the correspondences\.*The strongest links are to principal–agent theory and least privilege\. Other analogies—for example to BDI architectures or span\-of\-control theory—are possible, but they are interpretive rather than theorem\-bearing and are not needed for the main results of the paper\.

### Boundary conditions of the theory

The correspondences above are strongest where the paper’s own assumptions are strongest\. The current formulation is built on binary outcomes, memoryless node behavior, conditional independence across nodes, and product or min aggregation at merges\. Several extensions are plausible, but they are not yet proved here\. Continuous or multiclass outcomes would replace the Bernoulli manifold by a higher\-dimensional statistical manifold and alter both the Fisher volume term and the channel model\. Shared hidden state or correlated latent failures would weaken the conditional\-independence assumptions behind the current DAG analysis\. Adaptive governance policies with feedback beyond stationary symbolwise review would require extending Theorem 1 from the present governed channel to a richer controlled channel with memory\. The framework is therefore best read as a principled base case whose objects and limits are explicit, not as a universal theory of all delegation\.

## Related Work

The MSO draws on and extends work across several communities\. We position it relative to the most relevant strands\. The discussion is selective: we emphasize the strongest intellectual neighbors and keep the focus on frameworks that change how delegated autonomy should be designed or measured\.

### AI safety and scalable oversight

The masking problem is closely related to the*scalable oversight*challenge in AI alignment: as AI systems become more capable, human overseers struggle to evaluate outputs they cannot themselves produce\(Amodei et al\.,[2016](https://arxiv.org/html/2606.15563#bib.bib4)\)\. Reinforcement learning from human feedback \(RLHF\) uses human preferences as a training signal, but the preference model can itself drift or be gamed—a form of corrector drift in our framework\. Constitutional AI\(Bai et al\.,[2022](https://arxiv.org/html/2606.15563#bib.bib5)\)attempts to reduce dependence on human feedback by having the model critique its own outputs, but self\-critique is a dual\-role delegation: each unit of capacity spent on review is a unit not spent on generation, and the incentive structure biases toward the self\-rewarding generative role\. The MSO provides a formal criterion—M∗M^\{\*\}—for detecting when the oversight process is masking rather than revealing the agent’s true competence\. Recent work on scaling laws for scalable oversight\(Engels et al\.,[2025](https://arxiv.org/html/2606.15563#bib.bib9)\)derives empirical relationships between oversight success and capability gaps; the MSO provides the principled allocation theory that such empirical observations currently lack\. ALARA\-style agent harness engineering similarly emphasizes practical constraints in portable, composable multi\-agent teams\(Agostino and D’Souza,[2026](https://arxiv.org/html/2606.15563#bib.bib1)\); MSO supplies a quantitative governance objective for such harnesses\.

### LLM\-as\-judge and evaluation

The practice of using one LLM to evaluate another’s output\(Zheng et al\.,[2023](https://arxiv.org/html/2606.15563#bib.bib31)\)is a delegation with an autonomous corrector\. Recent work has documented systematic biases in LLM judges: position bias, verbosity bias, and self\-enhancement bias\. In the MSO framework, these biases manifest as a non\-uniform catch ratec​\(x\)c\(x\)across the scope, producing regions whereM∗M^\{\*\}is high \(the judge is lenient\) and regions whereM∗M^\{\*\}is low \(the judge is strict\)\. The dualσ\\sigmadiagnostic detects this heterogeneity: whenσraw​\(x\)\\sigma\_\{\\mathrm\{raw\}\}\(x\)varies across regions whileσcorr​\(x\)\\sigma\_\{\\mathrm\{corr\}\}\(x\)does not, the judge is applying different standards in different parts of the scope\.

### Delegated AI and agent orchestration

Research on delegated AI and agent orchestration has long addressed how to distribute tasks and coordination, from contract nets\(Smith,[1980](https://arxiv.org/html/2606.15563#bib.bib23)\)and BDI architectures\(Rao and Georgeff,[1991](https://arxiv.org/html/2606.15563#bib.bib20)\)to holonic architectures and classical span\-of\-control questions\(Urwick,[1956](https://arxiv.org/html/2606.15563#bib.bib26)\)\. What those frameworks usually lack is a governing optimality principle for autonomy allocation itself\. The MSO supplies such a principle and turns heuristic questions—where to place evaluation, how to size oversight, when to decompose scope—into computable quantities\.

### Agentic AI frameworks and orchestration

Recent orchestration frameworks make delegated pipelines easy to build: AutoGen\(Wu et al\.,[2023](https://arxiv.org/html/2606.15563#bib.bib28)\), CrewAI\(Moura,[2024](https://arxiv.org/html/2606.15563#bib.bib18)\), LangGraph\(LangChain,[2024](https://arxiv.org/html/2606.15563#bib.bib15)\), Google ADK\(Google,[2025](https://arxiv.org/html/2606.15563#bib.bib12)\), and task\-adaptive orchestration systems such as AdaptOrch\(Yu,[2026](https://arxiv.org/html/2606.15563#bib.bib30)\)\. They do not by themselves specify how much autonomy each node should receive, where correctors belong, or when intervention becomes necessary\.Tomašev et al\.\([2026](https://arxiv.org/html/2606.15563#bib.bib25)\)propose a comprehensive conceptual framework for AI delegation covering sub\-delegation, permission attenuation, and reputation systems, but explicitly without formal bounds or theorems\. Static workflow verification\(Xavier et al\.,[2026](https://arxiv.org/html/2606.15563#bib.bib27)\)and resource\-bounded agent contracts\(Ye and Tan,[2026](https://arxiv.org/html/2606.15563#bib.bib29)\)are complementary: they constrain or verify agent graphs, while the MSO allocates autonomy and oversight over those graphs\. The MSO is best read as a layer above such frameworks: it treats their graphs as governance objects and supplies the formal quantities—capacity ceilings, masking diagnostics, intervention timing—that conceptual and orchestration frameworks currently lack\.

### Variational and information\-processing viewpoints

Variational principles also appear in predictive\-processing and free\-energy formulations\(Friston,[2010](https://arxiv.org/html/2606.15563#bib.bib10)\)\. The MSO uses a variational form in a narrower engineering sense: minimize a governance functional subject to a delivery constraint\. It does not adopt the free\-energy principle as a cognitive or biological theory\.

### Information geometry

The Fisher information metric is the paper’s main geometric commitment\. Where information geometry usually appears in learning or estimation, including natural\-gradient learning\(Amari,[1998](https://arxiv.org/html/2606.15563#bib.bib2)\), we use it to allocate governance attention across a delegated task space\. We do not claim a full information\-geometric theory of delegated control; we use the Fisher geometry because it is the natural metric for the local cost of calibration\.

### Computational considerations

The Euler–Lagrange solution \([8](https://arxiv.org/html/2606.15563#S1.E8)\) is computable inO​\(N\)O\(N\)time per scope update, whereNNis the number of scope points: it requires computingσraw\\sigma\_\{\\mathrm\{raw\}\}at each point \(a running average,O​\(1\)O\(1\)per observation\) and the Fisher volume integral \(a sum over the scope,O​\(N\)O\(N\)\)\. The masking indexM∗M^\{\*\}is a single division per node\. For delegation DAGs, the principled improvement target is the local sensitivity∂Tauto∗/∂c​\(v\)\\partial T^\{\*\}\_\{\\mathrm\{auto\}\}/\\partial c\(v\)or its finite\-difference approximation; when exact sensitivities are not available, the proxy scoreS​\(v\)=DC​\(v\)×M∗​\(v\)×κ​\(v\)S\(v\)=\\mathrm\{DC\}\(v\)\\times M^\{\*\}\(v\)\\times\\kappa\(v\)remains computable after a single topological sort \(O​\(\|V\|\+\|E\|\)O\(\|V\|\+\|E\|\)\) to evaluateDC\\mathrm\{DC\}for all nodes\. The diagnostic differential\(Δ​σraw,Δ​M∗\)\(\\Delta\\sigma\_\{\\mathrm\{raw\}\},\\Delta M^\{\*\}\)is a comparison of two consecutive windows\. The theory is therefore tractable enough to support production monitoring and iterative redesign, not just offline analysis\.

### Identifiability and measurement

The theory is only useful operationally if its main quantities can be estimated from logs\. The distinction among what is directly observed, what is inferred, and what is only approximated is therefore part of the framework rather than an implementation detail\. Corrected output qualityσcorr\\sigma\_\{\\mathrm\{corr\}\}is usually directly observable from post\-review or post\-execution outcomes\. Raw competenceσraw\\sigma\_\{\\mathrm\{raw\}\}is harder: it requires either a pre\-correction slice, a shadow evaluation channel, or a model\-based estimator that reconstructs the counterfactual uncorrected outcome stream\. Catch rateccis not usually observed as a primitive quantity; it is inferred from reviewed\-error statistics and is therefore vulnerable to evaluator drift\. Process entropyH​\(W\)H\(W\)is estimated from routing, tool, and timing traces, and its quality depends on the completeness of those traces\. Drift and noise termsμeff\\mu\_\{\\mathrm\{eff\}\}andνeff\\nu\_\{\\mathrm\{eff\}\}are fitted from temporal windows and inherit the usual local\-stationarity assumptions\.

Table 4:Measurement status of the main quantities in the framework\. The point is not that every quantity is directly observed, but that the observational status of each is explicit\.This perspective also clarifies why dual tracking matters\. The theory requires two informational channels because raw competence and delivered quality are different observables\. Conflating them not only creates the masking pathology; it also destroys identifiability of the variable that the governance law actually needs\.

## 3Numerical Validation

All numerical results are reproducible via the companion packageminimal\-oversight\(delegation\-lab,[2026](https://arxiv.org/html/2606.15563#bib.bib8)\)\. The simulator uses a mean\-field ODE model with Bernoulli outcome noise overN=80N=80–144144scope points\. Unless otherwise stated,η=10\\eta=10,δ=2\\delta=2,σskill=0\.55\\sigma\_\{\\mathrm\{skill\}\}=0\.55–0\.800\.80,c=0\.65c=0\.65–0\.700\.70, andK/N=0\.50K/N=0\.50\. Each experiment runs for 200–400 steps overn=20n=20–3030independent seeds\. The corrector model is the expected\-value update

σcorr=σraw\+\(1−σraw\)​c​\(K/N\)\.\\sigma\_\{\\mathrm\{corr\}\}=\\sigma\_\{\\mathrm\{raw\}\}\+\(1\-\\sigma\_\{\\mathrm\{raw\}\}\)\\,c\\,\(K/N\)\.This is not a holdout measurement protocol: in production,σraw\\sigma\_\{\\mathrm\{raw\}\}andσcorr\\sigma\_\{\\mathrm\{corr\}\}should be measured on separate pre\- and post\-correction channels\.

### Semi\-real reconstructed workflow

We also evaluate a reconstructed software\-delivery workflow with 2,400 items over four windows\. Items pass through a generator, reviewer, three parallel checks \(test, requirements, security\), and a merge decision\. The reviewer has the highest masking and upstream leverage:σraw=0\.62\\sigma\_\{\\mathrm\{raw\}\}=0\.62,σcorr=0\.90\\sigma\_\{\\mathrm\{corr\}\}=0\.90, andM∗=1\.45M^\{\*\}=1\.45\. The workflow entropy isH​\(W\)=2\.3H\(W\)=2\.3bits; the estimated pipeline ceiling isCop≈0\.86C\_\{\\mathrm\{op\}\}\\approx 0\.86\. Forpmin=0\.75p\_\{\\min\}=0\.75andλ​H​\(W\)≈0\.046\\lambda H\(W\)\\approx 0\.046, the effective buffer isBeff≈0\.064B\_\{\\mathrm\{eff\}\}\\approx 0\.064\. Withμeff≈0\.012\\mu\_\{\\mathrm\{eff\}\}\\approx 0\.012, the implied autonomy time is about5\.35\.3time units\. The design prescription is upstream: reduce reviewer masking or simplify branch routing before spending more effort at the merge gate\.

### Controlled experiments

Table[5](https://arxiv.org/html/2606.15563#S3.T5)summarizes the controlled validation\. The experiments should be read as internal model checks rather than external field validation: they test whether the closed\-form quantities and scaling laws match the simulator generated from the same assumptions\.

Table 5:Summary of controlled validation results from the companion package\.Across grid sizes \(6×66\\times 6to20×2020\\times 20\), catch rates \(0\.300\.30–0\.900\.90\), and skill distributions \(uniform, Gaussian, bimodal\), the qualitative results persist\. Quantitative values shift as expected: higher catch rates increase masking, smaller grids increase trajectory noise, and heterogeneous skills increase variance across the scope\.

## 4Discussion

The framework has three practical implications\. First, delegated systems must separate competence from delivered quality\. A dashboard that records only corrected output quality can look healthy while the producing agent is weakening\. The immediately implementable prescription is to log both pre\-correction and post\-correction outcomes at every delegation boundary and trackM∗=σcorr/σrawM^\{\*\}=\\sigma\_\{\\mathrm\{corr\}\}/\\sigma\_\{\\mathrm\{raw\}\}\.

Second, topology is a governance variable\. In chains, masking and quality loss accumulate with depth\. In fan\-out structures, upstream correction is amplified across children\. In diamonds, average quality may hide conditional fragility caused by a shared parent\. The governance target should therefore be chosen by sensitivity—for example∂Tauto∗/∂c​\(v\)\\partial T^\{\*\}\_\{\\mathrm\{auto\}\}/\\partial c\(v\)or∂Cop/∂c​\(v\)\\partial C\_\{\\mathrm\{op\}\}/\\partial c\(v\)—rather than by local error rate alone\.

Third, autonomy has a finite buffer:

Beff=Cop−pmin−λ​H​\(W\)\.B\_\{\\mathrm\{eff\}\}=C\_\{\\mathrm\{op\}\}\-p\_\{\\min\}\-\\lambda H\(W\)\.When this quantity is positive, the pipeline has room to operate without continuous intervention; when it is near zero, small drift or added workflow complexity can push the system below target\. In the drift\-dominated approximation, the intervention cadence isTauto∗=Beff/μeffT^\{\*\}\_\{\\mathrm\{auto\}\}=B\_\{\\mathrm\{eff\}\}/\\mu\_\{\\mathrm\{eff\}\}\.

### Implementation

A minimal operational version of the theory has four steps:

1. 1\.Instrument each delegation boundary with raw and corrected outcome records\.
2. 2\.Estimateσraw\\sigma\_\{\\mathrm\{raw\}\},σcorr\\sigma\_\{\\mathrm\{corr\}\},M∗M^\{\*\},H​\(W\)H\(W\), and local drift over rolling windows\.
3. 3\.Check feasibility usingCopC\_\{\\mathrm\{op\}\}andBeffB\_\{\\mathrm\{eff\}\}before expanding autonomy\.
4. 4\.Allocate scarce human or model review to the node with the largest estimated gain inCopC\_\{\\mathrm\{op\}\}orTauto∗T^\{\*\}\_\{\\mathrm\{auto\}\}\.

This is not a replacement for safety evaluation or deployment testing; it is a calculus for deciding where such evaluation and oversight are most valuable\.

### Limitations and theoretical status

The results have different levels of rigor\. The water\-filling allocation, fixed points, masking index, capacity threshold, convergence time, and recursive operational ceiling are derived under the stated Bernoulli mean\-field assumptions\. The information\-theoretic capacity theorem is exact for the revealed\-action stationary symbolwise governed channel\. The process\-complexity law is a local Taylor approximation whose coefficientλ\\lambdais measured or estimated\. The autonomy\-time result is a drift\-dominated first\-passage scaling law; in the simulations it captures the1/μ1/\\muslope but overestimates absolute values by about 20%\.

The empirical validation is synthetic, not a production study\. The agents do not strategically adapt to the corrector; the corrector catch rate is fixed; the outcomes are binary; the nodes are conditionally independent; and the main experiments do not compare against strong baselines such as bandit allocation, queue\-aware control, active learning, or software\-testing heuristics\. These are not small caveats\. They define the current scope of the claim: this paper proposes and internally validates a base theory of governed delegation, not a universal theory of all agent coordination\.

### Conclusion

Delegated AI autonomy is governed by a three\-way tradeoff among competence, governance capacity, and workflow complexity\. The MSO makes that tradeoff computable: allocate governed delegation by a Fisher\-weighted variational principle, separate raw competence from corrected quality, check whether the system has enough capacity for the target quality, and schedule intervention according to the remaining autonomy buffer\. The central practical lesson is simple: when an AI system delegates decisions to models, tools, or evaluators, measure what the delegate can do before correction, measure what the system delivers after correction, and do not confuse the two\.

## Acknowledgments and AI Assistance Disclosure

This manuscript was written and revised with assistance from large language models and AI\-assisted coding tools, including Claude Code and Codex, using models identified by the author as Claude Opus 4\.7 and GPT\-5\.5\. These tools were used for drafting, editing, mathematical and conceptual consistency checks, LaTeX revision, simulation\-code support, and preparation of reproducibility materials\. They are tools, not authors\. The author reviewed, selected, and edited the resulting text, derivations, code, figures, and references, and takes full responsibility for the content of the paper\. Any errors, omissions, incorrect claims, or misinterpretations are the author’s alone\.

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