Autonomy Time

Paper reference: Section 1, "Autonomy time"; Proposition 3; Experiment 8

Once you know the pipeline is feasible (\(B_\text{eff} > 0\)), the operational question is: how long can it run without human intervention?

The formula

\[T_\text{auto}^* = \frac{B_\text{eff}}{\mu_\text{eff}}\]

where:

\(B_\text{eff} = C_\text{op} - p_\text{min} - \lambda H(W)\) is the effective autonomy buffer
\(\mu_\text{eff}\) is the drift rate (how fast the environment changes)

In words: autonomy time is the safety margin divided by how fast that margin erodes.

Verify the math (Motif 1 worked example)

from minimal_oversight._formulae import autonomy_time

t = autonomy_time(
    c_op=0.833,        # single-node capacity (η=10, δ=2)
    p_min=0.50,        # quality target
    lam=0.02,          # governance gap coefficient
    h_w=0.0,           # process entropy (bits)
    mu_eff=0.005,      # drift rate
)
print(f"T*_auto = {t:.0f} time units")  # 66.6

Five factors of autonomy

\(T_\text{auto}^*\) increases when:

\(C_\text{op}\) is high — better agents, better correctors, simpler topology
\(p_\text{min}\) is low — relaxed quality requirements give more margin
\(H(W)\) is low — simpler, more deterministic workflows
\(\lambda\) is small — better governance compresses the complexity tax
\(\mu_\text{eff}\) is small — stable models, large scope, frequent observations

The 1/μ scaling law

The paper's Experiment 8 confirms that \(T_\text{auto}^* \propto 1/\mu\) over two orders of magnitude of drift rate. The log-log slope is \(-0.99\) (predicted: \(-1.0\)).

This is a usable scheduling law: if you know the drift rate, you know how often to intervene.

The capacity cliff

There's a critical process entropy beyond which autonomous operation becomes impossible:

\[H_\text{crit} = \frac{C_\text{op} - p_\text{min}}{\lambda}\]

Below \(H_\text{crit}\): autonomous operation works (\(T_\text{auto}^* > 0\)). Above \(H_\text{crit}\): continuous human oversight required, regardless of governance policy.

This is a phase transition, not a gradual trade-off. As \(H(W) \to H_\text{crit}\), the buffer shrinks quadratically in \(T_\text{auto}^*\), so autonomy time collapses rapidly near the cliff.

Practical implication

Every tool-call decision, routing branch, and conditional path adds process entropy. The cliff means there is a hard limit on how complex an autonomous workflow can be within a fixed architecture. The path to more complexity runs through increasing \(C_\text{op}\) (better agents/correctors) and decreasing \(\lambda\) (better governance), not through ignoring the constraint.

Intervention scheduling

The minimum review frequency at each node is \(f(v) = 1/T_\text{auto}^*(v)\):

report = analyze_pipeline(pipeline, p_min=0.80)

for s in report.intervention_schedule:
    print(f"{s.node_name}: review every {s.t_auto:.0f} steps (rank {s.priority_rank})")

Nodes with short \(T_\text{auto}^*\) (high drift, high process entropy, low capacity) need frequent review. Nodes with long \(T_\text{auto}^*\) can operate autonomously for extended periods.