Companion instrument · Paper 6, version 6

Penalty Ceiling Explorer

The enforceable multiple of actual loss is not a legal datum. It is a fixed point of D(m) = 1 + λ̄ F(m): a clause is worth the doctrinal maximum when a court recognises the interest it protects, and courts recognise it in the share F(m) of cases whose separation threshold mcrit = 1/(θ0(1−ξκ̄)) lies below the multiple the market expects. F is the paper's own case population, computed exactly. No case separates at compensation, so m = 1 is always an equilibrium; move the parameters and watch a permissive equilibrium appear, split off its watershed, and fall off its fold.

Calibrated jurisdictions:
Positions in c:
Regime—
λ̄ / λ̄†—
Equilibria—
Watershed mu—
Folds c̱ / c̄—
Maxwell point c*—
Policy threshold c†(σ, 30)—
Recovery of the permissive doctrine—
Exit of the permissive doctrine—
Fuse t*(ℓ1)—

The doctrine map

Where the curve meets the 45° line, expectation and enforcement agree. Filled dots are stable, hollow dots are watersheds; the band marks the support of the case population.

D(m) 45° basin of compensation case population

The doctrinal potential

Wells are equilibria; the ridge between them is the barrier an accident must clear. Both wells sit on the edges of the doctrinal range when the population lies inside it.

U(m) barrier out of compensation barrier out of the permissive well

The fold in c

Equilibrium ceiling as the case population shifts. Between the folds the map carries two stable answers and history decides. Hatched: the population would contain cases below one, which Definition 1 rules out — the lower fold lies there.

greatest equilibrium least equilibrium current c Maxwell c* c†(σ, 30)

The fuse

Minimal duration after which a cover withdrawal of depth ℓ1 becomes irreversible, at adjustment speed η = 0.5. Two periods for a deep shock; without bound as ℓ1 approaches the watershed. A shock that keeps ℓ1 above the watershed is reversed at any duration.

t*(ℓ1) reversed at any duration current ℓ1

What is computed. F is the exact distribution of m_crit = 1/(θ0(1−ξκ̄)) with θ0 equiprobable on {0.78, 0.80, 0.82, 0.85} and ξκ̄ uniform on the sharing range, shifted so that its mean is c; the potential U is its closed-form integral. Recovery and exit times are exact first-passage quadratures of the perturbed adjustment with reflection at 1 and 1+λ̄ (not the Kramers approximation), in periods, to be read as orders of magnitude. Adjustment speed η = 0.5. The folds are computed from g0(u) = 1 + λ̄F0(u) − u; c* is 1 + λ̄/2 whenever the population lies inside the doctrinal range there, and found numerically otherwise; c† is where the expected exit time of the permissive doctrine equals 30 periods.

Version. Version 6 of the paper (September 2026) replaced the logistic stand-in of earlier versions by this population, which moved the watershed from 1.95 to 1.77 and made negotiated German contracting multiple rather than determinate; this page was rebuilt on it and reproduces the paper's numbers at printed precision. The page is a companion, not the source of record: every number in the paper comes from population.py, make_doctrine.py and make_stochastic.py in the replication package, doi:10.6084/m9.figshare.33212916.

Cite. Companion to Andreas Bauer, Doctrine as a Fixed Point: A Formal Model of the Enforceable Penalty Ceiling when Human Oversight of AI Must Remain Effective, working paper, version 6, September 2026. Source and verification script: github.com/Tafew/doctrine-fixed-point; licensed CC BY 4.0.