Intrinsic perturbation scale for certified oracle objectives with epigraphic information

Numerical optimal control is split between globally structured but dimensionally intractable Hamilton--Jacobi--Bellman (HJB) methods and scalable but local trajectory optimization. We introduce Featurized Occupation Measures (FOM), a finite-dimensional primal--dual interface where HJB subsolutions guide primal search and primal occupation data can inform certificate refinement. Explicit and implicit realizations, based on finite Liouville tests and rollout-sampled residuals, are proved asymptotically consistent with the exact occupation-measure LP. FOM also identifies when HJB-type certificates avoid full state-space representation. For factor graphs induced by compatible passivity-based interconnections, blockwise HJB inequalities yield block-additive OM-dual certificates, including blockwise approximate variants. The curse of dimensionality is then shifted from state dimension to interconnection topology. Approximate certificates remain reusable under time shifts and bounded model perturbations, with explicit degradation bounds. A static obstacle-avoidance benchmark illustrates how tighter certificates improve sample-based global search.

A sound runtime admission gate executes only actions it can certify, and certifies only what its observations support. This paper asks how observations should be designed to maximize the set of actions that can be safely admitted, and shows the question is not a re-vocabulary of classical design problems. First, a projection--estimability separation: decomposing a constraint normal as $c=c_{\mathrm{Range}}+c_{\ker}$ relative to an information matrix, two-point discrimination along $c$ becomes arbitrarily reliable as the budget grows whenever $c_{\mathrm{Range}}\neq 0$, while robust admission of an action with normal $c$ is impossible at every budget whenever $c_{\ker}\neq 0$; such mixed directions are generic at any deficient rank, and at full rank the decoupling is bounded by the Kantorovich ratio and diverges with the condition number. Discrimination-optimal designs, being corner solutions of a linear criterion, land in exactly this regime. Second, an active-face equivalence theorem: the permissiveness-optimal design is $L$-optimal for a target matrix generated endogenously by the action faces that become certification bottlenecks, weighted inversely by their remaining slack; single-face collapse recovers $c$-optimal and goal-oriented design exactly, and a Caratheodory argument yields a bottleneck certificate of at most $r(r+1)/2+1$ faces. Around these we assemble exact certifiability per convex contract mode, for which $κ\approx 3.29$ is derived rather than calibrated, the $\sqrt{r}$ price of contract-agnostic design, an information-to-slack transfer theorem with a curvature-budget corollary, and a two-part audit in which a design meeting every margin requirement still leaves an action face at a certification cost above ten times its testing cost, in every probe library tested.

We study the reuse of positive trial profiles under finite perturbations of nonsymmetric matrix pencils B-λG. The lower and upper cone Rayleigh levels need not coincide and are defined without requiring positive eigenvectors. In the positive orthant with positive diagonal $G$, we derive computable perturbation bounds and determine the exact worst-case trial gap over prescribed independent entrywise perturbation classes. This yields the largest uniform radius meeting a given trial-gap tolerance for fixed profiles, certifying trial-value accuracy without recomputation. Experiments on a nonnegative operator and five signed matrices compare sufficient and optimal radii, directional thresholds, and cold- and warm-start recomputation. Several signed cases exhibit severely limited uniform profile reuse despite the optimality of the radius. Exact rational checks verify the reported bounds and worst-case constructions for stored numerical inputs. A finite-budget linear program and differentiable constraints illustrate applications to verified control and learning.