Observation Design for Certified Control Authority: Projection--Estimability Separation and Active-Face Equivalence

Guangxi Wan; Hualong Du; Yuqi Liu; Qingwei Dong; Qingxin Li; Hongfei Bai; Peng Zeng

cs.ROSep 5, 2026

Observation Design for Certified Control Authority: Projection--Estimability Separation and Active-Face Equivalence

Authors: Guangxi Wan, Hualong Du, Yuqi Liu, Qingwei Dong, Qingxin Li, Hongfei Bai, Peng Zeng

Organizations: State Key Laboratory of Robotics and Intelligent Systems, Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang 110016, China · University of Chinese Academy of Sciences, Beijing 100049, China

Abstract

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.

Figures & tables

Literature	Object	Design objective	Output	Hard admission guarantee
Distinguishability (EIT) [ 3 , 4 , 6 ]	two states	maximize distinguishability	detectability	no
$T$ -optimal discrimination [ 7 , 8 ]	rival models	maximize discriminating power	model choice	no
Goal-oriented OED [ 9 , 10 , 11 ]	downstream QoI	minimize QoI uncertainty	prediction uncertainty	no
$D/G/c/L$ -optimal OED [ 12 , 13 , 14 , 15 , 17 ]	parameters, contrasts	estimation information	covariance	no
This paper	contract $+$ action set	information sufficient for admission	certified action set	yes; $(\alpha,\beta)$ interface of [ 1 ]

Table 1: Position. The last column is discharged in Sections 4 and 5 .

library	$n_{f}{=}1$	$2$	$3$	$4$	$5$	$6$	$7$	peak
mixed (original)	$0.00$	$0.19$	$0.34$	$0.37$	$0.29$	$0.14$	$0.00$	$4$
full-rank enriched ( $+0.1\%$ scan)	$0.00$	$0.00$	$0.00$	$0.00$	$0.00$	$0.00$	$0.00$	—
leave out slope sensing	$0.00$	$0.00$	$0.00$	$0.00$	$0.00$	$0.00$	$0.00$	—
leave out curvature	$0.00$	$0.19$	$0.34$	$0.37$	$0.29$	$0.14$	$0.00$	$4$
leave out full scan	$0.43$	$0.33$	$0.37$	$0.37$	$0.29$	$0.14$	$0.00$	$1$
leave out mean-gap probe	$0.00$	$0.19$	$0.34$	$0.37$	$0.29$	$0.14$	$0.00$	$4$

Table 2: Exact-failure rate

p_{\mathrm{fail}}(n_{f})

over all

127

face subsets, across library perturbations. The peak location is library dependent and the profile need not be interior-peaked; the rotated control preserves every probe’s rank and spectrum and changes only the alignment with the contract normals.

Figure 1: Exact-failure rate against contract size across library perturbations. The interior peak of the original library (black) is not reproduced by the narrowband-heavy library (monotone to one) or by the library without a full angular scan, and vanishes entirely once every probe carries

0.1\%

of a full scan—although near-inestimability does not vanish with it.

Figure 2: Budget to test a face against budget to certify an action on it, as the constraint normal rotates from a well-transmitted mode (

\theta=0

) into the attenuated subspace (

\theta=90^{\circ}

); slack

0.70

decision units,

D

-optimal design, displacement-only library. The decoupling ratio

\chi

(dashed, right axis) peaks at intermediate angles rather than at the extreme: a normal lying wholly in the weak subspace is also hard to test, so both budgets grow together, whereas a mixed normal is cheap to test and ruinous to certify.

Figure 3: Verification of the equivalence condition ( 2 ) at the permissiveness optimum: the generalized sensitivity meets the threshold exactly on the support of

\xi^{\star}

(to

10^{-8}

) and lies strictly below it on every other excitation.

Safety-Critical Control

Safety-critical control designs controllers that keep physical systems within hard safety constraints despite uncertainty, disturbances, and limited actuation.

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Sep 5, 2026

Generator-Independent Runtime Assurance under Partial Observation

Proposal-based controllers---learned policies, language-model planners, and other black-box generators---are increasingly deployed behind runtime verification gates. We ask when the closed-loop safety guarantee decouples from the generator. The prevailing per-candidate certification pattern does not compose: under retry or best-of-

k

selection a per-candidate false-admission level

α

can inflate to

1-(1-α)^{k}

. Our main theorem shows that simultaneous setwise soundness---certifying a set of admissible proposals containing no nonviable action---is necessary and sufficient for generator-independent admission soundness, the worst case over all generators of executing a nonviable proposal equalling the probability of setwise failure; together with a design-time certificate and a no-bypass rule it is sufficient for contract safety, with violation bound

Γ+\sum_t\varepsilon_t+η

invariant under arbitrary, even adversarial, replacement of the generator. A second theorem bounds every admission mechanism under partial observation: for a fixed probing and admission policy, if two state hypotheses whose information laws lie within total-variation distance

δ

require different safe decisions, then

\abar+β+δ\ge1

. A sequential risk ledger makes the guarantee implementable with time-uniform confidence tubes, and shows that deterministic admission computations concentrate all statistical risk in state estimation. Simplex-style runtime assurance and control-barrier-function filtering are recovered as degenerate cases.

Guangxi Wan, Yongbo Xie, Yuqi Liu +4

State Key Laboratory of Robotics and Intelligent Systems, Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang 110016, China · University of Chinese Academy of Sciences, Beijing 100049, China

Runtime Assurance Runtime Verification Safety-Critical Control

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Jun 7, 2026

A Joint Finite-Sample Certificate for Adaptive Selective Conformal Risk Control

Selective predictors answer on confident inputs and abstain elsewhere; deploying one safely needs a single finite-sample certificate that simultaneously upper-bounds the selected risk, lower-bounds the acceptance probability

\pacc

above a floor

\pmin

, and lower-bounds the deployment utility. This certificate must be valid under adaptive threshold selection from a finite grid of

m

pairs on

\ncert

samples. We give such a certificate for bounded, possibly non-monotone losses by treating the selected risk directly as a ratio rather than through a Hoeffding-style range bound. The construction couples three confidence bounds: a variance-adaptive empirical-Bernstein bound on the ratio risk, a Clopper--Pearson bound on acceptance, and a two-sided closeness bound on utility. Together they lower-bound the certified policy's utility absolutely and to within

2\gammau

of the best over the certified set, both non-vacuous whenever feasible; a regime-scoped third leg matches an external oracle, informative only where the risk margin

\gammar < α

and vacuous at the headline operating points. Relative to the range-only Hoeffding-ratio construction this sharpens the acceptance-floor dependence from

1/\pmin

1/\sqrt{\pmin}

, and a closed-form corollary identifies a per-pair regime in which our risk bound dominates a Hoeffding conformal risk control (Hoeffding--CRC) selective bound. Empirically, on ImageNet (three ResNets) and COCO val 2017 panoptic, the certificate opens a

+22

pp certified-acceptance frontier over Hoeffding--CRC and is

{\approx}10{\times}

tighter than a non-vacuous matched-valid baseline; these gains are regime-scoped, not universal, and absent on ADE20K. The certifier runs in

O(\ncert m)

time.

Xiaoli Yu, Jiamiao Liu

School of Cyber Security and Information Law Chongqing University of Posts and Telecommunications Chongqing 400065, China · Department of Information, Xinqiao Hospital Army Medical University (Third Military Medical University) Chongqing 400037, China

Selective Prediction Conformal Risk Control Generalization Bounds

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Aug 9, 2026

The Cost of Adaptivity: Matching Lower Bounds Across Learning Problems

Adaptive procedures must work without nuisance information an oracle may use, such as a gradient scale or smoothness index, and robust procedures may have to answer queries whose coordinate and inspection time are chosen only after the data are seen. Such comparisons are meaningful only when the oracle advantage and validity contract are stated explicitly. We formalize nuisance adaptation via a slice-normalized minimax ratio retaining the worst-case instance within each nuisance slice, and separately define the robustness cost of expanding from one preannounced Gaussian query to arbitrary post-hoc inspection. Our main result is a finite-horizon composition law for Gaussian certification: from M independent coordinates, a familywise certifier protecting every coordinate and time up to T pays optimal normalized squared half-width of order log(eM) + log log(e^eT), within the sample-mean-centered rectangular class. Epoch stitching gives the upper bound; independent Gaussian block increments across coordinates and geometric time scales give a matching lower bound, already holding on a geometric checkpoint grid, forcing quantiles of the realized maximum width so selection and stopping taxes add. Two benchmark regimes complete the picture: unknown gradient scale in online convex optimization has constant cost, while pointwise adaptation over nested Holder classes costs order (log n / log log n)^(s1/(2s1+1)). Cast as model monitoring, the law lets an analyst inspect any of M slice metrics at any data-dependent time: the naive fixed-query band's selected coverage degrades sharply, to 0.30 at M=1 and to zero for M>=10, while the epoch-stitched certifier holds familywise coverage at an additive iterated-logarithm width cost. Experiments put both sharp predictions at risk of refutation; both survive.

Ibne Farabi Shihab, Adria Binte Habib

Department of Computer Science, Iowa State University, USA · Department of Computer Science & Engineering, Independent University of Bangladesh, Bangladesh

Minimax Estimation

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Abstract

Safety-Critical Control

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