Certified Fidelity Interval

• Certified Fidelity Interval is a rigorously justified range that mathematically guarantees containment of a target quantity using methods like propagation, exact optimization, or posterior construction.
• It applies across domains such as quantum-state fidelity, function range estimation in neural networks, and optimization error certification in complex systems.
• Practical examples include interval-certified ReLU approximation, adaptive stabilizer-state fidelity certification, and Bayesian credible intervals for entangled-state fidelity under general noise.

A certified fidelity interval is a rigorously justified interval that contains a target quantity and whose endpoints are obtained from sound propagation, exact endpoint optimization, or a posterior construction rather than from an uncontrolled heuristic. In the most literal usage, it is an interval for a quantum-state fidelity. In closely related usages, it is an interval for the exact output range of a function over a region, for the optimum of an optimization problem, or for the suboptimality gap of an optimizer. Across these settings, the common structure is that the reported interval is accompanied by a guarantee that the true quantity lies inside it, together with an explicit notion of tightness or fidelity of the interval to the underlying object (Baader et al., 2019, Wang, 28 May 2026, Ruan et al., 2024, Tekeler et al., 19 Nov 2025, Montbrun et al., 2023).

1. General form of the concept

Across the cited works, a certified fidelity interval has three components: a target quantity, a feasible or reachable set consistent with the available information, and an interval whose containment properties are mathematically guaranteed. In interval-certified approximation, the target quantity is the exact image of a continuous function on an input box. In stabilizer-state certification, it is the overlap $F(\rho,\psi)=\langle\psi|\rho|\psi\rangle$. In general-noise entanglement estimation, it is the average fidelity of unsampled pairs. In optimization settings, it is either the optimum itself or the optimization error of a recommended point (Baader et al., 2019, Wang, 28 May 2026, Ruan et al., 2024, Tekeler et al., 19 Nov 2025, Montbrun et al., 2023).

A useful unifying pattern is that certification is expressed as a worst-case or posterior-valid enclosure. Representative forms appearing in the literature are

$$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$

$$[F_{\min},F_{\max}] = \left[ \min_{p\in\mathcal F} p(\boldsymbol 0), \max_{p\in\mathcal F} p(\boldsymbol 0) \right],$$

$\Pr[\bar F\in \mathcal C\mid \mathbf r]\ge \alpha,$

and

$f(x^\star)-f(x_t^\star)\le \xi_t.$

These are not interchangeable definitions, but they encode the same structural idea: the interval is certified because it is valid for every admissible function, state, or environment compatible with the assumptions and observations (Baader et al., 2019, Wang, 28 May 2026, Ruan et al., 2024, Montbrun et al., 2023).

Setting	Certified object	Certification form
Interval-certified ReLU approximation	True range of $f$ on a box $B$	$n^\sharp(B)$ lies within $\delta$ of $[\min f(B),\max f(B)]$
Stabilizer-state certification	Fidelity $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$0	Exact LP endpoints $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$1
Entangled-state estimation under general noise	Average fidelity of unsampled pairs	Credible interval $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$2 with posterior probability at least $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$3
SC-DCOPF bounding	Optimal objective value or violations	Certified lower/upper bounds enclosing the true optimum
Multi-fidelity zeroth-order optimization	Global suboptimality gap	One-sided certified bound $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$4

The central distinction from an ordinary estimate is that the interval is itself an object of proof. A narrow interval is useful only insofar as its soundness is preserved; conversely, a sound but loose interval may be operationally limited. Much of the technical work in the cited literature is devoted to tightening certified intervals without losing validity.

2. Interval-certified approximation and the $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$5-accurate range enclosure

In "Universal Approximation with Certified Networks" (Baader et al., 2019), the relevant object is the interval produced by interval bound propagation (IBP) for a ReLU network on an input box $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$6. For a continuous target function $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$7, the paper asks whether there exists a ReLU network $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$8 such that $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$9 approximates $$[F_{\min},F_{\max}] = \left[ \min_{p\in\mathcal F} p(\boldsymbol 0), \max_{p\in\mathcal F} p(\boldsymbol 0) \right],$$0 uniformly and, simultaneously, $$[F_{\min},F_{\max}] = \left[ \min_{p\in\mathcal F} p(\boldsymbol 0), \max_{p\in\mathcal F} p(\boldsymbol 0) \right],$$1 is arbitrarily close to the exact output range of $$[F_{\min},F_{\max}] = \left[ \min_{p\in\mathcal F} p(\boldsymbol 0), \max_{p\in\mathcal F} p(\boldsymbol 0) \right],$$2 on every box $$[F_{\min},F_{\max}] = \left[ \min_{p\in\mathcal F} p(\boldsymbol 0), \max_{p\in\mathcal F} p(\boldsymbol 0) \right],$$3. The scalar guarantee is

$$[F_{\min},F_{\max}] = \left[ \min_{p\in\mathcal F} p(\boldsymbol 0), \max_{p\in\mathcal F} p(\boldsymbol 0) \right],$$4

where $$[F_{\min},F_{\max}] = \left[ \min_{p\in\mathcal F} p(\boldsymbol 0), \max_{p\in\mathcal F} p(\boldsymbol 0) \right],$$5 and $$[F_{\min},F_{\max}] = \left[ \min_{p\in\mathcal F} p(\boldsymbol 0), \max_{p\in\mathcal F} p(\boldsymbol 0) \right],$$6. In vector form, the same statement holds coordinate-wise. This is the paper’s formal notion of interval-certified fidelity (Baader et al., 2019).

The result is presented as a universal approximation theorem for interval-certified ReLU networks. Classical universal approximation is recovered by taking degenerate boxes $$[F_{\min},F_{\max}] = \left[ \min_{p\in\mathcal F} p(\boldsymbol 0), \max_{p\in\mathcal F} p(\boldsymbol 0) \right],$$7, in which case the interval statement reduces to pointwise approximation. The novelty is that the theorem concerns not only values $$[F_{\min},F_{\max}] = \left[ \min_{p\in\mathcal F} p(\boldsymbol 0), \max_{p\in\mathcal F} p(\boldsymbol 0) \right],$$8 but also exact min-max behavior of $$[F_{\min},F_{\max}] = \left[ \min_{p\in\mathcal F} p(\boldsymbol 0), \max_{p\in\mathcal F} p(\boldsymbol 0) \right],$$9 over every box, as seen through the IBP transformer $\Pr[\bar F\in \mathcal C\mid \mathbf r]\ge \alpha,$0 (Baader et al., 2019).

The proof is constructive. A scalar continuous function is decomposed into an $\Pr[\bar F\in \mathcal C\mid \mathbf r]\ge \alpha,$1-slicing $\Pr[\bar F\in \mathcal C\mid \mathbf r]\ge \alpha,$2, where each slice has height $\Pr[\bar F\in \mathcal C\mid \mathbf r]\ge \alpha,$3. For each slice $\Pr[\bar F\in \mathcal C\mid \mathbf r]\ge \alpha,$4, the authors construct a ReLU network $\Pr[\bar F\in \mathcal C\mid \mathbf r]\ge \alpha,$5 with tightly controlled interval behavior: if $\Pr[\bar F\in \mathcal C\mid \mathbf r]\ge \alpha,$6 lies sufficiently below the slice threshold then $\Pr[\bar F\in \mathcal C\mid \mathbf r]\ge \alpha,$7; if it lies sufficiently above then $\Pr[\bar F\in \mathcal C\mid \mathbf r]\ge \alpha,$8; and in all cases $\Pr[\bar F\in \mathcal C\mid \mathbf r]\ge \alpha,$9. The final network is

$f(x^\star)-f(x_t^\star)\le \xi_t.$0

Only a bounded number of slices can contribute partial uncertainty under IBP, which yields the $f(x^\star)-f(x_t^\star)\le \xi_t.$1 enclosure (Baader et al., 2019).

A distinctive feature of the construction is the use of local bumps $f(x^\star)-f(x_t^\star)\le \xi_t.$2 over a fine grid $f(x^\star)-f(x_t^\star)\le \xi_t.$3. Each bump is exactly $f(x^\star)-f(x_t^\star)\le \xi_t.$4 on $f(x^\star)-f(x_t^\star)\le \xi_t.$5, exactly $f(x^\star)-f(x_t^\star)\le \xi_t.$6 outside $f(x^\star)-f(x_t^\star)\le \xi_t.$7, and has controlled linear transition in between. Their interval transformers satisfy

$f(x^\star)-f(x_t^\star)\le \xi_t.$8

$f(x^\star)-f(x_t^\star)\le \xi_t.$9

with $f$0 in general. This exactness away from transition regions is what makes IBP tight in the final network (Baader et al., 2019).

The paper’s Figure 1 isolates a common misconception: function equality does not imply equal certifiability. Two ReLU networks $f$1 and $f$2 represent exactly the same scalar function on $f$3, but

$f$4

Thus, certified fidelity depends on representation, not only on the realized function. The paper proves existence of accurate, interval-certified representations, but it is explicit that the result is existential and does not provide a practical training algorithm or detailed depth/width complexity bounds (Baader et al., 2019).

3. Exact fidelity intervals for stabilizer states

In "Adaptive Stabilizer State Fidelity Certification" (Wang, 28 May 2026), the term certified fidelity interval is used literally. For a prepared $f$5-qubit state $f$6 and a target stabilizer state $f$7, fidelity is taken in the linear form

$f$8

Using a stabilizer generator gauge, one measures expectation values of $f$9 generators at a time and retains only those generator expectations. The resulting information constrains the syndrome distribution $B$0 to a feasible polytope $B$1. The certified fidelity interval is then

$B$2

because $B$3 (Wang, 28 May 2026).

For a single gauge $B$4, the paper gives analytic formulas for both endpoints. The KKL lower bound is

$B$5

and the new matching worst-case upper bound is

$B$6

Accordingly, a one-gauge certified fidelity interval is

$B$7

These endpoints are optimal worst-case bounds relative to the retained one-gauge data (Wang, 28 May 2026).

The paper also gives a finite-shot upper certificate. If each generator is measured at least

$B$8

times, then with probability at least $B$9,

$n^\sharp(B)$0

and therefore

$n^\sharp(B)$1

This produces a finite-sample certified upper endpoint that is linear in $n^\sharp(B)$2 up to logarithmic factors (Wang, 28 May 2026).

A major structural result is gauge dependence. For any $n^\sharp(B)$3 and any $n^\sharp(B)$4, there exist syndrome distributions with identical measured generator expectations in that gauge but different fidelities. Hence no single gauge determines the fidelity exactly in general. The 3-qubit example in the paper shows this concretely: a canonical gauge yields the exact lower endpoint $n^\sharp(B)$5 and a loose upper endpoint $n^\sharp(B)$6, whereas a parity gauge yields a vacuous lower endpoint $n^\sharp(B)$7 and the exact upper endpoint $n^\sharp(B)$8 for the same state. This is why the interval is best viewed as data-relative rather than intrinsic to the state alone (Wang, 28 May 2026).

The adaptive contribution of the paper is to exploit gauge freedom to tighten the interval. After $n^\sharp(B)$9 rounds, the feasible set is

$\delta$0

with endpoints

$\delta$1

Using endpoint witnesses $\delta$2 and $\delta$3, the paper defines disagreement scores

$\delta$4

and selects the next gauge by a witness elimination policy maximizing the total disagreement over the gauge columns. The interval tightens monotonically because $\delta$5, and exact recovery occurs once all nontrivial stabilizers are covered. In the worst case, full coverage is necessary, requiring at least $\delta$6 full-gauge rounds (Wang, 28 May 2026).

4. Credible intervals for entangled-state fidelity under general noise

"Reliable Interval Estimation for the Fidelity of Entangled States in Scenarios with General Noise" (Ruan et al., 2024) studies a different notion of fidelity interval: a Bayesian credible interval for the average fidelity of unsampled entangled pairs that remains valid under arbitrarily general noise. Two remote nodes share $\delta$7 entangled qubit pairs, randomly sample $\delta$8 pairs, measure each sampled pair in a uniformly random basis $\delta$9, and record $[\min f(B),\max f(B)]$0. The quantity of interest is the average fidelity of the unsampled pairs,

$[\min f(B),\max f(B)]$1

The paper treats $[\min f(B),\max f(B)]$2 as a random variable $[\min f(B),\max f(B)]$3 and seeks an interval $[\min f(B),\max f(B)]$4 such that

$[\min f(B),\max f(B)]$5

without any i.i.d. assumption on the noise (Ruan et al., 2024).

The resulting interval is

$[\min f(B),\max f(B)]$6

where $[\min f(B),\max f(B)]$7 is an even positive integer, the center is

$[\min f(B),\max f(B)]$8

and the radius is

$[\min f(B),\max f(B)]$9

The theorem states that in cases with general noise, the average fidelity $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$00 lies in this interval with probability at least $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$01 conditioned on the observed sample QBER $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$02 (Ruan et al., 2024).

The construction relies on random sampling, a thought experiment, and Bayesian inference. The thought experiment imagines measuring all pairs first and only then designating a random sampled subset. Because measurements on different pairs commute, the order of measuring and sampling is irrelevant, which reduces the problem to a classical finite-population estimation problem. A hierarchical objective prior based on Jeffreys prior yields a beta-binomial posterior for the unsampled error count. The posterior mean of the unsampled QBER is

$$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$03

and its variance is

$$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$04

The fidelity interval is then obtained by bounding even central moments of $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$05 through those of $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$06 (Ruan et al., 2024).

A distinctive feature of this work is the use of all even moments up to order $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$07, rather than variance alone. The generalized Chebyshev argument gives

$$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$08

and optimizing over $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$09 yields a sharper interval. The paper gives the rule

$$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$10

with examples $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$11 for $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$12, $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$13 for $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$14 and $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$15, and $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$16 for $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$17 (Ruan et al., 2024).

The work also identifies a nontrivial measurement-ratio effect. Under general noise, increasing $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$18 reduces statistical noise but also makes the unsampled set smaller and potentially more atypical. For the second-moment interval, the radius becomes an increasing function of $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$19 when

$$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$20

For the full $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$21, the numerical evidence in the paper places the minimal radius near $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$22 as long as $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$23. This directly contradicts the common intuition that more measurements must always reduce uncertainty (Ruan et al., 2024).

5. Broader extensions: certified intervals for optimization and system objectives

Several recent works extend the same certification pattern from fidelity in the quantum-information sense to objective values, output ranges, and optimization errors. In "Fast and Certified Bounding of Security-Constrained DCOPF via Interval Bound Propagation" (Tekeler et al., 19 Nov 2025), the authors explicitly state that they use IBP to compute relaxed bounds for hard power grid optimization problems and refer to the resulting bounds as certified because they are guaranteed convex relaxations of the true solution bounds. For the soft-constrained SC-DCOPF objective $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$24, IBP yields numbers $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$25 and $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$26 such that

$$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$27

The paper defines the gap

$$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$28

reports mean gap below $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$29 for all tested systems up to 617 buses, maximum gap $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$30, and scalability to systems up to 8,316 buses with a runtime of approximately $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$31 seconds. It also uses a negative IBP upper bound on the penalized objective as a certificate of infeasibility or negative welfare (Tekeler et al., 19 Nov 2025).

This suggests a broader interpretation of certified fidelity interval as a rigorously valid interval $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$32 for a quantity of interest whose width measures the fidelity of the relaxation to the exact problem. The SC-DCOPF work makes this interpretation explicit for both objectives and constraint violations: if a residual $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$33 has IBP interval $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$34, then $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$35 certifies strict satisfaction and $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$36 certifies unavoidable violation over the whole input box (Tekeler et al., 19 Nov 2025).

An analogous one-sided version appears in "Certified Multi-Fidelity Zeroth-Order Optimization" (Montbrun et al., 2023). There, a certified algorithm must output a recommendation $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$37 and an error certificate $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$38 such that for every $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$39-Lipschitz $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$40, every admissible environment $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$41, and every round $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$42,

$$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$43

The certificate is therefore a one-sided interval on the global suboptimality gap. In the certified variant c.MF-DOO, the certificate is constructed as

$$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$44

combining a global upper bound on $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$45 with a lower bound on the value at the recommended point. The paper derives near-optimal cost complexity bounds for this certification problem and extends them to the stochastic setting, where the same inequality holds with high probability (Montbrun et al., 2023).

These optimization-oriented papers do not define fidelity as state overlap. Nonetheless, they instantiate the same formal template: a certified interval or bound encloses the true target quantity under worst-case or high-probability semantics, and its usefulness depends on the trade-off between soundness, tightness, and acquisition cost.

6. Assumptions, limitations, and recurring misconceptions

A first recurring limitation is existentiality. The interval-certified universal approximation theorem proves that for every continuous $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$46 and every $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$47, there exists a ReLU network whose IBP interval is within $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$48 of the exact range on every box. It does not provide a practical training algorithm, and the construction uses many slices, a fine grid, and many local bumps. The paper is explicit that difficulty in finding such networks in practice is an optimization or capacity issue rather than a proof of impossibility (Baader et al., 2019).

A second limitation is information insufficiency from restricted measurements. In stabilizer-state certification, one gauge is not enough in general: different syndrome distributions can agree on all generator expectations in a single gauge while having different fidelities. The interval therefore quantifies the information content of the measurement design, not merely experimental noise. The adaptive multi-gauge protocol addresses this, but the same paper proves a worst-case necessity result: exact certification requires querying all $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$49 nontrivial stabilizers in the worst case (Wang, 28 May 2026).

A third issue is that certification under general noise is not the same as i.i.d.-based uncertainty quantification. The entangled-state interval in (Ruan et al., 2024) is valid for arbitrary joint states because it relies on random sampling and a beta-binomial posterior for the unsampled error count, not on an i.i.d. likelihood. This robustness comes with assumptions of its own: uniformly random sampling of the measured subset, uniformly random basis choice in $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$50, and an objective-prior construction. The same work shows that standard i.i.d. intervals can under-cover under heterogeneous or correlated noise, while excessive measurement can enlarge the certified interval when $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$51 becomes too large (Ruan et al., 2024).

A fourth misconception is to equate certified bounds with direct solutions. In SC-DCOPF, IBP provides a mathematically sound bound on the optimal objective and can certify infeasibility drivers, but it does not directly produce a dispatch policy. In certified zeroth-order optimization, $$[l+\delta,u-\delta]\subseteq I_{\mathrm{cert}}\subseteq [l-\delta,u+\delta],$$52 certifies the recommendation’s global suboptimality gap, but it is one-sided rather than a symmetric confidence interval around the optimum. These are certified intervals in the sense of rigorous enclosure, not necessarily in the sense of full point estimation or solution recovery (Tekeler et al., 19 Nov 2025, Montbrun et al., 2023).

Taken together, these works establish certified fidelity intervals as a general methodology for replacing informal uncertainty statements with provable enclosures. The exact mathematical form varies by domain—interval propagation, linear programs over feasible polytopes, Bayesian credible intervals, or worst-case optimization certificates—but the underlying aim is constant: to quantify, in a certified way, how closely a computed interval tracks the true quantity of interest.