Interval Confidence Uncertainty

• Interval Confidence Uncertainty is defined as the minimal interval length that encloses a prescribed probability mass, emphasizing geometric support over variance-based metrics.
• A phase transition at θₓ + θₚ = 1 separates regimes with trivial and nontrivial uncertainty products, with sharp lower bounds derived via Slepian functions.
• Applications span quantum mechanics, statistical inference, and machine learning, enabling calibrated predictive intervals and improved uncertainty quantification.

Searching arXiv for recent and foundational papers on interval confidence uncertainty and related interval-based uncertainty quantification. “Interval confidence uncertainty” is introduced as a probability-mass notion of localization: uncertainty is measured by the minimal length of a single interval that contains a prescribed probability, rather than by variance or entropy (Lin et al., 6 May 2026). In this sense, the object of interest is not the spread of an entire distribution, but the smallest interval support needed to guarantee confidence level $\theta$. Closely related literatures in statistics, simulation, inverse problems, and machine learning likewise treat intervals, coverage, and interval-valued predictions as primary uncertainty objects, though the meanings of “confidence interval,” “uncertainty band,” and “interval-valued prediction” differ substantially across domains (Epstein et al., 30 Oct 2025, Shen et al., 2023, Li et al., 4 Feb 2026).

1. Formal definition and scope

In the quantum formulation, the paper distinguishes two quantities for position and momentum: the general confidence uncertainty $\Delta^{c}x(\theta_x)$ and the interval confidence uncertainty $\Delta^{I}x(\theta_x)$ (Lin et al., 6 May 2026). The general version allows any measurable support set,

$$\Delta^{c}x(\theta_x)=\inf\!\left\{\mu(X):\int_{X}|{\psi}(x)|^{2}\,d x\ge\theta_x\right\},$$

whereas the interval version restricts the support to a single interval,

$$\Delta^{I}x(\theta_x)=\inf\!\left\{x_2-x_1:\int_{x_1}^{x_2}|{\psi}(x)|^{2}\,d x\ge\theta_x\right\}.$$

The same definitions apply in momentum space, yielding $\Delta^{c}p(\theta_p)$ and $\Delta^{I}p(\theta_p)$ (Lin et al., 6 May 2026).

The interval restriction is geometrically significant. Because $\Delta^c$ may use unions of disjoint sets while $\Delta^I$ must use one interval, the paper states that

$\Delta^{c}x\le\Delta^{I}x,$

and that both quantities are monotone non-decreasing in the confidence level (Lin et al., 6 May 2026). The resulting uncertainty measure is therefore explicitly tied to containment probability and support geometry, rather than to second moments.

This usage differs from classical statistical confidence intervals. In the quantum paper, the interval is a support set for probability mass of a fixed state; in frequentist statistics, a confidence interval is a random interval intended to cover an unknown parameter with prescribed probability over repeated sampling. Related machine-learning papers use yet other interval notions, including predictive intervals, interval-valued scores, and uncertainty bands (Lin et al., 6 May 2026, Oala et al., 2020, Khaled et al., 26 Jan 2026).

2. Phase transition and uncertainty inequalities

The defining structural result is a phase transition at the line

$\Delta^{c}x(\theta_x)$0

For

$\Delta^{c}x(\theta_x)$1

the paper proves that no nontrivial uncertainty product lower bound exists: for every $\Delta^{c}x(\theta_x)$2, one can find a state such that both confidence uncertainties are smaller than $\Delta^{c}x(\theta_x)$3 (Lin et al., 6 May 2026). A special case is

$\Delta^{c}x(\theta_x)$4

which implies that position and momentum can be simultaneously localized with confidence at least $\Delta^{c}x(\theta_x)$5 (Lin et al., 6 May 2026).

For

$\Delta^{c}x(\theta_x)$6

a strictly positive lower bound appears. The general confidence-uncertainty product satisfies

$\Delta^{c}x(\theta_x)$7

derived by combining Lenard’s projection inequality with the Donoho–Stark operator-norm bound (Lin et al., 6 May 2026).

For the interval version, the paper gives the sharp implicit Landau–Pollak bound

$\Delta^{c}x(\theta_x)$8

equivalently

$\Delta^{c}x(\theta_x)$9

Here $\Delta^{I}x(\theta_x)$0 is the largest prolate-spheroidal eigenvalue (Lin et al., 6 May 2026).

The paper also records an earlier elementary interval bound, valid only when $\Delta^{I}x(\theta_x)$1,

$\Delta^{I}x(\theta_x)$2

and explicitly states that it is weaker than the Landau–Pollak result (Lin et al., 6 May 2026).

The statement that joint $\Delta^{I}x(\theta_x)$3 localization does not contradict Heisenberg is part of the paper’s interpretive framework: the Heisenberg–Kennard bound concerns variances, whereas confidence uncertainty concerns concentration of probability mass in chosen intervals or sets (Lin et al., 6 May 2026).

3. Sharpness, saturating states, and asymptotics

The sharp interval bound is reduced to classical Slepian–Pollak–Landau concentration theory (Lin et al., 6 May 2026). The defining operator is

$\Delta^{I}x(\theta_x)$4

whose principal eigenvalue is $\Delta^{I}x(\theta_x)$5 (Lin et al., 6 May 2026). The key identity is

$\Delta^{I}x(\theta_x)$6

with $\Delta^{I}x(\theta_x)$7 and $\Delta^{I}x(\theta_x)$8 the position- and momentum-space projectors for intervals $\Delta^{I}x(\theta_x)$9 and $$\Delta^{c}x(\theta_x)=\inf\!\left\{\mu(X):\int_{X}|{\psi}(x)|^{2}\,d x\ge\theta_x\right\},$$0 (Lin et al., 6 May 2026).

The interval bound is saturated. The saturating state is the leading Slepian function, namely the principal eigenfunction of the prolate-spheroidal kernel (Lin et al., 6 May 2026). The paper describes this as the standard Lenard-equality configuration, so the bound is not merely existential but attained by a concrete optimal state.

Asymptotically, the largest Slepian eigenvalue satisfies

$$\Delta^{c}x(\theta_x)=\inf\!\left\{\mu(X):\int_{X}|{\psi}(x)|^{2}\,d x\ge\theta_x\right\},$$1

and

$$\Delta^{c}x(\theta_x)=\inf\!\left\{\mu(X):\int_{X}|{\psi}(x)|^{2}\,d x\ge\theta_x\right\},$$2

Hence

$$\Delta^{c}x(\theta_x)=\inf\!\left\{\mu(X):\int_{X}|{\psi}(x)|^{2}\,d x\ge\theta_x\right\},$$3

(Lin et al., 6 May 2026). The high-confidence regime therefore diverges logarithmically. For $$\Delta^{c}x(\theta_x)=\inf\!\left\{\mu(X):\int_{X}|{\psi}(x)|^{2}\,d x\ge\theta_x\right\},$$4,

$$\Delta^{c}x(\theta_x)=\inf\!\left\{\mu(X):\int_{X}|{\psi}(x)|^{2}\,d x\ge\theta_x\right\},$$5

(Lin et al., 6 May 2026).

The numerical evaluation reported in the paper includes

• $$\Delta^{c}x(\theta_x)=\inf\!\left\{\mu(X):\int_{X}|{\psi}(x)|^{2}\,d x\ge\theta_x\right\},$$6: $$\Delta^{c}x(\theta_x)=\inf\!\left\{\mu(X):\int_{X}|{\psi}(x)|^{2}\,d x\ge\theta_x\right\},$$7,
• $$\Delta^{c}x(\theta_x)=\inf\!\left\{\mu(X):\int_{X}|{\psi}(x)|^{2}\,d x\ge\theta_x\right\},$$8: $$\Delta^{c}x(\theta_x)=\inf\!\left\{\mu(X):\int_{X}|{\psi}(x)|^{2}\,d x\ge\theta_x\right\},$$9,
• $$\Delta^{I}x(\theta_x)=\inf\!\left\{x_2-x_1:\int_{x_1}^{x_2}|{\psi}(x)|^{2}\,d x\ge\theta_x\right\}.$$0: $$\Delta^{I}x(\theta_x)=\inf\!\left\{x_2-x_1:\int_{x_1}^{x_2}|{\psi}(x)|^{2}\,d x\ge\theta_x\right\}.$$1 (Lin et al., 6 May 2026).

The paper compares the resulting product to several classical uncertainty relations. Heisenberg–Kennard gives

$$\Delta^{I}x(\theta_x)=\inf\!\left\{x_2-x_1:\int_{x_1}^{x_2}|{\psi}(x)|^{2}\,d x\ge\theta_x\right\}.$$2

the Białynicki-Birula–Mycielski inequality gives

$$\Delta^{I}x(\theta_x)=\inf\!\left\{x_2-x_1:\int_{x_1}^{x_2}|{\psi}(x)|^{2}\,d x\ge\theta_x\right\}.$$3

and the Donoho–Stark concentration bound gives

$$\Delta^{I}x(\theta_x)=\inf\!\left\{x_2-x_1:\int_{x_1}^{x_2}|{\psi}(x)|^{2}\,d x\ge\theta_x\right\}.$$4

The quantum confidence-uncertainty paper stresses that these control different objects: variance, entropy, and concentration are not interchangeable with confidence-mass localization (Lin et al., 6 May 2026).

4. Interval information and the partial order on uncertainty

A distinct statistical literature asks whether conditioning on a smaller interval necessarily reduces uncertainty. The paper "A Partial Order on Uncertainty and Information" shows that this proposition is not generally true (Chen, 2011). For an absolutely continuous or integer-valued random variable, the uncertainty measure is conditional variance under interval information,

$$\Delta^{I}x(\theta_x)=\inf\!\left\{x_2-x_1:\int_{x_1}^{x_2}|{\psi}(x)|^{2}\,d x\ge\theta_x\right\}.$$5

and the desired monotonicity for intervals $$\Delta^{I}x(\theta_x)=\inf\!\left\{x_2-x_1:\int_{x_1}^{x_2}|{\psi}(x)|^{2}\,d x\ge\theta_x\right\}.$$6 is

$$\Delta^{I}x(\theta_x)=\inf\!\left\{x_2-x_1:\int_{x_1}^{x_2}|{\psi}(x)|^{2}\,d x\ge\theta_x\right\}.$$7

(Chen, 2011).

The paper calls this property partial monotonicity of the conditional variance and provides necessary and sufficient conditions. For an absolutely continuous $$\Delta^{I}x(\theta_x)=\inf\!\left\{x_2-x_1:\int_{x_1}^{x_2}|{\psi}(x)|^{2}\,d x\ge\theta_x\right\}.$$8 with cdf $$\Delta^{I}x(\theta_x)=\inf\!\left\{x_2-x_1:\int_{x_1}^{x_2}|{\psi}(x)|^{2}\,d x\ge\theta_x\right\}.$$9, Theorem 2 states that $\Delta^{c}p(\theta_p)$0 is increasing in $\Delta^{c}p(\theta_p)$1 if and only if

$\Delta^{c}p(\theta_p)$2

is log-concave in $\Delta^{c}p(\theta_p)$3, and decreasing in $\Delta^{c}p(\theta_p)$4 if and only if

$\Delta^{c}p(\theta_p)$5

is log-concave in $\Delta^{c}p(\theta_p)$6 (Chen, 2011). If both hold on a convex set $\Delta^{c}p(\theta_p)$7, then conditional variance is partially monotonic for intervals $\Delta^{c}p(\theta_p)$8.

A broad sufficient condition is log-concavity of the cdf. Theorem 3 states that if $\Delta^{c}p(\theta_p)$9 is log-concave on an interval $\Delta^{I}p(\theta_p)$0, then

$\Delta^{I}p(\theta_p)$1

for any intervals $\Delta^{I}p(\theta_p)$2 (Chen, 2011). The paper also extends the framework to integer-valued variables via a continuous surrogate $\Delta^{I}p(\theta_p)$3, and derives an analogous monotonicity result for Shannon information of the difference variable $\Delta^{I}p(\theta_p)$4 under log-concavity assumptions (Chen, 2011).

This interval-conditioning theory is conceptually related to interval confidence uncertainty because both ask how uncertainty changes when one restricts admissible support. The underlying objects, however, are different: conditional variance under interval information on the one hand, and minimal interval length capturing prescribed probability on the other.

5. Interval-valued uncertainty in contemporary inference and machine learning

Outside the quantum setting, interval-centered uncertainty appears in several non-equivalent forms, including self-reported confidence intervals, interval-valued classifier scores, propagated solution bands, and interval outputs of neural or fuzzy systems (Epstein et al., 30 Oct 2025, Li et al., 4 Feb 2026, Shen et al., 2023, Stanley et al., 4 Feb 2025, Oala et al., 2020, Khaled et al., 26 Jan 2026).

Domain	Interval object	Reported role
LLM numerical estimation	$\Delta^{I}p(\theta_p)$5 in exponent form	Coverage and sharpness evaluation
Uncertainty-aware classification	$\Delta^{I}p(\theta_p)$6	Ranking with correct, incorrect, and overlap regions
Deterministic PDEs	Joint CI $\Delta^{I}p(\theta_p)$7 over $\Delta^{I}p(\theta_p)$8	Simultaneous domain-wide coverage
Constrained inverse problems	Four constraint-aware CIs	Coverage with reduced conservatism
Interval neural and fuzzy models	Learned lower and upper outputs; FOU	Width interpreted as uncertainty/confidence

In "LLMs are Overconfident: Evaluating Confidence Interval Calibration with FermiEval," models are prompted to output exponent-form intervals $\Delta^{I}p(\theta_p)$9 in JSON for 1000 Science Olympiad Fermi questions (Epstein et al., 30 Oct 2025). The benchmark evaluates coverage, sharpness, and the Winkler interval score,

$\Delta^c$0

The headline finding is that nominal $\Delta^c$1 intervals cover the true answer only $\Delta^c$2 of the time on average, indicating systematic overconfidence; a conformal prediction based adjustment restores accurate $\Delta^c$3 observed coverage and decreases the Winkler interval score by $\Delta^c$4 (Epstein et al., 30 Oct 2025).

In uncertainty-aware classification, interval-valued predictions

$\Delta^c$5

replace point scores (Li et al., 4 Feb 2026). A positive interval $\Delta^c$6 and negative interval $\Delta^c$7 can be confidently correct, confidently incorrect, or overlapping. The paper defines

$\Delta^c$8

with the decomposition

$\Delta^c$9

(Li et al., 4 Feb 2026). The overlap region functions as abstention, and the paper proves bounds on the optimal AUC $\Delta^I$0 under class-conditional coverage assumptions (Li et al., 4 Feb 2026).

In deterministic PDEs, PICProp formulates a joint confidence interval over the whole domain,

$\Delta^I$1

which is stronger than pointwise coverage (Shen et al., 2023). The method constructs a CI for the clean data and propagates it through a PDE solver or PINN by bi-level optimization. The paper proves that if the data CI is valid, then the propagated solution interval is also valid (Shen et al., 2023).

In constrained inverse problems, four interval constructions—Global/Sliced and Inverted/Optimized—use a Berger–Boos style data-adaptive reduction

$\Delta^I$2

to make functional uncertainty quantification computationally feasible and less conservative (Stanley et al., 4 Feb 2025). The paper proves nominal coverage for all four intervals and reports that, in a high-energy-physics unfolding example, Sliced Inverted reduces average length by about $\Delta^I$3 versus OSB in the smooth setting and about $\Delta^I$4 in the adversarial setting (Stanley et al., 4 Feb 2025).

In learned predictive systems, the interval itself often becomes the uncertainty score. Interval Neural Networks produce output bounds

$\Delta^I$5

and use the width

$\Delta^I$6

as an uncertainty map (Oala et al., 2020). In the interval type-2 neuro-fuzzy framework for wastewater energy prediction, lower and upper membership functions define a Footprint of Uncertainty; a narrow FOU means high confidence, whereas a wide FOU means lower confidence and greater epistemic uncertainty (Khaled et al., 26 Jan 2026). The paper explicitly states that its interval-valued prediction is not a classical statistical confidence interval but an uncertainty band induced by fuzzy antecedent ambiguity (Khaled et al., 26 Jan 2026).

6. Calibration, validity, and interpretive issues

A recurrent theme across interval-based uncertainty research is that nominal confidence, interval width, and actual validity are distinct quantities. In the LLM setting, increasing the requested nominal confidence from $\Delta^I$7 to $\Delta^I$8 does not appropriately increase observed coverage; the models lie below the calibration diagonal and observed coverage plateaus at high nominal levels (Epstein et al., 30 Oct 2025). The paper’s “perception tunnel” theory explains this as underestimation of tail uncertainty: the model behaves as if it reasons over a truncated slice $\Delta^I$9 of its inferred distribution rather than the full $\Delta^{c}x\le\Delta^{I}x,$0 (Epstein et al., 30 Oct 2025).

In rare-event estimation, the same distinction appears in classical confidence interval construction. For Bernoulli rare-event probabilities, the paper on naive rare-event estimators concludes that the normality and Wilson intervals are not always valid, the exact Clopper–Pearson interval is conservative, and the Chernoff-based and Berry–Esseen-based intervals are valid but conservative (Bai et al., 2023). Tightness and validity therefore trade off directly.

For quantiles of machine-learning performance distributions, the relevant uncertainty object is again an interval on a distributional functional rather than on a mean. The paper "Quantifying Uncertainty and Variability in Machine Learning: Confidence Intervals for Quantiles in Performance Metric Distributions" presents exact nonparametric, asymptotic nonparametric, and semiparametric bootstrap confidence intervals, with particular emphasis on small sample sizes in the order of $\Delta^{c}x\le\Delta^{I}x,$1 repeated runs (Lehmann et al., 28 Jan 2025). It concludes that $\Delta^{c}x\le\Delta^{I}x,$2 to $\Delta^{c}x\le\Delta^{I}x,$3 is often enough to get useful quantile CIs up to roughly the $\Delta^{c}x\le\Delta^{I}x,$4 quantile, while $\Delta^{c}x\le\Delta^{I}x,$5 can still be informative for central quantiles (Lehmann et al., 28 Jan 2025).

These results suggest that interval confidence uncertainty has at least three separable dimensions: the geometric or support-based object being measured, the probabilistic statement attached to that object, and the calibration mechanism used to validate the statement. In quantum localization, the primary quantity is minimal interval size at probability level $\Delta^{c}x\le\Delta^{I}x,$6; in statistical inference, it is repeated-sampling coverage of an unknown parameter; in predictive systems, it may be a calibrated predictive set, a propagated uncertainty band, or an interval-valued score. The common thread is that uncertainty is expressed as an interval endowed with an explicit probabilistic interpretation, but the semantics of that interpretation are domain-specific (Lin et al., 6 May 2026, Epstein et al., 30 Oct 2025, Shen et al., 2023).