Calibrated Radius-Sum-Rule Framework

• The Calibrated Radius-Sum-Rule Framework is a composite research pattern that integrates structural sum rules with empirical calibration to convert latent observables into testable constraints.
• It unifies methodologies from randomized smoothing, proton-radius extraction, femtoscopy, and SUSY studies, demonstrating versatile applications across physics domains.
• The approach emphasizes error correction via thresholding, normalization, and regulator-based adjustments to mitigate divergences and enhance predictive accuracy.

As an Editor's term, the “Calibrated Radius-Sum-Rule Framework” denotes a composite methodological pattern suggested by several otherwise distinct literatures in which a radius-sensitive, channel-weighted, or symmetry-constrained quantity is first expressed through a sum rule, a normalized additive law, or a monotone radius map, and is then sharpened by calibration against deployment data, nuisance-parameter measurements, detector thresholds, empirical structure functions, or regulated reference observables. The term is not native to any one paper. RRISE is explicit that it does not introduce an explicit “radius-sum rule,” and instead contributes a calibrated radius inference mechanism for randomized smoothing (Park et al., 1 Jun 2026). By contrast, the SUSY-Yukawa, forward two-photon-exchange, femtoscopic, threshold-detection, and Na-cluster works each provide one of the constituent ingredients: an experimentally testable sum rule, a radius-bearing observable, or a calibration layer (Perelstein et al., 2012, Gorchtein, 2014, Maj et al., 2019, Khrennikov, 2011, Raduta et al., 2010).

1. Conceptual structure

This literature suggests a recurring three-layer architecture. First, there is a structural relation: a randomized-smoothing certificate $R(x;\sigma)=\sigma \Phi^{-1}(p_A(x))$, a collider sum rule for $\Upsilon$, a dispersive relation for the forward TPE amplitude, a completeness-derived correlator identity, or an energy-weighted double-commutator rule. Second, there is a deformation or uncertainty mechanism: Monte Carlo noise, radiative thresholds, ultraviolet tails, detector thresholding, or finite-size radial corrections. Third, there is a calibration device that restores practical predictivity: conformal correction, external parameter measurements, empirical photoabsorption input, regulated subtraction, threshold scaling, or fitted diffuseness (Park et al., 1 Jun 2026, Perelstein et al., 2012).

Under this reading, “radius” does not have a single meaning across the sources. In randomized smoothing it is an $\ell_2$-certified robustness radius. In elastic $ep$ scattering it is the proton charge radius extracted from the slope of $G_E(t)$. In femtoscopy it is the Gaussian source scale $r_0$ encoded through $D_r(0)$. In the Na-cluster paper, radius enters through ground-state radial moments such as $\langle r^2\rangle$ and $\langle r^4\rangle$. The threshold-detector and SUSY-Yukawa papers do not introduce a geometric radius in the same sense, but they do supply additive or calibrated structures that are directly analogous to the “sum-rule” side of the phrase (Gorchtein, 2014, Maj et al., 2019, Raduta et al., 2010, Khrennikov, 2011).

A plausible implication is that the phrase names not a single theorem but a family resemblance among methods that convert latent quantities into testable lower bounds or integrated constraints. The sources differ sharply in physics, but they repeatedly combine a structural identity with a conservative correction.

2. Randomized smoothing as calibrated radius inference

In randomized smoothing, the base classifier is $f:\mathbb{R}^d\to\{1,\dots,K\}$, the smoothed class probabilities are

$\Upsilon$0

and the smoothed classifier is

$\Upsilon$1

If $\Upsilon$2, then the standard Gaussian certificate is

$\Upsilon$3

with positive certified radius only when $\Upsilon$4. Standard certification replaces $\Upsilon$5 by a one-sided Clopper–Pearson lower bound $\Upsilon$6, giving $\Upsilon$7 (Park et al., 1 Jun 2026).

RRISE replaces per-input Monte Carlo certification by an offline-learned surrogate $\Upsilon$8 trained against MC class-count targets via soft-label cross-entropy. The surrogate predicts the full smoothed class distribution, with deployment statistics

$\Upsilon$9

A held-out calibration set then produces residuals $\ell_2$0, where $\ell_2$1 is a Clopper–Pearson lower bound for the smoothed probability of the surrogate-selected class, and the global offset $\ell_2$2 is set to the $\ell_2$3-th smallest residual. The resulting guarantee is

$\ell_2$4

and the operational radius is

$\ell_2$5

When $\ell_2$6, then with probability at least $\ell_2$7, the surrogate prediction equals the smoothed classifier prediction and the smoothed classifier is constant on the corresponding $\ell_2$8 ball (Park et al., 1 Jun 2026).

QCRS addresses a different part of the same pipeline. It keeps the standard Gaussian certificate but chooses an input-specific scalar $\ell_2$9 by optimizing

$ep$0

Its key claim is that this per-input sigma-radius curve is usually quasiconcave rather than concave, so a bisection-like search on the sign of a finite-difference gradient can recover the optimal or near-optimal $ep$1 for most points. Under the ideal $ep$2-SQC condition, QCRS finds the same optimal solution as grid search, with interval shrinkage

$ep$3

This is not a sum rule, but it is a calibrated radius optimization rule in which the objective itself is built from confidence-bounded probabilities (Kung et al., 2023).

Experimentally, these two randomized-smoothing papers instantiate complementary versions of the same design pattern. RRISE amortizes the map from clean input to smoothed class distribution and reports matching fixed-budget MC certified accuracy within $ep$4 percentage points while replacing up to $ep$5 noisy base-model evaluations per query with a single surrogate forward pass, with break-even after $ep$6 deployment queries (Park et al., 1 Jun 2026). QCRS keeps online Monte Carlo but searches over $ep$7; on CIFAR-10 it reports average certified radius gains such as $ep$8 for a $ep$9 model, with runtime $G_E(t)$0 sec/image for Cohen and $G_E(t)$1 sec/image for QCRS (Kung et al., 2023). This suggests two distinct calibration regimes: offline amortization and online parameter optimization.

3. Dispersive and correlator sum rules for radius extraction

The most literal radius-sensitive sum rules in the source set appear in the proton-radius and femtoscopy papers. In forward elastic $G_E(t)$2 scattering, the proton charge radius is extracted from

$G_E(t)$3

The forward TPE paper shows that the leading inelastic TPE effect is a universal nonanalytic term of the form $G_E(t)$4, whose coefficient is fixed by the total real photoabsorption cross section. The absorptive relation is

$G_E(t)$5

and the real part is reconstructed by an unsubtracted fixed-$G_E(t)$6 dispersion relation. The resulting master formula gives $G_E(t)$7, with leading low-$G_E(t)$8 behavior

$G_E(t)$9

Since $r_0$0, omitting this term biases low-$r_0$1 radius fits; the paper states that subtracting it lowers the extracted $r_0$2 (Gorchtein, 2014).

In femtoscopy, the correlation function is

$r_0$3

and the original completeness-based sum rule is

$r_0$4

For a Gaussian source,

$r_0$5

so the integrated correlator is directly tied to the source radius $r_0$6. The central revision of the paper is that the original sum rule is often ultraviolet divergent, and practical use requires a regulated difference or sum of two suitably matched correlators. In the simplest improved case,

$r_0$7

provided the regulator is non-identical and has no bound state (Maj et al., 2019).

These two literatures share an important methodological feature. The forward-TPE paper calibrates a low-$r_0$8 correction by empirical $r_0$9, whereas the femtoscopy paper calibrates a finite integral by choosing a regulator with matched ultraviolet asymptotics. In both cases, a naive radius extraction becomes unreliable without a controlled correction to the structural relation.

4. Symmetry-based and operator-corrected sum rules

The SUSY-Yukawa and Na-cluster papers provide the clearest examples of calibrated sum rules in which the structural relation is exact or symmetry-driven, but practical testing requires correcting for additional parameters or radial moments. In the SUSY-Yukawa case, the tree-level relation among stop and sbottom observables is written in the proceedings version as

$D_r(0)$0

and the corresponding dimensionless observable is

$D_r(0)$1

In the large-$D_r(0)$2 limit the tree-level prediction is

$D_r(0)$3

The paper’s central point is that radiative corrections substantially deform this tree-level relation, so experimental testing must proceed in two steps: constrain loop-sensitive SUSY parameters outside the third-generation squark sector, then compare the measured stop/sbottom combination with the corrected prediction. The quoted predictive distributions are $D_r(0)$4 with no new LHC/ILC constraints, $D_r(0)$5 after LHC-14 measurements, and $D_r(0)$6 after ILC-500 measurements, with true benchmark value $D_r(0)$7 (Perelstein et al., 2012).

The Na-cluster paper modifies the ordinary TRK logic by replacing the dipole operator with a Schiff-type operator,

$D_r(0)$8

The resulting energy-weighted sum rule becomes

$D_r(0)$9

To express these radial moments, the paper uses a Fermi-type density

$\langle r^2\rangle$0

with moment expansion

$\langle r^2\rangle$1

The diffuseness is then fitted as

$\langle r^2\rangle$2

and the paper concludes that the RPA results for Na clusters obey the modified TRK sum rule (Raduta et al., 2010).

Taken together, these two papers show two versions of the same calibrated sum-rule logic. In one, loop-sensitive thresholds must be constrained before the sum rule becomes sharp; in the other, operator-induced radial corrections must be expressed through a calibrated density profile before the energy-weighted relation becomes numerically predictive.

5. Calibration as thresholding, normalization, and conservative correction

Khrennikov’s threshold-detection paper provides the clearest additive-normalization template. A detector does not read out instantaneous amplitude; it integrates collected energy until a threshold $\langle r^2\rangle$3 is reached,

$\langle r^2\rangle$4

with calibration

$\langle r^2\rangle$5

For the normalized field $\langle r^2\rangle$6, the click frequency scales as

$\langle r^2\rangle$7

but the normalized probability is invariant,

$\langle r^2\rangle$8

In the general covariance formulation,

$\langle r^2\rangle$9

Calibration changes the absolute click rate and suppresses double clicks, but not the relative probabilities (Khrennikov, 2011).

This supplies a useful abstract template for the broader framework. Positive local contributions are accumulated, a calibration constant fixes the operational scale, and normalization by a total sum or trace converts them into a conservative observable. The papers differ in what is being accumulated or normalized. In RRISE, the object is a surrogate top-class probability corrected by a global offset $\langle r^4\rangle$0, leading to the lower bound $\langle r^4\rangle$1 and radius $\langle r^4\rangle$2 (Park et al., 1 Jun 2026). In the TPE paper, the coefficient of the leading nonanalytic correction is anchored to measured total photoabsorption $\langle r^4\rangle$3 rather than to an arbitrary off-shell model (Gorchtein, 2014). In the SUSY-Yukawa paper, auxiliary measurements of $\langle r^4\rangle$4 and related quantities narrow the allowed radiative correction band for $\langle r^4\rangle$5 (Perelstein et al., 2012). In the femtoscopic paper, calibration takes the form of a matched regulator correlator, and in some long-range Coulomb cases an additional analytic ultraviolet subtraction (Maj et al., 2019).

This suggests that “calibration” is the most stable part of the composite framework. Across the source set it means conservative correction, not merely parameter fitting.

6. Validity conditions, limitations, and common misunderstandings

A recurrent misunderstanding is to treat the composite phrase as if it named a single established framework. The sources do not support that reading. RRISE explicitly states that it does not provide an explicit radius decomposition, summation principle for radii, or additive “radius-sum rule” theorem; it provides a calibrated radius inference mechanism and a conformal wrapper for one-pass certification (Park et al., 1 Jun 2026). QCRS likewise provides no additive radius law; its exactness is conditional on the ideal $\langle r^4\rangle$6-SQC assumption, while in practice quasiconcavity is an empirical prevalence claim rather than a universal theorem (Kung et al., 2023).

Another common error is to use structural formulas without their correction layer. In the SUSY-Yukawa case, the tree-level number $\langle r^4\rangle$7 is not by itself a sharp experimental target because $\langle r^4\rangle$8 is a cancellation observable and threshold corrections can be numerically large (Perelstein et al., 2012). In the femtoscopic case, the original single-correlator sum rule can be ultraviolet divergent; the improved regulated combination is the practically meaningful object, and even that may require an additional asymptotic subtraction in attractive Coulomb examples (Maj et al., 2019). In the forward-TPE case, the $\langle r^4\rangle$9 term is theoretically clean in forward low-$f:\mathbb{R}^d\to\{1,\dots,K\}$0 kinematics, but the paper is explicit that subleading analytic $f:\mathbb{R}^d\to\{1,\dots,K\}$1 pieces are not controlled model-independently (Gorchtein, 2014).

The threshold-detector paper has its own domain restrictions. The derivation is approximate and asymptotic, relying on ergodicity, threshold detection, quadratic energy collection, long integration times, and calibration to the average energy scale. The paper also states that if detectors respond to higher-order nonlinearities such as $f:\mathbb{R}^d\to\{1,\dots,K\}$2, then Born’s rule can be violated (Khrennikov, 2011). The Na-cluster paper is similarly conditional on the Schiff-type operator, the chosen RPA treatment, and a calibrated diffuseness profile $f:\mathbb{R}^d\to\{1,\dots,K\}$3 (Raduta et al., 2010).

The most defensible synthesis is therefore narrow. A “Calibrated Radius-Sum-Rule Framework” is best understood as a composite research pattern in which a structural radius- or sum-rule relation is made operational by a conservative correction layer. Where the sources are strongest, they are strongest about that second step: finite-sample calibration in randomized smoothing, nuisance-parameter calibration in SUSY spectroscopy, data-driven calibration in forward TPE, regulator-based calibration in femtoscopy, threshold calibration in classical-signal detection, and density-based calibration of radial moments in Na clusters.