Spectral-Curvature Index (SCI) Overview

• Spectral-Curvature Index (SCI) is a scalar measure that quantifies curvature in spectral data, defined differently across radio, X-ray, and imaging applications.
• SCI methods enhance feature localization by normalizing second derivatives with first-derivative information, reducing peak broadening and bias compared to pure second derivatives.
• In radio studies, SCI differentiates between ageing/remnant and restarted radio galaxies by comparing low- and mid-frequency spectral indices, supporting lifecycle diagnostics.

Spectral-Curvature Index (SCI) denotes a class of scalar measures used to quantify spectral curvature, spectral steepening, or local geometric curvature in spectral data. The label is used explicitly in radio-galaxy population analysis, while closely related curvature parameters or curvature transforms appear in ARPES visualization, blazar X-ray spectroscopy, and Galactic synchrotron modeling (White et al., 8 Dec 2025, Zhang et al., 2011, Goswami et al., 2018, Irfan et al., 2021, Kogut, 2012, Irfan et al., 9 Dec 2025). This suggests that SCI is best understood as a domain-dependent quantity rather than a single universal formula: in some settings it is a curvature transform of a spectrum or image, in others it is the coefficient of a log-parabolic term, and in others it is a difference between spectral indices measured in adjacent frequency ranges.

1. Terminology and scope

In the most explicit usage, the G4Jy analysis defines a Spectral-Curvature Index from radio spectral indices as

$\mathrm{SCI}_0=\alpha_{\rm low}-\alpha_{\rm mid},$

with $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$ and $$\alpha_{\rm mid}\equiv \alpha_{151\,\mathrm{MHz}}^{1400\,\mathrm{MHz}}$$ (White et al., 8 Dec 2025). In diffuse Galactic synchrotron work, curvature is instead parameterized by a coefficient $c$ or $C$ in a log-parabolic temperature law such as

$$T(\nu)=T_0\left(\frac{\nu}{\nu_0}\right)^{\beta_0+c\,\ln(\nu/\nu_0)},$$

or equivalently

$$\ln T(\nu)=\ln T_0+\beta_0\ln(\nu/\nu_0)+c\,[\ln(\nu/\nu_0)]^2,$$

so that the curvature parameter itself functions as an SCI (Irfan et al., 2021, Kogut, 2012, Irfan et al., 9 Dec 2025). In blazar X-ray spectroscopy, the analogous empirical quantity is the log-parabola curvature parameter $\beta$ in

$$F(E)=K\left(\frac{E}{E_*}\right)^{-\alpha-\beta\log(E/E_*)},$$

while a more physical curvature description is supplied by the pair $(\xi_0,\kappa)$ in an energy-dependent diffusion or escape model (Goswami et al., 2018).

A distinct usage arises in spectral image analysis. Zhang et al. do not name a Spectral-Curvature Index, but they derive curvature-based scalar fields for one-dimensional spectra and two-dimensional intensity images that were explicitly organized as SCI-like quantities in the supplied material (Zhang et al., 2011). By contrast, "Spectral Coherence Index" in protein-ensemble quality control is a different construct; that work states that SCI there means Spectral Coherence Index, not "Spectral-Curvature Index" (Bi et al., 26 Mar 2026).

2. Geometric-curvature formulations for spectra and spectral images

In spectroscopy and imaging, the motivation for replacing the second derivative by curvature is to improve the localization of extrema and reduce peak broadness. The stated problems with the pure second derivative are peak position bias, broad features, and sensitivity to scale. The curvature method addresses these by normalizing second derivatives with first-derivative information, so that regions with large slope are down-weighted and regions near extrema, where first derivatives vanish, are emphasized (Zhang et al., 2011).

For a one-dimensional spectrum $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$0, the central curvature formula is

$$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$1

with $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$2 arbitrary and recommended as

$$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$3

Two limiting behaviors are emphasized. When $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$4, curvature reduces to the second derivative. When $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$5,

$$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$6

which diverges at extrema and sharpens localization at true maxima and minima (Zhang et al., 2011). The supplied material then makes the SCI-like identification explicit: $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$7 or, when both maxima and minima are of interest, its absolute value.

For two-dimensional spectral images with inequivalent axes, such as ARPES $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$8-$$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$9 maps, the corresponding curvature field is

$$\alpha_{\rm mid}\equiv \alpha_{151\,\mathrm{MHz}}^{1400\,\mathrm{MHz}}$$0

with positive free parameters $$\alpha_{\rm mid}\equiv \alpha_{151\,\mathrm{MHz}}^{1400\,\mathrm{MHz}}$$1 and $$\alpha_{\rm mid}\equiv \alpha_{151\,\mathrm{MHz}}^{1400\,\mathrm{MHz}}$$2. The associated two-dimensional SCI is written as

$$\alpha_{\rm mid}\equiv \alpha_{151\,\mathrm{MHz}}^{1400\,\mathrm{MHz}}$$3

As $$\alpha_{\rm mid}\equiv \alpha_{151\,\mathrm{MHz}}^{1400\,\mathrm{MHz}}$$4, the expression reduces to a weighted Laplacian; as $$\alpha_{\rm mid}\equiv \alpha_{151\,\mathrm{MHz}}^{1400\,\mathrm{MHz}}$$5, the denominator vanishes at extrema of the intensity surface, producing highly localized peaks (Zhang et al., 2011).

The reported applications include simulated single-band ARPES dispersions, a kink in Ba$$\alpha_{\rm mid}\equiv \alpha_{151\,\mathrm{MHz}}^{1400\,\mathrm{MHz}}$$6K$$\alpha_{\rm mid}\equiv \alpha_{151\,\mathrm{MHz}}^{1400\,\mathrm{MHz}}$$7Fe$$\alpha_{\rm mid}\equiv \alpha_{151\,\mathrm{MHz}}^{1400\,\mathrm{MHz}}$$8As$$\alpha_{\rm mid}\equiv \alpha_{151\,\mathrm{MHz}}^{1400\,\mathrm{MHz}}$$9, complex multi-band ARPES in Sr$c$0V$c$1O$c$2Fe$c$3As$c$4, Fermi-surface mapping in Ba$c$5K$c$6Fe$c$7As$c$8, and a blurred Chinese-character test pattern. In these examples, curvature maps are described as producing peaks closer to real peak positions than second-derivative peaks, much sharper bands and contours, and a more consistent global picture than EDC- or MDC-derived second-derivative methods (Zhang et al., 2011).

3. Log-parabolic SCI in photon spectra

In blazar X-ray spectroscopy, curved spectra are commonly modeled with a log-parabola,

$c$9

where $C$0 directly quantifies curvature. In this setting, the natural empirical SCI is $C$1. The synchrotron SED peak for the log-parabola is given by

$C$2

so larger $C$3 moves the peak to lower energy for fixed $C$4. In the NuSTAR fits reported for MKN 421, $C$5, and Spearman analysis gives $C$6 versus $C$7 as $C$8, $C$9, with $$T(\nu)=T_0\left(\frac{\nu}{\nu_0}\right)^{\beta_0+c\,\ln(\nu/\nu_0)},$$0 versus $$T(\nu)=T_0\left(\frac{\nu}{\nu_0}\right)^{\beta_0+c\,\ln(\nu/\nu_0)},$$1 as $$T(\nu)=T_0\left(\frac{\nu}{\nu_0}\right)^{\beta_0+c\,\ln(\nu/\nu_0)},$$2, $$T(\nu)=T_0\left(\frac{\nu}{\nu_0}\right)^{\beta_0+c\,\ln(\nu/\nu_0)},$$3 (Goswami et al., 2018).

The same work introduces a physical model in which curvature arises from energy-dependent escape from the acceleration region: $$T(\nu)=T_0\left(\frac{\nu}{\nu_0}\right)^{\beta_0+c\,\ln(\nu/\nu_0)},$$4 This yields a synchrotron spectrum

$$T(\nu)=T_0\left(\frac{\nu}{\nu_0}\right)^{\beta_0+c\,\ln(\nu/\nu_0)},$$5

with curvature governed by $$T(\nu)=T_0\left(\frac{\nu}{\nu_0}\right)^{\beta_0+c\,\ln(\nu/\nu_0)},$$6, and SED peak

$$T(\nu)=T_0\left(\frac{\nu}{\nu_0}\right)^{\beta_0+c\,\ln(\nu/\nu_0)},$$7

For small $$T(\nu)=T_0\left(\frac{\nu}{\nu_0}\right)^{\beta_0+c\,\ln(\nu/\nu_0)},$$8, the model approaches log-parabolic form with

$$T(\nu)=T_0\left(\frac{\nu}{\nu_0}\right)^{\beta_0+c\,\ln(\nu/\nu_0)},$$9

Accordingly, the empirical SCI $$\ln T(\nu)=\ln T_0+\beta_0\ln(\nu/\nu_0)+c\,[\ln(\nu/\nu_0)]^2,$$0 is proportional to $$\ln T(\nu)=\ln T_0+\beta_0\ln(\nu/\nu_0)+c\,[\ln(\nu/\nu_0)]^2,$$1, while $$\ln T(\nu)=\ln T_0+\beta_0\ln(\nu/\nu_0)+c\,[\ln(\nu/\nu_0)]^2,$$2 functions as a physical curvature index tied to the energy dependence of escape (Goswami et al., 2018).

The same correlation analysis reports $$\ln T(\nu)=\ln T_0+\beta_0\ln(\nu/\nu_0)+c\,[\ln(\nu/\nu_0)]^2,$$3 versus $$\ln T(\nu)=\ln T_0+\beta_0\ln(\nu/\nu_0)+c\,[\ln(\nu/\nu_0)]^2,$$4 as $$\ln T(\nu)=\ln T_0+\beta_0\ln(\nu/\nu_0)+c\,[\ln(\nu/\nu_0)]^2,$$5, $$\ln T(\nu)=\ln T_0+\beta_0\ln(\nu/\nu_0)+c\,[\ln(\nu/\nu_0)]^2,$$6, $$\ln T(\nu)=\ln T_0+\beta_0\ln(\nu/\nu_0)+c\,[\ln(\nu/\nu_0)]^2,$$7 versus $$\ln T(\nu)=\ln T_0+\beta_0\ln(\nu/\nu_0)+c\,[\ln(\nu/\nu_0)]^2,$$8 as $$\ln T(\nu)=\ln T_0+\beta_0\ln(\nu/\nu_0)+c\,[\ln(\nu/\nu_0)]^2,$$9, $\beta$0, and $\beta$1 versus $\beta$2 as $\beta$3, $\beta$4. The paper interprets these results as indicating that the flux variations in MKN 421 may arise from a definite physical process related to escape and diffusion rather than from arbitrary fit-parameter variation (Goswami et al., 2018).

4. Galactic synchrotron SCI as a log-frequency curvature parameter

For diffuse Galactic synchrotron, several papers use a common log-parabolic framework. MeerKLASS pilot analysis parameterizes the temperature spectral index as

$\beta$5

with $\beta$6, and correspondingly

$\beta$7

In this notation, the curvature parameter $\beta$8 is explicitly the coefficient of the $\beta$9 term in $$F(E)=K\left(\frac{E}{E_*}\right)^{-\alpha-\beta\log(E/E_*)},$$0, and the second derivative obeys

$$F(E)=K\left(\frac{E}{E_*}\right)^{-\alpha-\beta\log(E/E_*)},$$1

The reported measurement in the target field is $$F(E)=K\left(\frac{E}{E_*}\right)^{-\alpha-\beta\log(E/E_*)},$$2 and $$F(E)=K\left(\frac{E}{E_*}\right)^{-\alpha-\beta\log(E/E_*)},$$3, with the statement that the spectral index changes from $$F(E)=K\left(\frac{E}{E_*}\right)^{-\alpha-\beta\log(E/E_*)},$$4 at 73 MHz to $$F(E)=K\left(\frac{E}{E_*}\right)^{-\alpha-\beta\log(E/E_*)},$$5 at 1050 MHz (Irfan et al., 2021).

Kogut adopts the same structure in the form

$$F(E)=K\left(\frac{E}{E_*}\right)^{-\alpha-\beta\log(E/E_*)},$$6

which implies

$$F(E)=K\left(\frac{E}{E_*}\right)^{-\alpha-\beta\log(E/E_*)},$$7

The quoted best-fit values are $$F(E)=K\left(\frac{E}{E_*}\right)^{-\alpha-\beta\log(E/E_*)},$$8 at $$F(E)=K\left(\frac{E}{E_*}\right)^{-\alpha-\beta\log(E/E_*)},$$9 GHz and $(\xi_0,\kappa)$0, corresponding to a steepening of $(\xi_0,\kappa)$1 every octave in frequency. The same paper gives local power-law indices from the fitted model, including $(\xi_0,\kappa)$2 at 22 MHz, $(\xi_0,\kappa)$3 at 408 MHz, $(\xi_0,\kappa)$4 at 3.3 GHz, $(\xi_0,\kappa)$5 at 23 GHz, and $(\xi_0,\kappa)$6 at 94 GHz (Kogut, 2012).

A later all-sky analysis between 45 and 2300 MHz generalizes the same idea to a per-pixel curvature map: $(\xi_0,\kappa)$7 or, in the notation of that work, $(\xi_0,\kappa)$8. In strict mathematical terms,

$(\xi_0,\kappa)$9

The full-sky average is reported as $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$00, and in the ARCADE2/Kogut region the value is

$$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$01

in excellent agreement with the earlier ARCADE2 result. The least-squares parametric model is identified as the most reliable product across radio frequencies, with average accuracies around 20 per cent when compared to external empirical data (Irfan et al., 9 Dec 2025).

Taken together, these results establish a stable radio-continuum usage of SCI: the curvature parameter $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$02, $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$03, or $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$04 measures the linear drift of spectral index with $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$05, and equivalently half the second derivative of $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$06 with respect to $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$07 (Irfan et al., 2021, Kogut, 2012, Irfan et al., 9 Dec 2025).

5. SCI as a differential spectral-index diagnostic in radio-galaxy evolution

In the G4Jy sample, SCI is defined directly from two broadband spectral indices: $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$08 The sign convention follows $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$09. A concave spectrum, flattening at higher frequencies, has $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$10 and therefore $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$11; this is interpreted as candidate restarted or renewed activity. A convex spectrum, steepening at higher frequencies, has $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$12 and therefore $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$13; this is interpreted as candidate ageing or remnant activity (White et al., 8 Dec 2025).

The sample is divided into five SCI bins:

• $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$14
• $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$15
• $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$16
• $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$17
• $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$18

The reported median values for the extreme bins are $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$19, $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$20, $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$21 for $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$22, and $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$23, $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$24, $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$25 for $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$26 (White et al., 8 Dec 2025).

This SCI is then incorporated into a P–D–(SCI) analysis, described as the first study of the radio-power–size diagram as a function of radio spectral curvature. The principal qualitative results are that candidate remnant radio galaxies with $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$27 show an interesting predominance at $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$28 kpc, although these may instead be young radio sources, and that candidate restarted radio galaxies with $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$29 span a vast range of linear sizes (White et al., 8 Dec 2025). The same work reports that there is no relation between the SCI of the radio source and its host-galaxy properties, including WISE colour-colour space and K–z behavior.

Operational thresholds are stated explicitly. Spectra with $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$30 are typically well-described by a power law from 72 MHz to 1400 MHz. Sources with $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$31 are candidate ageing or remnant radio galaxies, those with $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$32 are strong-curvature remnant candidates, those with $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$33 are candidate restarted sources, and those with $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$34 are likely restarted radio galaxies (White et al., 8 Dec 2025).

6. Interpretation, limitations, and related concepts

Across the literature, SCI functions either as a visualization operator, an empirical curvature parameter, or a phenomenological lifecycle proxy. These roles have different limitations. In ARPES and related imaging, curvature is explicitly described as a visualization and feature-tracking tool rather than a replacement for full spectral information; it loses information about spectral line shape, width, and subtle many-body effects, depends on parameters $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$35, $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$36, and $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$37, and can amplify noise if derivatives are poorly estimated (Zhang et al., 2011). In diffuse synchrotron modeling, the fitted $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$38 and $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$39 maps exhibit strong anti-correlation, the southern fine-scale solution is partially regularized, and the model is calibrated and validated only for 45–2300 MHz; at 11 GHz the residuals are substantially worse (Irfan et al., 9 Dec 2025). In radio-galaxy population studies, the G4Jy paper states that SCI alone cannot uniquely distinguish young peaked-spectrum sources, absorbed sources, and true remnants, and that improved core measurements or higher-resolution imaging can move sources in the SCI plane (White et al., 8 Dec 2025). In blazar fitting, the log-parabola curvature $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$40 is explicitly empirical, and only the diffusion or escape model attaches direct physical meaning to curvature through $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$41 (Goswami et al., 2018).

A related physical thread appears in pulsar radio emission. The analysis of coherent curvature radiation across pulsar profiles does not define an SCI, but it reports component-wise spectral-index differences of $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$42 and $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$43, interpreting them as a consequence of coherent curvature radiation from charged soliton bunches and relativistic beaming across field lines of different curvature (Basu et al., 2022). This suggests a physically motivated analogue of SCI based on spectral-index variation with field-line curvature, although that paper does not formalize the quantity under the SCI label.

The acronym itself is also non-unique. In protein structural ensemble quality assessment, SCI denotes Spectral Coherence Index, a bounded effective-rank summary derived from the eigenvalue spectrum of a distance-variance matrix. That work states explicitly that there is no "Spectral-Curvature Index" there, and any use of the same acronym in that context is unrelated to spectral curvature in spectroscopy, radio astronomy, or X-ray fitting (Bi et al., 26 Mar 2026).

The cumulative picture is therefore plural rather than singular. In spectral imaging, SCI-like quantities sharpen extrema by geometric curvature normalization; in curved continua they measure departure from a pure power law in $$\alpha_{\rm low}\equiv \alpha_{72\,\mathrm{MHz}}^{231\,\mathrm{MHz}}$$44; in radio-galaxy lifecycle work they compare low- and mid-frequency slopes. This suggests that "Spectral-Curvature Index" is best treated as a family resemblance term for curvature-sensitive scalars whose exact meaning is fixed by the spectral model, derivative operator, and physical regime under study.