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arxiv: 2604.25204 · v2 · pith:64KNJV7Nnew · submitted 2026-04-28 · ⚛️ physics.geo-ph

Accelerating unrest at Campi Flegrei signals a critical transition within the next decade

Davide Zaccagnino , Didier Sornette , Antonio Giovanni Iaccarino , Matteo Picozzi This is my paper

Pith reviewed 2026-05-25 06:34 UTC · model grok-4.3

classification ⚛️ physics.geo-ph
keywords Campi Flegreivolcanic unrestfinite-time singularitycritical transitionseismicitygeodetic deformationmagmatic volatilescaldera
0
0 comments X

The pith

The unrest at Campi Flegrei accelerates according to a regularised finite-time singularity that forecasts a critical transition around 2030-2034.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines data on seismicity and ground deformation at the Campi Flegrei caldera since 2005. It shows that the acceleration fits a regularised finite-time singularity model better than exponential growth. This points to progressive pressurization of the crust by deep magmatic volatile input as the driver. Independent analyses converge on a critical time between 2030 and 2034, with projected uplift of about 4 meters by the early 2030s. The system is approaching a mechanical threshold, though no sign of imminent eruption appears in the data.

Core claim

The acceleration of seismicity and geodetic deformation at Campi Flegrei is better described by a regularised finite-time singularity than by exponential growth. This indicates a different underlying process driven by deep magmatic volatile input that progressively pressurises the crust. Independent analyses converge on a critical time tc ≈ 2030-2034, with uplift projected to reach about 4 metres by the early 2030s. Although no evidence of imminent eruption is found, the system appears to be approaching a critical mechanical threshold whose outcome remains uncertain.

What carries the argument

Regularised finite-time singularity model fitted to seismicity and deformation time series, which captures the acceleration and yields a projected critical time tc.

If this is right

  • The system is approaching a critical mechanical threshold.
  • Uplift is projected to reach about 4 metres by the early 2030s.
  • Deep magmatic volatile input is the driver of the pressurization.
  • No evidence of imminent eruption is present in the current data.
  • Sustained high-resolution monitoring and continuously updated forecasts are required.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the singularity holds, the outcome at the threshold could include a bradyseismic peak or other regime change rather than steady growth.
  • The four-metre uplift projection would increase exposure for the population living near the caldera.
  • The same singularity approach could be applied to time series from other restless calderas to test for similar critical times.
  • Continuous model updates would allow hazard assessments to adjust as new data arrive.

Load-bearing premise

The regularised finite-time singularity model accurately captures the underlying physical process driving the unrest rather than serving only as an empirical fit.

What would settle it

Future measurements of uplift and seismicity rates that either continue to follow the projected singularity path toward 2030-2034 or deviate by leveling off or shifting pattern before that window.

Figures

Figures reproduced from arXiv: 2604.25204 by Antonio Giovanni Iaccarino, Davide Zaccagnino, Didier Sornette, Matteo Picozzi.

Figure 1
Figure 1. Figure 1: Recent evolution of seismic and volcanic activity at Campi Flegrei, Italy. (a) Spatial and temporal organization of seismicity from the high-quality, relocated machine-learning-augmented catalog of22. Most seismicity occurs within a thin brittle cap above shallow magma reservoirs, confined to depths of 2–3 km and distributed through a widely fractured volume. Topographic data are from23. A minority of even… view at source ↗
Figure 2
Figure 2. Figure 2: Probability density of the regularized finite-time singularity time (tc), defined as the time at which a change of regime is predicted from the current accelerating upscaling trend in seismic and volcanic activity. The estimate based on geodetic data from GNSS station RITE (2000–2026) is shown in light blue, while the output obtained by analyzing the Benioff strain computed from seismicity is in orange col… view at source ↗
Figure 3
Figure 3. Figure 3: Forecast of vertical soil displacement (in centimeters) at GNSS station RITE. Blue dots represent past data with zero offset set at January 2005. Light blue dots indicate the optimal data range for the forecast based on the Lagrange multiplier methodology (see Methods). The green line shows the singular model used to estimate the scaling exponent of the power law diverging trend. The red line is the regula… view at source ↗
Figure 4
Figure 4. Figure 4: Dependence between physical quantities describing seismic-volcanic activity at Campi Flegrei. This dependence is quantified using (a) Spearman’s rank correlation coefficient to assess monotonic similarities between time series, and (b) transfer entropy to infer directional causality. The analysis shows that, as expected, crustal deformation is positively correlated with seismicity (expressed as cumulative … view at source ↗
Figure 5
Figure 5. Figure 5: Vertical displacement recorded at GNSS station RITE for analysis windows with start dates ranging from 2000 to 2015 (panels a-p). The dashed blue line represents the best-fitting exponential model, the solid green line the finite-time singularity model, and the solid red line the regularized singularity model. The vertical dashed line in each panel marks the estimated critical time tc from the regularized … view at source ↗
Figure 6
Figure 6. Figure 6: Residuals of the vertical displacement recorded at GNSS station RITE for different fitting functions and different analysis windows with start dates ranging from 2000 to 2015 (panels a-p). The shadowed gray area represents the residuals of the exponential model, the solid green line the finite-time singularity residuals, and the solid red line the regularized singularity residuals view at source ↗
Figure 7
Figure 7. Figure 7: Inverse-time linearity diagnostic for GNSS vertical displacement at station RITE. The observable is plotted against the transformed time coordinate 1/(tc −t) for analysis windows with start dates ranging from 2000 to 2015. Colours progress from blue (earliest start dates) through light blue, yellow, and orange to red (latest start dates). The linear relationship confirms that the acceleration follows a fin… view at source ↗
Figure 8
Figure 8. Figure 8: Probability density of the critical time tc estimated from the finite-time singularity model for GNSS vertical displacement at station RITE. Panels a-p correspond to analysis windows with start dates progressing from 2000 to 2015. For detailed average values with uncertainty, see view at source ↗
Figure 9
Figure 9. Figure 9: Probability density of the critical time tc estimated from the finite-time regularised singularity model for GNSS vertical displacement at station RITE. Panels a-p correspond to analysis windows with start dates progressing from 2000 to 2015. For detailed average values with uncertainty, see view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of the finite-time singularity and regularized singularity models for GNSS vertical displacement at station RITE. (a) Estimated critical time tc as a function of analysis start date for both models. The regularized model yields tc values systematically closer to the present, with progressive stabilisation observed after 2008 and compatibility within uncertainty. (b) Root-mean-square error (RMSE… view at source ↗
Figure 11
Figure 11. Figure 11: Identification of the optimal analysis window for the finite-time singularity model applied to GNSS vertical displacement at station RITE. (a) Root-mean-square error (RMSE) as a function of analysis start date, with a linear fit capturing the systematic trend. (b) Residuals of the RMSE with respect to the linear fit. The minimum residual occurs at 2008 (red circle), indicating that the 2008–2026 window mo… view at source ↗
Figure 12
Figure 12. Figure 12: Identification of the optimal analysis window for the regularised finite-time singularity model applied to GNSS vertical displacement at station RITE. (a) Root-mean-square error (RMSE) as a function of analysis start date, with a linear fit capturing the systematic trend. (b) Residuals of the RMSE with respect to the linear fit. The minimum residual occurs at 2008 (red circle), indicating that the 2008–20… view at source ↗
Figure 13
Figure 13. Figure 13: Weighting procedure for the combined estimation of the geodetic critical time tc from the finite-time singularity model. (a) Probability density function of tc obtained via weighted bootstrap resampling. (b) Cumulative distribution function corresponding to the density in panel a. The weights are defined as the inverse of the total normalized uncertainty, which combines the bootstrap uncertainty of tc (25… view at source ↗
Figure 14
Figure 14. Figure 14: Weighting procedure for the combined estimation of the geodetic critical time tc from the finite-time regularised singularity model. (a) Probability density function of tc obtained via weighted bootstrap resampling. (b) Cumulative distribution function corresponding to the density in panel a. The weights are defined as the inverse of the total normalized uncertainty, which combines the bootstrap uncertain… view at source ↗
Figure 15
Figure 15. Figure 15: Correlation between the singularity exponent β and the regularization parameter a when both are fitted simultaneously for different analysis start dates (indicated by the colour bar). The parameters exhibit a strong positive linear relationship on a semi-logarithmic scale, demonstrating that they are highly covariant when left free to vary. This degeneracy motivates the sequential fitting strategy adopted… view at source ↗
Figure 16
Figure 16. Figure 16: Evolution of the singularity exponent β estimated from GNSS vertical displacement at station RITE. (a) β as a function of the number of years removed from the start of the analysis window (beginning in 2000). (b) β plotted against the inverse-time linearity diagnostic R 2 (see view at source ↗
Figure 17
Figure 17. Figure 17: Relative importance of the regularization term compared to the leading singular term, quantified by the dimensionless ratio R(t) = |a/A|·|ln(tc −t)|·(tc −t) −1 , as a function of analysis start date. The blue line (left axis) shows the maximum value of R(t) over the interval [tstart,tc), while the orange line (right axis) shows the mean value over the same interval. Both metrics reach a minimum for window… view at source ↗
Figure 18
Figure 18. Figure 18: Benioff strain for analysis windows with start dates ranging from 2000 to 2015 (panels a-p). The dashed blue line represents the best-fitting exponential model, the solid green line the finite-time singularity model, and the solid red line the regularized singularity model. 27/46 view at source ↗
Figure 19
Figure 19. Figure 19: Residuals of the Benioff strain for different fitting functions and different analysis windows with start dates ranging from 2000 to 2015 (panels a-p). The solid green line the finite-time singularity residuals, and the solid red line the regularized singularity residuals view at source ↗
Figure 20
Figure 20. Figure 20: Probability density of the critical time tc estimated from the finite-time singularity model for the Benioff strain. Panels a-p correspond to analysis windows with start dates progressing from 2000 to 2015. For detailed average values with uncertainty for intervals starting beyond 2015, see view at source ↗
Figure 21
Figure 21. Figure 21: Probability density of the critical time tc estimated from the finite-time regularised singularity model for the Benioff strain. Panels a–p correspond to analysis windows with start dates progressing from 2000 to 2015. For detailed average values with uncertainty for intervals starting beyond 2015, see view at source ↗
Figure 22
Figure 22. Figure 22: Identification of the optimal analysis window for the finite-time singularity model applied to the Benioff strain. (a) Root-mean-square error (RMSE) as a function of analysis start date, with an exponential fit capturing the systematic trend. (b) Residuals of the RMSE with respect to the linear fit. The minimum residual occurs at 2023 (red circle), indicating that the 2023–2026 window most faithfully refl… view at source ↗
Figure 23
Figure 23. Figure 23: Identification of the optimal analysis window for the regularised finite-time singularity model applied to the Benioff strain. (a) Root-mean-square error (RMSE) as a function of analysis start date, with an exponential fit capturing the systematic trend. (b) Residuals of the RMSE with respect to the linear fit. The minimum residual occurs at 2023 (red circle), indicating that the 2023–2026 window most fai… view at source ↗
Figure 24
Figure 24. Figure 24: Weighting procedure for the combined estimation of the Benioff strain-based critical time tc from the finite-time singularity model. (a) Probability density function of tc obtained via weighted bootstrap resampling. (b) Cumulative distribution function corresponding to the density in panel a. The weights are defined as the inverse of the total normalized uncertainty, which combines the bootstrap uncertain… view at source ↗
Figure 25
Figure 25. Figure 25: Weighting procedure for the combined estimation of the Benioff strain-based critical time tc from the regularised finite-time singularity model. (a) Probability density function of tc obtained via weighted bootstrap resampling. (b) Cumulative distribution function corresponding to the density in panel a. The weights are defined as the inverse of the total normalized uncertainty, which combines the bootstr… view at source ↗
Figure 26
Figure 26. Figure 26: Comparison of the regularized and pure finite-time singularity models for cumulative Benioff strain at Campi Flegrei as a function of analysis start date. (a) Estimated critical time tc for both models. The estimate remains nearly constant for windows starting before 2020, reflecting the limited seismic energy release during the early phase of the unrest. A rapid forward shift occurs in the most recent wi… view at source ↗
Figure 27
Figure 27. Figure 27: Benioff strain modelling for the optimal analysis window (2023–2026). (a) Cumulative Benioff strain with three competing model fits: exponential (dashed blue), finite-time singularity (solid green), and regularized singularity (solid red). (b) Residuals of the three fits relative to the observed data. (c) Regularized singularity fit plotted against the transformed time coordinate 1/(tc −t), demonstrating … view at source ↗
Figure 28
Figure 28. Figure 28: Evolution of the singularity exponent β estimated from Benioff strain. (a) β as a function of the number of the start of the analysis window (beginning in 2000). (b) β plotted against the inverse-time linearity diagnostic R 2 (see view at source ↗
Figure 29
Figure 29. Figure 29: Temporal evolution of the inverse logarithmic derivative of GNSS vertical displacement, 1/(d(lny)/dt), at station RITE. This quantity represents the characteristic timescale of deformation. Pronounced upward peaks correspond to rapid deformation bursts, each coinciding with episodes of heightened seismic activity. The overall trend reflects the interplay between transient accelerations and the secular evo… view at source ↗
Figure 30
Figure 30. Figure 30: Logarithmic derivative of GNSS vertical displacement, d(lny)/dt, plotted against displacement on a semi-logarithmic scale. This representation separates the deformation dynamics into distinct declining trajectories, with upward concavity reflecting the interplay between fast and slow deformation modes. The clustering of points along different curves indicates the coexistence of multiple timescales in the … view at source ↗
Figure 31
Figure 31. Figure 31: Mutual information matrix for the geophysical, geodetic and geochemical variables monitored at Campi Flegrei. Each cell quantifies the statistical dependence between variable pairs, capturing both linear and nonlinear relationships. The colour intensity (sky colormap) represents the mutual information magnitude, with brighter shades indicating stronger dependence. The matrix reveals a coupled system, with… view at source ↗
Figure 32
Figure 32. Figure 32: Granger causality matrix for the geophysical and geochemical variables monitored at Campi Flegrei. Each cell reports the F-statistic (and corresponding p-value) testing whether the past of the row variable significantly improves the prediction of the column variable beyond its own history (under the assumption of time series separability - for a more reliable assessment, see the matrix of transfer entropy… view at source ↗
Figure 33
Figure 33. Figure 33: Spectral coherence between GNSS vertical displacement and other geophysical and geochemical variables at Campi Flegrei. Coherence quantifies the frequency-domain linear correlation, with values ranging from 0 (no relationship) to 1 (perfect linear coupling). Significant coherence at periods of 2–4 years between deformation and seismic variables indicates common forcing mechanisms operating at these timesc… view at source ↗
read the original abstract

Campi Flegrei, a large caldera in southern Italy, is among the most hazardous volcanic systems on Earth, directly threatening over one million people. Since 2005, it has entered a phase of accelerating uplift accompanied by intensified seismicity, raising the key question of whether this evolution will culminate in eruption, a bradyseismic peak, or another regime change. Here, we show that the acceleration of seismicity and geodetic deformation is better described by a regularised finite-time singularity than by exponential growth, implying not just a better empirical representation but a different underlying process with potentially dire consequences for the system's subsequent evolution. Independent analyses converge on a critical time $t_c \approx 2030-2034$, with uplift projected to reach about 4 metres by the early 2030s. Geochemical and statistical evidence indicates that deep magmatic volatile input drives this evolution by progressively pressurising the crust. Although no evidence of imminent eruption is found, the system appears to be approaching a critical mechanical threshold whose outcome remains uncertain, requiring sustained high-resolution monitoring and continuously updated forecasts.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 0 minor

Summary. The manuscript claims that the accelerating uplift and seismicity at Campi Flegrei since 2005 is better described by a regularised finite-time singularity model than by exponential growth. This is interpreted as indicating a distinct underlying process driven by deep magmatic volatile pressurization, with independent analyses converging on a critical time tc ≈ 2030-2034 and projected uplift reaching ~4 m by the early 2030s. The system is said to be approaching a critical mechanical threshold without evidence of imminent eruption, necessitating sustained monitoring.

Significance. If the singularity model is demonstrated to be more than an empirical descriptor and the extrapolation is shown to be robust against parameter choices, the result would carry substantial implications for hazard assessment at one of Earth's most dangerous calderas. It would shift the interpretation from simple acceleration to an approaching critical transition, potentially guiding monitoring priorities. The work's value would be enhanced by any reproducible fitting code or falsifiable predictions, but these are not indicated in the provided text.

major comments (3)
  1. [Abstract] Abstract: the claim that the regularised finite-time singularity 'implies not just a better empirical representation but a different underlying process' is unsupported, as no derivation is given linking the functional form to the governing equations of crustal deformation or volatile-driven pressurization; only a comparison to exponential growth is stated.
  2. [Abstract] Abstract: the reported tc ≈ 2030-2034 is obtained by fitting the singularity model (with free parameters tc and regularisation parameter) to the observed acceleration, so the 'prediction' of the critical transition is the fitted value itself rather than an independent forecast; this circularity undermines the central claim that the model signals an approaching threshold.
  3. [Abstract] Abstract: no details are supplied on the seismicity and geodetic data sources, the precise fitting procedure, error analysis, model comparison metrics, or sensitivity of tc to the regularisation parameter, all of which are load-bearing for establishing superiority over exponential growth and for the uplift projection of ~4 m.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their detailed and constructive comments, which have helped us identify areas where the manuscript can be clarified and strengthened. We respond to each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the regularised finite-time singularity 'implies not just a better empirical representation but a different underlying process' is unsupported, as no derivation is given linking the functional form to the governing equations of crustal deformation or volatile-driven pressurization; only a comparison to exponential growth is stated.

    Authors: We agree that the abstract overstates the implication. The manuscript does not contain a derivation from the governing equations of crustal deformation or volatile pressurization to the regularised finite-time singularity form; the functional form is selected for its empirical superiority over exponential growth and its prior use in describing accelerating failure. The interpretation of a distinct process is instead supported by the separate geochemical evidence of deep magmatic volatile input presented in the paper. We will revise the abstract to remove the unsupported claim and rephrase it as indicating a better empirical representation that is consistent with a distinct driving process, while expanding the discussion section to clarify the distinction between empirical fit and physical derivation. revision: yes

  2. Referee: [Abstract] Abstract: the reported tc ≈ 2030-2034 is obtained by fitting the singularity model (with free parameters tc and regularisation parameter) to the observed acceleration, so the 'prediction' of the critical transition is the fitted value itself rather than an independent forecast; this circularity undermines the central claim that the model signals an approaching threshold.

    Authors: The referee is correct that tc is obtained by fitting. The manuscript's central claim, however, rests on two elements: (1) the regularised singularity model provides a statistically superior description of the acceleration compared with exponential growth, and (2) independent fits to separate observables (seismicity rate and geodetic uplift) converge on closely similar tc values. This cross-dataset consistency is presented as evidence that the data are approaching a threshold, not as an independent forecast. We will revise the text to make this distinction explicit, stating clearly that tc is a fitted parameter whose value is corroborated across observables, and that the projection beyond the observation window is an extrapolation under the fitted model. revision: partial

  3. Referee: [Abstract] Abstract: no details are supplied on the seismicity and geodetic data sources, the precise fitting procedure, error analysis, model comparison metrics, or sensitivity of tc to the regularisation parameter, all of which are load-bearing for establishing superiority over exponential growth and for the uplift projection of ~4 m.

    Authors: The full manuscript contains a dedicated methods section that specifies the data sources (INGV seismic catalogue and continuous GPS stations), the nonlinear least-squares fitting procedure, model comparison via AIC and residual variance, and a brief sensitivity check on the regularisation parameter. Nevertheless, these elements are not presented with sufficient explicitness or supporting figures to satisfy the referee's requirements. We will therefore expand the methods section with precise algorithmic details, report formal uncertainties on tc, include a supplementary figure showing tc sensitivity to the regularisation parameter, and make the fitting scripts available as supplementary material. These additions will also support the ~4 m uplift projection. revision: yes

Circularity Check

1 steps flagged

tc reported as convergence on critical transition is the fitted parameter of the singularity model

specific steps
  1. fitted input called prediction [Abstract]
    "we show that the acceleration of seismicity and geodetic deformation is better described by a regularised finite-time singularity than by exponential growth, implying not just a better empirical representation but a different underlying process with potentially dire consequences for the system's subsequent evolution. Independent analyses converge on a critical time $t_c ≈ 2030-2034$, with uplift projected to reach about 4 metres by the early 2030s."

    The regularised finite-time singularity model contains tc as a fitted parameter that controls the location of the singularity; the reported 'convergence' on 2030-2034 and the uplift projection are direct outputs of fitting this model to the existing data series, making the forecast equivalent to the fitted input rather than an independent prediction.

full rationale

The paper fits a regularised finite-time singularity model to the observed acceleration in seismicity and deformation, with tc as an explicit parameter of that model. The abstract then presents the fitted tc value (2030-2034) as the outcome of 'independent analyses' that 'converge' on a critical time, and projects future uplift from the same fit. This reduces the central forecast to the input fit by construction (pattern 2). No separate derivation from governing equations or external validation is quoted that would make the tc value independent of the fit itself. The interpretive step from 'better fit' to 'different underlying process' is not a derivation and does not alter the circularity of the tc claim.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The model relies on fitting parameters to data and assuming the singularity framework applies to this volcanic system; no new entities postulated. Assessment limited to abstract content.

free parameters (2)
  • critical time tc = 2030-2034
    Fitted parameter from the regularised finite-time singularity model to the seismicity and deformation data.
  • regularisation parameter
    Introduced to regularise the singularity model; specific value not given in abstract.
axioms (1)
  • domain assumption The observed acceleration in unrest is governed by a process that can be modeled as a regularised finite-time singularity.
    Central to the claim that it is better than exponential growth and leads to a critical transition.

pith-pipeline@v0.9.0 · 5734 in / 1552 out tokens · 35248 ms · 2026-05-25T06:34:07.371910+00:00 · methodology

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