Rigidity spectra and onset geometry of the two largest Forbush decreases of solar cycle 25 from visibility-graph curvature

D. Sierra-Porta

REVIEW 3 major objections 5 minor 29 references

Rigidity spectra and onset geometry of the two largest Forbush decreases of solar cycle 25 from visibility-graph curvature

T0 review · 3 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Curvature of the visibility graph built from neutron-monitor count rates collapses sharply and coherently at the onset of the two deepest Forbush decreases of solar cycle 25, and the signature survives a structure-preserving permutation tes

desk verdict Whole-series visibility-graph curvature collapses at FD onset and survives permutation testing, but the primary analysis is not time-local and some 'onset' minima occur before the shock; the sliding-window version that would fix this is only a robustness check. read the letter →

arxiv 2607.13236 v1 pith:F7ZCKRW7 submitted 2026-07-14 astro-ph.IM astro-ph.EPastro-ph.SR

D. Sierra-Porta This is my paper · ORCID

classification astro-ph.IMastro-ph.EPastro-ph.SR

keywords ForbushdecreasevisibilitygraphForman-Riccicurvatureneutronmonitorrigidityspectrumcosmic-raymodulationgeomagneticstormtime-seriesnetwork

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper is trying to establish that a geometric network measure—the nodal Forman–Ricci curvature of a natural visibility graph—turns the onset of a Forbush decrease into a sharp, statistically significant dip in a time series, and that this dip carries information about the event's amplitude and rigidity spectrum. The two test events are the largest of solar cycle 25: the fast, sheath-dominated May 2024 storm and the deeper, compound June 2025 event, observed by the same six neutron monitors. The claim matters because it offers a compact, single-station, real-time-computable descriptor of Forbush-decrease morphology—onset timing, depth, and spectral hardness—rather than just a detection of the decrease itself. The analysis anchors every graph-derived quantity to shock-based event phases and checks specificity with an amplitude-blind control.

What carries the argument

The machinery is the natural visibility graph (NVG)—nodes are 30-minute count-rate samples, and two samples are linked if every intermediate sample lies below the straight line joining them—combined with the nodal Forman–Ricci curvature F(u) = 4 − k_u − ⟨k⟩_N(u), where k_u is the node's degree and ⟨k⟩_N(u) is the mean degree of its neighbors. This quantity is a two-hop local measure: a node adjacent to a hub inherits strongly negative curvature even if its own degree is modest. In a Forbush decrease, the few extreme samples of the steep gradient become hubs visible to large stretches of the record, so the curvature collapses exactly where the count-rate gradient is steepest. Unit-weight, tem

What would settle it

Compute nodal curvature on a sliding window shorter than the main-phase gradient (e.g., 3 hours) for the May 2024 event; if the sharp onset minimum persists at the shock time, the signal is local, whereas if it shifts or disappears the whole-series result is contaminated by long-range visibility links. Alternatively, replace the FD interval in the record with quiet-time fluctuations, rebuild the visibility graph from the remaining samples, and check whether a curvature minimum still appears at the same time.

Watch

Extended reading notes

Core claim

The central discovery is that nodal Forman–Ricci curvature F(u) of the natural visibility graph built from 30-minute neutron-monitor records drops from quiet-time medians of roughly −10 to −14 to onset medians of −20 to −134, giving onset-to-quiet median ratios of 2.5–5.2 for the May 2024 storm and 3.3–8.2 for the June 2025 event, with the curvature minimum reached within hours of the shock arrival and before the count-rate minimum. A placement test that compares each observed phase-window mean with all contiguous windows of equal length in the pre-event interval finds the observed mean below every null window for all 24 station–phase combinations, so the attained p-value sits at the test fl

Load-bearing premise

The load-bearing premise is that the curvature value at a timestamp is a local descriptor of the series at that time, even though the natural visibility graph links any mutually visible pair across the entire record; if long-range links from the deep-decrease hub contaminate the quiet-time baseline, the claimed onset-to-quiet contrast and timing are not purely local.

Editorial extensions

If this is right

If the curvature collapse tracks onset, a single neutron monitor can timestamp the arrival of the driving interplanetary shock or sheath without waiting for the count-rate minimum.
The rigidity-ordered amplitudes yield FD spectral indices (0.80 and 0.50) that place both events in the canonical range and show the June 2025 event is both deeper and spectrally harder.
The |F_min| ∝ A^0.57 relation means curvature can serve as a proxy for FD amplitude once calibrated, including at stations where baseline normalization is difficult.
The HVG control implies the signal is tied to the depth, not just the ordering, of the decrease, so the method is aimed at resolving significant events rather than cataloging sub-percent Forbush effects.
Because the curvature is time-local and cheap to compute, it could be monitored in real time alongside established precursor indicators.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the curvature response is read as a geometric marker of the transition from fluctuation-dominated to gradient-dominated transport, the onset dip should also appear in other impulsive cosmic-ray modulations (such as solar energetic particle events, after masking) whenever a steep unidirectional gradient is present.
The authors' explicit caveat that the placement-test p-value is floored at about 2×10⁻³ by the length of the quiet interval suggests that longer pre-event baselines, or surrogate-based tests, could sharpen significance and allow phase-resolved comparisons between events of different morphology.
The sublinear exponent 0.57 hints that curvature saturates for very deep decreases; a calibration with synthetic profiles injected into real quiet-time records—which the authors propose as next work—would determine whether the relation is universal across station sizes, counting rates, and cutoff rigidities.
A catalog-scale application would require hourly or finer data and short event-matching windows, as the paper itself states; this is an extension, not an established result.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper does two things well. It is the first application of Forman–Ricci curvature on visibility graphs to Forbush decreases, and it documents the effect carefully: six stations, two large events, a placement test that preserves autocorrelation, an amplitude-blind HVG control, and a decomposition showing that the two-hop neighbor-degree term carries most of the signal. The qualitative result—curvature collapses at onset, with ratios of 2.5 to 8.2 and large effect sizes—is credible and consistent with a deep negative excursion creating visibility hubs. The spectral indices (γ = 0.80 and 0.50) and the amplitude–curvature scaling (A^0.57) are new numbers, even if the sample is small. I also credit the authors for being explicit about limitations: they say detection is not an open problem, they mask the GLE74 interval, and they acknowledge the placement test floor.

The soft spot is not the statistics or the graph construction per se; it is the claim that nodal curvature is time-local. Equation (1) connects mutually visible pairs across the entire record, so F(t) at a quiet-time node can be pulled down by a future FD hub. The paper's own Figure 2 shows Gannon curvature minima at 14:30–16:30 UT, hours before the 17:00 UT shock. Under a local construction, that should not happen. The authors mention this indirectly when they say curvature responds to the gradient rather than the floor, but the pre-shock minima are not explained by that. The sliding-window NVGs in §4.3 are exactly the right fix, but they are presented only as robustness checks, and the timing claim in §4.2 is driven by the whole-series graphs. The placement test does not resolve this, because the null windows are drawn from the same globally constructed graph and are themselves contaminated. The amplitude–curvature power law is fit to the same 12 points that define the amplitudes, so it is suggestive, not established. And no code is shipped.

This is not a fatal flaw. The central result—that the curvature collapses at onset and is statistically distinguishable from quiet-time fluctuations—likely survives a local reanalysis; the June 2025 event shows the effect in sliding windows too. But the paper overstates time-locality and the onset geometry claim needs rework. A serious referee should ask the authors to present the sliding-window analysis as primary, report where the minima fall under local construction, and ideally release the code. It is worth engaging with, and it should be sent to peer review rather than desk-rejected.

Referee Report

3 major / 5 minor

Summary. The paper applies nodal Forman–Ricci (FR) curvature of natural visibility graphs (NVG) to the two largest Forbush decreases of solar cycle 25, using 30-min NMDB data from six NM64 stations spanning cutoff rigidities 0–7 GV. It reports a sharp, coherent curvature minimum at FD onset in both events, with onset-to-quiet median curvature ratios of 2.5–5.2 (Gannon) and 3.3–8.2 (June 2025), supported by a block-permutation placement test at p below the attainable floor for all 24 station–phase combinations. The paper also fits FD rigidity spectra (γ = 0.80 and 0.50), reports an amplitude–curvature scaling |F_min| ∝ A^0.57, and includes robustness checks using edge-weighted curvature, sliding-window graphs, and an amplitude-blind horizontal-visibility control.

Significance. If the central timing claim survives scrutiny, the paper introduces a novel and computationally cheap geometric descriptor for FD morphology, with the attractive properties of single-station operation and sensitivity beyond the degree sequence. The study is commendably transparent: the placement test preserves autocorrelation, the HVG control is a sensible specificity check, the sliding-window experiments directly address non-stationarity, and the authors list several limitations explicitly. However, the main claim that nodal FR curvature marks FD onset in a time-local manner rests on whole-series NVG construction, whose acausal long-range edges can shift or create pre-onset minima. This issue is load-bearing for the paper's primary conclusion and must be resolved before the results can be accepted as stated.

major comments (3)

[§3.1, Eq. (1); §4.2, Fig. 2] The whole-series NVG makes F(t) depend on the entire record: Eq. (1) links any mutually visible pair across all n samples, so a node at time t can be connected to a future FD hub. In §4.2 five of six Gannon stations have their curvature minima between 14:30 and 20:30 UT on May 10, several hours before the 17:00 UT shock. Under a truly time-local construction, a 14:30 UT node should not respond to a shock arriving at 17:00 UT; the most natural explanation is long visibility edges to the future depression. The paper's primary 'onset geometry' claim therefore rests on an acausal descriptor. The sliding-window analysis in §4.3 mitigates this, but it is presented only as a robustness check, and the placement test does not resolve the issue because its null windows are drawn from the same globally constructed graph and are equally contaminated. The primary analysis should be repeated with a ca
[§4.4, Fig. 4b] The amplitude–curvature relation |F_min| ≈ 26.6 A^0.57 is fit to 12 station–event pairs, but both A and |F_min| are derived from the same count-rate series and the same NVG; the fit reports no uncertainties on the prefactor or exponent, no scatter plot residuals, and no test of sensitivity to excluding either event or individual stations. With 12 points clustered into two events, Spearman ρ = 0.75 (p = 5×10^-3) is suggestive but not a strong scaling law. The paper should provide confidence intervals on the power-law fit, show per-event fits, and discuss the effective number of independent samples.
[Eq. (3); §5] Part of the headline 'curvature collapse' is a mathematical consequence of the definition: Eq. (3) gives F(u) = 4 − k_u − ⟨k⟩_{N(u)}, and a deep negative excursion becomes a visibility hub by the NVG rule, so any node connected to that hub acquires strongly negative curvature. The paper acknowledges this mechanism in §5, but the framing in the abstract and conclusions presents the collapse as an empirical discovery rather than as an expected property. The empirical content lies in the timing, coherence, rigidity ordering, and amplitude scaling of the collapse. Once the acausality of the whole-series graph is addressed, the paper should explicitly separate the tautological component from the genuinely data-driven component.

minor comments (5)

[Title] The title contains a typo: 'CUR V ATURE' should be 'CURVATURE'.
[§5] The phrase 'locality in time (a curvature value per sample)' is misleading because the whole-series NVG curvature at a sample depends on the full record unless a sliding-window or causal construction is used. Please rephrase to avoid contradicting §3.1.
[§4.3] The text says 'Figure 5 compares...' but the figure appears after Fig. 4; please check the numbering and in-text references.
[Data and code availability] The statement 'Analysis code and derived products are available from the authors on request' is weaker than the reproducibility standards of most journals. Please consider depositing the code in a permanent repository, especially since the graph construction and permutation test are central to the paper.
[§3.2 and §5] The GLE74 masking by linear interpolation over 8 h at polar stations is acknowledged as a limitation, but the possible effect on curvature in the recovery phase is not quantified. A brief sensitivity test (e.g., masking a few hours more or less) would be useful.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the curvature collapse is a measured property of a well-defined graph statistic, onset phases are anchored to independent OMNI shock times, and the scaling relation is an empirical fit, not a prediction.

full rationale

The paper's central claim is that nodal Forman-Ricci curvature of natural visibility graphs exhibits a coherent minimum at FD onset. This is not circular: the curvature statistic is defined by Eqs. (2)-(3), but the FD onset phases against which it is compared are anchored to independent OMNI shock times ('Shock arrivals are taken at 2024-05-10 17:00 UT... and 2025-06-01 05:00 UT... both confirmed by the largest positive 30-min jumps of SYM/H'). The curvature collapse is therefore a measured property of the graph, not an input to the phase definition. The amplitude-curvature relation |F_min| ≈ 26.6 A^0.57 is an empirical least-squares fit to 12 station-event pairs, not a derived prediction; the spectral indices are similarly fitted through the Dorman response and reported with leave-one-out sensitivity, so they are not an instance of a fitted parameter being relabeled as a prediction. The only self-citations (Sierra-Porta 2024, 2025) appear as contextual justification that the method should be anchored to interplanetary context ('rather than applied as a physics-blind transformation'), not as load-bearing derivations. The skeptical concern that whole-series NVG links are acausal (a node can see future FD hubs) is a real robustness limitation, but the paper explicitly addresses it with sliding-window graphs (§4.3) that are 'immune to non-stationarity outside the window,' and the placement test is not used to define onset times. Acausality is a methodological correctness risk, not a case of the result being equivalent to its inputs by construction. No circular step satisfying the hard-rule requirement (explicit reduction via Eq. X = Eq. Y, or fitted parameter renamed as prediction) can be quoted.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central empirical claims rest on two fitted spectral parameters per event plus one fitted scaling law; all other inputs are standard definitions or literature parameters. No new physical entities are introduced.

free parameters (5)

FD spectral index gamma (Gannon) = 0.80 (leave-one-out 0.78-0.81)
Least-squares fit to six station amplitudes via Dorman coupling folding (§3.3, §4.4).
FD amplitude A10 (Gannon) = 11.2%
Normalization parameter in the same spectral fit (§4.4).
FD spectral index gamma (June 2025) = 0.50 (leave-one-out 0.37-0.59)
Least-squares fit to six station amplitudes (§4.4).
FD amplitude A10 (June 2025) = 18.8%
Normalization parameter in the same spectral fit (§4.4).
Amplitude-curvature power-law coefficient and exponent = 26.6 and 0.57
Fit to 12 station-event curvature extremes vs FD amplitudes; Spearman rho = 0.75 (§4.4, Fig. 4b).

assumptions (6)

standard math Forman-Ricci curvature with unit weights equals 4 - k_u - k_v, and nodal curvature F(u) combines own degree and neighbor mean degree.
Used in Eq. (3) as the definition of the diagnostic; the qualitative link between steep drops and negative curvature follows from it.
domain assumption The Dorman coupling function W(R) = a k R^{-(k+1)} e^{-a R^{-k}} with (a,k) = (8.123, 0.933) is a valid response function for all six NM64 stations.
Invoked in §3.3 to convert station amplitudes to FD spectral parameters; if inaccurate, especially for high-altitude BKSN, the fitted gamma values are biased.
domain assumption The assigned shock times (2024-05-10 17:00 UT and 2025-06-01 05:00 UT) define the event phases correctly.
§3.2: all phase statistics and 'onset' claims depend on these anchors, confirmed only by SYM/H jumps and literature values.
domain assumption Linear interpolation across the masked GLE74 interval does not materially alter visibility-graph curvature at polar stations.
§2, §3.2: the GLE74 mask replaces 8 h of polar data; this smoothing could affect OULU/TERA Gannon curvature during recovery.
domain assumption The FD rigidity spectrum is a single power law delta J/J = -A10 (R/10 GV)^-gamma over the 0-7 GV range.
§3.3: assumed before fitting; residuals under 1% support it, but it is not independently tested.
domain assumption Contiguous pre-event blocks provide a valid null distribution for onset-window mean curvature under the true autocorrelation.
§3.4: relies on quiet-time stationarity across the pre-event interval; contamination of quiet windows by long-range visibility edges would distort the null.

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Cite this review

Pith. "Pith review of Rigidity spectra and onset geometry of the two largest Forbush decreases of solar cycle 25 from visibility-graph curvature." pith.science (2026). https://pith.science/paper/F7ZCKRW7

@misc{pith2026260713236,
  author       = {Pith},
  title        = {Pith review of: Rigidity spectra and onset geometry of the two largest Forbush decreases of solar cycle 25 from visibility-graph curvature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F7ZCKRW7}},
  note         = {Machine review of arXiv:2607.13236}
}

abstract

We apply a geometric network diagnostic -- the nodal Forman-Ricci (FR) curvature of natural visibility graphs (NVG) -- to the two largest Forbush decreases (FDs) of solar cycle 25: the Gannon storm of 2024 May 10-11 and the 2025 June 1 event. Using 30-min NMDB records from six NM64 stations spanning cutoff rigidities $R_c \simeq 0$-$7\,$GV, we show that nodal FR curvature exhibits a sharp, coherent minimum at FD onset in both events, with onset-to-quiet median curvature ratios of 2.5-5.2 (Gannon) and 3.3-8.2 (June 2025). A placement (block-permutation) test that preserves the full autocorrelation structure confirms the onset signature at $p \lesssim 2 \times 10^{-3}$ (the floor of the test) for all 24 station-phase combinations, with $|z| = 4.7$-$23.7$. The signature is robust to edge weighting of the Forman curvature and to sliding-window (6-24 h) graph construction, and an amplitude-blind (horizontal-visibility) control confirms that it is specific to the amplitude geometry of the decrease. Nodal FR curvature nearly doubles the effect size obtained from the NVG degree sequence alone (mean Cliff's $\delta$ of 0.89 vs 0.54), because it encodes two-hop (neighbor-degree) structure. Folding the station amplitudes through the Dorman coupling function yields FD spectral indices $\gamma = 0.80$ (Gannon) and $0.50$ (June 2025) with $A_{10} = 11.2\%$ and $18.8\%$: the deeper event is also spectrally harder. Across the 12 station-event pairs the curvature extreme scales with FD amplitude as $|\mathcal{F}_{\min}| \propto A^{0.57}$ (Spearman $\rho = 0.75$). These results establish local graph curvature as a compact, rigidity-resolved descriptor of FD morphology.

Figures

Figures reproduced from arXiv: 2607.13236 by the authors.

**Figure 1.** — Normalized count-rate profiles for the six stations (top) and OMNI SYM/H (bottom) for the Gannon storm (left) and the June 2025 event (right). Shading marks the onset and minimum phases and the masked GLE74 interval. the moderate storm activity of May 28–31; the main decrease then proceeds in two steps after the 05:00 UT shock on June 1: a first sub-minimum at 13:30–15:30 UT (where BKSN, YKTK and TERA record their… view at source ↗

**Figure 2.** — Nodal FR curvature (top: 6-h running median over the full window; bottom: raw values around onset). pre onset main minimum recovery 150 125 100 75 50 25 0 Nodal Forman Ricci curvature Gannon storm (May 2024) pre onset main minimum recovery June 2025 event CALM (6.95 GV) BKSN (5.70 GV) NEWK (2.40 GV) YKTK (1.65 GV) OULU (0.81 GV) TERA (0.00 GV) [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** — Distributions of nodal FR curvature by event phase. 4.3. Robustness [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: — (a) Station FD amplitudes vs cutoff rigidity with the fitted Dorman-response power-law spectra. (b) Curvature extreme vs FD amplitude for the 12 station–event pairs, with the fitted power law. 0 1 2 3 4 5 6 7 Cutoff rigidity Rc [GV] 0.2 0.0 0.2 0.4 0.6 0.8 1.0 Cliff'…

**Figure 5.** Figure 5: — Onset effect size (Cliff’s δ, pre vs onset) as a function of cutoff rigidity for the three graph/curvature variants. degree term alone yields mean onset δ = 0.54 (range 0.33–0.82), the neighbor-degree term alone 0.89 (0.64–0.99), and the full nodal curvature 0.89 (0.…

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Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

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On the effects in cosmic-ray intensity observed during the recent magnetic storm , author=. Physical Review , volume=. 1937 , publisher=

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Coronal mass ejections and Forbush decreases , author=. Space Science Reviews , volume=. 2000 , publisher=

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From time series to complex networks: The visibility graph , author=. Proceedings of the National Academy of Sciences , volume=. 2008 , publisher=

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