A universal relation predicts the observed reversal of a supernova chirp

The power source of hydrogen-poor (Type I) superluminous supernovae (SLSNe-I), which outshine their ordinary supernova counterparts by at least an order of magnitude, remains an open question [1, 2, 3]. A leading explanation is that they are powered by a central engine, such as a newborn rapidly spinning magnetar, which deposits its rotational energy into the expanding ejecta [4, 5]. Despite this model predicting a monotonic, smooth decline post-peak, a majority (50–70%) of SLSNe-I show bumps in their light curves that the magnetar model cannot reproduce [6, 7, 8]. These bumps have been alternately attributed to either engine variability [e.g., 9, 10] or collisions with circumstellar material (CSM) [e.g., 11], with surveys unable to distinguish between the two competing mechanisms [7, 12].

The discovery of SN 2024afav—the first “chirping” supernova, displaying in its light curve (Figure 1) quasiperiodic modulations with an increasing frequency—presents a new trial for proposed mechanisms [13]. We attributed these modulations to a tilted fallback disk precessing around the magnetar, tying the bumps directly to the engine. However, interpreting the chirp in light of this model is challenging, due to the large number of degrees of freedom available. The neutron star’s spin, magnetic field, and moment of inertia, as well as the structure of and accretion rate through the disk, all set the modulation period in combinations the light curve cannot disentangle. A good fit thus can only show that the engine picture is consistent with the data, not that the data require it. Similarly, a sufficiently structured CSM could theoretically reproduce any finite set of bumps [11]. A stronger test of either hypothesis would therefore be a falsifiable prediction that can be made independently of the highly-degenerate degrees of freedom. Here, we reduce the engine model to the scalings that survive any choice of system parameters, derive a novel prediction from those alone, and test it against follow-up observations of SN 2024afav at late times.

A universal relation for the engine

Although the details of the engine configuration are challenging to measure individually, four requirements common to the engine picture are sufficient to uniquely determine how the chirp must follow the fading of the engine. First, the disk precesses at a rate which depends on a power law of inner-edge radius $r_{\mathrm{i}}$ and the magnetar’s angular momentum $J$, so that the precession period scales as $P \propto r_{\mathrm{i}}^{\kappa}J^{-q}$. The three physical mechanisms capable of producing this effect—Lense–Thirring, quadrupole-driven, and magnetic warp precession—all take this form, with differing exponents [15, 16]. Second, the inner edge evolves with the location where the pressure of the magnetar wind, of luminosity $L_{\mathrm{w}}$ (producing pressure $L_{\mathrm{w}}/r_{\mathrm{i}}^{2}$), balances the pressures of the disk [17, 18], which we take in the most general case to be $\propto\dot{M}r_{\mathrm{i}}^{-w}$ [19]. Third, we assume that the accretion through the disk $\dot{M}$ is driven by fallback which declines as a power-law in time, $\dot{M} \propto t^{-\chi}$ [20] ($t = 0$ assumed to be the explosion epoch, see Section A3). Finally, we assume the magnetar powers the wind extracting its rotational energy, with the rotational frequency $\Omega$ spinning down according to a braking index $n$, i.e., $\dot{\Omega} \propto\Omega^{n}$ [21]. We assume that the salient exponents stay constant during the chirp, and though we do not specify other parameters (e.g., disk $\alpha$) we assume they remain constant with time. Note that we do not require the assumption of a particular precession mechanism ($q$, $K$), disk pressure regime ($u$, $w$), braking mechanism ($n$), &c.

Intriguingly, under this relatively broad set of assumptions, the new observables provided by the chirp are linked by a universal relation which contains none of the unknown system parameters (see Section A3). If we consider a measure of the “chirping rate”, $C(t) \equiv-\mathrm{d}\ln P/\mathrm{d}\ln t$, and the diffusion-corrected “dimming rate” of the supernova, $D(t) \equiv-\mathrm{d}\ln L_{\mathrm{w}}/\mathrm{d}\ln t$, the four assumptions above enforce,

Regardless of any particular choice of system parameters, the measured evolution of the supernova in the $(D,C)$ plane must trace a straight line. This expected linearity is robust to most effects, including the choice of precession mechanism; the pressure regime of the disk; the spin-down mechanism, for any torque that scales as a power of the spin (magnetic dipole, magnetospheric, or gravitational-wave); a precession rate that also depends on the accretion rate, as in magnetic warping; and observational uncertainties (e.g., distance, explosion epoch, host extinction, &c.). The linearity demonstrated here is therefore a test of the proposed engine picture as a whole rather than of any particular realization of it. The same robustness extends beyond linearity, as any prediction that follows from Eq. (1) inherits its independence from the unknown parameters.

The predicted period minimum

Indeed, the consequences of Equation (1) make a concrete prediction about chirping systems that require no particular choice of the unknown parameters. The linear construction of the relation directly implies that the chirp has a zero-crossing determined solely by the closure, at $D_{\min} = (w-2)^{-1}K\chi u/(K/(w-2) + q/(n+1))$, independent of any specific properties of the magnetar or the disk. In fact, the zero-crossing implies that the chirp halts and reverses at the epoch $t_{\min}$ associated with $D_{\min}$, set by the magnetar spin-down. The physical origin of this reversal is the competing declines of the engine luminosity and the fallback accretion rate, whose interplay sets the disk radius (and thus the precession period), producing a turnaround when the magnetar engine shuts off after $\approx t_p$ (section A 3, Figure 7).

What makes this prediction remarkable is that the timing is largely driven by the spin-down time, which can be estimated from the light curve independently from the modulations. The system configuration $(K, w, \chi, \ldots)$ sets the overall level (offset from $P = 0$) and scale (magnitude between maximum and minimum period) of the chirp, but the shape (relative curvature at any two points; section A 3) of the periodicity evolution (and therefore the minimum) is essentially dictated by the smooth spin-down of the engine. An object observed to chirp past $t_{\min}$, or one that never executes a predicted reversal, falsifies the engine model outright. For SN 2024afav, whose published chirp ends just short of this regime, the prediction is unambiguous: continued follow-up should reveal a reversing chirp.

The universal relation of Equation (1) goes beyond predicting simply that the chirp will reverse, also forecasting characteristics such as the time of reversal and the spread of allowed period derivatives post-reversal (section A 4). The prediction uses only two inputs, neither of which depends on the late-time modulations: (i) the five chirp periods previously reported in [13], and (ii) the spin-down time measured from the smooth light curve. No extremum after $t \approx170$ d is included in the fit. To produce a prediction independent of any particular closure, we fit Equation (1) to the previously-reported chirp periods alone (determining the relation slope and intercept for the entire event without specifying the system configuration), with the spin-down time fixed to $t_p = 87 \pm11$ d by the unmodulated light curve (section A 2, Figure 5). The constraint on the relation slope and intercept from the chirp-only periods, in conjunction with the spin-down time fit from the overall light curve, significantly limit the allowed behavior of the antichirp. The universal relation predicts that the chirp rate zero-crossing should occur at $t_{\min} \approx133$ d (rest frame, mode of $t_{\min}$ distribution from universal relation fit to chirp, see Figure 3), before the period lengthens into an antichirp with an average period derivative of $\dot{P} = 0.22^{+0.24}_{-0.11}$ between $t = 157$ and $t = 202$ d, with this interval chosen to match follow-up observations of SN 2024afav. This prediction is not simply an extrapolation of the chirp. While a simple linear fit to the chirp periods alone favors continued shortening (Figure 3, dot-dashed line), the turnaround emerges organically once the chirp is fit with Equation (1), whose curvature is necessarily positive and whose minimum is timed by the spin-down. Equation (1) thereby fixes three predictions in advance: that $\dot{P}$ changes sign, when it does so, and how steeply the period subsequently rises.

Chirp reversal observed in SN 2024afav

The published chirp of SN 2024afav spans 7 extrema between about 50 and 170 days (times are from explosion in observer frame unless noted as rest frame) [13]. Over that interval, the period fell from $P = 39 \pm8$ d to $P = 17 \pm5$ d. The period evolution invites a simple linear extrapolation ($\dot{P} = -0.44 \pm0.05$), and phenomenologically predicts that the period would keep shortening until it reaches zero near $t = 180$ d. The universal relation forbids this, as did the model published with the chirp [13], of which Equation (1) is simply a generalization. In that model, the precession period decelerated, flattening toward $P \approx25$ d by $t = 150$ d, rather than continuing to fall. Extending the model of [13] past the previously reported chirp predicts a clear period reversal (Figure 3, panel A).

To test the prediction, we examine the data acquired during continued follow-up of SN 2024afav past the window analyzed for the published chirp. Through this follow-up, we obtained 109 further epochs in the $g,r,$ and $i$ band with the Las Cumbres Observatory (LCO) global telescope network [22] through MJD 60908 (262 days after explosion), alongside $c$ and $o$ photometry from the Asteroid Terrestrial-Impact Last Alert System (ATLAS; [23]). These epochs had been published earlier in [24] and were searched for modulations only after the precessing-disk model had been published [13]. We built the bolometric luminosity light curve following the same procedure as [24] (section A 4).

The late light curve continues to show modulations. Relative to a cubic spline smoothing baseline (section A 4), the residuals show five more extrema after $t = 170$ d (Table III), corresponding to roughly two and a half additional cycles (although we note that the modulations are also readily apparent in the non-subtracted light curve). The presence of a reversal is qualitatively visible in the spacings of the extrema (Figure 2, panel B); before $t \approx160$ d, the extrema grow closer together, whereas after $t \approx160$ d they drift apart. We identify identical modulations when constructing the residual using the (smooth post-peak) magnetar model (suggesting that the presence of the modulations is not sensitive to the flexibility in the smoothing spline baseline) and when using multiple methods of extrema identification. We further validate the behavior via a time-frequency decomposition (Figure 2 panel C; using the method of weighted wavelet $z$-transform, see section B 1 for details). The dominant ridge in the power spectrum climbs from $\approx0.025\,\mathrm{d}^{-1}$ at $t \approx90$ d to $\approx0.07\,\mathrm{d}^{-1}$ at $t \approx170$ d, then clearly descends, tracing a reflection of the published chirp. The modulation frequency unambiguously rises and then falls—the chirp reverses.

ID	Endpoints	Observer frame			Rest frame
		$t_{\mathrm{obs}}$ (d)	$P \pm\sigma_P$ (d)	$\Delta$ (d)	$t_{\mathrm{rest}}$ (d)	$P \pm\sigma_P$ (d)
P01	E02–E04	99.33	39.63 ± 7.58	7.76	84.90	36.97 ± 7.07
P02	E03–E05	117.20	31.07 ± 2.03	6.58	102.76	28.98 ± 1.89
P03	E04–E06	132.16	26.04 ± 1.66	5.83	117.45	24.29 ± 1.55
P04	E05–E07	145.03	24.58 ± 1.95	5.32	129.97	22.93 ± 1.82
P05	E06–E08	155.29	20.23 ± 3.48	4.96	139.90	18.87 ± 3.25
P06	E07–E09	166.00	17.36 ± 5.00	4.64	150.20	16.19 ± 4.66
P07	E08–E10	177.63	24.46 ± 3.60	4.34	161.36	22.81 ± 3.36
P08	E09–E11	190.98	32.62 ± 5.01	4.04	174.12	30.43 ± 4.67
P09	E10–E12	208.93	38.15 ± 2.39	3.69	191.21	35.58 ± 2.23
P10	E11–E13	225.05	35.51 ± 4.78	3.43	206.51	33.13 ± 4.46

ID	Type	MJD	$t$ (d)	$\sigma_{\rm fit}$ (d)	$\sigma_{\rm amp}$ (d)	$\sigma_t$ (d)
E02	trough	60725.59	79.51	7.34	1.51	7.49
E03	peak	60747.75	101.67	1.57	0.59	1.68
E04	trough	60765.22	119.14	1.13	0.31	1.17
E05	peak	60778.82	132.74	1.10	0.28	1.14
E06	trough	60791.26	145.18	1.12	0.37	1.18
E07	peak	60803.39	157.32	1.37	0.78	1.58
E08	trough	60811.48	165.40	2.64	1.94	3.28
E09	peak	60820.75	174.67	4.59	1.21	4.74
E10	trough	60835.94	189.86	1.22	0.86	1.49
E11	peak	60853.37	207.29	1.37	0.86	1.61
E12	trough	60874.08	228.00	1.61	0.94	1.87
E13	peak	60888.88	242.80	2.24	3.90	4.50

We tested the prediction on the antichirp periods extracted from the extrema of the modulations (section A 4, Table III), correcting for the observer frame and for photon diffusion through the ejecta using the best-fit MOSFiT diffusion kernel. After the interval spanning the turnover ($16 \pm5$ d at $t = 150$ d, rest frame), the period rises from $23 \pm3$ d at $t = 161$ d to $35 \pm5$ d at $t = 191$ d (rest frame). This behavior results in a slope of $0.28 \pm0.12$, at the $\approx48$th percentile of the predicted distribution, and excluding the continued decline of the linear extrapolation at $\gg5\sigma$. Fitting the chirp and antichirp together with the model of Equation (1) places the minimum at $t_{\min} = 132 \pm3$ d (rest frame), statistically consistent with the prediction. The behavior of the new modulations matches the prediction from the universal relation.

Circumstellar interaction does not set the timing

The primary alternative explanation for light curve undulations in SLSNe-I is interaction between the ejecta and CSM [7, 11, 27]. Indeed, spectra of SN 2024afav show several features consistent with CSM interaction [24], so CSM may well be present. However, the presence of a chirp-antichirp feature in the light curve invites the question of whether CSM is merely present or actively driving the observed modulations. We consider three features of the data that we argue indicate CSM is not the primary driver of the light curve modulations.

First, the modulations are inconsistent with the stochastic variability seen in other SLSNe-I, also proposed to be driven by CSM interaction. We modeled the post-peak modulations of six SLSNe-I as red noise and generated $\sim5 \times10^{4}$ realizations at the cadence and photometric uncertainties of SN 2024afav. Each realization was processed identically to the SN 2024afav dataset (see section A5 for additional details) and scored on two quantities: (i) the modulation power (measured via excess root-mean-square (rms) amplitude), and the chirp-antichirp parabola “contrast”, a measure of the strength of the chirp-antichirp feature present. SN 2024afav is an outlier in comparison to the variability present in other SLSNe-I, with $\lesssim0.1\%$ of realizations displaying behavior as extreme as SN 2024afav in both quantities combined (Figure 6). As bumps in similar SLSNe-I are commonly attributed to CSM interaction, this result implies that interaction-driven variability of the kind observed in SLSNe-I [7] typically does not produce a chirp followed by an antichirp. Even when the red noise process is calibrated to SN 2024afav exclusively, realizations with comparable power and parabola contrast occur $\lesssim5\%$ of the time.

Second, CSM shells placed at deliberate radii could potentially produce the extrema observed, but the physical scenario required to do so is contrived. A decelerating shock crossing shells should produce intervals that lengthen with time [28] unless the shell spacing itself shrinks outward. Further, while the chirp requires spacings that contract outwards over several shells, the antichirp requires the reverse—spacing that expands (or stays constant while the shock significantly decelerates). Previous mechanisms proposed to explain possible regularity in CSM shell placement (e.g., pair instability, [11]) do not easily explain the turnaround observed. Additionally, the data enforce that the transition must occur at a specific time, $t \sim150\,\mathrm{d}$. While this turnover time is predicted by our model, there is nothing in the progenitor’s mass loss history that ties the spacing of the shells to this specific epoch.

Finally, the shape of the modulations is inconsistent with previous models of CSM interaction. Forward and reverse shock collisions with shells produce highly asymmetric bumps with a rapid rise and slow decay [11, 29]. The modulations of SN 2024afav—particularly during the chirp—are consistent with a sinusoid with a dynamic envelope [13], which is inconsistent with the CSM interaction profile even before any attempt to match timing is performed. In the engine picture, by contrast, the period sequence, magnitude of the turnover, and reversal epoch all follow from Equation (1) with no additional freedom. The proposed relation also makes testable predictions for events beyond SN 2024afav.

Discrete properties of the universal relation

A further property of the universal relation presented in Equation (1) is that its coefficients are quantized, which provides a meaningful diagnostic for future events. The parameters of the system $(K, \omega, \chi, u, \eta)$ are fixed by discrete physical choices (see section B 2), as opposed to continuous tunable parameters. The precession parameters sets $K$ and $q$; the spin-down of a magnetar with a dipole field sets $n = 3$; the pressure and opacity regime of the disk sets $(u, w)$; and the assumption of fallback sets $\chi= 5/3$. The consequence of the quantization is that no physical parameter interpolates between the discrete values, and each combination as a result yields an isolated line, forming a “comb” in the $D$–$C$ plane. The line constructed by measuring $C$ and $D$ from the light curve can therefore identify the active precession mechanism and disk regime, and an observed track that falls between or beyond the lines effectively excludes the entire menu.

Our application of this method to SN 2024afav demonstrates the discriminating power of the discreteness test (Figure 4). The chirp alone lies within the allowable range spanned by thin-disk lines, consistent with the reported [13] ability of the published thin-disk model to reproduce it. However, the chirp spans too narrow a range in $D(t)$ to effectively constrain the slope without the reversal (Figure 4). The chirp and antichirp combined constrain the slope to $dC/dD = 11.6 \pm1.6$, significantly steeper than any line on the thin-disk comb. As a result, no thin-disk closure is consistent with the measured relation, the best-fitting closure (QDNP/gas–Kramers/gas) gives reduced $\chi^2 = 9.8$ ($p = 3 \times10^{-15}$). The track is linear (curvature consistent with zero at $\lesssim1\sigma$), consistent with the relation, but the inconsistency with the comb suggests that no radiatively-cooled thin disk can reverse the chirp as sharply as observed.

Estimates of the fallback rate at late times exceed the neutron star’s Eddington rate by orders of magnitude [19, 30], even in the thin-disk fit to the chirp alone [13]. As a result of the high accretion rate, the inner disk should be advective, cooling inefficiently and losing mass to outflows inside the photon-trapping radius [31, 32]. Adopting the self-similar structure of such a flow, with the inflow rate declining inward as $\propto r^p$ and the trapping radius scaling with the fallback supply, the disk pressure scales as $\dot{M}^{1-p}r^{p-5/2}$, giving $u = 1-p$ and $w = 5/2-p$ with $p$ continuous (though a chirp-antichirp requires $p < 1/2$ by construction). Remarkably, the universal relation retains its discriminating power despite the presence of a continuous parameter. Because the same $p$ sets both the slope and the intercept, every line belonging to a given precession mechanism passes through one fixed point in the $(D,C)$ plane (section B 2), at $D = \chi/2$ and $C = K\chi- q\chi/[2(n+1)]$. This point is set by the precession mechanism, fallback index, and braking index; the outflow strength only rotates the line about the pivot point. Discreteness survives as a pivot point for each mechanism instead of a comb of lines, as the measured line must pass through one of them. We find that SN 2024afav passes almost directly through the Lense–Thirring point (offset $\lesssim0.2\sigma$), is inconsistent with the quadrupole point ($\gtrsim1\sigma$), and rules out magnetic-warping (by $\gtrsim5\sigma$). The slope as a consequence then reports the outflow index, $p \sim0.46$ for Lense–Thirring precession, at the weak-outflow end of the values found in simulations [33]. This test is selective for different model choices; for example, had the outflow zone ended at a fixed radius rather than at the trapping radius, no mechanism would reproduce the modulations (Figure 8).

Summary of the observed chirp reversal

From the universal relation and continued follow-up of SN 2024afav, we conclude that the chirp reverses where the engine model predicts. The relation holds for any precession mechanism, disk pressure regime, or braking index, and requires the chirp to halt at a time set by the spin-down of the magnetar. Fit to the published chirp alone, the new relation predicts a period minimum at $133 \pm20$ d (rest frame). The late-time light curve shows five new extrema with a period minimum at $132 \pm3$ d (rest frame), after which the period lengthens from $23 \pm3$ d to $35 \pm2$ d in a clear antichirp. SLSN red noise cannot reproduce this behavior to any reasonable probability ($\lesssim0.1\%$ of simulated realizations). The sharpness of the reversal (relation slope $dC/dD = 11.6 \pm1.6$) excludes every radiatively-cooled thin disk and points to a hot, advective inner flow. Applied to future chirping events, the same test requires no model fit, as each event will either land on the thin-disk menu, identifying its precession mechanism directly, or, like SN 2024afav, constrain the structure of its disk. We conclude that the unprecedented chirp and antichirp of SN 2024afav are driven by the magnetar engine, whose output is modulated by a precessing disk supported by the magnetar’s wind.

References

Acknowledgments

We gratefully acknowledge D. A. Howell for acting as an internal referee for this paper. We thank D. Kasen, W. Lu, R. Margutti, D. Gates, R. Chornock, J. Delgado, R. Narayan, and D. Pesce for helpful discussions and feedback. We thank M. Haller, E. Abate, and A. Mei for reviewing the manuscript and providing comments. This work makes use of data from the Las Cumbres Observatory (LCO) global telescope network and the Asteroid Terrestrial-impact Last Alert System (ATLAS) project, both of which make use of instruments on Haleakala. We respectfully acknowledge the profound cultural significance and enduring reverence of the summit of Haleakala to the indigenous Hawaiian community and we are grateful for the opportunity to study the heavens from this mountain.

Funding: J.R.F. is grateful to the Miller Institute for Basic Research in Science at UC Berkeley for supporting this work. L.J.P. acknowledges support from The Flatiron Institute, which is supported by the Simons Foundation. H.Y. is supported by NSF through Awards PHY-2308415, PHY-2541579 (CAREER Award), and AST-2606125, and Montana NASA EPSCoR Research Infrastructure Development under award No. 80NSSC22M0042. The LCO group is supported by NSF grants AST-1911225 and AST-1911151. The ATLAS project is primarily financed to search for near-Earth asteroids through NASA grants NN12AR55G, 80NSSC18K0284 and 80NSSC18K1575; by-products of the near-Earth object search include images and catalogues from the survey area. The ATLAS science products have been made possible through the contributions of the University of Hawaii Institute for Astronomy, the Queen’s University Belfast, the Space Telescope Science Institute, the South African Astronomical Observatory and the Millennium Institute of Astrophysics (MAS), Chile.

Author contributions: (following CRediT system) Conceptualization: J.R.F. Data curation: J.R.F. Formal analysis: J.R.F., L.J.P. Investigation: J.R.F. Methodology: J.R.F., L.J.P., H.Y. Project administration: J.R.F. Software: J.R.F., L.J.P. Validation: L.J.P., H.Y. Visualization: J.R.F., L.J.P. Writing (original draft): J.R.F. Writing (review & editing): J.R.F., L.J.P., H.Y.

Competing interests: There are no competing interests to declare.

Data, code and materials availability: The data used in this analysis are made publicly available on WiseREP. Code used to run the analysis is available on GitHub.

Materials and methods

Observations of SN 2024afav

SN 2024afav (RA 12h49m12.050s, Dec−18°06′12.61″, J2000) was discovered by L-GOTO on 2024 December 12.7 UTC [35] and classified as a SLSN by ePESSTO+ in January 2025 [36]. We adopt $z = 0.072$, measured from the narrow host lines. Assuming flat cosmology ($\Omega_{\mathrm{M}} = 0.286$, $H_0 = 69.6$ km s$^{-1}$ Mpc$^{-1}$), we adopt a distance to SN 2024afav of 327 Mpc. Following [13, 24], we correct all photometry for Milky Way extinction with $E(B - V)_{\mathrm{MW}} = 0.0527 \pm0.0029$ mag. As in [13], we apply no host extinction correction, as the host offset combined with an upper limit of $E(B - V)_{\mathrm{H}} \lesssim0.04$ mag suggest the host contribution is negligible.

The additional (i.e., beyond the chirp reported in [13]) photometry analyzed in this paper has already been reported in [24] and was obtained via ATLAS and LCO. ATLAS observed the field on alternate nights in $c$-band ($\approx420 - 650$ nm) from before discovery through MJD 60853, and roughly once per week in $o$-band ($\approx560 - 820$ nm). We obtained forced photometry [23] on ATLAS difference images in both bands. Multiple exposures from the same night were combined to increase signal-to-noise on the detections for each epoch. ATLAS photometric data points were calibrated to the Pan-STARRS catalog [37].

LCO monitoring began near the middle of the “plateau” present in the early light curve, on MJD 60711, as part of follow-up with the Global Supernova Project. We used the 1m and 2m network [22] in $g$-, $r$-, and $i$-band on an approximately daily cadence. Images were reduced with the lcogtsnpipe pipeline and calibrated to the Sloan Digital Sky Survey catalog AB magnitudes.

Fit to the magnetar model

The peak luminosity of SN 2024afav ($M \approx-20.5$) and the $\sim50$ day rise time are typical of magnetar-powered SLSNe-I. We fit the magnetar model of [4] using the Modular Open-Source Fitter for Transients (MOSFiT; [14]) to all 811 ATLAS ($c$ and $o$) and LCO ($g$, $r$, and $i$) photometric measurements acquired during observations of the target. As the model declines smoothly post-peak and cannot reproduce modulations of any kind, we use this fit to estimate engine parameters independently of the chirp-antichirp and to demonstrate that the modulations are not a property of the smooth central engine. To perform the fit, we vary the explosion epoch, spin period, dipole field, ejecta mass, ejecta velocity, gamma-ray opacity, and photospheric temperature floor, fixing all remaining parameters (e.g., neutron star mass, optical opacity, &c.) to reasonable values.

The magnetar model is sensitive to post-peak behavior, but the light curve is significantly impacted by the presence of the modulations at $t \gtrsim50$ d. The modulations are a correlated astrophysical signal, not measurement noise, and thus we wish to prevent them from driving the fit. To achieve this, we add a fixed uncertainty of $\sigma_{\mathrm{mod}} = 0.515$ (equal to $\approx1/2$ the peak-to-trough amplitude of the largest modulation) in quadrature to the photometric uncertainties associated with measurements after the first appearance of the modulations (approximately MJD 60703). We additionally incorporate an intrinsic scatter term $\sigma_{\mathrm{eff}}$ which will be dominated by $\sigma_{\mathrm{eff}} \ll\sigma_{\mathrm{mod}}$ at $t > \mathrm{MJD}\ 60703$ but impacts the reliability of the fit on the smooth rise. The favored intrinsic scatter term favors $\sigma_{\mathrm{eff}} = 0.059 \pm0.008$ mag, a largely negligible scatter. We find this approach is ultimately conservative; for the median model, $\chi^2 = 0.43$ for all 811 points ($\chi^2 = 1.14$ for pre-modulation points whose uncertainties are dominated by $\sigma_{\mathrm{eff}}$, and $\chi^2 = 0.29$ for modulation-impacted points dominated by $\sigma_{\mathrm{mod}}$).

Having adjusted the uncertainties to minimize the impact of the modulations, we condition the magnetar model on the photometric dataset. We use Markov-Chain Monte Carlo (MCMC) to sample the posterior with 72 walkers for $\approx10^4$ steps and discard the first 40% as burn-in. Convergence is assessed following [38]; the split-chain Gelman–Rubin statistic is $\leq1.03$ for every parameter. We quote medians of the posterior distributions for each parameter with uncertainties drawn from the 16th–84th percentile intervals. This simple approach is justified as the posterior is well-behaved, with fast convergence and relatively symmetric posteriors. The results are shown in Figure 1 and the corresponding posterior is shown in Figure 5. We find that the fit can reproduce the rise, peak, and mean decline in all five bands with $P_{\mathrm{spin}} = 3.9 \pm0.2$ ms, consistent with the previously reported fit up to $1\sigma$; and magnetic field strength $B = (0.51 \pm0.05) \times10^{14}$ G, about a factor of $\approx2$ smaller than previously reported. The ejecta mass and ejecta velocity are similar to previously reported results as well. The fit favors an explosion epoch of MJD 60646.1 $\pm0.5$, which is 50.3 $\pm0.5$ rest-frame days prior to maximum light. We adopt this explosion epoch as the reference epoch for all light curves in this analysis. The best-fit parameters imply a rotational energy of $1.9 \times10^{51}$ erg and a spin-down time (as input into the period law of Equation (1)) of $t_{\mathrm{p}} = 87 \pm11$ d.

Construction of the universal relation

We consider a disk (any regime) suspended on the wind of a central magnetar, and assume that the precession period scales with the radius of the disk’s inner edge and the angular momentum of the central engine,

with $A$, $K$, and $q$ constant over the window. This form covers a broad range of precession mechanisms, including quadrupole-driven nodal precession [39], magnetic warp precession (MWP; [16]), and Lense–Thirring precession [15, 40], for which $K > 0$ and $q \geq0$; we neglect spin-up of the magnetar. We further assume that the inner edge evolves with the balance point between the magnetar wind $L_{\mathrm{w}}$ and the combined disk pressures, which regardless of regime are power laws in radius and accretion rate [19],

with $u$ and $w$ constant (their non-constancy has a direct observable consequence, discussed below). The accretion rate follows fallback, $\dot{M} = \dot{M}_0(t/t_0)^{-\chi}$, with $\chi= 5/3$ at late times [20]; although $\chi$ and $u$ enter only as a product, they are drawn from different menus (fallback physics and the disk response, respectively). Finally, the magnetar spins down with braking index $n$, $\Omega= -K\Omega^n$ [21, 41], driving a wind $L_{\mathrm{w}} = -I\Omega\dot{\Omega}$ with $J = I\Omega$ [4]. Times $t$ are measured from magnetar birth ($\approx$ explosion), and the observables are $C \equiv-d\ln P/d\ln t$ for the engine-frame period and $D \equiv-d\ln L_w/d\ln t$ for the wind luminosity prior to diffusion. We define $s \equiv1/(w-2)$, $A_{\rm UR} \equiv\chi K s u$, and $B_{\rm UR} \equiv K s + q/(n+1)$.

Rearranging Eq. (3) (for $w \ne2$) isolates $r_i = (C\dot{M}^{u}/L_w)^s$. Integrating the spin-down law for $n>1$ gives $\Omega^{1-n} = \Omega_0^{1-n}(1+\tau)$, where $\tau\equiv t/t_p$ and $t_p \equiv[(n-1)K\Omega_0^{n-1}]^{-1}$ is the doubling time of $\Omega^{1-n}$, so that $\Omega= \Omega_0(1+\tau)^{-1/(n-1)}$ and

with $L_0 \equiv I K\Omega_0^{n+1}$. $D$ is simply a function of the braking index and spin-down time, both of which we expect to be constant, along with $t$, and so serves as an almost specifics-independent clock of the system. Likewise $d\ln J/d\ln t = -D/(n+1)$, which holds simply because $L_w \propto\Omega^{n+1}$ and $J \propto\Omega$, so long as $I$ and $K$ are constant over the chirping window. Substituting the fallback rate and Eq. (4) into $r_i$, and $r_i$ and $J = J_0(1+\tau)^{-1/(n-1)}$ into Eq. (2), we obtain

and therefore

The relation is linear, and the line communicates only the exponents setting the disk, precession, and magnetar physics; it is independent of the specific values of the system. Explicitly, none of the clock constant $\mathcal{A}$ ($G$, $c$, geometric factors), the disk constant $C$ ($\alpha$, opacity, mean molecular weight, the fraction of the wind intercepted by the rim), the fallback normalization $\dot{M}_0$, the engine constants $K$ and $\Omega_0$ (equivalently $B$-field, birth period, and radius), the moment of inertia, the tilt, or any constant multiplicity of bumps per precession cycle ($P \to P/N$ rescales only $P_{\rm ref}$) enters $A_{\rm UR}$ or $B_{\rm UR}$. A fixed ratio of outer to inner disk radius $r_o/r_i$ is absorbed into $\mathcal{A}$, and if $r_o/r_i \gg1$ with $r_o$ tracking a power of $r_i$, it is absorbed into an effective $K$. The linearity therefore serves as a universal and meaningfully-falsifiable statement about supernovae powered by engine-clocked disk systems.

Deviations from linearity arise trivially if any exponent ($K$, $s$, $u$, $\chi$, or $q$) evolves over the chirp window. An offset $\delta$ in the explosion epoch (typically $\pm0.5$ d) gives $t^\prime= t-\delta$ and $d\ln t/d\ln t^\prime= 1-\delta/t$, so that $C^\prime= (1-\delta/t)C$, $D^\prime= (1-\delta/t)D$, and $C^\prime= (1-\delta/t)A_{\rm UR}-B_{\rm UR}D^\prime$: the slope is unchanged and the intercept is rescaled, with the time dependence of the rescaling introducing curvature only at $O(\delta/t) \lesssim1\%$ for $t \sim10^2$ d. Precession mechanisms whose normalization depends on the disk surface density, e.g. MWP with $\Omega_{\rm prec} \propto\Sigma^{-1}$ and $\Sigma\propto\dot{M}^{-m}$ [16], generalize Eq. (2) to $P \propto r_i^K\dot{M}^{-m}J^{-q}$, which yields $C = \chi(Ksu-m)-B_{\rm UR}D$ and leaves the linearity unchanged; because MWP has no intrinsic dependence on the magnitude of the spin vector, however, such a chirp would not probe $J$.

The framework further implies a falsifiable period minimum. Setting $C=0$ gives $D_{\min}=A_{\rm UR}/B_{\rm UR}$, and with $\Phi\equiv\tau/(1+\tau)$ and Eq. (4),

Using $d\Phi/d\ln t = \tau/(1+\tau)^2$, the curvature

is necessarily positive for $n>1$, $B_{\rm UR}>0$ (i.e. $K s > -q/(n+1)$), and $\tau>0$, so the extremum is a minimum. A chirp observed with a period continuing to decrease beyond $\tau_{\min}$ therefore falsifies the model for that system. In the edge case $B_{\rm UR}(n+1)<A_{\rm UR}(n-1)$, $\tau_{\min}<0$, there is no minimum at $t>0$, and the period declines monotonically; for $A_{\rm UR}<0$, $\tau_{\min}\in(-1,0)$ and the period rises monotonically. To apply the test, $n$ and $t_p$ are taken from a magnetar fit to the light curve ignoring the modulations (following the uncertainty-inflation method in section A2; a joint magnetar+chirp fit is circular by construction), and $\ln P=-A_{\rm UR}\ln t+\frac{n+1}{n-1}B_{\rm UR}\ln(1+t/t_p)+\mathrm{const}$ is fit directly to the periods, with the full covariance of the correlated $A_{\rm UR}$ and $B_{\rm UR}$ propagated into $\tau_{\min}$. If the period is observed shortening up to some last epoch $t_{\rm last}$, the model requires $t_{\min}>t_{\rm last}$; a posterior placing $t_{\min}<t_{\rm last}$ at significance is direct falsification.

The predictive power of this test comes from fixing $t_p$ using data unrelated to the chirp. By Eq. (8), the ratio of the curvature at any two epochs cancels $\frac{n+1}{n-1}B_{\rm UR}$ and depends only on $\Phi$, itself only a function of $t_p$; the model thus states how the chirp’s deceleration must change across the window before any period is fitted, while $A_{\rm UR}$ and $B_{\rm UR}$ set only the level and scale and a single pair must serve the entire window. If the observed chirp decelerates much harder early than the curvature allows (the period “collapses” and then “coasts”), the fit undershoots $C(t)$ and reports a minimum-period epoch while the period is still decreasing: an inconsistency between chirp and light curve that falsifies the model even when the fit is satisfactory by eye (i.e. with a good $\chi^2$). Conversely, since a poor $\chi^2$ does not distinguish an incorrect engine from a correct engine described by an oversimplified model, the minimum-epoch check provides a parameter-light self-consistency assessment beyond goodness-of-fit.

Assessment of modulations

We characterize the modulations in the bolometric light curve. We reproduce the blackbody bolometric luminosity evolution reported in [24] from the LCO ($g$, $r$, $i$) and ATLAS ($o$, $c$) bands with the open-source package extrabol [42]. In this section, we report times in observer-frame days since explosion (MJD 60646, from the magnetar fit, section A 2) and do not yet correct for diffusion or the rest-frame.

First, we separate the modulations from the overall evolution of the supernova by fitting a smooth baseline to the light curve. We choose a cubic regression spline with five interior knots spaced 34 d apart following [24]. The choice of knot spacing can impact the modulation recovery, as a more flexible baseline could absorb modulations, while a less flexible baseline may not properly detrend the data. We vary the number of knots and their spacing around the identified optimal choice and find that the residual changes by on average $\lesssim0.04$ and at most $\sim0.08$, well below the modulation amplitudes ($\sim0.1 - 0.5$ mag).

We identified 12 extrema (six peaks and six troughs) after $t = 63$ d in the residual. The epoch associated with the extremum is first located by eye and then refined by a parabola fitted to the nearby modulation ($\approx35\%$ of the local half-cycle on each side). The modulations recovered from the bolometric luminosity using this procedure agree with the previously-reported chirp modulations (which were extracted directly from the per-filter photometric data) with a mean absolute offset of $\sim0.3$ d. In addition to the uncertainty from the quadratic fit, we add systematic uncertainties in quadrature based on (i) scatter in extraction using our pipeline as a result of changes to fitting window width and the smoothing baseline as well as (ii) the strength of the modulation relative to the local noise. Once the modulations are constructed, we measure periods as intervals between consecutive extrema (peak-to-peak and trough-to-trough), resulting in ten periods. The uncertainty on each period is constructed by combining the uncertainties of each component endpoint in quadrature. We note that periods which share an endpoint are anticorrelated and therefore any fits performed using the periods employ the full covariance matrix.

Red noise analysis

A counter to the claim that the modulations represent a coherent physical process is that stochastic variability with a steep (red) power spectrum can produce several cycles of apparently periodic behavior (see e.g., the problem of identifying periodic signals in red-noise-contaminated active galactic nuclei light curves, [43]). The modulations in other SLSNe-I are potentially driven by CSM interaction, so in principle, if SN 2024afav is also driven by CSM, its light curve modulations should be similar to other objects in its object class. We evaluate this claim for SN 2024afav by simulating red noise light curves at the photometric cadence and uncertainties of SN 2024afav and characterizing how often a chirp-antichirp appears in the randomly generated light curves. We calibrate our red-noise model to six objects from the SLSN-I population (SN 2007bi, SN 2017cgi, SN 2018kg, SN 2018ibb, SN 2020qlb, and SN 2020wnt; data taken from [44]) which have modulations as well as well-sampled $r$-band photometry beyond 150 d post-peak. We subtracted from each the same smooth baseline used for SN 2024afav and fitted a power-law spectrum $P(f) \propto f^{-\alpha_{\rm ps}}$ to the residuals, and adopted the median amplitude and slope ($\alpha_{\rm ps} = 2.85$). For each model, we generated red-noise realizations with the method of [45]. We added each realization to the smoothed baseline of SN 2024afav at the observed $r$-band epochs and incorporated white noise whose per-epoch magnitude was drawn from SN 2024afav’s photometric errors. Each simulated light curve was then passed through the identical pipeline used on SN 2024afav (nightly binning, baseline subtraction per band, automated extremum extraction). We retained only realizations whose number of extracted extrema lay within $\pm2$ of SN 2024afav, up to a total of $\sim5 \times10^{4}$ realizations of the model.

Each light curve (real and simulated) was then characterized by the following two quantities: (i) modulation power, measured as the excess rms of the residual over the span of the measured extrema, after subtracting the photometric error variance; and (ii) parabola “contrast”, a measure of how cleanly the period sequence traces a chirp followed by an antichirp. To construct the parabola contrast, we constructed the “period” evolution from the extracted extrema and fitted a parabola to it with Cauchy loss. The contrast is then the rise of the shallower arm of the parabola above the vertex, divided by the scatter of the points about the fit. We rejected realizations which produced parabolas with non-positive or insignificant curvature, or a vertex that lay outside the period sequence, with at least two points pre- and post-minimum. Under this prescription, SN 2024afav has a modulation power of 0.22 mag and a parabola contrast of 13.0 (corresponding to a 14.7 d rise in period on the antichirp arm against a scatter of 1.1 d). For each quantity, we computed the fraction of realizations at least as extreme as SN 2024afav and combined them through their product, and calculated a combined $p$-value as the fraction of noise realizations whose product $p_{\rm power}p_{\rm contrast}$ is at least as small as SN 2024afav. Against the population-calibrated noise, red noise matches or exceeds SN 2024afav’s modulation power (0.22 mag) with just 74 of $5 \times10^{4}$ realizations ($p_{\rm power} = 1.5 \times10^{-3}$) and its parabola contrast (13.0) in 1,280 ($p_{\rm contrast} = 0.026$). Just 31 realizations are as or more extreme than SN 2024afav in the combined test, a combined $p$-value of $6.4 \times10^{-4}$ ($\sim3.2\sigma$). SN 2024afav is therefore an extreme outlier even among red-noise light curves calibrated on other bumpy SLSNe-I.

To perform a more conservative test, we run the same analysis as above but using a red-noise power spectrum calibrated only to the modulations of SN 2024afav. By construction, this method negates the rarity based on modulation power (as amplitude is calibrated to the mean amplitude of SN 2024afav, the only object in the sample). Despite this, red noise calibrated specifically to SN 2024afav’s power spectrum yields extrema with comparable parabola contrast $\ll5\%$ of the time.

Supplementary Text

Validation of chirp-antichirp behavior

To validate the presence of the quasiperiodicity (particularly, the chirp-antichirp) further, we investigate the period evolution without reference to individual extrema, using a time-frequency decomposition of the residual. We first scale the residual by a smooth envelope to ensure every cycle contributes equally, then computed the weighted wavelet Z-transform, suitable for unevenly-sampled data, following the method of [25], using 240 logarithmically-spaced frequencies between periods of 19 and 48 d. The resulting map is then refined via the method of synchrosqueezing [26, 46], which reassigns each cell’s power to the instantaneous frequency implied by how the fitted phase advances with time. Finally, each time column is normalized to its maximum, to assist with interpretability and visualization.

The resulting map shows two clear ridges which evolve with time. The dominant ridge rises from $\sim0.025\ \mathrm{d}^{-1}$ near $t \sim90\ \mathrm{d}$ to $\sim0.07\ \mathrm{d}^{-1}$ near $t \sim163\ \mathrm{d}$, then turns over and sharply descends. The ridge is identified (up to $\sim7\%$ in frequency) with the locations of the extrema, confirming that the visible extrema track the dominant mode in the power spectrum. The time-frequency decomposition recovers the chirp and subsequent reversal independently of how the extrema were selected or extracted. We also note the presence of a lower ridge at periods of $\sim33 - 39\ \mathrm{d}$, which traces a slower variation observable in the modulations, namely the alternating deep and shallow troughs.

Selection of discrete menu for the universal relation

Disk prescriptions

Both prescriptions share the assumptions of the universal relation, $L_{\mathrm{w}}/r^{2} \propto P_{\mathrm{disk}}(r)$, $\dot{M} \propto t^{-\chi}$, $L_{\mathrm{w}}=-I\Omega\dot{\Omega}$, and precession period $P \propto J^{-q}r_{\mathrm{K}}$. The disk enters only through $P_{\mathrm{disk}} \propto\dot{M}^{u}r^{-w}$, with $s \equiv1/(w-2)$, so that the full relation reads

a. Thin disk. We adopt the steady, geometrically-thin $\alpha$-disk relations ²: $\rho=\Sigma/H$, $H=c_sR^{3/2}(GM)^{-1/2}$, $T^4=\frac{9\tau G M\dot{M}R^{-3}}{32\pi\sigma}$, $c_s^2=\gamma P/\rho$, $\tau=\Sigma\kappa$, $\nu\Sigma=\dot{M}/3\pi$, and $\nu=\alpha c_sH$. Here $\dot{M}$ is independent of radius (no outflow), and we work at $R\gg R_*$, where the inner-boundary factor $1-(R_*/R)^{1/2}\to1$. We close the system with a power-law equation of state, $P_{\mathrm{disk}}\propto\rho^AT^B$, and opacity, $\kappa\propto\rho^CT^D$. Gas (radiation) pressure has $(A,B)=(1,1)$ [(0,4)], and Kramers’ (electron-scattering) opacity has $(C,D)=(1,-7/2)$ [(0,0)] ³. For any $(A,B,C,D)$ the system can be solved algebraically for $T,\rho,\Sigma$, and $v_r=-\dot{M}/(2\pi R\Sigma)$ as pure power laws in $\alpha,\dot{M},M$, and $R$. Each pressure component, $P_{\mathrm{gas}}=(k_B/\mu m_p)\rho T$, $P_{\mathrm{rad}}=(4\sigma/3c)T^4$, and $P_{\mathrm{ram}}=\frac{1}{2}\rho v^2$, then scales as $P_{\mathrm{disk}}\propto\dot{M}^{u}R^{-w}$. The quantities $\alpha$, $M$, $\mu$, and all numerical coefficients appear only in the normalization, which cancels in the universal relation. Because $\dot{M}$ is constant with radius, each regime and component fixes a single $(u,w)$ with no continuous parameter (Table I).

Regime	Component	$u$	$w$	$s$
Gas + Kramers’	$P_{\mathrm{gas}}$	$17/20$	$21/8$	$8/5$
	$P_{\mathrm{rad}}$	$6/5$	$3$	$1$
	$P_{\mathrm{ram}}$	$23/20$	$19/8$	$8/3$
Gas + e.s.	$P_{\mathrm{gas}}$	$4/5$	$51/20$	$20/11$
	$P_{\mathrm{rad}}$	$8/5$	$18/5$	$5/8$
	$P_{\mathrm{ram}}$	$6/5$	$49/20$	$20/9$
Rad + Kramers’	$P_{\mathrm{gas}}$	$2/25$	$9/5$	$-5$
	$P_{\mathrm{rad}}$	$16/25$	$12/5$	$5/2$
	$P_{\mathrm{ram}}$	$34/25$	$13/5$	$5/3$
Rad + e.s.	$P_{\mathrm{gas}}$	$-2$	$-9/8$	$-8/25$
	$P_{\mathrm{rad}}$	$0$	$3/2$	$-2$
	$P_{\mathrm{ram}}$	$2$	$7/2$	$2/3$

b. Thick disk. We assume a self-similar advective flow, which implies constant $H/R\sim c_s/v_K$ ⁴. That solution was derived for optically thin, low-$\dot{M}$ flows, but gravity is the only scale in both cases, so the specific scalings for mass loss apply. The scalings with mass loss are therefore those of ⁵ (eqs. 8–11). Mass is lost via an outflow that depends on radius, $\dot{M}=\dot{M}_0(r/r_{\mathrm{out}})^p$ with $p\in[0,1)$ ⁶. We propagate this into the surface density, $\Sigma\propto\dot{M}/(r|v_r|)\propto\dot{M}(r)r^{-1/2}$, and the density, $\rho\propto\Sigma/H\propto\dot{M}(r)r^{-3/2}$. The total disk pressure is then $P_{\mathrm{disk}}\propto\rho c_s^2\propto\dot{M}(r)r^{-5/2}\propto\dot{M}_{\mathrm{out}}r_{\mathrm{out}}^{p}r^{p-5/2}$.

The final assumption we require is the radius $r_{\mathrm{out}}$ at which the outflow begins. In a super-Eddington disk the flow becomes radiatively inefficient inside the photon-trapping (spherization) radius, so $r_{\mathrm{out}}=r_{\mathrm{tr}}\propto\dot{M}_0$ ⁷. With this choice, the pressure at fixed radius scales as $\dot{M}_0^u$ with $u=1-p$. Similarly, $w=5/2-p$, which implies $s=1/(1/2-p)$; a constant $r_{\mathrm{out}}$ instead gives $u=1$. This introduces a continuous parameter $p$. Simulations of advective flows with outflows find $p\approx0.4$–$0.7$ ⁸. Because the chirp + antichirp configuration is only supported by $s>0$, we must have $p<1/2$.

Substituting into Eq. (B1),

The choice of $p$ rotates the relation around a single point. For given $K$, $\chi$, $q$, and $n$, every $p$ yields a line through

The linearity of the relation, combined with this fixed point, lets us keep its discreteness even though $p$ is a continuous parameter. This holds whether the precession radius is set by the inner edge or by the entire zone precessing rigidly.

Supplementary Figures and Tables