Statistical Reevaluation of the USP Classification Boundary: Smaller Planets Within 1 Day, Larger Period Ratios Below 2 Days

Armaan V. Goyal; Songhu Wang

REVIEW 3 major objections 5 minor 119 references

Statistical Reevaluation of the USP Classification Boundary: Smaller Planets Within 1 Day, Larger Period Ratios Below 2 Days

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

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Pith's one-line read The paper shows that the two defining traits of ultra-short-period planets—small radius and wide companion spacing—cease to be statistically significant at 0.97 and 2.09 days respectively, giving the usual 1-day cutoff an empirical basis.

desk verdict Genuine empirical search for the USP boundary that finds a 1-day size transition and a 2-day spacing transition, but the error bars are optimistic and the 3–5 day control sample deserves a sensitivity test before the boundaries are canonized. read the letter →

arxiv 2502.07773 v1 pith:BF7INUCT submitted 2025-02-11 astro-ph.EP

Armaan V. Goyal , Songhu Wang This is my paper · ORCID

classification astro-ph.EP

keywords ultra-short-periodplanetsUSPclassificationboundaryperiodratioproto-USPAnderson-DarlingtestexoplanetarchitecturesKeplerK2TESSeccentricmigration

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

Ultra-short-period planets (USPs), worlds orbiting in under a day, are conventionally defined by a 1-day cutoff that has never been tested against data. This paper asks whether that boundary, or a different one, falls out of the observations themselves. Using 376 multi-planet systems from Kepler, K2, and TESS, it finds that the two hallmark USP signatures disappear at different periods: the tendency to be smaller than other short-period planets fades at about 0.97 days, while the tendency to be widely separated from neighboring planets persists until about 2.09 days. If the result holds, USPs are genuinely special below 1 day, and the interval from 1 to 2 days hosts a distinct proto-USP population that is detached but not especially small. The finding also gives formation theories concrete period ranges to reproduce.

What carries the argument

The analysis runs a permutation-based two-sample Anderson–Darling test, which is more sensitive than the Kolmogorov–Smirnov test to tail differences, on moving bins of 30 systems with innermost period below 3 days, each compared against a fixed control sample of 177 systems with innermost periods between 3 and 5 days. Bootstrap resampling of radius uncertainties propagates into p-values for the size comparison, while the period-ratio comparison uses a single run because period uncertainties are negligible. The critical periods are defined as the last bin centers where the p-value crosses above 0.05, with the bin width adopted as the uncertainty range.

What would settle it

Recompute $P_R$ and $P_{\mathcal{P}}$ using a control sample drawn from longer periods, such as 5–10 days, from the same catalogs; if the inferred boundaries move by more than the quoted uncertainties or the p-value curves lose their sharp single crossings, the control-sample assumption is violated.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the USP population's two empirical signatures—reduced planet size and enlarged period ratio to the nearest companion—are real against a 3–5 day control sample, and that each signature has its own sharp transition period: $P_R = 0.97^{+0.25}_{-0.19}$ days for size and $P_{\mathcal{P}} = 2.09^{+0.16}_{-0.22}$ days for spacing. The transitions are abrupt, roughly 2–3 orders of magnitude in p-value over less than 0.3 days, and insensitive to candidate status, radius uncertainties, giant companions, host spectral type, and simulated undetected planets. The paper interprets the 1-day size boundary as the signature of refractory mass loss and tidal decay acting on planets delivered close to the star, and the 2-day spacing boundary as the imprint of eccentric migration that strands proto-USPs in detached orbits. The authors explicitly present these results as evidence that the conventional 1-day cutoff has an astrophysical basis and that an additional 2-day boundary should be considered in future studies.

Load-bearing premise

The control sample of systems with innermost periods between 3 and 5 days is assumed to contain none of the small-size or wide-spacing USP signatures, so a failure to reject the null against that control marks the end of the USP regime.

Editorial extensions

If this is right

The conventional 1-day USP boundary is astrophysically meaningful: below roughly 1 day planets are statistically smaller than the 3–5 day control population, while between 1 and 2 days they are not.
A distinct proto-USP population exists at 1–2 days: planets there are not systematically small but are still widely spaced from companions, supporting a two-step formation history of eccentric migration followed by tidal decay.
Formation models must reproduce a sharp spacing transition near 2 days, not just the pile-up below 1 day, so the 2-day boundary becomes a new constraint for dynamical simulations.
If applied jointly, the two boundaries suggest a classification scheme where USPs are both small and detached, proto-USPs are detached but not small, and planets beyond 2 days are neither.
The 2-day transition provides a target for future transit surveys to enlarge the proto-USP sample and test whether the spacing boundary persists with better statistics.

Reading between the lines

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

The control-sample assumption could be checked by extending the same moving-bin analysis with a control drawn from longer periods, say 5–10 days; if the inferred boundaries shift beyond the quoted uncertainties, the choice of control population is driving the result.
If the 2-day spacing transition reflects recent eccentric migration, proto-USP systems should show elevated mutual inclinations or missing non-transiting companions compared with similar-period systems that lack wide spacing; this is a testable prediction for follow-up observations.
The size and spacing boundaries may be two manifestations of a single migration-plus-mass-loss channel; correlating within individual systems whether the smallest planets are also the most detached near 1 day could distinguish formation models.
Because single-transiting USPs share the size signature but cannot be tested for the spacing transition, a joint statistical model of single- and multi-transiting systems could extend the conclusions to the full USP population, which is dominated by single-transiting configurations.

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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

Worth your time. This is the first blind, multi-mission moving-bin search for where USP-like behavior in radius and period ratio actually ends, and it returns quantitative answers: size differences vanish at ~0.97 days, spacing differences at ~2.09 days. That is a real step beyond the usual “USPs are small and detached” qualitative statement. The validation is thorough: confirmed-only, precise radii, no giant companions, FGK hosts, and a mock-observation scheme for undetected planets all reproduce the two transitions. The detection-bias simulation in particular is more careful than most papers in this area.

The soft spots are statistical, and they are not fatal but they are real. The quoted ± uncertainties are just the moving-bin widths, not resampling or bootstrap uncertainties on the crossing period, so they overstate precision. Multiple testing across 170 overlapping bins is uncorrected, though the authors argue—fairly—that the monotone p-value curves make the crossing largely threshold-insensitive. The bigger issue is the control sample. Everything is compared against systems with 3 ≤ P1 < 5 days, and the assumption that USP-like signatures vanish by ~3 days is asserted, not demonstrated. The stress-test note is right to point at Table 1: the 1–2 day bin has pR = 0.057, just above 0.05, so a slightly contaminated control could push the size boundary from ~1 day toward ~2 days. The paper even notes that photoevaporation still operates within 10 days, which makes the control choice less innocent than it first appears.

I still think the central claim holds up in outline. The spacing transition at ~2 days is robust across every subsample, and the size transition at ~1 day, while more fragile, is consistently present even if its exact location moves. The qualitative conclusion—USPs are small and detached, proto-USPs are detached but not small—is a reasonable reading of the data. What this paper needs is an honest sensitivity analysis that varies the control sample (say 4–6 or 5–7 days) and reports resampled crossing uncertainties. Without that, the 1-day boundary is a good hypothesis, not a measured constant.

Who should read it: people working on USP formation, planetary architecture, or classification schemes. It deserves a serious referee and likely publication after the control-sample test is added. If I were the editor, I would send it out.

Referee Report

3 major / 5 minor

Summary. This paper presents a statistical search for an empirically motivated boundary of the ultra-short-period (USP) planet population. Using 376 multi-planet systems from Kepler, K2, and TESS, the authors compare the radius and period-ratio distributions of the innermost planets with P<1 day against a 'non-USP' sample (1-5 days) using a permutation-based Anderson-Darling test with bootstrapped radii. They then compare moving bins of systems with P1<3 days against a fixed control sample with 3<=P1<5 days, locating the periods where the USP-like small-size and large-spacing signatures become statistically insignificant: PR = 0.97+0.25/-0.19 days and PP = 2.09+0.16/-0.22 days. Robustness checks include confirmed-only samples, precise radii, no giant companions, FGK hosts, and a mock detection framework for missing planets. The paper interprets these boundaries as evidence for a 1-day USP size transition and a distinct 'proto-USP' population between 1 and 2 days.

Significance. If the central result holds, it provides an astrophysical, rather than purely conventional, basis for the 1-day USP cutoff, and it identifies a new transitional regime at about 2 days. The paper is careful in several ways: it uses a nonparametric test tailored to tail differences, propagates radius uncertainties through bootstrap, and runs an unusually broad set of validation subsamples that all preserve the main signal. The main caveats are that the control sample is an assumed baseline rather than a demonstrated one, and that the quoted uncertainties on the transition periods are bin widths rather than statistical intervals.

major comments (3)

[Section 4 and Table 1] The choice of the 3-5 day control sample is load-bearing and is not validated within the paper. The text calls it the 'sole assumption' that USP size/spacing signatures vanish within P<=3 days, but the only check in Section 4 compares USPs to this same control; that confirms a difference, not that the control is free of USP-like objects. Moreover, Section 6.1 states that photoevaporation remains effective within P<=10 days and Roche-lobe overflow can persist to P<=2 days, so the control may itself contain partially processed small planets. In Table 1, the full-sample pR for the 1-2 day bin is 0.057, just above the p=0.05 threshold; with a less contaminated control the pR for this bin would likely be below 0.05, moving PR from about 1 day toward about 2 days. The authors should repeat the analysis with alternative control ranges (e.g., 4-6 days) or explicitly inject a range of USP-like contamination in the control and show how PR and PP change.
[Section 4 and Figure 3] The quoted uncertainties on PR and PP are the widths of the moving bins at the crossing, not statistical uncertainties. The bootstrap procedure produces a median pR and a pP from a single permutation run, but it does not produce a distribution of crossing periods. As written, 'PR = 0.97+0.25/-0.19' can be read as a 1-sigma confidence interval, which is not what was computed. Please provide resampling-based confidence intervals (e.g., bootstrap over systems) or explicitly label these ranges as bin-width resolutions.
[Section 5.4.2] The treatment of geometric biases is explicitly qualitative and does not quantitatively bound their effect on PR. Because the sample is restricted to multi-transiting systems, a highly inclined innermost planet in a 1-5 day system could be missed by construction, and the mock-detection framework in Section 5.4.1 assumes edge-on orbits (b=0) for hypothetical planets and does not sample mutual inclinations. The paper should either implement a geometric correction or sensitivity test, or soften the claim that detection biases do not affect the identified size boundary.

minor comments (5)

[Section 5.1 and Table 1] The text says there are 40 USP systems with delta_Rp/Rp <= 0.20, but Table 1 lists 35; please correct the inconsistency.
[Section 2] The phrase 'to reduced selection bias' should be 'to reduce selection bias'.
[Section 5.4.1] The symbol eP is used for the hypothetical orbital period, which is confusing because e conventionally denotes eccentricity; consider P_hyp or similar.
[Section 1] The sentence 'USPs only occur around ≲ 1% of stars' lacks a citation at the end; please add a reference for the occurrence rate.
[Section 6.2] Verb forms such as 'Lee & Chiang (2017) posits' should be made plural or rephrased for consistency.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the USP boundary periods are empirical p-value crossings against a disjoint control sample, with no load-bearing self-citations.

full rationale

The derivation is empirical rather than deductive. The central quantities PR and PP are not predicted from a model fitted elsewhere; they are read directly from p-value crossings of permutation-based Anderson-Darling tests comparing moving bins to a disjoint control sample with 3 <= P1/days < 5 (Section 4, Figure 3). This is a standard comparison of disjoint subsets of the same catalog data, so no quantity is defined in terms of the target result. The only assumption, that USP-like signatures vanish for P < approximately 3 days, is stated explicitly and supported by external citations (Lee & Chiang 2017; Petrovich et al. 2019); it conditions the result but does not make the derivation circular. Self-citations (Goyal & Wang 2022, 2024; Goyal et al. 2023) are used only to caution about statistical bias and sample-size effects and are not load-bearing. The validations in Section 5 repeat the same comparison under restricted subsamples or an injected-missing-planet model; they are robustness checks, not redefinitions. The potential fragility of the fixed control sample is a correctness or assumption-risk issue, but no equation reduces the claimed boundary to its input by construction.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The analysis relies on several domain assumptions: that USP signatures vanish by 3 days, that the AD test with permutation is appropriate, that the detection-bias heuristic is adequate, and that the multi-transiting sample is representative. No new physical entities are introduced; the proto-USP label is adopted from Schmidt et al. (2024).

free parameters (3)

Bin size N_bin = 30
Chosen to ensure AD test power (N >= 30) and sensitivity; transition values and their quoted uncertainties depend directly on bin width.
Control sample period range = 3 to 5 days
Selected as the fixed non-USP baseline for the moving-bin comparison; if USP signatures persist beyond 3 days, the inferred boundaries would shift.
Critical p-value threshold = 0.05
Used to define when the null hypothesis of no difference is no longer rejected; authors argue results are insensitive for thresholds between 0.01 and 0.1.

assumptions (4)

domain assumption USP-like signatures in radius and period ratio vanish by P < 3 days
Invoked in Section 4 to justify restricting the blind search to P1 < 3 days and using 3-5 day systems as a control sample. If false, the control is contaminated.
standard math Two-sample Anderson-Darling test with permutation and bootstrap reliably quantifies distribution differences at these sample sizes
Section 3; the method is standard but has low power for N < 30, so the bin size is fixed at 30.
domain assumption The simple SNR heuristic (Pont et al. 2006) adequately models transit detectability of hypothetical missing planets
Section 5.4.1; the detection-bias correction relies on this approximation and on the Neil and Rogers (2020) period distribution.
domain assumption The sample of multi-transiting systems is representative of the intrinsic multi-planet population and is not biased by mutual inclinations
Section 5.4.2 acknowledges that geometric biases are not rigorously assessed; the central results are interpreted for multi-transiting systems only (Section 6.3).

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Cite this review

Pith. "Pith review of Statistical Reevaluation of the USP Classification Boundary: Smaller Planets Within 1 Day, Larger Period Ratios Below 2 Days." pith.science (2026). https://pith.science/paper/BF7INUCT

@misc{pith2026250207773,
  author       = {Pith},
  title        = {Pith review of: Statistical Reevaluation of the USP Classification Boundary: Smaller Planets Within 1 Day, Larger Period Ratios Below 2 Days},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BF7INUCT}},
  note         = {Machine review of arXiv:2502.07773}
}

abstract

Terrestrial worlds with $P < 1$ day, known as ultra-short period planets (USPs), comprise a physically distinct population whose origins may be attributed to various possible formation channels within multi-planet systems. However, the conventional 1 day boundary adopted for USPs is an arbitrary prescription, and it has yet to be evaluated whether this specific cutoff, or any alternatives, may emerge from the data with minimal assumptions. We accordingly present a statistical evaluation of the USP classification boundary for 376 multi-planet systems across Kepler, K2, and TESS. We find that USPs are smaller in size ($p = 0.004$) and exhibit larger period ratios with their immediate neighbors ($\mathcal{P} = P_{2}/P_{1}$; $p < 10^{-4}$) when compared to non-USP short-period ($1 < P/\text{days} < 5$) worlds, and that these discrepancies rapidly transition towards statistical insignificance ($p > 0.05$) at respective orbital periods of $P_{R} = 0.97^{+0.25}_{-0.19}$ days and $P_{\mathcal{P}} = 2.09^{+0.16}_{-0.22}$ days (see Figure 3). We verify that these results are not driven by imprecise planetary parameters, giant companions, low-mass host stars, or detection biases. Our findings provide qualitative support for pathways in which proto-USPs are detached from companions and delivered to $P \lesssim 2$ days via eccentric migration, while a subset of these objects near $P \sim 1$ day experience subsequent orbital decay and refractory mass loss to become USPs. These results lend evidence towards an astrophysical basis for the 1 day USP cutoff and encourage consideration of an additional 2 day boundary within future investigations of USP architectures and evolutionary dynamics.

Figures

Figures reproduced from arXiv: 2502.07773 by the authors.

**Figure 1.** Orbital architectures of the 49 USP systems considered in this work. Marker sizes correspond to planetary radius, filled markers represent objects listed as confirmed within the NEA (Akeson et al. 2013), and hollow markers are associated with Kepler candidates from Lissauer et al. (2024), TESS candidates from from the revised TESS Input Catalog (Stassun et al. 2019), or the NEA’s K2 Planets and Candidates Table (Ak… view at source ↗

**Figure 2.** Compared to non-USPs (black/gray), USPs (red) are smaller in size and more architecturally detached from their neighbors in a statistically significant manner. Left: The distribution of 49 USP radii exhibits a relative overdensity at very small sizes (∼ 1R⊕) and lacks extension beyond Rp ≈ 2R⊕ in a manner inconsistent with the 327 non-USP radii and likely representative of refractory mass loss (see Section 6.1). We … view at source ↗

**Figure 3.** Planets are systematically smaller in size for P ≲ 1 day and more detached from their companions for P ≲ 2 days. Pooling our USP and non-USP samples to remain agnostic to the conventional 1 day USP classification boundary, we use the permutation-based AD test to assess the level of statistical discrepancy present between the 199 systems with innermost period P1 < 3, sorted into moving bins of 30 systems each, and a … view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Probability (fmiss) of an undetected innermost or second innermost planet within each USP and non-USP system. USP systems exhibit a greater likelihood of hosting an undetected companion, in accord with previous findings (e.g. Steffen & Coughlin 2016; Qian & Wu 2021), t…

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