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Planetary Radius as a Regime-Dependent Relational Quantity: Evidence for a Paradigm Defect in Exoplanet Radius Measurement——A Cross-Disciplinary Empirical Test in Astronomy Based on Factor Hierarchy Theory and the Four-Test Method

This cross-disciplinary study challenges the universality of the planetary radius concept by demonstrating, through exhaustive analysis of 39,913 exoplanet records and Factor Hierarchy Theory, that transit depth is a regime-dependent relational quantity modulated by stellar properties rather than an intrinsic planetary attribute, thereby necessitating a paradigm shift toward regime-specific modeling and the elevation of Factor Hierarchy Theory to a fundamental scientific law.

Original authors: Shuiping Tang

Published 2026-07-15

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Shuiping Tang

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to measure the size of a tiny, invisible marble floating in front of a giant, glowing flashlight. For forty years, astronomers have believed that if they just built better rulers and sharper cameras, they could measure that marble's size perfectly. They thought the "size" was a fixed, unchangeable fact about the marble itself.

But a new study suggests something wild: The size you measure isn't actually a property of the marble at all. Instead, it's a "relationship" that changes depending on which flashlight you use, what kind of light it shines, and how your camera sees it.

Here is the story of how this discovery happened, using the tools of a detective who doesn't trust the usual suspects.

The Mystery: Why the Ruler Keeps Breaking

Astronomers have been collecting data from two giant space telescopes, Kepler and TESS, which watch stars for tiny dips in brightness caused by planets passing in front of them (like a moth flying past a porch light). This dip is called the "transit depth."

The problem? Even though the telescopes are incredibly precise, the measurements don't agree. When one telescope says a planet is a certain size, the other says something different. The usual fix has been to tweak the math models, adding more complex rules to "correct" the errors. It's like trying to fix a crooked picture by adding more tape to the wall, hoping it will eventually look straight.

But this paper asks a different question: What if the picture isn't crooked? What if the wall itself is made of different materials in different rooms?

The Detective's Toolkit: The "Four-Test" Method

The author, Shuiping Tang, didn't use standard astronomy tools. Instead, they brought in a "foreign" method from finance called the Four-Test Method, combined with a management checklist called 5W2H and a cycle called PDCA. Think of this as bringing a master locksmith to a house where everyone else has been trying to pick the lock with a butter knife.

They didn't guess which factors mattered. They ran a massive, computerized "exhaustive search" through 39,913 raw records from the NASA Exoplanet Archive. They tested every possible combination of factors to see what actually explained the data.

The Big Reveal: It's Not the Marble, It's the Light

The computer found the best combination of factors to explain the transit depth: the ratio of the planet's size to the star's size, the planet's orbital period, and the distance to the star. This model explained 69.1% of the data (an adjusted R² of 0.691).

But here is the kicker: 31.7% of the data was still unexplained.

The authors then tested if the "discovery facility" (Kepler vs. TESS) was the culprit. They found that the facility did change the results significantly. It looked like the telescope itself was the problem.

But then, the plot twist:
When the researchers controlled for the type of star (its spectral type, like whether it's a hot blue star or a cool red one), the "telescope effect" vanished completely.

Before controlling for star type: The difference between Kepler and TESS was huge (a statistical F-statistic of 1,597).
After controlling for star type: The difference disappeared (F-statistic dropped to 0.05, with a p-value of 0.99).

This means the telescopes weren't the problem. The telescopes were just "proxies" (stand-ins) for the type of star they were looking at. Kepler mostly looked at one type of star, and TESS looked at another. The "size" of the planet changed because the star's light changed, not because the telescope was broken.

The Real Culprits: The "Regime Factors"

The paper identifies three real "Regime Factors" that determine how the measurement works:

Stellar Spectral Type: The color and temperature of the star.
Metallicity: How many heavy elements (like iron) are in the star.
Surface Gravity: How tightly the star's atmosphere is packed.

The study found specific "switching points" where the rules change:

Temperature: At 4,762 K, the behavior of the light changes.
Surface Gravity: At log g = 4.64, the rules shift again.

It's like realizing that a ball bounces differently on concrete than on a trampoline. If you try to measure the ball's "bounciness" by mixing data from both surfaces without separating them, your results will be a mess. The paper found that the "bounciness" (the measured radius) is a relational quantity—it depends on the relationship between the planet, the star's light, and the detector.

The "Zero Overlap" Proof

Here is the most dramatic piece of evidence: The authors checked if any single planet had been measured by both Kepler and TESS.

The result: Zero.
Why? Kepler looked at a specific patch of sky with distant, faint stars. TESS looks at the whole sky with nearby, bright stars. They never looked at the same planet.

This means astronomers have been trying to compare "apples" (Kepler's planets) with "oranges" (TESS's planets) and calling them the same fruit. The paper argues that we cannot simply merge these datasets or assume a planet's radius is the same number in both catalogs.

What This Means (And What It Doesn't)

The paper does not say we can't measure planets anymore. It says we have been asking the wrong question. We have been trying to find a single, universal "Planetary Radius" that exists independently of the measurement. The paper suggests this concept lacks physical universality when comparing different types of stars.

Instead of trying to "fix" the data by adding more correction factors, the paper proposes a new rule for the future:
Before you mix data from different telescopes or different types of stars, you must run a "Chow Test" and an "Interaction Test." If the results show a "regime switch" (like the ones found at 4,762 K or log g = 4.64), you must stop and model them separately. You cannot force them into one single equation.

The Bottom Line

This study uses a method originally designed for financial markets to solve an astronomy puzzle. It suggests that the "Transit-Depth Precision Problem"—where measurements get more precise but the errors get bigger—isn't a failure of our tools. It's a sign that we are trying to measure a relationship as if it were a fixed object.

The authors propose elevating their "Factor Hierarchy Theory" to the status of a Law, because it has now been proven to work in two completely different fields: finance and astronomy. But for now, the main takeaway is simple: The size of a planet isn't just a number written in stone; it's a conversation between the planet, the star, and the telescope. And if you change the star, you change the conversation.

Technical Summary: Planetary Radius as a Regime-Dependent Relational Quantity

Problem Statement
The paper addresses the "Transit-Depth Precision Problem" (TDPP), a paradox in exoplanet science where measurement uncertainty for transit depth fails to converge with improved instrumentation. Instead of converging toward photon-noise limits, discrepancies between facilities (e.g., Kepler, K2, TESS) systematically exceed theoretical expectations by over tenfold. The prevailing astronomical response has been to treat these discrepancies as correctable systematic errors (e.g., via complex limb-darkening models). This study challenges the fundamental premise of that approach: it questions whether "planetary radius" is an intrinsic, cross-regime physical property or if the observed discrepancies are structural, irreducible "regime switches" caused by the coupling of the planet, the stellar radiation field, and the detector response.

Methodology
The study employs a cross-disciplinary framework derived from finance, specifically the Principle of Testability and Factor Hierarchy Theory (Tang, 2026g), augmented by management science frameworks (5W2H and PDCA) to ensure research completeness and process closure.

Data Source: The analysis utilizes 39,913 raw records from the NASA Exoplanet Archive. No pre-screening was applied to the raw file; all filtering (e.g., retaining only transit-method planets, correcting unit inconsistencies in radius ratios) was performed transparently within the code. The final sample for the optimal model consisted of 15,540 planets with complete data.
The Four-Test Method:
- Exhaustive Testing: All 1- to 4-factor combinations of 12 candidate "executing factors" (e.g., radius ratio, orbital period, stellar distance) were traversed to identify the optimal model for explaining transit depth, constrained by Variance Inflation Factor (VIF) < 5.
- Interaction Effect Testing: Candidate "regime factors" (discovery facility, stellar spectral type, metallicity, surface gravity) were tested to see if they modulate the exposure coefficients of executing factors.
- Chow Test: Used to detect structural breaks in the transit-depth equation across different groups defined by regime factors.
- Auxiliary Evidence: Includes permutation tests, quantile regressions, pseudo-out-of-sample predictions (Kepler training vs. TESS testing), and placebo tests to rule out statistical artifacts.
Completeness Frameworks:
- 5W2H: Ensures the research question is answered across seven dimensions (What, Why, Who, Where, When, How, How Much).
- PDCA: Ensures the research process follows a closed loop of Plan, Do, Check, and Act.

Key Results

Optimal Executing Factors: Exhaustive testing identified radius_ratio + pl_orbper + sy_dist as the optimal combination, yielding an adjusted $R^2$ of 0.691. Crucially, radius_ratio alone explains only 68.3% of the variance, leaving a significant 31.7% residual space that is not random noise but systematically modulated.
Discovery Facility as a Proxy Regime Factor: Initial tests showed the discovery facility (Kepler vs. TESS) significantly modulates the radius_ratio coefficient ( $\Delta R^2 = 0.048$ , $p = 6.59 \times 10^{-7}$ ). However, when controlling for stellar spectral type, this facility effect completely vanished (Chow $F$ dropped from 1,597 to 0.05, $p=0.99$ ). This identifies the discovery facility not as a true regime factor, but as a proxy regime factor for the stellar radiation field.
True Regime Factors: Stellar spectral type, metallicity, and surface gravity were confirmed as significant regime factors.
- Metallicity: Regime coupling is strongest in metal-poor stars ( $\Delta R^2 = 0.0313$ ) and disappears in metal-rich stars.
- Surface Gravity: A threshold of $\log g = 4.64$ marks a regime switch between main-sequence and evolved stars.
- Temperature: A threshold of 4762 K constitutes a critical point for regime switching, corresponding to transitions in molecular absorption bands and convection zone depth.
Structural Breaks: Chow tests confirm structural breaks across all spectral type comparisons and facility comparisons (before controlling for spectral type).
Zero Overlap: A critical empirical finding is that zero planets in the archive have been observed by both Kepler and TESS. The two missions observe physically non-overlapping populations (Kepler: distant/faint; TESS: nearby/bright), meaning there is no empirical basis for treating their derived radii as directly comparable.
Robustness: Results hold across permutation tests (empirical $p=0.002$ ), quantile regressions (effects stable across all quantiles), and pseudo-out-of-sample predictions (RMSE amplification of 4.4× when applying Kepler-trained models to TESS data).

Key Contributions

Redefinition of Planetary Radius: The paper argues that transit depth is not an intrinsic property of a planet but a "relational quantity" arising from the coupling of the planet, the stellar radiation field, and the detector response. Consequently, the concept of a universal "planetary radius" lacks physical universality in cross-regime comparisons.
Identification of the Proxy Regime Factor: The study introduces the concept of a "proxy regime factor" within Factor Hierarchy Theory. It demonstrates that in physically driven systems (like astronomy), observable variables (facilities) may act as proxies for deeper physical drivers (stellar radiation fields), a distinction not found in the previously studied institution-driven financial systems.
Methodological Validation: This work provides the first independent cross-disciplinary verification of Factor Hierarchy Theory outside of finance. By replicating the same hierarchical structure and testing procedures in astronomy, the paper proposes elevating the theory to the status of Factor Hierarchy Law.
Procedural Criterion: The paper proposes a specific criterion for the astronomical community: before merging transit data from multiple facilities or stellar types, researchers must conduct interaction and Chow tests. If significant, separate modeling should be adopted rather than forced uniform calibration.

Significance and Claims
The paper claims to resolve the TDPP by diagnosing it not as an error to be corrected, but as a structural feature of the measurement regime. The "more precision, more discrepancy" paradox arises because higher precision reveals the underlying regime coupling that was previously masked by noise.

The study asserts that the Factor Hierarchy Theory, having now been independently verified in five financial markets and exoplanet transit measurements, represents a universal law governing complex systems driven by multiple quantifiable factors. Furthermore, it claims to be the first successful implementation of the 5W2H and PDCA frameworks as "completeness companions" to the Four-Test Method, establishing a rigorous, presupposition-free operational system for scientific discovery.

The authors conclude that the ultimate root of the precision problem is the physical limit of "precise measurement" in the face of regime coupling, necessitating a paradigm shift from viewing planetary radius as an intrinsic quantity to viewing it as a regime-dependent relational quantity.