Atmospheric-loss regime boundary

### PDF page 8 Radius valley as atmospheric-loss geometry into radius–period space, and comparing that predicted curve to the planet distribution in a forward model, would replace the empirical labelling function with a first-principles boundary. Most importantly, the under-performance of the single face motivates the multi-face extension the geometry of Section 2 was built for: a coefficient-locked core-powered face Λ_CP, evaluated as Λ_atm = min(Λ_PE, Λ_CP), together with time-integrated XUV histories and ages and a formation/composition face for water-rich or migrated planets (Burn et al. 2024). The framework would be falsified if, with these additions and provenance-clean masses, no locked lower-envelope coordinate retained the discriminative power of a free fit. 9 CONCLUSIONS A finite set of power-law atmospheric-loss channels defines a piecewise-linear zero-slack boundary in logarithmic planet coordinates, and the observed radius valley can be read as the projected trace of that boundary. Tested on a provenance-clean sample of 493 planets with measured masses, the central empirical finding is a constraint result: a coefficient-locked photoevaporative coordinate Λ_PE = 1.19 ln M_p − ln S loses no measurable discriminative power relative to a free fit (paired ΔAUC = +0.001; 95 per cent CI −0.005 to +0.006, stable under host-grouped and repeated cross-validation), but a negative-control test and a power analysis together show that the present data cannot identify the mass–flux exponent or the loss mechanism — coefficients well above the energy-limited value perform comparably, and the test has power only against exponents far from it. The standalone coordinate moreover underperforms a mass-free demographic baseline (0.795 versus 0.865). Within the measured-mass subset, then, the empirically labelled valley is consistent with a low-dimensional separation whose locked photoevaporative coordinate is sufficient but not uniquely selected; it is not, on present evidence, a demonstration that photoevaporation sets the valley. The lower-envelope framework gives the route to a sharper test: add a coefficient-locked core-powered face and ask whether Λ_atm = min(Λ_PE, Λ_CP), with XUV histories and a larger unbiased mass sample, both retains the free fit’s power and begins to separate the mechanisms. DATA AVAILABILITY This work is based on the publicly available NASA Exoplanet Archive Planetary Systems Composite Parameters table (Akeson et al. 2013), which is permanently archived under DOI 10.26133/NEA13. The catalogue was retrieved on 2026 May 31 (the analysis date); because pscomppars is updated as new measurements are published, the exact Table Access Protocol query and column list given in Section 4 should be run together with this date, or against the archived DOI version, to regenerate the analysed sample in full without any intermediate data product. The analysis and figure-generation code, the out-of-fold model predictions, and the model-comparison, power, and robustness metrics are available from the author on reasonable request.

valley (Burn et al. 2024). This paper does not attempt to replace detailed escape models. Its narrower purpose is to test a geometric prediction common to them: if atmospheric retention is governed by demand/capacity inequalities, the population boundary should simplify when expressed as signed distance to an atmospheric-loss surface. Page 1 of 8

allowed to choose that direction. F0 (free native) and L1 (locked plus context) carry identical information except that L1 constrains the M_p:S coefficients. The headline statistic is a paired bootstrap of AUC(F0) − AUC(L1) on identical out-of-fold predictions, which — unlike two marginal confidence intervals — directly answers whether the lock costs anything. All performance is five-fold stratified out-of-fold logistic classification: predictors are standardized within each training fold, the fitted scaling and model are applied to the held-out fold, and AUC is computed on the assembled out-of-fold predictions. Confidence intervals are nonparametric bootstraps (2000 resamples) over those fixed out-of-fold predictions, and the paired difference is bootstrapped on identical resamples. A robustness variant replaces stratified folds with folds grouped by host identifier (Section 6) so that planets around the same star do not appear in both partitions; uncertainty propagation uses 200 split-normal Monte Carlo draws respecting the asymmetric catalogue error bars. Model Predictors n AUC (95% CI) N0 no-mass ln P + ln S + ln M★ 2591 0.819 (0.802–0.835) F0 free native ln M_p + ln S + ln P + ln M★ 2591 0.964 (0.954–0.972) Page 3 of 8

retains mass–radius-derived masses, its mass-dependent statistics are partly circular. (i) The coefficient lock is not penalised by the present data. The paired bootstrap difference AUC(F0) − AUC(L1) on identical out-of-fold predictions is +0.001 in the gold subset (95 per cent CI −0.005 to +0.006), an interval spanning zero. Constraining the mass–flux combination to the fixed direction removes no resolvable discriminative power relative to a free fit. This result is robust on three axes: repeating the cross- validation with folds grouped by host star, so that planets sharing a host cannot appear in both training and test partitions, changes the gold AUCs by at most 0.006 (F0 0.902→0.897, L1 0.901→0.895); the F0–L1 expected when radius-derived masses sharpen the free fit artificially. Figure 1. Paired bootstrap of the AUC difference between the free model (F0) and the coefficient-locked model (L1) on identical out-of-fold predictions. In the provenance-clean gold subset the interval spans zero. Page 4 of 8

collinearity between mass and flux in the sample, not evidence for any particular value. To quantify what this null can and cannot exclude, I ran a power analysis: synthetic above/below labels were generated on the real gold predictors with a known true mass:flux exponent and noise calibrated to the observed AUC (≈0.90), and the same F0-versus-L1 comparison was applied. The paired ΔAUC the test would register remains ≤0.001 for true exponents from 0.5 to 2.0, rises to +0.005 at an exponent of 3, +0.008 at 5, and +0.012 at 8. The observed gold ΔAUC of +0.001 is therefore not vacuous — it is inconsistent with true exponents well above ≈3 at the sensitivity this sample affords — but the test genuinely sample is thus carried substantially by the demographic context variables, and the descriptor ‘mass- dominated’ should be understood to mean that mass enters the most compact sufficient coordinate, not that mass alone is the strongest single predictor. That a mass-based physical coordinate underperforms a mass- free baseline on its own is a negative result for the single-face model, reported here without mitigation; Page 5 of 8

asks whether a physical coordinate aligns with that boundary, not whether the boundary itself is correct. Mechanism discrimination requires extending the single face. Both photoevaporation and core-powered mass loss predict a broadly similar mass–flux trade-off, and the wide recovered interval cannot separate them; the under-performance of the single face is itself evidence that more than one channel shapes the boundary. The lower-envelope geometry of Section 2 gives the structure for this: add a coefficient-locked core-powered face Λ_CP and test whether Λ_atm = min(Λ_PE, Λ_CP) sharpens the boundary, together with time-integrated XUV histories and ages, and a formation/composition face for water-rich or migrated

Radius valley as atmospheric-loss geometry into radius–period space, and comparing that predicted curve to the planet distribution in a forward model, would replace the empirical labelling function with a first-principles boundary. Most importantly, the under-performance of the single face motivates the multi-face extension the geometry of Section 2 was built for: a coefficient-locked core-powered face Λ_CP, evaluated as Λ_atm = min(Λ_PE, Λ_CP), together with time-integrated XUV histories and ages and a formation/composition face for water-rich or migrated planets (Burn et al. 2024). The framework would be falsified if, with these additions and provenance-clean masses, no locked lower-envelope coordinate retained the discriminative power of a free fit. 9 CONCLUSIONS A finite set of power-law atmospheric-loss channels defines a piecewise-linear zero-slack boundary in logarithmic planet coordinates, and the observed radius valley can be read as the projected trace of that should be run together with this date, or against the archived DOI version, to regenerate the analysed sample in full without any intermediate data product. The analysis and figure-generation code, the out-of-fold model predictions, and the model-comparison, power, and robustness metrics are available from the author on reasonable request. ACKNOWLEDGEMENTS This research has made use of the NASA Exoplanet Archive, operated by the California Institute of Technology under contract with NASA under the Exoplanet Exploration Program. The analysis used the