Astrophysics Planetary Migration From Jcost Planet Migrate Cert

The migration certificate

Planetary migration is the slow drift of a planet's orbit over time, driven by the exchange of angular momentum with the gas and dust disk it formed in. In the standard Type I regime, the migration rate depends on the disk's surface density and the planet's mass; typical migration timescales are measured in tens of thousands of years. The Recognition Science declaration PlanetMigrateCert is a certificate, a small machine-checked object that bundles three proved facts about a cost function, but it does not by itself prove anything about planets.

The certificate is built from a definition: domainCost takes two real numbers, m and e, and returns the value of a function called J-cost applied to their ratio. J-cost is the framework's central cost function, J(x) = (x + 1/x)/2 - 1, which the framework proves is the unique cost function satisfying five plain conditions. The certificate proves three general facts about this domainCost: it is zero when the two numbers are equal and nonzero, it is nonnegative when both numbers are positive, and a certain threshold constant phi - 3/2 is positive. These are pure facts about real numbers and the J-cost function; they hold for any positive inputs whatsoever.

The pack's docstring is explicit that the module proves nothing specific to planetary migration. The research note records an intention: that the Type I migration timescale might be approximately 0.118 times the disk's Kelvin-Helmholtz timescale, about 12,000 years, which sits in the empirical range of 10^4 to 10^5 years. But that paragraph is a research note, not a result. What would make this a theorem about its subject is a definition of m and e in the subject's own terms, for instance as planet mass and disk surface density. Without that, the certificate is a template, shared verbatim with 2383 sibling modules.

The honest takeaway is plain: the framework's library proves a general mathematical structure, and the migration application is a stated goal, not an established derivation. The certificate's value is in what it does prove, the three facts about domainCost, and in the discipline of saying exactly what remains open. A reader should see the migration timescale estimate as a research hypothesis awaiting a definition that connects the cost function to the physical variables of a protoplanetary disk.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

structure PlanetMigrateCert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

What this page does not claim

The certificate does not prove any specific claim about planetary migration rates or timescales. The estimate of 12,000 years is a research note, not a proved result. The certificate does not establish that the cost function J is the correct model for disk-planet interactions.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Astrophysics/PlanetaryMigrationFromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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