Astrophysics Planetary Migration From Jcost

Migration timescales

Planetary migration is the drift of a planet's orbit as it exchanges angular momentum with the disk of gas and dust around a young star. In the standard Type I picture, a low-mass planet embedded in such a disk feels a net torque from the density waves it raises, and its semimajor axis evolves on a timescale that scales with the disk's surface density and the planet's mass. The classical result is that a planet like a super-Earth can move inward by a factor of several over a few hundred thousand years, fast enough to matter for the final architecture of a planetary system.

In Recognition Science, the framework models the same physical situation by a cost, a number that measures how expensive a given ratio of masses is to recognize. The framework's library, a machine-checked collection of formal theorems, defines the domain cost as J(m/e), where J is the unique cost function forced by the framework's axioms and m and e are positive real numbers. The framework proves three general facts about this cost: it vanishes when the two masses are equal, it is never negative for positive inputs, and the threshold phi - 3/2 is positive, where phi is the golden ratio. These are facts about the cost function itself, not about planetary disks.

The framework's own docstring is explicit that it proves nothing specific to planetary migration, because the definition of the cost does not reference any property of disks or planets. The research note attached to the framework records an intended application: a migration timescale tau_I approximately equal to J(phi) times the Kelvin-Helmholtz timescale of the disk, which evaluates to about 0.118 times 10^5 years, or roughly 12,000 years. That estimate falls within the empirically measured range of about 10^4 to 10^5 years. The comparison is a numerical check, not a theorem: the framework does not derive the disk timescale from the framework, and the identification of the ratio m/e with a physical mass ratio is a modeling choice, not a proved result.

The value of the framework is therefore structural, not physical. It shows that the framework's cost function has the basic analytic properties one would want of a migration rate: zero when there is no mass contrast, nonnegative for all positive inputs, and a positive threshold that could mark a boundary between regimes. What would turn this into a theorem about planetary migration is a definition of m and e in terms of disk surface density, planet mass, and the disk's thermal structure. Until that definition is supplied and proved, the framework remains a template, not a migration model.

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)

def canonicalThreshold : ℝ := phi - 3 / 2

What this page does not claim

The framework does not prove any specific planetary migration timescale or rate. The empirical agreement of 12,000 years with the observed range is not a theorem. The framework does not derive the disk's Kelvin-Helmholtz timescale from its axioms.

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)

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