Astrophysics Accretion Disk From Jcost Accretion Regime

Accretion regimes

An accretion disk is a rotating flow of gas falling toward a compact object such as a black hole or neutron star. As the rate of infalling matter rises, the disk's internal structure changes. Astrophysicists distinguish a thin disk, a thick disk, a slim disk, a photon-trapped state, and a super-critical state. The Recognition Science library packages these five as an inductive declaration called AccretionRegime, with each regime a named constructor. A companion theorem, accretionRegimeCount, proves by direct computation that the number of regimes is exactly five.

The declaration does more than count. It ties the five regimes to a transition threshold drawn from the framework's cost function J. In Recognition Science, J(x) = (x + 1/x)/2 - 1 is the forced cost of recognition, proved unique in the library's core. The docstring predicts that the slim-disk-to-photon-trapping transition occurs when the mass accretion rate ratio crosses J(φ), where φ is the golden ratio, placing the threshold in the interval (0.11, 0.13). This is a prediction with a named falsifier: a measured transition outside that interval would refute it.

The library also defines a certificate structure, AccretionDiskCert, that bundles the five-regime count with the transition threshold. A noncomputable definition, accretionDiskCert, supplies the certificate. The file reports zero axioms and zero sorries, meaning the counting theorem is checked by the machine. The physical claim about where the transition occurs, by contrast, is a prediction, not a theorem.

What the declaration does not claim is just as important. It does not derive the five regimes from first principles; it names them as a definitional choice. It does not prove that real astrophysical disks exhibit these regimes; that is an empirical matter. And it does not assert that the threshold J(φ) is measured; it predicts it. The certificate's existence guarantees internal consistency, not correspondence with observations.

theorem accretionRegimeCount : Fintype.card AccretionRegime = 5 := by decide

inductive AccretionRegime where
  | subEddingtonThin | thick | slim | photonTrapped | superCritical
  deriving DecidableEq, Repr, BEq, Fintype

theorem accretionRegimeCount : Fintype.card AccretionRegime = 5 := by decide

What this page does not claim

The five regimes are derived from first principles rather than named by definition. Real astrophysical disks have been observed to exhibit all five regimes. The transition threshold J(φ) has been measured and confirmed.

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

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