Astrophysics Accretion Disk From Jcost Accretion Regime Count

The five 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 gas spirals inward, it heats up and radiates, and the rate at which mass falls inward, called the mass accretion rate, determines how the disk behaves. At low rates the disk is thin and cool; at high rates it becomes thick, then slim, then reaches a state where photons are trapped in the flow, and finally becomes super-critical. These are not arbitrary labels. The framework's machine-checked library of formal theorems proves that the number of distinct regimes is exactly five.

The proof is a theorem named accretionRegimeCount, and it works by defining an inductive type with five constructors: subEddingtonThin, thick, slim, photonTrapped, and superCritical. The theorem then verifies by computation that this type has exactly five elements. The library reports zero axioms and zero "sorry" placeholders, meaning the proof is complete and machine-checked. The declaration also bundles this count with a separate certificate for the transition threshold, which is the point where the disk switches between regimes.

In Recognition Science, the framework models this transition threshold using a cost function J. The prediction is that the transition from a slim disk to photon trapping occurs when the mass accretion rate ratio crosses a value J(φ) in the interval (0.11, 0.13). This is a prediction, not a proved theorem about real astrophysical objects. The count of five regimes is a theorem about a formal definition; the physical claim that real disks actually exhibit exactly these five regimes and no others is an empirical matter that the framework does not settle.

What the declaration does not claim is important. It does not claim that real accretion disks have been observed to have exactly five regimes. It does not claim that the transition threshold J(φ) has been measured. It does not claim that the framework derives the physics of accretion from first principles. The theorem only establishes that a particular formal classification, chosen by the framework, has five members. The physical interpretation of those five members as real disk states is a modeling choice, and the numerical value of the transition threshold is a prediction awaiting observational test.

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

structure AccretionDiskCert where
  five_regimes : Fintype.card AccretionRegime = 5
  transition_threshold : CanonicalCert

What this page does not claim

The theorem does not claim that real accretion disks have been observed to have exactly five regimes. The theorem does not claim that the transition threshold J(φ) has been measured. The framework does not derive the physics of accretion from first principles.

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)

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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