Astrophysics Accretion Disk Stability3 From Jcost Disk Instab3 Cert

A certificate, and its limits

Accretion disks are the swirling gas and dust that feed black holes and young stars, and their stability is a classic problem in astrophysics. The Shakura-Sunyaev model, developed by Nikolai Shakura and Rashid Sunyaev in 1973, describes how such a disk can suddenly switch from a cool, dim state to a hot, bright one when its surface density crosses a critical value. The Recognition Science declaration DiskInstab3Cert is a machine-checked certificate, a formal object that bundles three proved facts about a particular cost function, but its name is a promise the underlying code does not yet keep.

The certificate's three facts are general properties of the function domainCost m e = Jcost (m / e), where Jcost is the framework's forced cost function and m and e are real numbers. The first fact is that the cost vanishes when the two inputs are equal: domainCost r r = 0 for any nonzero r. The second is that the cost is never negative when both inputs are positive. The third is that a constant called canonicalThreshold, defined as the golden ratio minus 3/2, is positive. These are all proved in the machine-checked library of formal theorems, and the certificate simply collects them into a single structure.

What the certificate does not do is connect those facts to accretion disks. The docstring attached to the module records a research note, not a result: the idea was that at a critical surface density, the cost function would hit a threshold and trigger an outburst, but the code defines domainCost with no reference to surface density, temperature, or any disk quantity. The same three facts are stated once, universally, in a shared template module, and this file copies them verbatim. To make this a theorem about disks, someone would need to define m and e in terms of disk physics, for example by setting m to the surface density and e to a critical value, and then prove the relevant properties hold. That step has not been taken.

The practical consequence is a clear division of labor. The certificate is a genuine, checked piece of mathematics about a cost function, and it is a template for what a disk stability proof could look like. But a reader who wants to know whether the framework predicts a specific outburst threshold in X-ray novae must look elsewhere: the certificate itself is silent on that question, and the research note in the docstring is explicitly marked as an intention, not an achievement.

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)

What this page does not claim

DiskInstab3Cert does not prove that any real accretion disk becomes unstable at any particular density. The certificate does not establish that the golden ratio appears in disk physics; it only shows that a constant built from phi is positive. The docstring's research note about Shakura-Sunyaev instability is an intention, not a proved result.

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/Accretion_Disk_Stability3_FromJCost.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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