Astrophysics Accretion Luminosity From Jcost Accretion Lum Cert

What the certificate proves

Accretion luminosity is the light and heat released when matter spirals into a compact object such as a black hole or neutron star. The efficiency of that process, usually written eta, is the fraction of the infalling rest energy that emerges as radiation. For a thin accretion disk, standard models put eta between about 0.1 and 0.42, depending on the black hole's spin. A common benchmark is roughly 0.1 for a non-spinning black hole.

Within Recognition Science, a machine-checked library of formal theorems contains a declaration named AccretionLumCert. The declaration bundles three proved facts about a function called domainCost, which is defined as Jcost(m/e), where Jcost is the framework's forced cost function and m and e are real numbers. The three facts are: domainCost equals zero when m equals e; domainCost is nonnegative for positive m and e; and the quantity phi minus 3/2 is positive, where phi is the golden ratio. Each fact is proved in the library, and the certificate simply packages them together.

In Recognition Science, the cost function J(x) = (x + 1/x)/2 - 1 is forced by five plain conditions and is proved unique in the machine-checked library. The certificate's first two facts are general properties of that cost function: it vanishes when its input is 1, and it never goes below zero for positive inputs. The third fact, that phi is greater than 1.5, is a numerical threshold. None of these facts mentions accretion, mass, energy, or luminosity.

The library's own documentation is explicit: the module proves nothing specific to accretion, because domainCost is defined as Jcost(m/e) without linking m and e to any physical quantity. The paragraph in the source that suggests an efficiency formula, eta = J(phi) * phi^2 / 2 = 0.155, is a research note, not a theorem. For the certificate to become a statement about accretion, one would need a definition of m and e in terms of the accreting system, such as mass and energy. That definition does not exist in the module.

What the certificate does establish is a small, clean result: a certain cost function has three elementary properties, and those properties are machine-checked. It does not establish an accretion efficiency, nor does it predict a value for eta. The empirical comparison to thin-disk efficiency, where 0.155 falls within the 0.1 to 0.42 range, is a research aspiration, not a derived result. A reader should take the certificate as a proof of general cost-function facts, and as a placeholder for a future physical definition that has not yet been supplied.

structure AccretionLumCert 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

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

The certificate does not prove any specific accretion efficiency value. The certificate does not define m and e in physical terms. The research note's formula eta = 0.155 is not a theorem.

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/AccretionLuminosityFromJCost.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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