Astrophysics Accretion Luminosity From Jcost

Accretion luminosity

Accretion luminosity is the light a body emits as gravity pulls surrounding matter inward. When gas falls toward a black hole or a young star, it forms a rotating disk, heats up, and radiates. The standard measure of how well that process turns infalling mass into radiation is the accretion efficiency eta = L / (dot(M) * c^2), where L is the emitted luminosity, dot(M) is the rate of mass inflow, and c is the speed of light. For a thin disk, the efficiency depends on how close the matter gets before it is swallowed; measured values run from about 0.1 for Schwarzschild black holes up to roughly 0.42 for maximally spinning ones.

In Recognition Science, the framework models the same quantity through its cost function. The framework's cost, a forced measure of the price of recognizing one event relative to another, takes the form J(x) = (x + 1/x)/2 - 1. The astrophysics module defines its domain cost as J(m / e), where m and e stand for two positive quantities in the accretion problem. The machine-checked library of formal theorems proves three general facts about this cost: it vanishes when m equals e, it is never negative for positive inputs, and the threshold phi - 3/2 is positive. Those facts are general properties of J, not results specific to accretion, because the module never defines m and e in the subject's own terms.

The framework's research note goes further and computes an efficiency from the golden ratio phi: eta = J(phi) * phi^2 / 2 = 0.118 * 1.309 = 0.155. That number falls comfortably inside the empirical thin-disk range of 0.1 to 0.42, near the middle. The note labels itself a plan, not a result, and the Lean proofs confirm none of it. What the module actually establishes is narrower: the cost function's zero, nonnegativity, and threshold properties, all stated once and universally quantified in a shared template.

The consequence is a clean separation between what the framework proves and what it hopes. The proved facts guarantee the cost behaves like a sensible measure: it costs nothing when the two quantities match, it never rewards a mismatch with negative cost, and it has a positive threshold tied to phi. The accretion efficiency remains a research note, an open target that needs a definition of m and e in the accretion problem's own terms before it becomes a theorem.

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

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]

What this page does not claim

The module does not prove the accretion efficiency estimate of 0.155. The module does not define m and e in physical terms. The framework does not derive the fine-structure constant.

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)

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Derived articles

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