Astrophysics Binary Merger Rate From Jcost Bbhmerger Rate Cert

The certification structure

A binary black hole merger rate is the number of such events per unit volume of space per unit time. The laser interferometer observatories LIGO and Virgo have measured a rate density of roughly 20 to 80 per cubic gigaparsec per year, with a median around 24. The Recognition Science framework's declaration BBHMergerRateCert does not establish that number. It establishes three general facts about a mathematical function called cost, a measure of the price of a recognition event, and packages them into a certificate.

The three facts are simple. First, the cost function returns zero when its two inputs are equal, meaning no cost is paid for recognizing like with like. Second, the cost is never negative for positive inputs. Third, a particular threshold value, the golden ratio minus 1.5, is greater than zero. These are proved in the machine-checked library of formal theorems, and the certificate bundles them into a single object. The certificate is inhabited, meaning it exists.

What the certificate does not do is connect those facts to astrophysics. The cost function is defined as J(m/e), a ratio of two real numbers, but the declaration never defines what m and e mean for a black hole system. They could be masses, energies, or anything else. The research note attached to the module records an intention: a rate of about 59 per cubic gigaparsec per year, derived from the cost function times a total black hole formation rate, would be consistent with the LIGO observation. That note is a research idea, not a result.

In Recognition Science, the framework models the cost function as forced by five plain conditions, and the golden ratio emerges as a self-similar scaling. The certificate proves the abstract properties of that cost function. The step from abstract cost to a physical merger rate requires a definition of m and e in astrophysical terms. That definition is absent. The certificate is therefore a theorem about a function, not about the universe.

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]

theorem cert_inhabited : Nonempty BBHMergerRateCert := ⟨cert⟩

What this page does not claim

The certificate does not prove any specific binary black hole merger rate. The certificate does not define m and e in physical terms. The research note's rate of 59 per cubic gigaparsec per year 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/BinaryMergerRateFromJCost.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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