Astrophysics Binary Merger Rate From Jcost

Binary merger rates

Binary black hole mergers are among the most violent events observed in modern astronomy. When two black holes spiral together and collide, they release energy as gravitational waves, ripples in spacetime first detected by the LIGO and Virgo observatories in 2015. The rate at which these mergers happen across the universe is a key measurement: the LIGO collaboration's O3 observing run estimated a median merger rate density of about 24 events per cubic gigaparsec per year, with a plausible range of 20 to 80.

Within the Recognition Science framework, a research note attached to one module proposes a way to estimate this rate. The idea is to take a cost function, a measure of how expensive a recognition event is, and evaluate it at a special ratio. The note suggests that the merger rate equals the cost evaluated at the golden ratio, about 0.118, multiplied by the total black hole formation rate of roughly 500 per cubic gigaparsec per year, giving about 59 events per cubic gigaparsec per year. That number sits comfortably inside the LIGO range.

The machine-checked library of formal theorems in the framework proves something much more modest. The file defines a function called domainCost, which applies the framework's cost function to the ratio of two real numbers m and e. Three facts are proved about this function: it returns zero when m equals e, it never returns a negative value when both inputs are positive, and a related threshold constant, phi minus 1.5, is greater than zero. These are general properties of the cost function, not statements about black holes.

In Recognition Science, the framework models the merger rate by treating the ratio m over e as standing for some physical comparison in the binary system. But the module never defines what m and e mean in astrophysical terms. The docstring itself says the paragraph above is a research note recording where the idea was meant to go, not a result. The formal proofs establish only the general mathematical facts, which are shared verbatim across 2383 sibling modules in the library.

What the module does establish, in plain language, is that the cost function has the basic properties one would want for any physical quantity: it is zero at the identity point, it is never negative for positive inputs, and a certain threshold is positive. The astrophysical prediction, by contrast, is a hypothesis. It could be tested by future gravitational wave observations, but the current module does not prove it.

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

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]

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

What this page does not claim

The framework proves the astrophysical merger rate prediction. The number 59 events per cubic gigaparsec per year is a measured value. The module's theorems are specific to binary black hole mergers rather than general properties of the cost function.

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)

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