Astrophysics Gamma Ray Burst3 From Jcost

The afterglow slope

A gamma-ray burst afterglow fades as a power law: the brightness at a given frequency drops with time as t raised to some negative exponent. For the optical band, that exponent, usually called alpha, measures around 0.8 to 1.2 in observed bursts. The canonical simple model predicts a single power-law decay, and the measured values cluster near that prediction.

In Recognition Science, the framework's cost function J(x) = (x + 1/x)/2 - 1, which is proved to be the unique cost satisfying five plain conditions, produces a specific value for this slope. The framework identifies the exponent as alpha = 1 - J(phi), where phi is the golden ratio, giving 0.882, or roughly 0.9. This sits inside the observed range of 0.8 to 1.2 and matches the canonical simple power-law afterglow slope.

What the machine-checked library actually proves is narrower and fully general. The module defines a cost for a ratio m/e, then proves three facts: the cost vanishes when m equals e, it is never negative for positive inputs, and the threshold phi - 3/2 is positive. These are properties of the cost function itself, not of gamma-ray bursts. The library states this plainly: the definition of domainCost as Jcost (m / e) makes no reference to any astrophysical quantity, so the afterglow connection is a research note, not a theorem.

The gap is the definition of m and e in burst terms. If someone supplied those, the general facts would become statements about afterglows. Until then, the module certifies the cost's behavior and records where the idea was meant to go, which is honest groundwork rather than a completed derivation.

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 module proves any specific fact about gamma-ray bursts. The afterglow slope is derived from the cost function without additional assumptions. The observed range 0.8 to 1.2 is a precise prediction of the framework.

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/Gamma_Ray_Burst3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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