Astrophysics Grb Afterglow3 From Jcost

A gamma-ray burst timer

A gamma-ray burst is the brightest flash in the sky, a short blast of high-energy light from a distant explosion. After the initial flash, a slower afterglow fades over days. Astronomers watch for a jet break, a bend in the fading light when the burst's narrow jet slows and its edge becomes visible. For many bursts that break comes about one day after the flash.

The module GRB_Afterglow3_FromJCost in the framework's machine-checked library of formal theorems tries to connect that one-day break to a recognition cost. In Recognition Science, a ledger, a discrete record of events, assigns a cost to each pair of compared quantities. The cost function J(x) = (x + 1/x)/2 - 1 measures how far two numbers are from matching: it is zero when they are equal and grows as they diverge. The module defines a domain cost as J applied to the ratio of two quantities, m over e.

The library proves three general facts about that cost. It vanishes when the two quantities are equal, it never goes negative for positive inputs, and the golden-ratio threshold phi - 3/2 is positive. Those three facts assemble into a certificate structure, a formal record that the cost behaves as expected. The certificate is inhabited, meaning a proof exists.

What the module does not do is prove anything specific to gamma-ray bursts. The definition of m and e never refers to burst physics; they are just real numbers. The research note attached to the module sketches an intended path: if m and e were defined as burst times, then powers of phi times a 30-second burst duration would give 2 minutes, 14.5 minutes, 1 hour, and 1.67 hours, stepping toward the observed one-day break. That sketch is a research note, not a result. The formal theorems would become burst-specific only if a future definition tied m and e to measured burst quantities.

The practical lesson: the framework's cost function is a general tool for comparing any two positive numbers, and the library has proved its basic properties. Applying that tool to a real astrophysical clock requires a separate step, defining the burst times in physical terms. Until then, the one-day jet break remains an observed pattern awaiting a framework derivation.

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 GRBAftglow3Cert := ⟨cert⟩

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

What this page does not claim

The module proves no statement about gamma-ray bursts, jet breaks, or the one-day timescale. The research note's phi-power sequence is not a formal derivation. The framework does not claim that the golden ratio governs all burst afterglows.

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/GRB_Afterglow3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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