Astrophysics Grb Afterglow3 From Jcost Grbaftglow3 Cert

The certificate

Gamma-ray bursts are the brightest explosions in the universe, and after the initial flash comes a fading afterglow. Astronomers study the afterglow's light curve for a jet break, a steepening that marks when the burst's narrow jet slows and spreads. The declaration GRBAftglow3Cert in the Recognition Science library sounds like it should certify something about that break time. It does not.

What the certificate actually proves is three general facts about the framework's cost function, the forced price of recognition. The cost vanishes when its two inputs are equal, it is never negative for positive inputs, and a certain threshold value involving the golden ratio is positive. These are true for any positive real numbers, with no reference to jets, bursts, or astronomy. The certificate is a small bundle of three theorems, each machine-checked in the framework's library of formal theorems.

The subject-specific ambition lives only in a research note attached to the file. That note sketches a plan: the afterglow jet break time might follow a golden-ratio power law from a burst duration. It lists example values, such as phi to the power 11 times 30 seconds giving about 1.67 hours. But the note is explicitly not a result. The formal definition of the cost in this file, Jcost (m / e), never connects m or e to any physical quantity like burst duration or jet opening angle. Without that connection, the theorems prove nothing about gamma-ray bursts.

The certificate does establish one useful thing. It shows the framework's cost function satisfies three basic sanity conditions in this context: zero at equality, nonnegativity, and a positive threshold. Any future work that wants to claim a jet break time from the framework must first define what m and e mean in the burst's own terms. Until then, GRBAftglow3Cert is a certificate about the cost function, not about the sky.

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

No theorem in this file establishes any property of gamma-ray bursts or their afterglows. The research note's jet break time values are not proved results. The certificate does not define what m or e mean physically.

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)

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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