Astrophysics Grbjet Angle From Jcost Grbjet Angle Cert

What the certificate proves

Gamma-ray bursts are the brightest explosions in the universe, and their jets of energy are narrow, typically opening only a few degrees. In Recognition Science, the cost function, a measure of the forced price of recognition, takes the form J(x) = (x + 1/x)/2 - 1. The declaration GRBJetAngleCert is a machine-checked certificate, a formal package of three proved facts about this cost function applied to a ratio of two positive numbers.

The three facts are general properties of the cost function, not results about astrophysics. First, the cost vanishes when the two inputs are equal: J(m/e) = 0 when m = e. Second, the cost is never negative for positive inputs. Third, the golden ratio phi is greater than 3/2, so the quantity phi - 3/2 is positive. These are proved in the framework's machine-checked library of formal theorems, and the certificate packages them together as a single structure.

The research note attached to the declaration records an intended application: it suggests the gamma-ray burst jet opening angle might equal J(phi) * pi/2 radians, which computes to about 10.6 degrees, consistent with the observed range of 3 to 15 degrees. That note is a plan, not a result. The certificate itself proves no statement about jets, angles, or bursts, because the inputs m and e are never defined in astrophysical terms.

In Recognition Science, the framework models the cost function as a universal object, and its library shows that J is forced by five plain conditions. The certificate illustrates how a general theorem can be packaged, but the bridge from the abstract cost to a physical jet angle remains open. What the certificate establishes is the general mathematical skeleton; what it does not establish is any claim about gamma-ray bursts.

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

The certificate proves no statement about gamma-ray bursts, jets, or angles. The 10.6 degree value is a research note, not a proved or measured result. The inputs m and e are not defined in astrophysical terms in the certificate.

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/GRBJetAngle_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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