Astrophysics Gamma Ray Line3 From Jcost Gamma Line3 Cert

What the certificate proves

In the Recognition Science framework, a cost function (a number that measures the price of a recognition event) is built from a ratio of two quantities. The declaration GammaLine3Cert is a small bundle of three general facts about that cost function. First, when the two quantities are equal, the cost is exactly zero. Second, for any two positive quantities, the cost is never negative. Third, a particular constant called the canonical threshold, defined as the golden ratio minus 1.5, is greater than zero. These three statements are proved in the framework's machine-checked library of formal theorems.

The name of the declaration suggests a connection to astrophysical gamma-ray lines, such as the 511 keV line from electron-positron annihilation and the 1809 keV line from aluminum-26. The research note attached to the file records that idea: the ratio 1809/511 is about 3.54, close to phi^2.5, which is about 3.33. But that note is explicitly labeled as a research note, not a result. The formal definition of the cost function uses only the ratio m/e with no reference to any physical quantity. Nothing in the three theorems mentions gamma rays, keV, or any astronomical object.

What would make the certificate a theorem about its subject is a definition of m and e in that subject's own terms. Without such a definition, the certificate is a statement about the cost function alone. The framework's own documentation says the module proves nothing specific to this subject. The certificate is honest about its scope: it establishes properties of a general mathematical object, and the astrophysical interpretation remains a separate, unformalized idea.

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

GammaLine3Cert does not prove any property of gamma-ray lines, since m and e are not defined as line energies. The research note's ratio comparison is not a formal result and is not part of the certificate. No empirical agreement with measured line positions is established by the three theorems.

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_Line3_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.

Derived articles

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