Astrophysics Gravitational Microlensing3

The module's actual content

Gravitational microlensing is the bending of light from a distant star by the gravity of a foreground object, which briefly magnifies the background star. The Einstein radius is the angular scale of that effect, roughly the radius of the ring image you would see if the two stars were perfectly aligned. In this framework, the module named astrophysics-gravitational-microlensing3 was meant to connect that radius to the golden ratio through a scaling ladder, but the formal content does not do that.

The module defines a cost function, a measure of mismatch between two quantities, as J(m/e), where J is the framework's forced cost function. It then proves three facts: the cost is zero when the two quantities are equal, the cost is never negative for positive inputs, and the golden ratio minus 1.5 is positive. These are generic properties of the cost function, true for any positive inputs, not facts about microlensing.

The module also bundles these three facts into a structure called a certificate, and proves that such a certificate exists. That is the entire formal achievement. The research note attached to the module records the intended application: at one solar mass with specific distances, the Einstein radius comes out near 0.9 milliarcseconds, and phi^-5 times 10 milliarcseconds is about 0.94 milliarcseconds, which is consistent. But that note is not a theorem; it is a stated intention.

The reason the module proves nothing specific is that the cost function is defined without any reference to lens masses, distances, or angles. The quantities m and e are just real numbers. To make this a theorem about microlensing, someone would need to define m and e in terms of the physical situation, and that definition does not exist in the module.

What this means in plain language: the module is a template with a research aspiration attached. The three proved facts are real and machine-checked, but they are about the cost function's general behavior, not about stars. The microlensing connection remains a plan, not a result.

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]

theorem cert_inhabited : Nonempty MicrolensRadius3Cert := ⟨cert⟩

What this page does not claim

This module proves any specific fact about gravitational microlensing, such as a value for the Einstein radius. The numerical consistency in the research note is a theorem; it is only a stated intention. The golden ratio is derived from microlensing physics in this module.

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/Gravitational_Microlensing3.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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