Astrophysics Gravitational Microlensing3 Microlens Radius3 Cert

A certificate's honest scope

Microlensing is the bending of light from a distant star by the gravity of a foreground object, which briefly brightens the background star. The Einstein radius is the angular size of the ring this bending produces, and it scales roughly as the square root of the lens mass. A common estimate puts it near one milliarcsecond for a solar-mass lens halfway to a source.

The declaration MicrolensRadius3Cert is a machine-checked certificate, a formal object that bundles three proved facts about a cost function. The cost function is recognition cost, a measure of mismatch between two quantities, here written as J(m/e) where m and e are real numbers. The three facts are: the cost is zero when the two quantities are equal, the cost is never negative for positive inputs, and a certain threshold value phi minus 1.5 is positive. The certificate is a structure that packages these three theorems together, and the library proves such a certificate exists.

The key limitation is explicit in the source. The cost function is defined as J(m/e) with no reference to any physical mass or Einstein radius. The docstring states plainly that the module proves nothing specific to microlensing, and that it is shared verbatim with 2383 sibling modules. What would make this a theorem about microlensing is a definition of m and e in that subject's own terms, for example m as lens mass and e as some expected mass scale. Without that definition, the certificate is a general statement about the cost function, not about astronomy.

The research note attached to the module does record a numerical coincidence: for a solar-mass lens at 4 kiloparsecs with a source at 8 kiloparsecs, the Einstein radius is about 0.9 milliarcseconds, and phi to the minus fifth times 10 milliarcseconds is about 0.94 milliarcseconds. This is a consistency check, not a derivation. The note says the idea was meant to go in this direction, but the formal content does not follow it.

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 certificate does not prove any specific claim about microlensing or Einstein radii. The numerical agreement between 0.9 milliarcseconds and 0.94 milliarcseconds is not a theorem, only a research note. The certificate does not define what m and e mean in physical terms.

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.

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