Astrophysics Gravitational Lensing From Jcost

The lensing cost module

Gravitational lensing is the bending of light by a massive object, such as a galaxy or black hole, which acts like a lens. The deflection angle α for a point mass is given by α = 4GM/(c²b), where G is the gravitational constant, M is the mass, c is the speed of light, and b is the impact parameter, the closest distance the light would pass if no mass were present. A famous special case is the Einstein ring, which appears when the source, lens, and observer are perfectly aligned.

The framework's machine-checked library of formal theorems defines a cost function for lensing. The cost, domainCost, is a discrete record of the mismatch between a measured deflection and an expected deflection. It is defined as J(measured / expected), where J is the framework's universal cost function. The library proves three general facts: the cost is zero when the measured and expected values are equal, the cost is never negative for positive inputs, and a threshold constant φ - 3/2 is positive.

These facts are general properties of the cost function, not specific results about lensing. The library does not define what "measured" and "expected" mean in terms of the physical parameters of a lensing event, such as the mass or the impact parameter. It proves nothing about the deflection formula itself. The research note in the library records the intended direction, that maximum deflection occurs at an impact parameter related to the golden ratio φ, but this is not a proved result.

In plain language, the library establishes that the framework's cost function behaves sensibly when applied to lensing: it is a measure of error that is zero at perfect agreement and positive otherwise. This is a necessary first step, but the physical content of lensing, the connection between the cost and the actual deflection angle, remains to be defined in the library's own terms.

theorem domainCost_at_equilibrium (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 library does not prove any specific result about gravitational lensing deflection angles. The connection between the cost function and the physical parameters of a lensing event is not established. The maximum deflection at b = φ × r_s is a research note, not a theorem.

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/GravitationalLensingFromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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