Astrophysics Gravitational Lensing V2 Gravitational Lensing V2 Cert

The certification record

Gravitational lensing is the bending of light from a distant source by the gravity of an intervening mass, so the source appears shifted or distorted. A recognition ledger, a discrete record of events, can model the cost of matching an observed image to a source. The declaration GravitationalLensing_v2Cert is a small formal record that bundles two consistency checks for such a model.

The first check states that the cost of matching an object to itself is zero. The cost function is defined as J(m/e), where J is the framework's forced cost function and m and e are real numbers representing the measured and expected values. The theorem proves that for any nonzero value r, the cost of matching r to r is exactly zero. This is the basic sanity condition that a match with itself costs nothing.

The second check states that a certain threshold value is positive. The threshold is defined as phi minus 3/2, where phi is the golden ratio, approximately 1.618. The theorem proves this threshold is greater than zero. The threshold itself is a model choice, not a derived quantity, and its physical meaning is not established by the declaration.

The declaration bundles these two facts into a single structure, and the library proves that this structure is inhabited, meaning the two checks are consistent with each other and can be satisfied together. In Recognition Science, this declaration is a minimal certificate that a lensing model built on the forced cost function passes these two basic checks. It does not claim to derive the deflection angle, the Einstein radius, or any observable lensing prediction.

The value of this declaration is architectural. It establishes a named, machine-checked record that future lensing theorems can reference as a precondition. A later theorem about lensing can require a GravitationalLensing_v2Cert as an input, and the library can verify that the certificate exists. This is how the framework builds larger claims on smaller checked foundations.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

theorem cert_inhabited : Nonempty GravitationalLensing_v2Cert := ⟨cert⟩

What this page does not claim

The declaration does not derive the Einstein radius or any deflection angle. The threshold's positivity does not establish its physical meaning as a lensing boundary. The certificate does not prove that recognition cost governs actual photon paths.

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/GravitationalLensing_v2.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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