Gravity Gravitational Lens3 From Jcost

A lens formula at phi-ladder distances

Gravitational lensing occurs when a massive object bends light from a background source, producing rings or arcs. For a point mass, the Einstein ring radius is theta_E = sqrt(4GM/c^2 * D_LS/(D_L D_S)), where M is the lens mass and the D terms are distances between observer, lens, and source. This classical formula was derived from general relativity and is a standard tool in astronomy.

In Recognition Science, the framework models the cost of recognition between two quantities. The cost, a measure of how hard it is to recognize one value from another, is forced by a proved theorem to equal J(x) = (x + 1/x)/2 - 1. The framework then places physical constants on a ladder of powers of the golden ratio phi, about 1.618.

The module GravitationalLens3FromJCost applies this cost function to the ratio of two masses, m and e. It defines domainCost(m, e) = J(m/e). The machine-checked library proves three general facts: this cost is zero when m equals e, it is nonnegative for positive inputs, and the threshold phi - 3/2 is positive. These facts are true for any positive masses, not specific to lensing.

The research note attached to the module states the intended physical step: at a distance D = phi^k times the Schwarzschild radius, the Einstein ring radius would be theta_E = phi^(k/2) times the Schwarzschild angle. This is a structural conjecture, not a proved result. The module itself does not define m and e in terms of lens masses or distances, so it does not yet establish the lens formula.

What the module does establish is a template: if one can define m and e for a lens system in the framework's terms, then the cost function's properties apply. The positivity and zero-at-equality facts are the groundwork. The phi-power ring radius remains a target for future work, requiring a definition that connects the cost ratio to actual lens parameters.

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]

What this page does not claim

The Einstein ring radius formula theta_E = phi^(k/2) * Schwarzschild_angle is not proved in this module. The module does not define m and e in terms of lens mass or distance parameters. No claim is made that the framework's cost function replaces general relativity for lensing calculations.

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/Gravity/GravitationalLens3FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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