Gravity Gravitational Lensing Ilg Correction Enhances

A stronger lensing signal

Gravitational lensing is the bending of light by mass, one of the main observational tools in modern astronomy. The standard general relativistic prediction for the deflection angle of a photon passing a mass M at a distance b is θ = 4GM/(c²b). This formula was famously confirmed by Arthur Eddington's 1919 solar eclipse expedition, which measured a deflection of about 1.75 arcseconds for starlight grazing the Sun. The factor of 2 relative to the Newtonian prediction, where a photon is treated as a particle, comes from both the temporal and spatial components of the metric contributing equally. The machine-checked library of formal theorems in the Recognition Science framework reproduces this standard result, including a theorem that the general relativistic deflection is exactly twice the Newtonian value.

The same library then goes further and studies a proposed modification to the standard lensing convergence, which is a measure of how much the apparent density of background galaxies is distorted by a foreground mass. The proposed modification, called the Information Ledger Gravity correction, multiplies the standard convergence by a scale-dependent factor: κ_RS(ℓ) = κ_GR(ℓ) × (1 + α_t × (ℓ/ℓ₀)^(−β)). Here ℓ is a scale, ℓ₀ is a reference scale, α_t is a positive coefficient, and β is a small positive exponent. The theorem named ilg_correction_enhances proves that whenever the standard convergence κ_GR is positive, the corrected value is strictly larger than the uncorrected one. In plain language, the correction always enhances the lensing signal, never reduces it, under the stated conditions that the mass, scale, and reference scale are all positive.

This is a conditional statement about a definitional choice, not a claim that the correction exists in nature. The theorem proves an inequality about a defined function, given that the coefficient α_t is positive. It says nothing about whether the Information Ledger Gravity model is true, whether the coefficient actually has that sign, or whether the correction matches any astronomical observation. The framework's library also contains a theorem that the solar deflection is positive, consistent with the standard result, but the ILG correction itself is a hypothesis to be tested against data, not a derived law of physics.

/-- **KEY THEOREM**: GR deflection is exactly twice the Newtonian value.
    The factor of 2 arises because both temporal AND spatial metric
    components contribute equally to photon deflection. -/
theorem gr_is_twice_newton (M b : ℝ) (hb : b ≠ 0) :
    deflection_GR M b = 2 * deflection_newtonian M b := by
  unfold deflection_GR deflection_newtonian
  ring

/-- The ILG correction is positive for α_t > 0. -/
theorem ilg_correction_enhances (κ_GR α_t ℓ ℓ₀ β : ℝ)
    (hκ : 0 < κ_GR) (hα : 0 < α_t) (hℓ : 0 < ℓ) (hℓ₀ : 0 < ℓ₀) :
    κ_GR < ilg_convergence_correction κ_GR α_t ℓ ℓ₀ β := by
  unfold ilg_convergence_correction
  have hterm : 0 < α_t * (ℓ / ℓ₀) ^ (-β) := by
    apply mul_pos hα
    apply rpow_pos_of_pos
    positivity
  nlinarith

/-- Key structural fact: solar deflection is positive. -/
theorem solar_deflection_positive : 0 < solar_deflection := by
  unfold solar_deflection
  apply deflection_positive <;> norm_num

What this page does not claim

The ILG correction is a measured or confirmed physical effect. The theorem proves that the correction applies to any real astronomical system, since it assumes positive mass, scale, and reference scale. The framework derives the fine-structure constant or any other coupling constant from the lensing formalism.

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/GravitationalLensing.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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