Cosmology Gravitational Lensing From Rs Deflection Pos

A positive deflection angle

Gravitational lensing is the bending of light from a distant source by a massive object in between, like a galaxy or a cluster of galaxies. In the Recognition Science framework, the deflection angle for each of five canonical lensing regimes is defined as a power of the golden ratio: deflectionAngle(k) = φ^k, where k is a natural number labeling the regime. The framework's machine-checked library of formal theorems proves that this angle is always positive: for every k, 0 < φ^k. The proof is a direct consequence of the fact that φ is positive and that a positive number raised to any natural power remains positive.

The declaration deflection_pos is the formal statement of this positivity. It is a theorem in the framework's library, proved with no gaps and no axioms beyond the standard logical ones. Its content is modest but load-bearing: it guarantees that the deflection angle never becomes zero or negative, which is a sanity condition for any physical quantity meant to represent a bend. The same library also proves that the ratio of consecutive deflection angles is exactly φ, so the angles form a geometric progression with the golden ratio as its common ratio.

The framework groups lensing into five regimes: weak lensing, strong lensing, microlensing, cluster lensing, and time-delay. The positivity theorem applies uniformly to all of them, because it holds for every natural number k. A certificate structure in the library packages these facts together, but the positivity claim itself is the simplest piece.

What deflection_pos does not claim is important. It does not say that the deflection angle formula matches observed gravitational lensing measurements; that would be an empirical question, and the pack provides no such comparison. It does not say that the golden ratio is the measured deflection angle of any real lens. It does not even say that the five regimes are physically exhaustive or that the labeling by k is physically meaningful. The theorem is purely internal: given the definition, the angle is positive. That is all.

theorem deflection_pos (k : ℕ) : 0 < deflectionAngle k := pow_pos phi_pos k

theorem deflection_ratio (k : ℕ) :
    deflectionAngle (k + 1) / deflectionAngle k = phi := by
  unfold deflectionAngle
  have hpos : (0 : ℝ) < phi ^ k := pow_pos phi_pos k
  rw [div_eq_iff hpos.ne', pow_succ]
  ring

theorem lensingRegime_count : Fintype.card LensingRegime = 5 := by decide

What this page does not claim

The deflection angle formula matches any observed lensing measurement. The golden ratio is the measured deflection angle of any real gravitational lens. The five regimes are physically exhaustive or the k labeling is physically meaningful. The framework derives the deflection angle from general relativity or any conventional theory of gravity.

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

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