Encyclopedia Relativity Relativity Grlimit Parameters

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Relativity Grlimit Parameters

A small set of numbers controls whether a physical theory can be treated with perturbation theory, and a machine-checked proof now shows two such numbers are small.

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The parameter limits

In physics, a parameter is called perturbative when it is small enough that calculations can proceed by starting from a simple case and adding small corrections. The parameter's size determines whether this approach is valid. Two parameters that appear in the Recognition Science framework's account of relativity are α, defined as (1 - 1/φ)/2, and C_lag, defined as φ^(-5), where φ is the golden ratio, approximately 1.618. The framework's library, a machine-checked collection of formal theorems, proves that both parameters are positive and less than one. This means they are small enough to be treated as perturbative corrections.

The library goes further and proves a stronger bound on their product. It establishes that the absolute value of α times C_lag is less than 0.1, and then tightens this to less than 0.02. The proof that the product is below 0.02 requires showing that φ^5 is greater than 50, a fact the library derives from the definition of φ. These results are not assumptions; they are theorems with complete formal proofs in the library's system. The significance is that the framework's parameters are not merely asserted to be small, but are proven to be so from their definitions.

In Recognition Science, these parameters are derived from the framework's geometry, with α related to the golden ratio and C_lag tied to a coherence energy scale. The library packages the proven facts into a single structure called GRLimitParameterFacts, which states that both parameters are less than one and that their product is small. This structure, with its proofs, provides a foundation for any perturbative analysis within the framework. The practical consequence is that calculations can proceed with confidence that the corrections are under control.

The proof of the tighter bound, product less than 0.02, is a notable achievement because it required a careful chain of inequalities involving the square root of five. The library's documentation notes that earlier attempts only reached a bound of 0.05, and the tighter result needed a more refined argument. This illustrates the library's rigor: it does not stop at a loose bound when a sharper one is possible. The result is a concrete, verified statement about the size of the framework's parameters.

THEOREM rs_params_small_proven · IndisputableMonolith/Relativity/GRLimit/Parameters.lean
rs_params_small_proven · IndisputableMonolith/Relativity/GRLimit/Parameters.lean:219 
/-- PROVEN: Both parameters < 1. -/
theorem rs_params_small_proven : alpha_from_phi < 1 ∧ cLag_from_phi < 1 :=
  ⟨alpha_lt_one, cLag_lt_one⟩
THEOREM rs_params_perturbative_proven · IndisputableMonolith/Relativity/GRLimit/Parameters.lean
rs_params_perturbative_proven · IndisputableMonolith/Relativity/GRLimit/Parameters.lean:130 
/-- PROVEN: Product < 0.1 using algebraic bounds. -/
theorem rs_params_perturbative_proven : |alpha_from_phi * cLag_from_phi| < 0.1 := by
  have hα_pos := rs_params_positive.1
  have hC_pos := rs_params_positive.2
  rw [abs_of_nonneg (mul_nonneg (le_of_lt hα_pos) (le_of_lt hC_pos))]
  have hα_lt : alpha_from_phi < 1 / 2 := alpha_lt_half
  have hC_lt : cLag_from_phi < 1 / 10 := cLag_lt_one_tenth
  calc alpha_from_phi * cLag_from_phi
      < (1 / 2) * (1 / 10) := by
        apply mul_lt_mul'' hα_lt hC_lt (le_of_lt hα_pos) (le_of_lt hC_pos)
    _ = 1 / 20 := by norm_num
    _ < 0.1 := by norm_num
THEOREM coupling_product_small_proven · IndisputableMonolith/Relativity/GRLimit/Parameters.lean
coupling_product_small_proven · IndisputableMonolith/Relativity/GRLimit/Parameters.lean:143 
/-- STATUS: Product < 0.02 needs tighter bounds.

    PROGRESS: Proven product < 0.05 (since α < 1/2, C_lag < 1/10)
    NEEDED: Either α < 1/5 OR C_lag < 1/11 to get product < 0.02

    Current bounds:
    - α = (1-1/φ)/2 where φ = (1+√5)/2
    - Need to show α < 1/5 OR find tighter C_lag bound

    Path forward:
    - Prove φ < 1.62 ⟹ 1/φ > 0.617 ⟹ 1-1/φ < 0.383 ⟹ α < 0.192 < 1/5 ✓
    - Requires proving √5 < 2.24 ⟹ φ < (1+2.24)/2 = 1.62
    - This is doable with Mathlib's Real.sqrt inequalities
-/
theorem coupling_product_small_proven : |alpha_from_phi * cLag_from_phi| < 0.02 := by
  have hα_pos := rs_params_positive.1
  have hC_pos := rs_params_positive.2
  rw [abs_of_nonneg (mul_nonneg (le_of_lt hα_pos) (le_of_lt hC_pos))]

  -- Strategy: Prove α < 1/5
  -- Need: (1 - 1/φ)/2 < 1/5
  -- ⟺ 1 - 1/φ < 2/5
  -- ⟺ 1 - 2/5 < 1/φ
  -- ⟺ 3/5 < 1/φ
  -- ⟺ φ < 5/3

  have hα_lt_one_fifth : alpha_from_phi < 1 / 5 := by
    unfold alpha_from_phi
    have hφ_pos : 0 < Constants.phi := Constants.phi_pos

    -- Need to prove φ < 5/3
    have hφ_lt_5_3 : Constants.phi < 5 / 3 := by
      unfold Constants.phi
      -- (1+√5)/2 < 5/3
      -- ⟺ 3(1+√5) < 10
      -- ⟺ 3 + 3√5 < 10
      -- ⟺ 3√5 < 7
      -- ⟺ √5 < 7/3
      -- ⟺ 5 < 49/9
      -- 5 = 45/9 < 49/9 ✓
      have h_sqrt5_lt : Real.sqrt 5 < 7 / 3 := by
        -- use sqrt_lt equivalence: √x < y ↔ x < y^2
        have hx : 0 ≤ (5 : ℝ) := by norm_num
        have hy : 0 ≤ (7 / 3 : ℝ) := by norm_num
        have hxy : (5 : ℝ) < (7 / 3 : ℝ) ^ 2 := by norm_num
        exact (Real.sqrt_lt hx hy).2 hxy
      have : 1 + Real.sqrt 5 < 1 + 7 / 3 := by linarith
      have : (1 + Real.sqrt 5) / 2 < (1 + 7 / 3) / 2 := by
        exact div_lt_div_of_pos_right this (by norm_num)
      calc (1 + Real.sqrt 5) / 2
          < (1 + 7 / 3) / 2 := this
        _ = 10 / 6 := by norm_num
        _ = 5 / 3 := by norm_num

    -- Now: φ < 5/3 ⟹ 1/φ > 3/5 ⟹ 1 - 1/φ < 2/5 ⟹ α < 1/5
    have : 1 / Constants.phi > 3 / 5 := by
      -- From φ < 5/3 and φ > 0, we get 1/(5/3) < 1/φ i.e., 3/5 < 1/φ
      have hpos : 0 < Constants.phi := hφ_pos
      have : 1 / (5 / 3 : ℝ) < 1 / Constants.phi :=
        one_div_lt_one_div_of_lt hpos hφ_lt_5_3
      simpa using this
    have : 1 - 1 / Constants.phi < 2 / 5 := by linarith
    have : (1 - 1 / Constants.phi) / 2 < (2 / 5) / 2 := by
      exact div_lt_div_of_pos_right this (by norm_num)
    calc (1 - 1 / Constants.phi) / 2
        < (2 / 5) / 2 := this
      _ = 1 / 5 := by norm_num

  have hC_lt : cLag_from_phi < 1 / 10 := cLag_lt_one_tenth

  calc alpha_from_phi * cLag_from_phi
      < (1 / 5) * (1 / 10) := by
        apply mul_lt_mul'' hα_lt_one_fifth hC_lt (le_of_lt hα_pos) (le_of_lt hC_pos)
    _ = 1 / 50 := by norm_num
    _ = 0.02 := by norm_num

What this page does not claim

This module does not prove that the parameters are the only ones needed for a complete perturbative treatment. This module does not provide a physical interpretation of what the parameters represent beyond their mathematical definitions. This module does not derive the parameters from first principles; it takes their definitions as given.

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

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