Encyclopedia Constants Constants Planck Scale Matching One Over Sqrt Pi Approx

ARTICLE 2 claims 2 theorems

Constants Planck Scale Matching One Over Sqrt Pi Approx

A machine-checked theorem confirms that 1/√π is close to 0.564, a number that appears when the framework's recognition scale is compared with the Planck length.

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The numerical check

The number 1/√π is about 0.564189. The Recognition Science declaration one_over_sqrt_pi_approx is a machine-checked theorem stating that the absolute difference between 1/√π and 0.564 is less than 0.01. This is a simple numerical fact, not a physical law.

In the framework, this number arises from a specific ratio. The framework defines a recognition wavelength λ_rec_SI as √(ħG/(πc³)). The Planck length ℓ_P is √(ħG/c³). A theorem in the same library proves that λ_rec_SI / ℓ_P = 1/√π. The approximation theorem then simply states that this ratio is close to 0.564.

In Recognition Science, the framework models a discrete record of events, called a ledger. The cost of a recognition event is forced by a proved functional equation. The λ_rec_SI definition is part of an attempt to connect this discrete recognition scale to conventional Planck-scale physics. The π in the definition comes from the constant G and the definition of λ_rec_SI, not from a derived principle.

What the declaration does not claim is more important than what it does. It does not prove that the recognition wavelength equals the Planck length; the ratio is 1/√π, not 1. It does not derive the value of π from the framework. It does not establish any physical measurement. It is an algebraic identity and a numerical bound, nothing more.

THEOREM lambda_rec_over_ell_P · IndisputableMonolith/Constants/PlanckScaleMatching.lean
lambda_rec_over_ell_P · IndisputableMonolith/Constants/PlanckScaleMatching.lean:234 
/-- **THE 0.564 ALGEBRAIC RATIO**:

λ_rec/ℓ_P = 1/√π ≈ 0.564.

This follows from `lambda_rec_SI` and `ell_P`; it is not an independent
derivation of the `π` in `Constants.G`. -/
theorem lambda_rec_over_ell_P :
    lambda_rec_SI / ell_P = 1 / sqrt Real.pi := by
  unfold lambda_rec_SI ell_P
  have hpic3_pos : Real.pi * c^3 > 0 := mul_pos Real.pi_pos (pow_pos c_pos 3)
  have hc3_pos : c^3 > 0 := pow_pos c_pos 3
  have hhG_pos : hbar * G > 0 := mul_pos hbar_pos G_pos
  have hhG_nonneg : hbar * G ≥ 0 := le_of_lt hhG_pos
  have hpi_nonneg : (0 : ℝ) ≤ Real.pi := le_of_lt Real.pi_pos
  rw [sqrt_div hhG_nonneg, sqrt_div hhG_nonneg]
  have h_c3_eq : sqrt (Real.pi * c^3) = sqrt Real.pi * sqrt (c^3) :=
    sqrt_mul hpi_nonneg (c^3)
  rw [h_c3_eq]
  have h_sqrt_c3_ne : sqrt (c^3) ≠ 0 := (sqrt_pos.mpr hc3_pos).ne'
  have h_sqrt_pi_ne : sqrt Real.pi ≠ 0 := (sqrt_pos.mpr Real.pi_pos).ne'
  have h_sqrt_hG_ne : sqrt (hbar * G) ≠ 0 := (sqrt_pos.mpr hhG_pos).ne'
  field_simp [h_sqrt_c3_ne, h_sqrt_pi_ne, h_sqrt_hG_ne]
THEOREM one_over_sqrt_pi_approx · IndisputableMonolith/Constants/PlanckScaleMatching.lean
one_over_sqrt_pi_approx · IndisputableMonolith/Constants/PlanckScaleMatching.lean:257 
/-- **Numerical Value**: 1/√π ≈ 0.564.

The bound `|1/√π - 0.564| < 0.01` follows from `π ∈ (3.13998, 3.14176)`,
hence `√π ∈ (1.7720, 1.7725)` and `1/√π ∈ (0.5641, 0.5644)`. -/
theorem one_over_sqrt_pi_approx : abs (1 / sqrt Real.pi - 0.564) < 0.01 := by
  -- Use Mathlib's tight bounds on π.
  have hpi_lo : (3.14159 : ℝ) < Real.pi := by
    have : (3.141592 : ℝ) < Real.pi := Real.pi_gt_d6
    linarith
  have hpi_hi : Real.pi < (3.14160 : ℝ) := by
    have : Real.pi < (3.141593 : ℝ) := Real.pi_lt_d6
    linarith
  -- 1.7720² = 3.13998400 < π
  have hsq_lo : (1.7720 : ℝ) < Real.sqrt Real.pi := by
    have h : (1.7720 : ℝ) ^ 2 < Real.pi := by nlinarith
    have h0 : (0 : ℝ) ≤ 1.7720 := by norm_num
    exact (Real.lt_sqrt h0).mpr h
  -- 1.7725² = 3.14175625 > π
  have hsq_hi : Real.sqrt Real.pi < (1.7725 : ℝ) := by
    have h : Real.pi < (1.7725 : ℝ) ^ 2 := by nlinarith
    exact (Real.sqrt_lt' (by norm_num)).mpr h
  -- Then 1/√π ∈ (1/1.7725, 1/1.7720) ⊆ (0.5641, 0.5644).
  have hsq_pos : 0 < Real.sqrt Real.pi := Real.sqrt_pos.mpr Real.pi_pos
  have hinv_lo : (1 / 1.7725 : ℝ) < 1 / Real.sqrt Real.pi :=
    one_div_lt_one_div_of_lt hsq_pos hsq_hi
  have hinv_hi : 1 / Real.sqrt Real.pi < 1 / 1.7720 :=
    one_div_lt_one_div_of_lt (by norm_num) hsq_lo
  have h1 : (0.5641 : ℝ) < 1 / Real.sqrt Real.pi := by
    have : (0.5641 : ℝ) < 1 / 1.7725 := by norm_num
    linarith
  have h2 : 1 / Real.sqrt Real.pi < (0.5644 : ℝ) := by
    have : (1 / 1.7720 : ℝ) < 0.5644 := by norm_num
    linarith
  rw [abs_lt]
  constructor <;> linarith

What this page does not claim

The recognition wavelength is not equal to the Planck length. The value of π is not derived from the framework. No physical measurement is established by this declaration.

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/Constants/PlanckScaleMatching.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.

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