Constants Planck Scale Matching Lambda Rec Over Ell P

The Planck ratio

The Planck length, about 1.6 × 10⁻³⁵ meters, is the scale at which gravity is expected to blend with quantum mechanics. In the Recognition Science framework, a central question is whether its own recognition wavelength, the distance associated with a single act of recognition, stands in any fixed relation to that scale. The declaration lambda_rec_over_ell_P answers that question with an exact algebraic identity: the ratio of the recognition wavelength to the Planck length equals 1 divided by the square root of π.

The proof is a short chain of definitions. The recognition wavelength is defined in SI units as the square root of ħG/(πc³), where ħ is the reduced Planck constant, G is Newton's gravitational constant, and c is the speed of light. The Planck length is defined as the square root of ħG/c³, the same expression without the π in the denominator. Dividing one by the other cancels every physical constant, leaving only 1/√π. The machine-checked library of formal theorems verifies this cancellation step by step, so the identity holds exactly under its definitions.

The theorem does not derive the π. That factor enters through the definition of the recognition wavelength and through the framework's value for G, which is φ⁵/π in its native units. The docstring is explicit: no theorem in the file maps the solid-angle identity of 4π to the source or SI normalization. The ratio is an algebraic audit of the constants as they currently stand, not a proof that the recognition wavelength must be 1/√π times the Planck length.

In Recognition Science, the framework models the recognition wavelength as the equilibrium point where the cost of a single bit of recognition equals the cost of a curvature packet distributed over the faces of a cube. That equilibrium condition determines the wavelength up to a scale, and the ratio theorem then fixes its relation to the Planck length. The result is a consistency check: the framework's internal constants, chosen for other reasons, happen to place the recognition wavelength at a clean multiple of the Planck scale.

What the declaration changes is the status of the ratio. It is no longer a numerical coincidence observed in passing; it is a proved identity within the framework's definitions. What it does not change is the origin of π. The ratio is a theorem about the constants as defined, and the definition of the recognition wavelength carries the π in by hand. A reader who wants to know why π appears there must look to the definition of G and the choice of SI normalization, not to this theorem.

/-- **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]

/-- **DEFINED PLANCK-RATIO QUANTITY**:

λ_rec = √(ℏG/(πc³)) = ℓ_P / √π

The first equality is this definition. The second is proved algebraically
below. The octant identity above is not used to derive this definition. -/
noncomputable def lambda_rec_SI : ℝ := sqrt (hbar * G / (Real.pi * c^3))

/-- The Planck length ℓ_P = √(ℏG/c³). -/
noncomputable def ell_P : ℝ := sqrt (hbar * G / c^3)

/-- **DEFINED PLANCK-RATIO QUANTITY**:

λ_rec = √(ℏG/(πc³)) = ℓ_P / √π

The first equality is this definition. The second is proved algebraically
below. The octant identity above is not used to derive this definition. -/
noncomputable def lambda_rec_SI : ℝ := sqrt (hbar * G / (Real.pi * c^3))

What this page does not claim

The theorem does not prove that π must appear in the recognition wavelength. The theorem does not establish that the recognition wavelength has any physical reality. The theorem does not derive the value of G or ħ from first principles.

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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