Constants Planck Scale Matching Planck Gate Identity

The Planck gate identity

The Planck gate identity is a theorem in the Recognition Science framework's machine-checked library of formal theorems. It states that π times the framework's reduced Planck constant ℏ times its gravitational constant G equals the speed of light c cubed times the square of a recognition wavelength λ_rec. In symbols: π ℏ G = c³ λ_rec². The declaration proves this equality holds exactly, given the definitions of those constants inside the framework.

The identity is best understood as an algebraic audit. It checks that the framework's internally defined constants, which include ℏ = φ⁻⁵ and G = φ⁵/π, are mutually consistent under this particular product. The proof unfolds the definitions and simplifies, showing the equality is a consequence of how the constants are set up. The name "gate" suggests a checkpoint: the identity certifies that the constants pass a consistency test, not that they are measured or derived from experiment.

What the identity does not claim is equally important. It does not derive the value of π from recognition principles; the factor π enters through the definition of G and through the definition of λ_rec_SI, as the framework's own documentation states. It does not assert that the recognition wavelength equals the Planck length ℓ_P. A separate theorem in the same file proves that λ_rec_SI / ℓ_P = 1/√π, which is a ratio, not an equality. And the identity does not map the solid-angle fact that eight octants cover 4π steradians to any physical normalization; that remains an open identification.

In plain terms, the Planck gate identity is a bookkeeping check. It confirms that the framework's constants hang together algebraically. The reader should take it as a statement about internal consistency, not as a physical law connecting recognition to the Planck scale.

/-- In RS-native units where c = ℓ₀ = τ₀ = 1, λ_rec = ell0 = 1.
    The physical content is the relationship λ_rec/ℓ_P = 1/√π.

    The Planck gate identity: π · ℏ · G = c³ · λ_rec². -/
theorem planck_gate_identity :
    Real.pi * hbar * G = c^3 * lambda_rec^2 := by
  unfold G lambda_rec hbar c ell0 cLagLock tau0 tick
  simp only [one_pow, mul_one]
  have hpi : Real.pi ≠ 0 := Real.pi_pos.ne'
  have hphi5 : phi ^ (-(5 : ℝ)) ≠ 0 := (Real.rpow_pos_of_pos phi_pos _).ne'
  field_simp [hpi, hphi5]

/-- In RS-native units where c = ℓ₀ = τ₀ = 1, λ_rec = ell0 = 1.
    The physical content is the relationship λ_rec/ℓ_P = 1/√π.

    The Planck gate identity: π · ℏ · G = c³ · λ_rec². -/
theorem planck_gate_identity :
    Real.pi * hbar * G = c^3 * lambda_rec^2 := by
  unfold G lambda_rec hbar c ell0 cLagLock tau0 tick
  simp only [one_pow, mul_one]
  have hpi : Real.pi ≠ 0 := Real.pi_pos.ne'
  have hphi5 : phi ^ (-(5 : ℝ)) ≠ 0 := (Real.rpow_pos_of_pos phi_pos _).ne'
  field_simp [hpi, hphi5]

/-- **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]

What this page does not claim

The identity does not derive the Planck scale from recognition principles. The identity does not assert that the recognition wavelength equals the Planck length. The identity does not map the solid-angle fact to any physical normalization.

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.

Derived articles

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