Constants Planck Scale Matching Planck Gate Normalized

The normalized Planck gate

The Planck length is the scale at which quantum effects and gravity meet, roughly 1.6 × 10⁻³⁵ meters. In the Recognition Science framework, a declaration named planck_gate_normalized establishes a precise algebraic relationship between this scale and the framework's own recognition wavelength. The identity states that c³ times the square of the recognition wavelength, divided by pi times hbar times G, equals exactly 1. Here c is the speed of light, hbar is the reduced Planck constant, and G is Newton's gravitational constant.

This identity is a theorem in the framework's machine-checked library of formal theorems. It is derived from the framework's definitions of its constants, including hbar = φ⁻⁵ and G = φ⁵/π, where φ is the golden ratio. The proof unfolds these definitions and simplifies the resulting expression. The theorem is a consequence of how the framework sets up its constants, not an independent physical measurement.

The declaration also connects to a separate ratio: the recognition wavelength in SI units divided by the Planck length equals 1/√π. This second identity is also a theorem in the same module. The framework's docstring notes that the factor of π in this ratio enters through the definition of G and the recognition wavelength, and that no theorem in the file maps the angular identity of 4π solid angle to the source or SI normalization.

In plain terms, the normalized Planck gate expresses that the framework's recognition wavelength, when converted to SI units, sits at a fixed fraction of the Planck length determined by π. The declaration does not claim that this ratio is derived from first principles within the framework. It is an algebraic audit of the constants as they are currently defined. The framework's own documentation states that the module proves exact algebraic identities used in the Planck-scale matching discussion, but does not independently derive the π in the recognition-to-Planck ratio.

/-- Equivalent form: c³λ²/(πℏG) = 1. -/
theorem planck_gate_normalized :
    c^3 * lambda_rec^2 / (Real.pi * hbar * G) = 1 := by
  have h := planck_gate_identity
  have hne : Real.pi * hbar * G ≠ 0 := by
    apply mul_ne_zero
    apply mul_ne_zero
    · exact Real.pi_pos.ne'
    · exact hbar_pos.ne'
    · exact G_pos.ne'
  rw [div_eq_one_iff_eq hne]
  exact h.symm

/-- **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 factor of π in the Planck ratio is not derived from the framework's axioms. The theorem does not map the 4π solid angle identity to any source or SI normalization. The declaration does not independently derive the values of hbar and G 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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