Constants Planck Scale Matching

The Planck Ratio

The Planck length is the scale where quantum effects and gravity meet. It is built from the speed of light c, the gravitational constant G, and the reduced Planck constant ℏ. The Planck length ℓ_P is defined as √(ℏG/c³), about 1.6 × 10⁻³⁵ meters. It is a natural unit of length that arises from combining these three constants.

In Recognition Science, the framework models recognition as a discrete record of events with a forced cost. The cost function J(x) = (x + x⁻¹)/2 - 1 is proved in the framework's machine-checked library of formal theorems. At the self-similar scale φ, the golden ratio, this cost gives J_bit = J(φ), which is about 0.118. The framework then defines a curvature cost J_curv(λ) = 2λ². At equilibrium, the bit cost equals the curvature cost, which determines a recognition wavelength λ_rec.

The module proves an algebraic identity connecting this recognition wavelength to the Planck length. It defines λ_rec_SI = √(ℏG/(πc³)). The central theorem, lambda_rec_over_ell_P, proves that λ_rec_SI / ℓ_P = 1/√π. This is an exact algebraic statement in the framework's formal library. The π in the denominator enters through the definition of λ_rec_SI and the framework's constant G, not through a derived theorem about solid angles.

The module also proves supporting facts. The eight octants of a sphere cover the full solid angle 4π, a theorem named octants_cover_sphere. The number 8 matches the eight vertices of a 3-cube, proved in Q3_vertices. These are structural identities in the framework, but no theorem in the file maps the solid angle identity to the source of the π in the Planck ratio. That connection remains a definitional choice, not a proved result.

What this establishes in plain language: within the framework, the recognition wavelength is not an independent scale. It is pinned to the Planck length by a factor of 1/√π, about 0.564. The module certifies this ratio as a theorem. It also certifies that J_bit lies between 0.11 and 0.12, and that the extremum condition uniquely determines λ_rec from J_bit. The framework's library proves these identities exactly, with no numerical approximation in the main ratio.

The consequence is a clean algebraic bridge between the framework's recognition scale and conventional Planck-scale physics. A reader can now see that the framework's λ_rec is not arbitrary; it is locked to ℓ_P by a fixed constant. This does not explain why π appears, but it makes the relationship precise and checkable.

/-- **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]

/-- At λ_rec_from_Jbit, the extremum condition holds. -/
theorem extremum_condition : J_curv lambda_rec_from_Jbit = J_bit_val := by
  unfold J_curv lambda_rec_from_Jbit
  have h : J_bit_val / 2 ≥ 0 := le_of_lt (div_pos J_bit_pos (by norm_num))
  rw [sq_sqrt h]
  ring

/-- Verification: 8 × (π/2) = 4π. -/
theorem octants_cover_sphere :
    (num_octants : ℝ) * solid_angle_per_octant = total_solid_angle := by
  simp [num_octants, solid_angle_per_octant, total_solid_angle]
  ring

/-- The 3-cube Q₃ has 8 vertices (= 8 ticks in the Gray cycle). -/
theorem Q3_vertices : cube_vertices 3 = 8 := rfl

/-- **Numerical Bound**: J_bit ≈ 0.118.
    Since 1.61 < φ < 1.62, we have 0.11 < J_bit < 0.12. -/
theorem J_bit_bounds : 0.11 < J_bit_val ∧ J_bit_val < 0.12 := by
  rw [J_bit_eq_phi_minus]
  constructor
  · have h := phi_gt_onePointSixOne
    linarith
  · have h := phi_lt_onePointSixTwo
    linarith

What this page does not claim

The module does not derive the value of π from recognition principles. The module does not prove that the recognition wavelength has a direct physical measurement. The module does not map the solid angle identity to the SI normalization of the constants.

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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