Constants Planck Scale Matching Lambda Rec From Jbit Pos

The recognition wavelength

The Planck length, about 1.6 × 10⁻³⁵ meters, is the scale at which quantum effects of gravity become strong. Recognition Science (RS) proposes that this scale emerges from a balance between two kinds of cost. The first cost, J_bit, is the cost of a single bit of recognition, evaluated at the golden ratio φ ≈ 1.618. The second cost, J_curv, is a curvature cost. The framework models this as a quadratic function of a length λ, written J_curv(λ) = 2λ². Setting these two costs equal, J_bit = J_curv(λ), determines a specific length, which the framework calls the recognition wavelength λ_rec.

The declaration lambda_rec_from_Jbit_pos is a theorem in the framework's machine-checked library of formal theorems. It proves that this recognition wavelength, defined as λ_rec = √(J_bit / 2), is strictly greater than zero. This is a necessary sanity check: a length scale must be positive to be physically meaningful. The theorem follows from the fact that J_bit itself is positive, which is proved separately. The proof is a direct consequence of the definitions and the positivity of the golden ratio.

In Recognition Science, this positive length is then connected to the Planck scale. The framework defines a separate quantity, λ_rec_SI, with units, by the formula λ_rec_SI = √(ℏG/(πc³)). A separate theorem in the same module proves that the ratio of this quantity to the Planck length ℓ_P is exactly 1/√π. This is the algebraic identity that the framework uses to argue that its recognition wavelength matches the Planck scale up to a factor of √π.

What this specific theorem does not claim is just as important. It does not prove that the recognition wavelength equals the Planck length; the ratio 1/√π is a separate result. It does not derive the value of π or the constants ℏ, G, and c; those enter through definitions. It does not establish the physical interpretation of the recognition wavelength or its connection to actual measurements. The theorem is a precise, narrow statement about the positivity of a defined quantity, a building block in a larger framework, not a standalone physical prediction.

/-- λ_rec_from_Jbit > 0 since J_bit > 0. -/
theorem lambda_rec_from_Jbit_pos : lambda_rec_from_Jbit > 0 := by
  unfold lambda_rec_from_Jbit
  exact sqrt_pos.mpr (div_pos J_bit_pos (by norm_num : (2 : ℝ) > 0))

/-- λ_rec_from_Jbit > 0 since J_bit > 0. -/
theorem lambda_rec_from_Jbit_pos : lambda_rec_from_Jbit > 0 := by
  unfold lambda_rec_from_Jbit
  exact sqrt_pos.mpr (div_pos J_bit_pos (by norm_num : (2 : ℝ) > 0))

/-- **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

This theorem does not prove that the recognition wavelength equals the Planck length. This theorem does not derive the values of π, ℏ, G, or c. This theorem does not establish the physical reality of the recognition wavelength.

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: