lambda_rec_SI
The definition supplies the recognition wavelength in SI units as the square root of ħG over πc³. Researchers deriving Planck-scale relations in Recognition Science cite this to obtain the exact 0.564 factor relative to the Planck length. It is a direct algebraic definition obtained by restoring SI dimensions to the extremum condition that equates bit cost to curvature cost on the Q3 hypercube.
claimThe recognition wavelength in SI units is defined by $λ_{rec} = √(ℏG/(πc³))$.
background
In the Planck-Scale Matching module the recognition wavelength is obtained by equating the bit cost J_bit = J(φ) = cosh(ln φ) - 1 to the curvature cost J_curv(λ) = 2λ² that arises from a ±4 curvature packet distributed over the eight faces of the Q3 hypercube. The upstream constants supply G = λ_rec² c³/(π ℏ) from the Constants module and ℏ = cLagLock · τ₀ from the same module; the module document states that face-averaging over the eight-face geometry introduces the factor 1/π. The derivation chain therefore restores SI dimensions to the equilibrium condition J_bit = J_curv(λ_rec) and yields the stated square-root expression.
proof idea
This is a direct definition that unfolds the square-root expression involving the constants hbar, G, pi, and c. No lemmas are applied; the body is a single noncomputable assignment that serves as the base for the positivity theorem and the ratio theorem that follow it.
why it matters in Recognition Science
This definition completes the derivation chain for Conjecture C8 by supplying the SI form of λ_rec. It is used by the theorem lambda_rec_over_ell_P to establish λ_rec/ℓ_P = 1/√π ≈ 0.564 and by the PlanckScaleMatchingCert structure that certifies the full chain from J_bit_val > 0 through the extremum condition. In the Recognition framework it realizes the face-averaging principle that connects the unique cost functional J to the Planck scale and the eight-tick octave geometry.
scope and limits
- Does not derive the values of G or ℏ from scratch.
- Does not prove the ratio λ_rec/ℓ_P = 1/√π; that equality is established in a separate theorem.
- Does not address numerical approximations or higher-order corrections to the curvature cost.
- Does not incorporate quantum corrections beyond the leading J-cost extremum.
formal statement (Lean)
208noncomputable def lambda_rec_SI : ℝ := sqrt (hbar * G / (Real.pi * c^3))
proof body
Definition body.
209
210/-- λ_rec_SI > 0. -/