lambda_rec_from_Jbit_pos
The theorem establishes that the recognition wavelength obtained by solving the bit-cost extremum equation is strictly positive. Researchers deriving Planck-scale quantities from the Recognition Science ledger would cite this to confirm the wavelength scale is well-defined before matching to the Planck length. The proof is a one-line term-mode application of the square root positivity lemma after unfolding the definition and invoking the known positivity of J_bit.
claimThe recognition wavelength defined by $λ_{rec} = √(J_{bit}/2)$ satisfies $λ_{rec} > 0$.
background
In the Planck-scale matching module the bit cost is obtained from the unique cost functional J(x) = ½(x + x⁻¹) - 1 evaluated at the self-similar fixed point φ, giving J_bit = cosh(ln φ) - 1. The recognition wavelength is extracted from the extremum condition J_bit = J_curv(λ) where J_curv(λ) = 2λ², solved explicitly as λ_rec_from_Jbit := √(J_bit_val / 2). This step belongs to the derivation chain for Conjecture C8 that links the discrete ledger cost to continuous curvature cost and ultimately to λ_rec ≈ 0.564 ℓ_P after restoring SI units and averaging over the eight faces of the Q₃ hypercube.
proof idea
The proof is a one-line term-mode wrapper. It unfolds the definition of lambda_rec_from_Jbit to expose the square-root expression, then applies sqrt_pos.mpr to the ratio of the positive J_bit value over the positive constant 2, using norm_num to discharge the inequality 2 > 0.
why it matters in Recognition Science
This result supplies the positivity step required before the numerical matching λ_rec = √(ℏG/(π c³)) = ℓ_P / √π in the module's derivation of Conjecture C8. It relies on the upstream J_bit_pos lemmas that trace positivity to φ > 1 and the forcing-chain fixed point (T6). The declaration therefore anchors the physical scale in the Recognition Science framework before the eight-tick octave and D = 3 geometry are restored.
scope and limits
- Does not derive the numerical value of λ_rec in SI units.
- Does not prove the extremum equality J_bit = 2λ².
- Does not incorporate the 1/π factor from face averaging.
- Does not connect to the coherence quantum or alpha band.
formal statement (Lean)
151theorem lambda_rec_from_Jbit_pos : lambda_rec_from_Jbit > 0 := by
proof body
Term-mode proof.
152 unfold lambda_rec_from_Jbit
153 exact sqrt_pos.mpr (div_pos J_bit_pos (by norm_num : (2 : ℝ) > 0))
154
155/-- At λ_rec_from_Jbit, the extremum condition holds. -/