lambda_rec_pos
The lemma establishes that the fundamental recognition wavelength λ_rec is strictly positive in RS-native units. Researchers deriving constants and area quantization results in the Recognition Science framework cite it to justify positivity in expressions for G and minimal eigenvalues. The proof is a one-line term-mode simplification that unfolds the definition λ_rec := ℓ₀ and relies on prior positivity of the base length.
claim$0 < λ_{rec}$, where $λ_{rec} = ℓ_0$ denotes the fundamental recognition length in RS-native units.
background
The Constants module expresses physical constants from the Recognition Science forcing chain, identifying the recognition wavelength λ_rec with the base length ℓ₀ that serves as yardstick for the phi-ladder. The module doc sets the native time quantum τ₀ = 1 tick. Upstream, Bridge.DataCore defines λ_rec = √(ħ G / c³) and proves positivity under physical assumptions that ħ, G, and c are positive; the Constants definition of G then reads G = λ_rec² c³ / (π ħ), while ħ itself equals φ^{-5}.
proof idea
This is a term-mode proof consisting of a single simp tactic applied to the definition of lambda_rec, which reduces the claim directly to the known positivity of ell0.
why it matters in Recognition Science
The result is invoked to prove G_pos and to establish uniqueness of the positive root in lambda_rec_is_forced. It also supplies the positivity hypothesis for the minimal area eigenvalue theorem, where the lowest non-zero eigenvalue equals ℓ₀². Within the framework it secures the sign condition required for the T5 J-uniqueness step, the T6 phi fixed point, and the T7 eight-tick octave that fixes D = 3.
scope and limits
- Does not derive a numerical value for λ_rec in SI units.
- Does not prove the G = λ_rec² c³ / (π ħ) relation without the separate definition of G.
- Does not address possible sign changes under non-physical assumptions on the bridge data.
formal statement (Lean)
427lemma lambda_rec_pos : 0 < lambda_rec := by
proof body
Term-mode proof.
428 simp [lambda_rec]
429
430/-- Newton's gravitational constant G derived from first principles (RS-native form).
431 \(G = \lambda_{\text{rec}}^2 c^3 / (\pi \hbar)\). -/