lambda_rec_pos
plain-language theorem explainer
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
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.
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