lambda_rec_over_ell_P
plain-language theorem explainer
The theorem establishes that the SI recognition wavelength divided by the Planck length equals one over the square root of pi, yielding the numerical factor 0.564. Researchers deriving fundamental constants from recognition cost functionals in discrete informational models would cite this as the explicit outcome of Conjecture C8. The proof is a direct algebraic reduction that unfolds the definitions of lambda_rec_SI and ell_P, invokes positivity lemmas for the constants, and simplifies via field operations.
Claim. The ratio of the recognition wavelength in SI units to the Planck length satisfies $λ_{rec,SI}/ℓ_P = 1/√π$.
background
In the Planck-Scale Matching module the recognition wavelength arises from setting the bit cost J_bit = J(φ) = cosh(ln φ) - 1 equal to the curvature cost J_curv(λ) = 2λ² at equilibrium. Restoring SI dimensions via c³λ²/(ℏG) and averaging over the eight faces of the Q₃ hypercube introduces the factor 1/π, producing λ_rec = √(ℏG/(π c³)). The upstream constants are taken from IndisputableMonolith.Constants: c is the speed of light in RS-native units (voxel/tick), ħ = φ^{-5}, and G = λ_rec² c³/(π ħ) by definition. Positivity lemmas c_pos, hbar_pos, and G_pos guarantee that all square-root arguments remain positive.
proof idea
The term-mode proof unfolds lambda_rec_SI and ell_P, then records positivity of π c³, c³, ħ G, and π. It rewrites each sqrt of a quotient with sqrt_div, factors sqrt(π c³) via sqrt_mul, confirms the three square roots are nonzero with sqrt_pos.mpr, and finishes with field_simp.
why it matters
This supplies the exact Planck ratio required by Conjecture C8 and is referenced directly in the planck_scale_matching_cert definition as the planck_ratio field. Within the Recognition Science framework it realizes the face-averaging step that converts the eight-face Q₃ geometry into the observed 1/√π factor, closing the derivation from the J-cost functional through the eight-tick octave to the constant ratio.
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