one_over_sqrt_pi_approx
plain-language theorem explainer
The declaration bounds the absolute difference between 1 over square root of pi and 0.564 by 0.01. Researchers deriving the recognition wavelength from the ledger-curvature extremum in Recognition Science cite this bound to link 8-face geometry to the Planck length. The proof applies interval arithmetic in one step to confirm the inequality.
Claim. $|1/√π - 0.564| < 0.01$
background
In the PlanckScaleMatching module the recognition wavelength λ_rec is obtained by setting bit cost J_bit = J(φ) = cosh(ln φ) - 1 equal to curvature cost J_curv(λ) = 2λ² at equilibrium. This fixes λ_rec in RS-native units where ell0 = 1. Restoring dimensions via the 8-face average over the Q3 hypercube introduces the factor 1/π, yielding λ_rec = ℓ_P / √π ≈ 0.564 ℓ_P.
proof idea
The proof is a one-line wrapper that invokes the interval tactic to perform verified numerical computation confirming the bound holds.
why it matters
This numerical check anchors the parent derivation of λ_rec from the curvature extremum argument and the Planck gate identity π ħ G = c³ λ_rec². It fills the step from face-averaging geometry to the ratio λ_rec = ℓ_P / √π inside the eight-tick cycle and Conjecture C8. No open questions are directly addressed.
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