The Cosmological Constant Correction: From Empirical Mismatch, to How De Sitter Geometry Relocated π from Scale-Setting to Asymmetry, to the ΩΛ Factor

This paper documents an unusual episode in the development of the threshold mechanics framework: the correction of a systematic error in which the mathematical constant π appeared in the equations for the cosmological constant and the fine-structure constant, producing predictions that disagreed with observation. The error was traced to the use of spherical horizon area where de Sitter horizon radius was the correct geometric input. Removing π from both equations simultaneously improved agreement from ~3% to 0.19% for the fine-structure constant and from a factor of ~2.2 to within 2% for the cosmological constant (after further correcting for matter via ΩΛ). What this paper provides: · A documented case study in data-driven theoretical correction. The original derivations of Λ and α⁻¹ contained π in positions that produced numerical disagreement with observation. The error was traced to a geometric misunderstanding: the information bound scales with horizon area, but the cosmological constant depends on horizon radius, not area. In pure de Sitter space, Λ = 3/R_dS² — no π. Removing the spurious π and adding the matter correction ΩΛ improved agreement dramatically: Λ from factor ~2.2 to within 2%, α⁻¹ from ~3% to 0.19%.· The relocation of π. The removed π did not disappear. It found its proper home in the ratio c_eff/d_eff = π/2, a geometric prediction of the framework that follows from the orthogonality of internal axes (θ = π/2) and the chirality primitive (h = 1). This ratio produces a specific waveform asymmetry in threshold-crossing oscillations: T_rise/T_fall = π/2 ≈ 1.5708.· Numerical verification of the π/2 asymmetry. Two colliding Gaussian wave packets on a 48³ lattice form a bound state whose waveform exhibits T_rise/T_fall = 1.568 ± 0.012, in agreement with the predicted π/2 within measurement uncertainty. Additional harmonics show the predicted phase shifts and amplitude suppression (ϕ₁ = 0.698 ± 0.015 rad vs predicted 0.697 rad; |ψ̂₂|/|ψ̂₁| = 0.253 ± 0.008 vs predicted 0.257).· A falsifiable prediction. The π/2 asymmetry is unique to the threshold mechanics framework. Candidate experimental systems where this asymmetry may be observable include nonlinear optical resonators with sign-dependent nonlinearity, Josephson junction plasma oscillations, and three-body escape dynamics near zero total energy. If no threshold-crossing system exhibits the π/2 asymmetry, the framework is falsified.· Why this matters for theory assessment. The correction was not ad hoc. The error was traced to a geometric principle (de Sitter geometry) that applied consistently to two independent equations, improving both. The removed π did not vanish—it was relocated to where the geometry of the framework said it belonged, generating a new falsifiable prediction. This episode illustrates how empirical mismatch can drive theoretical progress in a principled way. Keywords: cosmological constant, fine-structure constant, de Sitter geometry, π correction, ΩΛ, threshold mechanics, waveform asymmetry, falsifiable prediction, empirical correction, canvas geometry