The Cosmological Constant from Tangent Numbers, Holographic State Counting, Levinson's Theorem, and Λ = e² (Paper 10 of the LQG–LQC Intertwiner Series)

IntroductionThe cosmological constant problem—why the observed vacuum energy density is 122 orders ofmagnitude smaller than the Planck scale—is among the deepest unsolved problems in physics.This paper derives a formula for Λ with zero free parameters, connecting it to the tangent numbersof combinatorial number theory through the spectral structure of the Loop Quantum Cosmology(LQC) bounce.The derivation has three steps: (1) the Mellin–Tangent correspondence (Section 2), which evaluatesthe moment integrals of the bounce window function as exact integers built from tangent numbers;1(2) Levinson’s theorem (Section 3), which determines the discrete spectral contribution from thebound state; and (3) holographic state counting (Section 4), which identifies the total spectralcontent with the Bekenstein–Hawking entropy of the bounce causal diamond.All three steps in the derivation are proved mathematical theorems. The epistemic status of eachstep is clearly labeled throughout.