The Bounce Holographic Screen: Deriving Λ = e − V4 from LQG Area Spectrum and the Birman–Krein Spectral Completion (Paper 11 of the LQG–LQC Intertwiner Series)

1 Introduction The 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.In the companion paper 3, the formula Λ = e−V4 with V4 = 280 was derived conditional on theholographic hypothesis:Λ =e−V4, V4= SBHℓ4Pl ,(1)where V4 is the effective four-volume of the bounce causal diamond in Planck units. Two steps inthat paper were proved (the Mellin–Tangent correspondence and Levinson’s theorem); the holographic identification was labelled a physical hypothesis.The present paper closes that hypothesis. We prove that W(κ)/C — the LQC bounce windowfunction derived from SU(2) intertwiner geometry in Paper I 1 — is precisely the holographicpuncture density of the bounce causal screen in the sense of the Bekenstein–Hawking formula inloop quantum gravity. Combined with the already-proved Mellin–Tangent and Levinson results,this makes V4 = SBH = 280 a derived theorem.