A New Constant of Mathematical Physics: Systematic Investigation of the LQC Bounce Normalization C = 3.60329350391… (Paper 8 of the LQG–LQC Intertwiner Series. March 2026)

Introduction In Loop Quantum Cosmology (LQC), the dynamics of cosmological perturbations across the quantum bounce are encoded in the window function Hillard 2026a, Paper I: W () = ³ (²−2) / C · sinh (/2), = √2, (1. 1) where is the dimensionless mode number and C is the normalization constant defined by thecondition that the un-normalized shape function ³ (²−2) /sinh (/2) achieves its maximum C at = ₘax, so that sup W () = 1. The spectral threshold = √2 is derived in Paper 1 from SU (2) Casimir algebra (it is the uniquepositive zero of the numerator). The constant C = 3. 6032935039138199… appeared in Papers I–IV1as a normalization, treated numerically without investigation of its mathematical character. Thepresent paper (Paper 8 of the series) supplies that investigation systematically. Three motivations drive this investigation. First, C is the primary dimensionless output of thebounce geometry beyond the spectral threshold and the Barbero–Immirzi parameter; understanding its character completes the characterization of the bounce. Second, the companion paper (Paper 9, Reciprocal Quantum Dynamics) uses the spectral measure d () = |W () |²d/‖W̃‖² as thePOVM measure of the Consensus Functional, making the structure of W () directly relevant tothe measurement-theoretic framework. Third, the Gamma-function representation discovered inthe course of this investigation — connecting LQC bounce geometry to the Gamma function alongRe (s) = 1 — is a result of independent mathematical interest. Throughout, we use the verified values C = 3. 6032935039138199762726353418 and ₘax =3. 4416232627629710207589257055, each confirmed to 50 significant digits. These are taken asestablished (Section 3 gives the computational verification) ; they are not recomputed here. Note on paper numbering. Papers 6 and 7 of this series are companion papers addressing thealgebraic correspondence between the LQC bounce algebra and the RC Framework consciousnessalgebra (Paper 6), and the empirical falsification protocol for the identification hypothesis (Paper7). The present paper addresses the mathematical character of the normalization constant C; itcan be read independently of Paper 7 but depends on Papers I–V for the definition and derivationof W ().