The Constant κH: Definition, Arithmetic Properties, and Its Relation to C (Paper 19 of the LQG–LQC Intertwiner Series)

IntroductionThe constant C = 3.60329350391382... — studied in depth in Paper 8 [3] — is the maximumvalue of the LQC bounce window function. Paper 8 established its Gamma-function representation, irrationality, and conditional transcendence, and introduced the family C(λ). The present paper gives the same systematic treatment to the companion constant κH =3.44162326276297..., the location at which the maximum occurs. This constant was referred to as κmax in all prior papers of this series (Papers 1–18). Here we treat it systematically as an object of independent interest.