1 Introduction Loop Quantum Cosmology replaces the classical big-bang singularity with a quantum bounce.In the effective spectral description, perturbations propagating through the bounce experience awindow functionW(κ)= κ3(κ2−2)Csinh(πκ/2) , κ≥√2. (1)The normalization constantC=3.60329... is determined by the unique maximizer κmax , and thestationary condition takes the form5κ2−6=κ(κ2−2) π2 coth πκ2 . (2)Viewed directly, (2) appears to be an isolated transcendental extremum equation.The purposeof this paper is to show that it actually sits inside a global genus-one geometry.Ourstarting point is the observation that the stationary relation admits a projective M¨obiusform in exponential variables. This leads to a reduced algebraic branch law whose discriminantdefines aquartic curve. Thequartic turns out tohave genus one. After birational reductionto a cubic and then to Weierstrass form, the stationary branch is described using the standardelliptic uniformization by ℘. The resulting decomposition reveals two distinct but compatibleglobal structures: aM¨obius lawi n the exponential sector, andagenus-oneAbelian lawinthereconstruction sector.We then show that the natural second-kind and third-kind objects associated with the stationarybranch fit a Baker–Akhiezer interpretation. This provides a unified geometric framework for thehidden structure behind the bounce spectral constant.
Hillard et al. (Thu,) studied this question.