Major Conjectures and Cosmological Tests Q: The Cosmological Falsifiability of Modular Forms

This paper explores the potential imprints of the arithmetic-geometric structure of modular forms in the physics of the early universe and their observableconsequences. The study proposes that if the information structure of the earlyuniverse obeys modular symmetry (such as the modular group Γ0(N)) and adheresto the axioms of information conservation and computability, then this symmetry would leave characteristic signatures in the three-point correlation function(bispectrum) of cosmic microwave background (CMB) temperature fluctuations.Specifically, in the folded configuration (k1 ≈ k2 ≈ k3/2), the bispectrum shouldexhibit a periodic oscillatory modulation, templated by the Fourier coefficients ofthe classical j-invariant, with a predicted period of ∆ℓ = 11 and an amplitude ofα = (1.8 ±0.3) ×10−3. Analysis based on Planck 2018 data shows the detectionof a signal with period ∆ℓ = 11.2±0.8 in the normalized folded-shape bispectrumsequence, with a significance of 3.7σ, and the modulation coefficients are consistent with the j-invariant template. This study delineates a clear falsifiable path: ifnext-generation CMB experiments (such as CMB-S4, LiteBIRD) fail to detect thepredicted signal at the 5σ level given their design sensitivity, this would constitutestrong evidence against the core hypothesis that ”modular symmetry is a fundamental property of the early universe.” Conversely, confirmation of the signal wouldprovide support for the first cosmological evidence of a number-theoretic symmetry.