New SubmissionQuantum Gravity Flux Modularity and Explicit Exact solution

We found closed continuous solution families of quantum gravity flux associated with the Quantum Speed Limit, which construct a theoretical framework of modern quantum gravity modularity theory. We present invariance of elliptic curves: j(E(λ))= 27λ³((8- λ³)/ (1+ λ³))³. We compute the Quantum Speed Limit = 4, the Universe Dilation rate Hubble one-parameter H₀ = √3 -1 = 0.7320..., more important, we compute the dark energy Ω_Λ= 3 + 2√3 = 6.464101615... or 65% of the Universe energy content is in a form which cannot associated to ordinary matter. The recent physical measurement proved Ω_Λ ≈ 68.5 % ( see Planck Collaboration et al., 2013).

One of the crowning achievements of twentieth century mathematics was the modularity theorem, which implies that every elliptic curve defined over Q can associated to a special automorphic form called a weight-two eigenform, more recently, the modularity has been extended to higher-dimensional Calabi-Yau three folds. Our results will extent these mathematical achievements to  theoretical physics and quantum physics, which mean that the quantum gravity flux in the Universe is also modular, gave interesting evidence - in the form of j-invariant.

This study casts quantum gravity flux modularity theory, which visually deeply carves all dynamical behaviour of quantum gravity flux in the Universe.

The classical Newton's experimental physics will enter into an One-Parameter quantum flux modularity time. The Quantum Speed Limit 4 and the Hubble parameter H₀= √3 -1 are fundamental modular parameters, which originate from the reduced Planck parameter h= 2π√3; the Boltzmann parameter k_B = 8 √3; gcd(h, k_B) = 2√3, finally lead to H₀²= 4 - 2√3  = QSL - gcd(h, k_B). We also present a complete invariant solution image of j(E(λ)) = 0, λ∈ (0,2); j(E(λ)) = 1728, λ= √3-1, where the Quantum Speed Limit =4 is a maximal value point. The new quantum modularity theory links the Number Theory, Analysis, Geometry and Quantum Gravity theory, closely.