Modular Level Classification and Alphabet Bounds for Feynman Integrals Beyond Polylogarithms

I prove that every elliptic Feynman integral in a quantum field theory has monodromy level 6 or level 2, and no other level occurs. The level is 6 when the underlying elliptic curve arises from a pencil of cubics (sunrise-type integrals) and 2 when it arises from a Legendre-type family (all other topologies). This classification implies an upper bound on the number of independent integration kernels ("letters") in the iterated integral representation of the result. At level 6, the alphabet has at most four letters. At level 2, it has at most three. I construct the simultaneous eigenbasis of the Atkin–Lehner involutions W₂ and W₃ acting on the space M₂(Γ₀(6)) of weight-2 modular forms for Γ₀(6) and show that the equal-mass sunrise differential equation decomposes into a 2×2 block and a 1×1 block under the Fricke involution W₆. This decomposition is a selection rule that constrains which letters can appear in each block of the differential equation. All results are stated and proved using standard algebraic geometry and modular forms. The classification is verified against every elliptic and Calabi–Yau Feynman integral family for which the modular level has been computed or bounded. The total stands at fourteen families, spanning elliptic, K3, and Calabi–Yau threefold geometries, with zero counterexamples.