The Remainder Program

The modulus/remainder decomposition is not a calculation technique. It is the structural principle of the soliton hierarchy. At every level of the hierarchy — from sub-femtometer QED loops to the Hubble-scale cosmic budget — physical quantities decompose into two components. The modulus carries the geometry: β = π/4, the L1/L2 spherical conversion, setting boundaries and determining when transitions occur. The remainder carries the inertia: the specific content that resists state change, accumulates with depth, and drives transitions when it exceeds the modulus boundary. In framework language, mass is inertia. Stored energy is inertia. Entropy is inertia. Coupling strength is inertia. These are not competing descriptions — they are different measurement angles on the same quantity. The remainder is this quantity, computed at each hierarchy level. The paper tests this claim at seven levels by computing the decomposition and comparing to measurements. Each level either supports or weakens the claim. Negative results are reported alongside positive ones.