This framework scaffolds spectral geometry, operator theory, higher-dimensional and fractal spacetime structures, recursive algebra, and topological invariants—all aiming to capture a deeply recursive and self-modulating cosmos. In many areas of mathematics and physics, the concept of an annihilator—commonly used in affine subspace or dual spaces—captures the idea of “zeroing out” a structure. In our work, we propose an extension of this idea via the Inverse Zero Operator (IZO), a projection operating kernel that recursively extracts the stable, invariant core (the fixed-point) from a given operator sequence. Rather than simply annihilating a structure, the IZO acts as a memory-extracting mechanism, preserving the recursive influence inherent in the system. This paper introduces the IZO, formalizes its properties, and demonstrates its utility through several examples. Our goal is to provide both a rigorous mathematical foundation and an accessible exposition for further exploration in mathematical physics.