Three I11 Reformulations of the Schrödinger Equation: Coordinate-Normalized, Mirror-Centered, and Fractal-Discrete Dynamics

Three complementary reformulations of the time dependent Schrodinger equation within a structured interval framework called I11, defined on the finite range from zero to eleven. The first formulation introduces a minimal coordinate normalization that embeds standard quantum dynamics into the I11 interval without changing the underlying continuum operator structure. The second formulation expresses the same dynamics in a mirror centered representation around the midpoint 5.5, allowing quantum states to be classified by mirror symmetry. The third formulation replaces the continuous spatial second derivative by a scale dependent, non uniform discrete operator derived from an I11 fragment partition, leading to a k level Hamiltonian suitable for multiscale and resolution dependent modeling. Together, these three variants form a coherent hierarchy from simple coordinate embedding to structurally modified quantum dynamics, with potential applications in numerical quantum mechanics and adaptive simulations.