Building on the recently published framework of advanced tensor theories—where tensors are rigorously defined via standard and fractional derivatives—this work extends the formalism to foundational mathematics, fractional calculus, and fractional geometry. We introduce these advanced tensors as generalized objects that unify differential structures across integer and non-integer orders, enabling seamless interpolation between classical and fractional regimes.

Central to the presentation is the construction of transformation laws for mathematical objects under advanced tensor actions, revealing that possibly novel algebraic and geometric invariants are preserved across fractional dimensions. These transformations are shown to induce natural fractional differential geometries on manifolds, yielding fractional manifold–metric pairs with curvature expressions involving Caputo–Fabrizio or Riemann–Liouville operators.

Real-world applications are explored in depth: modeling anomalous diffusion in heterogeneous media, designing fractional physical and engineering systems with memory- and environment-dependent dynamics, and developing scale-invariant image processing algorithms using fractional tensor convolutions. We further demonstrate how advanced tensors facilitate multi-scale physics simulations—bridging quantum microstates to macroscopic continuum behavior—through dimensionally hybrid tensor fields.

By establishing rigorous links between abstract fractional structures and actionable mathematical, physical, and engineering paradigms, this study positions advanced foundational tensors as a unifying language for next-generation mathematical modeling of sophisticated and complex systems, such as comology, gravity, and high-energy physics.