A fractional de Rham complex for coframe-attached Maxwell equations

Abstract

We develop a coordinate-anchored pointwise theory of fractional tangent functionals on bounded rectangles. By working on the range of the coordinatewise Riemann-Liouville integral, we define anchored fractional operators by exact inversion and prove a representation theorem: every coordinate fractional tangent functional is a scalar multiple of evaluation of the inverse operator at the base point, and the normalized one is unique. At interior points, the resulting fractional tangent space acts faithfully on the common anchored space. We then construct an exterior algebra over the polynomial algebra generated by the fractional coordinate primitives. Its fractional exterior differential defines a nilpotent algebraic chain complex, admits an explicit polynomial Poincare homotopy, and is related to the ordinary polynomial de Rham complex by a positive diagonal rescaling. Since this fractional differential is not a graded derivation for the ordinary wedge product, the global object is a de Rham-type chain complex. We also prove a rigidity theorem for positive-cone-preserving linear coordinate changes preserving the fractional coframe and formulate a Lorentzian coframe-attached Maxwell-type system, deriving charge conservation and fractional wave equations within the polynomial coefficient class.