Operational Mathematics Extended to Exterior Differential and Exterior Integral Operators: A Rigorous Theory of Fractional, Real, and Complex Orders of the Exterior Derivative and Its Inverse

This paper systematically extends the framework of Operational Mathematics to the exterior derivative d and the exterior integral I acting on differential forms over a compact Riemannian manifold. The core challenge is the nilpotency of the exterior derivative (d²=0), which renders classical integer-order iteration trivial beyond order one. We overcome this obstruction by recognizing that d simultaneously increments the degree of a differential form by one while acting on its components. This double-tier structure---operation count α and form degree p---enables the construction of a non-trivial analytic semigroup α\α satisfying dαdβ=dα+β on an appropriately constructed extended form space. We establish a complete axiomatic system of seven independent axioms tailored to the exterior differential setting, rigorously accommodating the nilpotency condition as a boundary case of the analytic continuation. Integer-order operations are shown to form an Abelian group on the subspace of exact forms, isomorphic to (Z,+), with the full infinite group structure realized on the extended space. Fractional-order iterations are constructed via three complementary and mutually equivalent methods: (i) the polar decomposition $d=UP$ yielding dα=UαPα via spectral calculus; (ii) an integral representation via the Euler formula dα= Γ(1-α)⁻¹∫₀∞t-αUαPe-tPd\,dt with analytic continuation by Hankel contour deformation; and (iii) a Dirac-type operator approach defining A=d+d* and relating dα to ΠoddAαΠₑᵥₑₙ via explicit spectral factors. Real-order exterior derivatives are constructed through the Hodge--Seeley method, wherein the spectral resolution of the Laplace--Beltrami operator provides the analogue of the Kneser construction for tetration. Uniqueness is established under the spectral regularity condition of commutation with the Laplace--Beltrami operator. The extension to complex orders reveals a rich singularity structure intimately connected to the de Rham cohomology of the manifold: the operator family α\ possesses simple poles at negative integers α=-1,-2,-3,…, and the negative real axis constitutes a natural boundary arising from cohomological obstructions. The Riemann surface of dα is constructed as an infinite-sheeted branched cover of C\-1,-2,…\ with monodromy group Z. We further prove that fractional exterior calculus and fractional variational calculus on manifolds emerge as special cases of this extended theory, fulfilling Axiom~6. The infinitesimal generator L=log d of the semigroup is characterized via spectral theory, and the associated exterior differential flow equation ∂αω(α)=Lω(α) is analyzed. A Hille--Yosida type theorem for the exterior semigroup is proved on the Fr\'echet space Ω(M). A profound duality is established: the additive group (C,+) parametrizing the operation count is categorically equivalent to the group of iteration shifts on the space of exact forms (Theorem~10.1, Theorem~10.3). Moreover, the operators J₊=d, J₋=I, and J₃=N generate a graded Lie algebra whose irreducible modules are two-dimensional on each eigenspace of Δ, with the Casimir element $C=dI+Id$ providing spectral invariants of the manifold. The extended space decomposes as a direct integral of these irreducible modules over the harmonic basis. The full exterior hyperoperation spectrum Σₑₓₜ(α,p)= dα\!_Ωp,• is constructed as an analytic family indexed by complex operation count α and integer form degree p, satisfying a double semigroup property. Numerical algorithms for computing fractional exterior derivatives on triangulated manifolds are developed using spectral Galerkin methods and finite element exterior calculus, with rigorous exponential convergence rates established for analytic forms and polynomial rates for forms of finite Sobolev regularity. The dependence of the convergence constants on the complex order α is explicitly quantified. Applications to mathematical physics are explored: fractional gauge field theory with field strength Fα=dαA+1/2[A,A]α satisfying the fractional Bianchi identity DαFα=0; fractional Yang--Mills equations; a fractional Atiyah--Singer index theorem establishing the invariance of the index under continuous deformation of the order; and fractional Chern--Simons theory on three-manifolds, where metric independence is rigorously proved. Fractional de Rham cohomology is developed using a nilpotent operator dα=dΔ(α-1)/2 and shown to be stably isomorphic to the classical cohomology for all α≠0, with the isomorphism realized by harmonic representatives. A fractional Lefschetz fixed-point theorem, fractional Poincar\'{e} duality, and a fractional Chern--Weil theory are proved. Transcendence results for values of fractional exterior derivatives at algebraic arguments are established for flat tori, with the general case formulated as a precise conjecture. All open problems are either transformed into rigorously proven theorems or explicitly identified as conjectures with supporting evidence and partial results. The paper establishes exterior calculus as a canonical subbranch of Operational Mathematics, fully integrating the nilpotency condition d²=0 as an essential structural feature rather than an obstruction.