A Unified System of Tensor Invariants and Their Integral Reduction Theorems

This paper organizes the complete operator set and the integral theorems of classical vector analysis into a single unified system. The starting point: every differential operator is an invariant slice of one object — the derivative tensor of the field — and every integral theorem is one rung of a single dimensional ladder. On the first floor stands the operator quadruple div–rot–dev–jad (the irreducible decomposition of the derivative tensor, 9=1+3+5 in components), held together by jet, the full second-order tensor invariant; each carries an exact, first-order integral reduction theorem, valid in identical form in R³, R⁴ and R⁵. The second floor is built from compositions: the Hessian as jet∘grad, the Laplacian as div∘jet, the Green identities and the Monge-Ampere theorems. Converse theorems (Poincare lemmas, Helmholtz decomposition) sharpen the system into equivalence criteria. As a highlighted application, under the w=ict embedding the jad operator applied to the four-potential yields the electromagnetic field tensor: the homogeneous Maxwell equations and the flux theorem arise as identities, the wave equation and Poynting's theorem as compositions — within this system the sourcelessness of the magnetic field is not a postulate but a theorem. The paper includes the complete R⁵ invariant table, jump (boundary-condition) and transport extensions, worked verification examples, the Kaluza-Klein tower of the five-potential, and six research directions. The theorems of the system were also verified symbolically for arbitrary (general polynomial) fields: the published R⁴/R⁵ invariant tables term by term, the ladder theorems from R³ to R⁵, the Piola family and the null identities — 23 checks, all passed.