Noise Geometry Determines the Ito Correction: A Structural Correspondence Between Generators, Quadratic Variation, and Heat Equation

his manuscript establishes a structural uniqueness principle for Itô-type correction terms under a causal derivation–divergence factorization framework. We consider operator triples (D,δ,Π)(D, \delta, \Pi)(D,δ,Π) satisfying a representation of the formF−E[F]=δ(ΠDF)F - \mathbb{E}[F] = \delta(\Pi D F)F−E[F]=δ(ΠDF),where DDD is a closable derivation, Π\PiΠ denotes predictable projection in a Hilbert module, and δ\deltaδ is the adjoint divergence operator. Under this structural assumption, we prove that the finite-variation correction term in Itô-type decompositions is uniquely determined by the underlying projection geometry. In particular, if two decompositions are compatible with the same operator triple and deterministic increasing clock, their correction terms coincide as deterministic signed measures. When the correction is represented by differential operators with continuous coefficients, the coefficients are identified uniquely. The results show that, within this operator framework, the second-order correction is not an arbitrary modeling choice but is forced by the geometry encoded in the causal projection structure. Classical diffusion and Lévy-type generators arise as special cases.