Title Bismuth Calculus: A Domain-Centric Integral, the Bismuth Series, and Curvature–Geometry Coupling Description This repository introduces Bismuth calculus, a novel domain–centric integral framework together with its associated expansion, the Bismuth series. The central object is the Bismuth integral, defined via locally symmetric (cell-wise) averaging rather than global symmetry of the integration domain. Unlike classical Riemann or Lebesgue integration, the Bismuth integral incorporates an intrinsic symmetrization at the level of the integrator itself, leading to systematic cancellation of odd-order contributions independently of the domain’s geometry. A distinguished feature of the theory is the definition of the Bismuth centroid, which serves as the natural expansion center for the calculus. Around this point, the Bismuth series is constructed as a scalar expansion of the integral value, not of the function itself. The coefficients of the series are given by derivatives of the integrand evaluated at the centroid, coupled with central Bismuth moments of the domain. A fundamental structural result is the vanishing of all first-order Bismuth moments, implying exactness of the Bismuth integral for affine functions and forcing the first nontrivial correction to appear at second order. Consequently, the leading correction term involves a curvature–geometry coupling: the Hessian of the integrand at the centroid interacts with a second-order Bismuth moment matrix of the domain. This feature sharply distinguishes Bismuth calculus from classical moment methods, where moments are purely geometric and independent of the integrand’s differential structure. The theory establishes: A rigorous definition of the Bismuth integral via symmetric partitions. Linearity and centroid-based structure. Exactness of the Bismuth series for polynomials of bounded degree. A quadratic (second-order) Bismuth approximation driven by the Hessian–moment interaction. A conceptual separation between Bismuth moments and classical geometric moments. Overall, Bismuth calculus provides a new integral formalism in which symmetry is internal, odd-order effects vanish structurally, and domain geometry influences integration only through interaction with curvature. The framework is self-contained and does not rely on classical quadrature rules or stochastic interpretations, though it exhibits conceptual connections to symmetric cubature and variance-reduction ideas.
Tsuff Bismuth (Sun,) studied this question.