Integroiduilles

Integroiduilles are a theoretical construct in mathematics and computational modelling used to describe discrete representations of integral accumulation. In this view, a system's continuous integral behavior is approximated by a collection of localized units embedded in a lattice.

The term blends elements of integration and tiling, and the concept appears in speculative literature on discrete calculus and digital physics. It is presented as an abstraction rather than an empirically observed phenomenon.

Conceptually, each integroiduile carries a value related to a local contribution to a global integral and occupies a position on a regular or irregular grid. Values are updated by local rules that transfer and combine contributions from neighboring units, aiming to converge toward the integral of a scalar field as the grid is refined.

Key properties include locality of interaction, scalability with grid resolution, and the ability to encode boundary conditions within the unit rules. The framework emphasizes convergence in the limit of finer discretization and offers a way to visualize integration as a distributed process.

Applications are largely theoretical or computational, including models of diffusion, resource flow in networks, and explorations of how continuous operations can be approximated in discrete systems. It is sometimes used in thought experiments about the computability of calculus on finite structures.

Reception among researchers is mixed. Proponents cite intuitive clarity and compatibility with lattice-based methods; critics point to the lack of unique definitions for update rules and to incomplete convergence guarantees. See also: discrete calculus, cellular automata, lattice gas automata.