The $\infty$-Oreo$^{^\circledR}$

TL;DR

This paper models the self-referential process of creating an infinite Oreo cookie through affine recursions, extending the concept to interconnected foods and classifying infinite food structures.

Contribution

It introduces a mathematical framework for self-referential foods, computes the limit of the infinite Oreo, and generalizes to bi-foods and cyclic recursive structures.

Findings

The limit creme fraction of the $ abla$-Oreo is approximately 95.8%.

A coupled recursion for bi-$ abla$ foods is derived and analyzed.

A classification scheme for $ abla$-foods based on recursive complexity is proposed.

Abstract

What happens when a food product contains a version of itself? The Oreo Loaded -- a cookie whose filling contains real Oreo cookie crumbs -- can be viewed as the result of mixing a Mega Stuf Oreo into a Mega Stuf Oreo. Iterating this process yields a sequence of increasingly self-referential cookies; taking the limit gives the $\infty$-Oreo. We model the iteration as an affine recurrence on the creme fraction of the filling, prove convergence, and compute the limit exactly: the stuf of the $\infty$-Oreo is approximately $95.8\%$~creme and $4.2\%$~wafer. We then extend the framework to pairs of foods that reference each other, deriving a coupled recursion whose fixed point defines a \emph{bi-$\infty$ food}, and illustrate the construction with M\&M Cookies and Crunchy Cookie M\&M's. Finally, we classify $\infty$-foods by the number of foods in the recursion and introduce \emph{homological foods}, whose recursive structure is governed by cycles in a directed graph of commercially available products. We close with a conjecture. All products used in this paper can be purchased at a supermarket.