The Recursive Adic Number Field: Construction Analysis and Recursive Depth Transforms

This preprint introduces a new mathematical construction called the Recursive-Adic Number Field, which defines a number system based on recursive structure rather than traditional divisibility.

The core of the paper is a recursive function R(n) that measures how efficiently a number can be built from smaller parts using a process called Recursive Division Tree (RDT). This function behaves differently from standard number-theoretic functions like the number of divisors or prime exponents. Instead, it captures a form of recursive compressibility, or how “deep” a number is when broken down through optimal recursive splits.

From this, the paper builds two related systems:

• A recursive metric on the integers, defining a new kind of distance based on R(n), and leading to an ultrametric completion of the integers.

• A valued field, where numbers are embedded into formal power series and their magnitude is given by recursive depth rather than by primes, as in p-adic numbers.

The paper includes:

• Formal definitions and proofs of the metric, valuation, and structural properties

• A saturation theorem showing that for certain parameters, the recursive depth function levels off at a finite value

• Code implementations in Wolfram Language to compute R(n), generate plots, and visualize recursive trees

• Definitions of recursive Dirichlet and Laplace transforms

• A discussion of complexity, with the main algorithm running in O(n2)O(n^2) time with memoization

• An appendix with a potential machine learning application, using recursive depth to weight attention mechanisms

This work  draws connections to p-adic numbers, non-Archimedean geometry, and formal power series fields.

The construction is independent, original, and designed to explore how recursion can define new notions of magnitude, distance, and structure in number systems.