Fractional-Residue_Encoding_of_Integers_via_D-Type_Divisions

For version 1.1 - Version 1.1 adds worked numerical examples illustrating the fractional-residue encoding and explicit integer reconstruction using D-type divisions.New material includes a detailed example with divisor sets D= 2,3,5,7 a varsity-style reconstruction exercise, and an explicit averaged reconstruction identity.No changes have been made to the core definitions or theoretical results; this update is intended solely to improve clarity, reproducibility, and exposition. Fractional-Residue Encoding of Integers via D-Type Divisions We introduce a fractional-residue encoding of integers based on the fractional parts of division by a fixed finite set of divisors. For an integer N and a divisor set D, this encoding represents N as a vector of fractional components, forming a continuous arithmetic signature. Our main result proves that this fractional-residue vector uniquely determines integers modulo lcm(D) . Thus, fractional parts alone encode complete modular information, establishing a precise correspondence between continuous fractional data and discrete modular structure. We further show that vanishing fractional components provide a necessary compositeness detector relative to the chosen divisor set, while a simple weighted identity reconstructs integers exactly. Although the framework does not generate primes or test primality, it offers a novel representation of integers that bridges fractional arithmetic and modular congruence. This perspective provides a foundation for experimental number theory, geometric visualization of residue classes, and alternative encodings of arithmetic structure.