Abstract

This paper presents a rigorous and philosophically grounded framework for understanding how small differences can be meaningfully articulated without contradicting classical real analysis, particularly the identity 0.999…=1. The thesis develops along three axes: 

• The exactness and completeness of the real number system in which 0.999… equals 1 
• Extended number systems (hyperreals and related frameworks) admitting infinitesimal magnitudes ε≠0 smaller than any reciprocal of a standard natural number, which formalize the notion of "nonzero but negligible" differences 

• A scale-relative metaphysics of identity that reconciles macroscopic sameness with microscopic divergence, showing how infinitesimal or sub-resolution differences can accumulate to produce macroscopic decay. 

  We provide formal statements and proofs, analyze decimal expansions with base-dependence (correcting the misconception that even denominators always yield terminating decimals while odd denominators do not), and include worked physical models (moisture diffusion, surface abrasion, thermal exchange, and digital signal error accumulation) that demonstrate how micro perturbations aggregate over time. The philosophical consequence is that mathematical identity at one layer can coexist with meaningful distinctions at a finer layer, thereby unifying rigor with a theosophy of impermanence.

Keywords

Decimal expansions, Hyperreal numbers, Infinitesimals, Non-standard analyses, Standard-part maps

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