Mirror-Pair-Function, A Number Theoretical Exploration

A finite, exact approach to digit symmetry and self-similarity with implications for the foundations of mathematics, stochastics, and digital signal processing. Every finite natural number other than one-digit numbers has a Mirror-Pair. If a number is plugged into the function M_b(x), it becomes its Mirror-Pair; if its Mirror-Pair is plugged back into the function, it returns to the original value, establishing an involution with z_2 symmetry.

Zero and infinity are excluded as numerical pairs because they function as conceptual Mirror-Pairs in a separate domain (M_conc(x)).
In different bases, the spread in the x and y directions shifts relative to the respective base. In base 10, a Mirror-Pair always appears in transitions from n+1 in 10^(n+1), mapping structural bounds across expanding orders of magnitude. The function illustrates how humans perceive numbers through cognitive scaffolding, collapsing singular entities into fractional baselines. When plotted on a log-log scale, self-similarity reappears dynamically at any scale, proving that bound-fractals can be fully generated and explained finitistically without relying on infinite sets.

This work naturally relates to the Lychrel-numbers.
The structural precision of this framework owes a significant debt to Professor Norman J. Wildberger. Upon reviewing an early iteration of the manuscript, Wildberger critiqued its initial depth, urging a rigorous further development of the underlying discrete definitions. This push directly catalyzed the subsequent algebraic refinement of the system within strict time constraints, centering the math on strict finitism and discrete z_2 symmetry. To verify this plane-bound fractal behavior, the D_0 Hausdorff dimensions were computationally analyzed via the box-counting method, demonstrating stable scaling towards a theoretical limit of 2.0 (achieving D_0 Hausdorff = 1.7236 at N = 50 × 10^6 with a highly consistent mean R^2 of 0.999647 ± 0.00004). Crucially, this updated framework bridges pure number theory directly into applied engineering by addressing the classic bit-reversal permutation bottleneck in Fast Fourier Transforms. By formalizing string-reversal as an explicit algebraic difference operation, the positional inversion is integrated directly into the phase rotation of a Discrete Fourier Transform (DFT). In specific test environments, this unified algebraic method removes the need for memory-heavy lookup tables and reduces bit-reversal execution time ten-fold, though results may vary across alternative computational environments. This offers a high-utility optimization pathway for telecommunication architectures.