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, on p. 13, 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.
An orthogonality test first confirmed that the method mathematically works as described, meaning that the energy levels before and after the transformation were perfectly identical according to Parseval's theorem. When comparing execution times on a controlled dataset of 5000 points, our sequential memory access was 38 times faster. As we scaled up the number of datapoints to 1 million points, the algorithm was tested to be 68000 times faster. At our computational limit, testing of 16 million points, the algorithm achieved a result, running 284481 times faster than traditional memory shuffling. Of course this varies with the tests used, code written and test-environment used. But it is applicable in others as well. Our calculations, based on known annual values of power usage in diffrent fields, it could, globally, contribute to save several gigawatt hours electricity annually. Being a realistic and positive contribution to how we all together tackle climate change, while maintaing modern living-standards as our power demands are increasing in other sectors. We acknowledge that this will not happen tomorrow, or in a year. But if researchers enable the algorithms implementation, something we as independent thinkers can't, we collectively are able to do so in 10 to maybe 15 years. A goal far more important than the paper itself.