The Mirror Harmony Mod-9 Law: Residue Classification of Brahmam Mirror Numbers

This paper develops the concept of Brahmam Mirror Numbers (BMNs), a class of integers defined through a mirror-product operation in which a number is multiplied by its digit-reversed counterpart. When this product forms a decimal palindrome, the number is classified as a BMN. This operation connects digit reversal, reflective symmetry, and multiplicative structure, creating an interesting foundation for studying digit-based transformations. In this work, we identify a central modular identity governing these behaviours, referred to as the Mirror Harmony Mod-9 Law. Through digit-sum properties and congruence arguments, we show that for any integer, the mirror product is always congruent to the square of the number modulo nine. As a result, every mirror product must lie within the set of quadratic residues modulo nine: {0, 1, 4, 7}. This restriction holds universally and remains valid whether or not the number itself is a BMN. To examine the frequency and structure of BMNs, we conducted a complete computational scan of integers up to 200 million and identified 1246 BMNs. These findings confirm their rarity and demonstrate the residue pattern predicted by the modular identity. Additional observations highlight how mirror products behave under mod-11 alternating-digit rules, suggesting deeper interactions between digit symmetry and modular behaviour. We also explore a polynomial interpretation in which digit strings are treated as self-reciprocal forms, offering an algebraic viewpoint on palindromic mirror products. Overall, the results presented here provide a unified framework for understanding Brahmam Mirror Numbers, combining modular theory, computational evidence, and structural analysis. This study opens pathways for further research in digit-based number theory, density heuristics, modular classification, and potential applications in reflective arithmetic and digital computation.