Universal Balance Law for Power Sums — Complete Monograph with Examples and Applications

Complete version of this work is available in full on Zenodo: https://zenodo.org/records/15662383, DOI: 10.5281/zenodo.15662383. Author's contact: [e-mail] introducing a symbolic identity I call the Universal Balance Law for Power Sums.This identity constructs symmetric sequences of numbers (natural, real, or complex) whose power sums cancel — either exactly or in the limit. The construction is based on recursive mirror-symmetric patterns that eliminate terms globally rather than by local pairings. It applies uniformly for all integer powers, and even for non-integer ones (in the case of convergence to zero). The method may introduce a novel mechanism within number theory and symbolic discrete mathematics.The equilibrium law answers the following core questions:Can we construct two equilibrium blocks of numbers, composed of subsequences of consecutive natural numbers of varying lengths, such that their power sums are equal for any natural power m?With additional constraints:– all numbers must be distinct (no repetitions),– both blocks contain the same number of subsequences,– and both have the same total number of terms.Can we balance any complex number with a set of other numbers so that both sides form blocks with equal power sums for any natural m?Constraints:– all numbers are distinct,– both blocks contain the same number of terms.Can we systematically balance any finite sequence of numbers with a set of other sequences of the same lengths and structure, ensuring equal power sums for all natural m?Again requiring:– all numbers are distinct,– equal number of subsequences on both sides,– and equal number of total terms.The Universal Balance Law deterministically constructs solutions to each of these problems — not just one, but infinitely many for every natural power.Complete version of this work is available in full on Zenodo: https://zenodo.org/records/15662383, DOI: 10.5281/zenodo.15662383."This material is shared for non-commercial, academic purposes only. No derivatives are permitted without author consent."