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

<i>Complete version of this work is available </i>in full on Zenodo: https://zenodo.org/records/15662383, DOI: 10.5281/zenodo.15662383.<i> Author's contact: [e-mail].</i>Summary introducing a symbolic identity I call the <b>Universal Balance Law for Power Sums</b>.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:<b>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 </b><b>m</b><b>?</b><br>With additional constraints:<br>– all numbers must be distinct (no repetitions),<br>– both blocks contain the same number of subsequences,<br>– and both have the same total number of terms.<b>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 </b><b>m</b><b>?</b><br>Constraints:<br>– all numbers are distinct,<br>– both blocks contain the same number of terms.<b>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 </b><b>m</b><b>?</b><br>Again requiring:<br>– all numbers are distinct,<br>– equal number of subsequences on both sides,<br>– and equal number of total terms.The <b>Universal Balance Law</b> deterministically constructs solutions to each of these problems — not just one, but infinitely many for every natural power.<br><i>Complete version of this work is available </i>in full on Zenodo: https://zenodo.org/records/15662383, DOI: 10.5281/zenodo.15662383.<i>"This material is shared for non-commercial, academic purposes only. No derivatives are permitted without author consent."</i><br>