Structural Method for Primality and Factorization From the Polegato positional sieve to the index-based formulation, internal recurrence, and quadratic compression

Abstract This document consolidates, in a single technical narrative, the evolution of the structural method associated with the name Marlon Fernando Polegato. The text begins with the original positional sieve, based on the sequence 12k ± {1,5,7,11} in symmetric ordering, and follows the transformation of that idea into a local index-based formulation capable of deciding compositeness and, when a structural witness exists, extracting factors. The main goal is to explain the logic of the method from the beginning, show how the GMP-based derivations emerged, present the internal identities that organize the search space, and evaluate honestly both the innovative reach and the limitations of the approach. The exposition shows that the original code has the nature of an interval-based structural sieve, because it builds an explicit list of candidates and applies a crossing-out procedure by position inside a nonstandard ordering. The three derived methods then cease to operate as global sieves and begin to act on a single integer n, searching for compositeness witnesses in closed families parameterized by u and v. The first derivation presents exact scanning with a structural bound; the second preserves the same mathematics but reorganizes the loop through internal recurrence; the third adds compression through quadratic forms, interpreting the witness found as an additional algebraic certificate. The main conclusion is that the most plausible innovation does not lie in the simple fact that a sieve can find factors, since that is already known in several families of algorithms, but rather in the bridge between a bespoke positional sieve and an index algebra that rewrites the multiplication of candidates coprime to 6. Once demonstrated rigorously, that bridge provides an original narrative: from positional pattern to local operator, from local operator to structural compositeness test, and from there to quadratic compression. The text also makes explicit, without grandstanding, that the current formulation still does not remove the general square-root-type barrier in the worst case, which places it in a conceptually promising position, though still distinct from the state of the art in general factorization of large integers.