Title: Structured Subfamilies of Congruent Numbers via Recursive Power Operators and Autoreferential Networks Abstract: This research introduces a transformative approach to the Congruent Number Problem, moving beyond traditional modularity-based detection towards a deterministic constructive framework. The author formalizes the Recursive Power Method (RPM) through the operator Eₖ (x) = x (x-1) P₊-₂ (x) + (x-1) ^k-1, uncovering its canonical block decomposition and its link to the "Oblong Nucleus" n (n-1). The work demonstrates that numerical congruence is not an isolated Diophantine occurrence but an structural property of tuned recursive structures. By establishing a parametric operator Dₙ^ (d) = n (n-1) (dn - d + 1), the research unveils an Autoreferential Network organized around a bidirectional nucleus at parametric order d=2. This network is anchored by rare integral solutions on a Rank 2 elliptic curve (y² = 4n³ - 6n² + 2n + 1), where the mapping y = 2j - 1 bridges the area parameter j with the curve’s geometry. Supported by a computational validation of 450, 000 cases, this preprint provides explicit Euclidean parametrizations and unifies classical figurate families (triangular, square, oblong). It offers a novel research avenue at the intersection of power combinatorics, discrete calculus, and arithmetic geometry, providing a structural explanation for the density and persistence of congruent numbers. This second version (submitted for review in ANTS XVII) represents a revised and optimized iteration of the original research, specifically corrected in scope and format for technical submission. This update shifts the focus from the general development of the Eₖ (x) operator to a specialized analysis of the Dₙ (d) operator, which serves as the fundamental engine for constructing and characterizing the Autoreferential Congruent Network. By restricting the algebraic development to this specific operator, the work provides a more precise description of the 'nucleus' at d=2 and the recursive propagation of solutions within the graph. This version also includes refined data on the constant solution generation rate (^* 0. 152) and the formalization of 'anchor pairs' (n, j), establishing a more direct link between polynomial generators and the structural topology of congruent numbers. Funding and Future Work: The author is an independent researcher currently seeking funding and institutional support to publish these findings in open-access journals and to extend this research program focused primarily on reverse engeneering protocol (from elliptic curves to parametric operators), its theoretical foundations and algorithmic applications.
Francisco Javier Lucero Bravo (Mon,) studied this question.