Structured Subfamilies of Congruent Numbers via Recursive Power Operators and Autoreferential Networks

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 E_k(x) = x(x-1)P_{k-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 D_n^{(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 (y^2 = 4n^3 - 6n^2 + 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. 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.