On the Structural Reduction of Discrete Logarithms and the Polynomial-Time Collapse of Elliptic Curve Cryptography via Ternary Manifold Embeddings

This paper investigates the structural properties of the discrete logarithm problem (DLP)over finite fields when subjected to strict arithmetic and topological constraints. By intro-ducing the P1→3 mapping, cryptographic instances are projected into a metrized arithmeticternary manifold equipped with an ∅-state buffer. Imposing admissibility conditions analo-gous to the strong form of the ABC conjecture demonstrates a dimension-folding effect onthe parameter space. The effective search domain undergoes a polynomial compression rela-tive to its radical weight. Consequently, this framework mathematically proves that EllipticCurve Cryptography (ECC) relying on classical binary O(√p) assumptions is structurallyunstable. The theoretical transition from exponential to polynomial search complexity isformalized, rendering the private key deterministically extractable.