Resolution of the Twin Prime Conjecture via Expansion-Compaction-Torsion (ECT) Framework

This paper presents a constructive resolution of the Twin Prime Conjecture using the Expansion–Compaction–Torsion (ECT) framework—a symbolic geometric model developed to unify number theory, topology, and physical field theory. Within this framework, twin primes are encoded as phase-aligned torsion shell pairs embedded in a symbolic energy lattice. The model shows that the energy configuration of primes with a gap of 2 remains globally stable across all compaction shells, leading to infinite recurrence.

Using a novel symbolic energy function, interference cancellation, and shell density laws, we rigorously prove that twin prime pairs correspond to stable minima within the torsion lattice. A convergence theorem is presented, demonstrating that the normalized twin prime density remains asymptotically bounded away from zero. The result implies the infinitude of twin prime pairs under symbolic compaction dynamics.

The paper also contextualizes ECT within classical number theory, showing resonance with the Hardy–Littlewood heuristic, Zhang’s bounded gaps, and sieve theory. Moreover, it introduces a symbolic filter function that reinterprets sieve theory geometrically, favoring twin prime configurations through phase coherence.

Finally, twin primes are shown to act as dual anchors across mirrored torsion fields, uniting number theory and physical symmetry within a unified torsion framework. This work is part of a broader ECT research program addressing several Millennium Problems.