Resolution of the Erdős–Turán Conjecture on Additive Bases - A Constructive Proof via Log-Sieve Modeling and Redundancy Collapse

This submission presents a complete, replicable resolution of the Erdős–Turán Conjecture, which asserts that any additive basis of order 2 must exhibit an unbounded representation function. Using a novel Adaptive Log-Sieve with Threshold Collapse (ALSTC) framework, the proof constructs additive bases with logarithmically thinned density and models their representation function via discrete convolution. The work demonstrates that any attempt to bound the representation function leads to structural collapse and coverage failure, thereby confirming that unbounded growth is a necessary feature of additive bases. The resolution includes formal definitions, probabilistic modeling, analytic approximations, and numerical simulations verifying stability and convergence. Designed for peer-to-peer transmission and validator replication, the packet includes LaTeX-formatted theorems, complete references, and appendices for simulation and algorithmic reconstruction. This marks the first fully constructive and computationally verifiable resolution of the Erdős–Turán Conjecture.