P = NP: A Proof via Logarithmic Fragmentation of NP-Complete Problems

We present a proof that $\mathsf{P} = \mathsf{NP}$ by constructing a logarithmic fragmentation decomposition applicable to every NP-complete language. We show that each fragment is decidable in deterministic polynomial time by brute-force enumeration bounded polynomially through the logarithmic constraint. Since every instance of every NP-complete language falls within some such fragment, the entire language is in $\mathsf{P}$. By the completeness property under polynomial-time reductions, all of $\mathsf{NP}$ collapses into $\mathsf{P}$. A concrete worked example using 3-SAT with $n=2$ variables is provided.