Part I: Exact Algebraic Identification of K3 String Vacua: Resolving Fuzzy Dark Matter Tensions via Automated Phenomenological Sieving

This preprint formalizes the exact algebraic identification of <strong>K3 String Vacua</strong> as a solution to current tensions in the Fuzzy Dark Matter (FDM) paradigm, particularly concerning the James Webb Space Telescope (JWST) high-redshift galaxy mass observations. By employing the <em>SocrateAI Scientific Agora</em> ecosystem and adhering to strict "Zero Simulation Flottante" and Lean 4 mechanical verification rules, we demonstrate that a mass-varying axion, driven by the expansion of the topological $T^2$ torus in a $K3 \times T^2$ compactification, naturally yields a ~19% heavier initial dark matter mass at the epoch of recombination ($z = 1100$). This mathematically and empirically resolves the anomalously dense galaxies observed by JWST without requiring modifications to standard $\Lambda$CDM structural formation timelines. <strong>Note:</strong> All proofs are fully verified in the Lean 4 kernel with zero unverified axioms or <code>sorry</code> stubs. Phenomenological alignments are backed by raw physical execution logs utilizing Astroquery/VizieR data.