The Twelve-Color Case of the Erdős–Gyárfás Balanced-Coloring

A balanced r-edge-colouring of a complete graph is an edge-colouring with r colours such that every set of r+1 vertices spans at least one edge of each colour. The Erdős–Gyárfás balanced-colouring conjecture asserts that, for every integer r ≥ 3, the complete graph K_(r²+1) has no such colouring. This preprint studies the fixed twelve-colour case, corresponding to K₁₄₅, and develops a structural approach based on minimum-degree reductions, induced-core bounds, support-set classifications, equality-case analysis, and repivot arguments. It presents the current proof architecture and the associated computational verification framework. This manuscript is a preprint and has not undergone formal peer review. Independent mathematical review and Lean formalization are ongoing. Readers are welcome to examine the arguments, reproduce the computational checks, and provide mathematical comments or corrections.