Height Functions and Yang-Baxter Inequalities for Octad Weights in M24 A Computational Framework Formalized in Lean 4

We develop a height function theory for octad weights in the Mathieu group M24 and binary Golay code G24, fully formalized and verified in Lean 4. Our central result is a Yang-Baxter-type inequality: h(gcd(m, n)) ≤ min(h(m), h(n)) for all 0 < m, n ≤ 24. This is accompanied by 48 theorems, all verified computationally with zero axioms or sorry statements. We introduce a discrete height on Golay weights W = {0, 8, 12, 16, 24}, establishing that distinct weights are separated by at least 4 / 3. Additionally, we formalize an Iwasawa-style identity (Theorem 4) showing that the height function preserves logarithmic multiplication. We discuss connections to representation theory and p-adic ideas as directions for future work.