The inter-universal Teichm\"uller theory and new Diophantine results over the rational numbers. I

By applying inter-universal Teichm\"uller theory and its slight modification over the rational number field, we prove new Diophantine results towards effective abc inequalities and the generalized Fermat equations. For coprime integers $a, b, c$ satisfying $a + b = c$ and log(|abc|) ≥ 700, we prove log|abc| ≤ 3(abc) + 8√log|abc| · loglog|abc|. This implies for any 0 < ε ≤ 1/10, |abc| ≤ maxexp(400 · ε⁻² · log(ε⁻¹)),\ rad(abc)3+3ε\, reducing the constant in effective abc bounds from 1.7 · 10³⁰ (Mochizuki-Fesenko-Hoshi-Minamide-Porowski) to $400$. For positive primitive solutions $(x, y, z)$ to the generalized Fermat equation xʳ + yˢ = zᵗ (r, s, t ≥ 3), define h = log(xʳ yˢ zᵗ). We prove explicit bounds: {gather*} h ≤ 573\ \ (r, s, t ≥ 8); \; h ≤ 907\ \ (r, s, t ≥ 5); \; h ≤ 2283\ \ (r, s, t ≥ 4); \\ h ≤ 14750\ \ (min\{r, s\} ≥ 4\ or\ t ≥ 4); \; h ≤ 24626\ \ (r, s, t ≥ 3). {gather*} These imply Fermat's Last Theorem (FLT) holds unconditionally for prime exponents ≥ 11. Combined with classical results for FLT with exponents $3, 4, 5, 7$, this yields a new alternative proof of FLT. Computational verification confirms no non-trivial primitive solution exists when r, s, t ≥ 20 or $(r, s, t)$ is a permutation of $(3, 3, n)$ (n ≥ 3).