Fujita's freeness conjecture

Theorem 1.1: for every smooth connected complex projective variety $X$ of dimension $n\ge1$ and every ample line bundle $L$, $K_X+(n+1)L$ is generated by global sections; hence $K_X+mL$ is globally generated for all $m\ge n+1$, even when $L$ itself is not. The bound is sharp. The method minimises an exponential average of normalised section orders over bases and weighted monomial valuations, then lifts sections with Fujita's discrepancy-and-vanishing mechanism. It does not prove the very-ampleness conjecture ($m\ge n+2$) or point separation.

Theorem 1.1: for every smooth connected complex projective variety $X$ of dimension $n\ge1$ and every ample line bundle $L$, $K_X+(n+1)L$ is generated by global sections; hence $K_X+mL$ is globally generated for all $m\ge n+1$, even when $L$ itself is not. The bound is sharp. The method minimises an exponential average of normalised section orders over bases and weighted monomial valuations, then lifts sections with Fujita's discrepancy-and-vanishing mechanism. It does not prove the very-ampleness conjecture ($m\ge n+2$) or point separation.