Finite Balance Identities and Weighted Error Bounds

Abstract

Telescoping and the triangle inequality determine how much local information survives in a finite total. The note proves exact finite balance identities, characterizes endpoint equality and strictness under nonnegative losses, and derives weighted local-to-global error bounds. For quadratic local-width errors it retains the nonuniform sum of squared widths before obtaining a sharp bound linear in the common mesh size.