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Theoretical analysis demonstrates exact finite balance identities and weighted error bounds in discrete partitions, indicating sharp linear convergence with mesh size.
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.
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Ritsuki Kawada (2026) studied this question.
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