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We introduce two independent numerator exponents into the Zeilberger–Zudilin integral and specialize them to The resulting integer linear forms in and prove This lowers the Zeilberger–Zudilin upper bound by more than ; the difference between the unrounded bounds is , approximately . The same parameter point is a strict two-dimensional local minimizer of the explicit auxiliary upper-bound function in its admissible arithmetic chamber. This is a local statement about that function, not a claim that the point is a global optimizer among all constructions.

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