A Formal Refutation of the Hypergeometric Parametric Extension for Reciprocal Binomial Sums

Recent work by Pain [1] proposed a systematic approach to evaluating binomial sums involving reciprocals of binomial coefficients via Beta integrals. In particular, a parametric extension (Proposition 6.1) was introduced and claimed to admit a closed-form representation in terms of a terminating 2F1 hypergeometric function. Through a combination of internal logical consistency checks, integral derivation analysis, and exact symbolic computation, we definitively prove that this parametric identity is false.