A candidate proof of the Han–Xiong 1/2-conjecture

Abstract

We present a candidate proof that the integer-exponent trace of the fractional Gaussian binomial coefficient is coefficientwise largest at r = 1/2, for every positive rational parameter r and every positive integer lower index k. An exact trace expansion reduces the comparison to r = 1/(2m). Same-index pairing leaves explicit deficits, which are bounded by adjacent even-index summands. These pairs are disjoint for m ≥ 3; a weighted identity handles their overlap for m = 2. A counting estimate covers the upper index boundary, and denominator comparisons reduce the remaining range to three rational kernels with denominator (q;q)₄. The classical restricted-partition formula gives a cubic approximation with an explicit linear error bound, yielding uniform coefficient bounds for all three kernels. The proof uses no finite parameter cutoff or symbolic case enumeration as a premise.