Every Nontrivial Finite MRB Partial Sum Is Irrational Lebsir Version

Abstract

Let SN= summation symbol (-1)Nnn=1(n1/n-1) denote the Nth partial sum of the defining series for the MRB constant. We prove that every nontrivial finite partial sum is an irrational algebraic number: SN element of Q (nAngle brackets2). The proof is entirely finite and algebraic. The proof uses the vanishing field trace of each radical n1/n, together with the exact contribution of the rational terms in the alternating sum. We also interpret the fractional-part representation SN= summation symbol(-1)Nnn=1 Fractional Part (n1/n) within Habib Lebsir’s congruence framework modulo 1/n. The MRB sequence corresponds to the diagonal choice x=n in Lebsir’s family of partitions, for which [n1/n] = 1 (nAngle brackets2).Thus, rationality never occurs at a finite stage of the defining MRB process. This does not determine the arithmetic nature of the infinite MRB constant. Still, it shows that any rationality of the limit, were it to occur, could arise only through the infinite limiting process and not through a finite algebraic cancellation among its defining radicals.