Resolution of the St. Petersburg Paradox via Asymptotic Capital-Growth Optimization

For nearly three centuries the divergence between the St. Petersburg lottery's infinite mathematical expectation and the finite valuation placed on it by rational agents has been treated with subjective modifications of the objective function, beginning with Daniel Bernoulli's logarithmic utility. Utility repairs are ad hoc and, as Menger showed, defeated by a super-St. Petersburg construction that inflates the payoffs. We give an objective resolution that introduces no psychological primitive. Modelling wealth as a repeated multiplicative process and applying the discrete Kelly (growth-optimal) criterion to the lottery offered at price C, we first prove an exact characterization of the growth-optimal fraction, sum_{i>=1} [C(1-f) + f 2^i]^{-1} = 1, together with the solvency (ruin) bound f < C/(C-2). From this identity we derive, by an Euler-Maclaurin evaluation whose half-integer midpoint correction is the exact source of the constant, the closed-form asymptotics f* ~ (C/sqrt2) 2^{-C} and G(f*) ~ 2^{-C}/(sqrt2 ln 2) = f*/(C ln 2), confirmed numerically to four significant figures. The resolution then follows in two parts: (i) because the expectation diverges, the game is favourable at every finite price, so a growth-optimal strategy always exists; and (ii) the maximal growth rate collapses geometrically in the price, so the wealth-doubling time grows like 2^C and any agent with a finite horizon rationally refuses tickets above a modest threshold (C ~ 10-15). No concave utility is required, and none would suffice. We state the relation to the Bernoulli-Kelly-Peters line and to the growth-optimal treatments of Cover-Thomas and MacLean-Thorp-Ziemba, and isolate the contribution as the explicit price-dependent optimum and its role in the two-part resolution.