Convergence of damped best responses in Tullock contests with no minimum effort

Unrefereed research candidate. In a Tullock (lottery, proportional-prize) contest each player wins with probability equal to its share of total effort and pays a private convex cost. Elkind, Ghosh and Goldberg (2024) proved that simultaneous damped best responses converge when every player must keep effort above a positive floor, and left the case without a floor open. This package proves convergence with no floor: high-cost players may withdraw, costs may be nonsmooth, and marginal cost at zero may vanish. The result covers sufficiently small constant steps (geometric convergence) and every vanishing step sequence with divergent sum, from any finite start with at least two positive efforts. Harmonic steps give a Nikaido–Isoda gap of order 1/k, exactly of that order when a strictly inactive player starts positive. The key new step is that every player's combined rival effort stays above an explicit constant from an explicit time. Corollaries cover concave impact functions (with damping in impact units) and unit-elastic Cournot partial adjustment; local step thresholds 4/n and 8ab/(a+b)2 show that any uniform step must shrink with the number of players and cost heterogeneity. The certified step is extremely conservative. The archive contains the manuscript (PDF and Markdown source), a Python verification program with its result records and figure, a replay driver with claim checks, semantic negative controls, CI and SHA-256 sums. Assurance boundary: written proofs; sampled exact and numerical checks; a producer-side replay from a clean copy; internal AI-assisted review (five reviewer perspectives, an independent-model adversarial review and a verification pass with proof audit). This is not peer review, independent reproduction or formal verification. Creator: Anonymous. Prose, manuscript and data CC0-1.0; code MIT.