A certified feedback-timing contrast in a logit duopoly

This technical note compares deterministic projected current-profit-gradient price updates in a two-firm multinomial logit market with an outside option, net qualities 3/2, markup box [1/2,3]^2 and fixed gains (261/50,259/50). An integer-interval certificate proves that the fresh-quote Gauss-Seidel map sends every initial state in this box into an explicit contraction region within 60 blocks and then converges to the unique Nash markups (3/2,3/2), with weighted contraction rate 199/200. At the same gains, the stale-quote Jacobi map has an interior attracting two-cycle with an open local basin. Changing holding-time proportions on that fixed cycle reverses mean profit across firms relative to Nash while mean markup remains unchanged and consumer and total surplus remain lower. The source archive includes the proof programs and saved finite certificates. The note specifies its differences from the read Guo et al. author manuscript and Conjeaud et al. preprint; exact overlap with the key model and payoff definitions of Askar (2020) and the 2012/2013 versions of Anufriev, Kopanyi and Tuinstra remains unresolved.