FOLLOW-THE-REGULARIZER-LEADER ROUTES TO CHAOS IN ROUTING GAMES

The paper proves chaos for sufficiently large a via a clean period-3 criterion (and supplies stability and coexistence results) without asserting an explicit universal threshold; see the derivative test (b attracting iff a in (0, -2Ψ′(b))) and Theorem 7.1/Cor. 7.2 on period-3 and chaos for large a . By contrast, the model’s key step—claiming a 2-horseshoe on a neighborhood of b for every a > -2Ψ′(b)—is incorrect: for a strictly decreasing map near b with f(b)=b, each side maps onto only one half of the neighborhood, so no two subintervals can each map onto the whole J (the model even incorrectly asserts J ⊂ f([b−δ,b]) ∩ f([b,b+δ])). Hence the claimed full two-shift conjugacy and immediate entropy bound log 2 do not follow.