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The geometric symbolic potential for the Feigenbaum period-doubling map has uniformly negative bounded values and exponentially decreasing variations. Potential sums describe logarithmic lengths of odd-address period-doubling intervals up to a uniform additive error. This is a geometric scaling potential, distinct from an unbounded derivative potential at a critical point. Its pressure is continuous, strictly decreasing and convex by the almost-additive partition-function limit and finite-volume inequalities.

The Feigenbaum period-doubling map is quadratic-critical at zero: $g^{\prime }(0)=0$ and $g^{\prime \prime }(0)\ne 0$. Differentiating the renormalization equation once gives $g^{\prime }(x)=g^{\prime }(g(\beta x))g^{\prime }(\beta x)$. Differentiate again and set $x=0$:Cancellation of the nonzero second derivative therefore givesThe quadratic critical-point property matters. The normalization and functional equation alone, without specifying the Feigenbaum solution class, would not justify cancelling $g^{\prime \prime }(0)$.