of innovation in ${\mathsf{C}}^{1024}_{\SySize,\pFfork}$ chain dictionaries, with symbol number $\SySize$ varying on a log scale from $2$ to $32$, and fork probability $\pFfork$ from $1/11$ to $10/11$.\
[]{data-label="fig-chain-parscaling"}](Chain00000-Nonl-par "fig:"){width="\plotthreewidth"}

Fig. \[fig-chain-scaling\] compares the innovation measures for two symbol discovery possibilities: (a) discovery the frequency-order of symbols in the world dictionary, or, (d) discovery in random-order. The two sets of results here span a range of symbol list and dictionary sizes, but are constrained by a fixed word fork probability $\pFfork=10\%$. For low word counts, and large enough symbol lists, some symbols might not be chosen for use in words in the dictionary; the average number of unused symbols is indicated by the floating contours on the figure.

There is no strong trend with dictionary size, but in a new feature not present for the previous dictionary types, discovery order now plays a very strong role: discovery in frequency-order shows the expected innovation increase with $\SySize$, whereas for random-order discovery the trend is weak (for $\delta_r$), or negligible (for $\delta_\omega, \delta_\chi$). Once might, for example, even hazard a comparison between the behaviour of F-dictionary behaviour and this for the random-order discovery; despite the fact that the frequency-order discovery looks more like 