this section is an average taken from an ensemble of at least $16$ test dictionaries, with more test dictionaries generated if necessary so as to keep the relative standard deviation in the aggregate quantities below $5\%$. In the rest of this section all data plotted is summed over the entire discovery process, from first symbol to last.

Fake random dictionaries: “null” or N-dictionaries
--------------------------------------------------

These “null” dictionary are not real dictionaries – no word list exists, just a nominal word count of $\DcSize$ and a nominal number of symbols $\SySize$. They are defined solely as an algorithm intended to test the normalization of the $\delta$ quantities, and to provide a baseline of comparison that is unaffected by any specifics of a dictionary generation algorithm. If such an N-dictionary is interrogated, it returns information that the dictionary size is scaled in proportion to the number of known symbols (i.e. that $\DcSsub = \SySsub \DcSize / \SySize$, and that the allowed symbols have randomly ordered frequencies. Note that any re-interrogation of an N-dictionary will return a different randomised set of frequencies.

Fig. \[fig-null-scaling\] compares the innovation measures for two symbol discovery possibilities: (a) discovery in order of frequency (or usefulness) in the 