3Kn vs 4Kn

Actually I think it has something to do with the performance of 3Kn and 4Kn during the Crusades.
The Byzantine princess Anna Comnena (1083 to 1153) mentions in her history that ‘Frankish’ Knights
“could charge through the very walls of Constantinople”.
Obviously an exaggeration, but she is trying to convey the superiority of the Western Knights to her
own ponderous heavy Byzantine Cataphracts.

Writing a set of Ancient Wargaming rules is like trying to assemble a jigsaw puzzle with many of the
pieces missing, so we would be fools to ignore any juicy little titbits like this that give us insight into
how those ancient warriors actually did behave and perform.

The next time you ask a question - and no I won't hold my breath - the answer is '42'.

All I can say Tony is “I didn’t write the army lists”.

“The native kataphraktoi were to be found in the imperial oikos, some imperial guards units and the personal guards of generals,
 but the largest numbers were found within the provincial tagmata. The level of military effectiveness, especially the quality of the
 armour and mount, of the individual provincial kataphraktos probably varied considerably, as both John II and Manuel I are recorded
 as employing formations of “picked lancers” who were taken from their parent units and combined. This approach may have been
 adopted in order to re-create the concentration of very effective heavy cavalry represented by the Imperial Tagmata of former times.
 The kataphraktoi were the most heavily armoured type of Byzantine soldier and a wealthy kataphraktos could be very well armoured.
 The Alexiad relates that when the emperor Alexios was simultaneously thrust at from both flanks by lance wielding Norman knights his
 armour was so effective that he suffered no serious injury.
 In the reign of Alexios I (1081-1118) the Byzantine kataphraktoi proved to be unable to withstand the charge of Norman knights, and
 Alexios, in his later campaigns, was forced to use stratagems which were aimed at avoiding the exposure of his heavy cavalry to such
 a charge. Contemporary Byzantine armour was probably more effective than that of Western Europe therefore reasons other than a
 deficit in armour protection must be sought for the poor performance of the Byzantine cavalry. It is probable that the Byzantine heavy
 cavalry traditionally made charges at relatively slow speed, certainly the deep wedge formations described in Nikephoros II Phokas’ day
 would have been impossible to deploy at anything faster than a round trot. In the course of the late 11th century the Normans, and other
 Westerners, evolved a disciplined charge at high speed which developed great impetus, and it is this which outclassed the Byzantines.
 The role of the couched lance technique, and the connected development of the high-cantled war saddle, in this process is obscure but
 may have had considerable influence.”
(Source: en.wikipedia.org/wiki/Byzantine_army_(Komnenian_era)#Cavalry )