To tell someone YOU DON'T KNOW to 'read the sources', someone who HAS PRODUCED a body of work based solely on primary sources is the height of dumb condescension. The crux of the matter of simple: where is that mysterious source I haven't read that yields the scientific 2:1 rule? You base your entire argument on something that may exist but that you yourself can't locate.
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Athos needs not apply here.
I have yet to see a single documented argument for the 1:2 ratio, only 'conclusion' that we are all supposed to believe as gospel. There was a thread not long ago about Scottish light cavalry and it was argued that although they undoubtedly existed, there was no battle record of them. The evidence in battle of a Scottish army dominated by archers is non existent.
The Scottichronicon (
https://en.wikipedia.org/wiki/Scotichronicon) has a LOT to say about English archers at Verneuil, but nothing about Scottish archers:
"The number of dead on all sides, English and the others, was 6,000, with as many English killed as were Scots and French. It is said that it was the vain arrogance and reckless haste of the Scots which was the reason for their fall and ruin. When they had four times the number of the English, they rushed against their enemies neglecting the principles of sensible military practice; then after some fighting the English archers were pierced by Scottish spears and as one man took to flight. On observing this the English lancers would for their part have surrendered had it not been that a little before the start of the battle they had learned for certain that the duke of Touraine had had it proclaimed that there would be certain penalties if anyone took Englishmen as prisoners, and that they were to be killed indiscriminately without any [hope of] ransom. It was another misfortune that when a large and fearsome band of Lombards (their war horses protected just with broom, while they themselves were splendidly armed) observed the English archers fleeing from the fight, they quickly forced them to return to the field. These Englishmen, seeing that flight was now out of the question, took their lives in their own hands. They sent such a vast cloud of sharp spears and arrows against their enemies that those at the receiving end were both frightened and distressed. Hence the reluctant turn-around by the archers was an unfortunate turn for the party on the other side, an irreparable misfortune, a deadly destruction." (Book XV)
I have seen it argued that the Scots in France was a heavily archers force because the French needed them to offset the English archery advantage. Yet this is what Thomas Basin (
https://en.wikipedia.org/wiki/Thomas_Basin) wrote :
"The French king assembled for his service a great help from the Scots, who indeed were warlike and strong, but were wont to be too rash and proud. There were about ten thousand combatants, whose chief leaders were the Earl of Douglas and the Earl of Buchan. There was also a large number of French soldiers, but most of them without discipline and military order, and even worse from the point of view of the number of weapons, badly equipped. Their leaders were the duke of Alencon, Jean, count of Aumale and several other barons and captains from the kingdom of France itself. And because the English, fighting with their archers, had often prostrated the French, it was believed that providence would come their aid, if it was opposed to the forces of English archers four hundred or five hundred lances of the Italians. The latter who in arms, as well the men, as well as their horses, being well protected, are in no way afraid of the blows of the arrows, and would burst upon the English foot archers, the shield of their army, and disturb their order with lances." (Rerum Gestarum Temporibus Karoli Septimi)
The French have thought about how to counter the English archers, and their answer wasn't the Scots. If the latter were an archers army in the English mould, nobody noticed on the battlefield!
So the question is one of nomenclature. Does Scottish 'archers' have the same meaning as English 'archers'? Many communal documents undoubtedly recorded a great number of Scottish archers (see OP). They may well be equipped with a bow but are they primarily bowmen? Were their bow mostly used in skirmish/siege/garrison duties, which is 99,99% of the time? In other words, it is a secondary weapon: in battle the Scots were just not trained to use it in mass formation like the English so they reverted to mass shock tactics which they know.