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I'll get more into the question's details, but the reason I ask this is the Kensei Monk. Kensei monks cannot use Heavy weapons, which results in them being unable to use all Two-Handed weapons, but the Double-Bladed Scimitar is a two-handed weapon that is not heavy.
Since a Double Bladed Scimitar usually does 2d4 damage, if a level 5 monk replaced the die using Martial Arts to a d6, would the Double Scimitar change into 2d6 damage or 1d6 damage?
The monk's Martial Arts ability says (PHB, p. 78; emphasis mine):
You can roll a d4 in place of the normal damage of your unarmed strike or monk weapon. This die changes as you gain monk levels, as shown in the Martial Arts column of the Monk table.
The normal damage of a Double Bladed Scimitar is 2d4, which you can choose to replace with 1d4 (1d6, 1d8, or 1d10, as you level up) if you want. Or rather, you could, if the Double Bladed Scimitar did not have the special property (as V2Blast pointed out). Way of the Kensei allows you to use a weapon with the two-handed property, but still prevents you from using one that has the special property.
Putting the special property aside for a moment, each die you roll for damage from a weapon is referred to as a "weapon damage die". For example, the barbarian's Brutal Critical feature (PHB, p. 49; emphasis mine):
Beginning at 9th level, you can roll one additional weapon damage die when determining the extra damage for a critical hit with a melee attack.
If Martial Arts was intended to apply to each weapon damage die individually, we could expect it to be worded something like this:
<fake> You can roll a d4 in place of the normal damage of your unarmed strike or each weapon damage die of your monk weapon. This die changes as you gain monk levels, as shown in the Martial Arts column of the Monk table. </fake>
The description of the monk's Martial Arts feature says, in part (emphasis mine):
At 1st level, your practice of martial arts gives you mastery of combat styles that use unarmed strikes and monk weapons, which are shortswords and any simple melee weapons that don’t have the two-handed or heavy property.
You gain the following benefits while you are unarmed or wielding only monk weapons and you aren’t wearing armor or wielding a shield:
- You can use Dexterity instead of Strength for the attack and damage rolls of your unarmed strikes and monk weapons.
- You can roll a d4 in place of the normal damage of your unarmed strike or monk weapon. This die changes as you gain monk levels, as shown in the Martial Arts column of the Monk table.
- [...]
As it says, you can roll a single d4 as your damage die (changing to a d6, d8, etc. as you level up). It does not keep the number of your existing weapon; it just replaces it. (The druid cantrip shillelagh works the same way.) This is true regardless of the weapon's original damage dice or number of damage dice.
Red Orca's answer further elaborates on this fact.
The "Kensei Weapons" potion of the Way of the Kensei's Path of the Kensei feature expands the list of weapons that count as monk weapons for you (XGtE, p. 34; emphasis mine):
Kensei Weapons. Choose two types of weapons to be your kensei weapons: one melee weapon and one ranged weapon. Each of these weapons can be any simple or martial weapon that lacks the heavy and special properties. The longbow is also a valid choice. You gain proficiency with these weapons if you don’t already have it. Weapons of the chosen types are monk weapons for you.
Unfortunately, the double-bladed scimitar (E:RftLW, p. 21; WGtE, p. 73-74) doesn't qualify as a kensei weapon, because it has the "special" property (in addition to the "two-handed" property), and describes it below:
Special. When you take the [A]ttack action and make a two-handed attack with a double-bladed scimitar, you can use a bonus action to make a melee attack with the blade at the opposite end of the weapon. This attack uses the same ability modifier as the primary attack. The weapon’s damage die for this attack is a d4, and it deals slashing damage.
Because it has the special property, the double-bladed scimitar can not be chosen as a kensei weapon. As such, the Martial Arts feature doesn't apply to it (nor do any other features that are tied to monk weapons).
