Mage Archon Vyr's Build
Storm Armor Shocking Aspect
Arcane Torrent Static Discharge
Black Hole Absolute Zero
Archon Slow Time
Slow Time Time Warp
Magic Weapon Deflection
The only Legendary Gem that is default included in all variants is Bane of the Trapped (BotT), which is a nice separate multiplier and has some great synergies with Vyr-Chantodo. Specifically the Cold Wave of Destruction applies a slow before the damage instance is calculated, which means that
BotT has permanent uptime on all targets; this means that it's not a necessity to be within 15 yards of an enemy.
For Solo Greater Rift Pushing, Bane of the Stricken must be included to kill Greater Rift Guardians within a reasonable timeframe. As a rule of thumb, one should incorporate Stricken once Rift Guardian fights take longer than two minutes.
The third viable damage option is Zei's Stone of Vengeance, which, in contrary to its description, grants its first stage benefit at 0 yards and not 10 yards.
Bane of the Powerful is an acceptable addition for Speed Greater Rifts or Nephalem Rifts, however, it's not recommended for pushing variants due to its subpar uptime.
For Nephalem Rifts, Boon of the Hoarder is always a default include specifically due to its amazing synergy with Avarice Band.
Lastly, Gogok of Swiftness is a good choice at season start due to Vyr-Chantodo's significant dependency on Cooldown Reduction; however, Gogok should be rotated out for the other options once enough CDR is acquired as long as you're not playing the Elemental Swap variant.
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The idea of the build revolves around entering and re-entering Archon as soon as possible to build up Archon stacks and overlap them with The Swami. In Greater Rifts you want to click Archon as soon as possible, make sure to activate Storm Armor and Magic Weapon first though, without caring too much for building up Chantodo's Resolve stacks. You are very squishy in your first rotation, and each time you drop out of Archon, which is when Slow Time as well as Black Hole come in real handy as defensive abilities.
After your first rotation you should be fairly comfortable and tanky meaning it should be fairly easy to get up to 20 Chantodo's Resolve stacks without the feel of being threatened. Keep in mind to repeatedly press Button 1 while in Archon(600ms cooldown) due to it granting additional stacks via the Vyr's Amazing Arcana (6) Bonus as well as some decent shielding via Magic Weapon.