The Forgotten Necklace is a harmode accessory that is obtained as a drop from the Lunatic Cultist and his "nightmare" version at a 10% rate in expert and 5% rate in normal.
When your HP is from 100% to 60%:
You gain more life regen, mining speed and movement speed.
The rune will be blue in colour and shoot small projectiles that do not pierce and deal ~60 damage, exploding into small shards that do not pierce.
"It looks calm"
When your HP is from 60% to 15%:
You gain life regen, but less than the previous stage, more movement than before, +5% damage (all) and 5 more defence.
The rune will be red in colour and shoot slightly bigger projectiles that explode on impact and release multiple piercing shards that do not obey gravity.
"It looks enraged"
When your HP is below 15%:
You will become immune to projectile damage, gain even more movement speed, 15% damage (all) and 15 more defence.
The rune will be black in colour and shoot the same projectiles that the Nebula Arcanum shoots.
"It no longer holds its power" (sic)
[Template yet to be made]
The design of the necklace looks like a mix between the N and Re in Hiragana, perhaps Lord AFK used that as a basis?