Phantom Edge
An equipment class parallel to weapons. Phantom Edges can be Tempered and upgraded; Summon-type ones call an ally.
Mechanics
Phantom Edges always appear alongside weapons in the achievements: 10 weapons Tempered and 10 Phantom Edges Tempered are two separate achievements, and a weapon at Level 30 and a Phantom Edge at Level 30 are another two. The gear system can therefore be read as two parallel upgrade lines — weapons and Phantom Edges.
Phantom Edges are divided into types. The achievements refer explicitly to a Summon-type Phantom Edge, where using any one of them summons an ally. Summoning is thus the effect of one category, not a property of every Phantom Edge.
The Phantom-side action is Chain Assassination and the Blade-side action is Chain Execution, each with its own achievement, 「Way of the Phantom」 and 「Way of the Blade」. Those are combat techniques, a different layer from equipment upgrading.
Achievement
「Phantom Arsenal」 — Perform Tempering on 10 Phantom Edges
「Skill Over Strength」 — Upgrade a Phantom Edge to Level 30
「Way of the Phantom」 — Perform a Chain Assassination
「Way of the Blade」 — Perform a Chain Execution
「A Friend from Thin Air」 — Use any Summon-type Phantom Edge to summon an ally