[G9] Equipped badge showcase on the profile

Split out of [G5] Achievements screen + showcase (post_01m1f97znnexyrvzqmkdwn0eet) so G5 can ship the achievements page alone. G5 builds the page where badges are browsed; this ticket builds the one place a badge is worn.

Depends on: [G3] (catalogue + badge glyphs), [G5] (achievements screen — the natural place to pick what you equip).

Scope

• Equipping. A user picks one badge (or a small fixed number — to settle here) among the achievements they have actually unlocked. Stored per user; unequipping is always possible.
• Display on /app/profile. The equipped badge shows on the owner's own profile and on the profile as seen by other visitors.

• Gating, non-negotiable — the showcase inherits the rules already in place, it does not invent new ones:

  • hidden entirely when GAMIFICATION_ENABLED is off;
  • hidden per the viewer/target relationship exactly like xp already is — reuse withXp()/fetchXpByUser (apps/api/src/gamification/xp-lookup.util.ts) and the hideProgression flag rather than adding a parallel visibility path;
  • a Figurant (GHOST) shows nothing, same as every other progression signal.
• Zero-footprint when empty. If nothing is equipped, or the viewer isn't allowed to see it, the profile must render exactly as it does today — no empty slot, no placeholder, no reserved space. This is an explicit requirement, not a nicety.

Open questions to settle at design time

• One badge or a small showcase of 2-3? A single one is stronger as an identity signal; several turn the profile into a second achievements page, which G5 already is.
• Where exactly on the profile — next to the pseudonym (like the existing LevelBadge/StreakBadge chips), or as its own block near LevelCard?
• Does the equipped badge travel with the author line on reviews/comments, like LevelBadge and StreakBadge already do? That's a much bigger surface — probably a separate decision, possibly a "no".
• Does a tiered achievement equip as its current tier (ring colour follows the tier reached), and what happens visually when the user later reaches the next tier?

Out of scope

The achievements page itself (G5), the unlock bubble (G6), any cosmetic that isn't an already-unlocked achievement — paid or unlockable cosmetics are [B1].