[Bug]: Lineage panel hides which account a subagent runs on (no account badge) #24831

Before submitting

• I searched existing issues and did not find a duplicate.
• I included enough detail to reproduce or investigate the problem.

Area

apps/web

Steps to reproduce

1. Add two accounts for the same provider, e.g. two Claude instances with accent colours (badges C1 and VY).
2. In a thread on one account, start a subagent or workflow so it shows in the right panel's Lineage list.
3. Compare the lineage row's icon with the same thread's sidebar row, the composer model picker, and the context-handoff row.

Expected behavior

The lineage row shows the same account badge (accent colour + initials) as every other place that shows a provider icon, so you can tell which account a subagent runs on without hovering.

Actual behavior

The lineage row shows a plain Claude icon. With two Claude accounts, every subagent looks the same; the account appears only as text inside the hover card. Workflow member rows have the same gap.

The mobile Agents sheet (and the workflow/subagent rows in the feed that reuse SubagentRow) has the same problem: the account name is in the metadata text, but the icon has no badge, unlike mobile thread rows, the composer and handoff rows.

Cause

ThreadRelationshipIcon (apps/web/src/components/chat/ThreadRelationshipIcon.tsx) renders ProviderInstanceIcon with only driverKind/displayName and never receives the provider list, so it can't run shouldShowInstanceBadge or pass accentColor/showBadge. Its status dot also sits in the bottom-right corner, where the account badge goes everywhere else:

https://github.com/pingdotgg/t3code/blob/c79ef8c454/apps/web/src/components/chat/ThreadRelationshipIcon.tsx#L57-L83

The hover card (SubagentTooltipContent) already computes the badge from providers. Mobile's SubagentRow renders ProviderIcon instead of the badge-capable ProviderInstanceIcon that thread rows and handoff rows use.

The earlier badge fixes (#5980 sidebar, #9899 mobile thread rows, #15477 handoffs, #21406 mobile composer) each covered their own surface; the lineage list was never wired up.

Suggested fix

• Web: pass providers to ThreadRelationshipIcon (lineage rows and workflow member rows), derive the entry the same way the hover card does, and show the compact badge. When the badge shows, the status dot moves to the top-left corner.
• Mobile: SubagentRow uses ProviderInstanceIcon with the badge when shouldShowInstanceBadge says so.
• The hover card keeps its accent dot plus account name, matching the sidebar hover card.

Single-account users see no change. No contract, server or adapter change.

Impact

Minor bug or occasional failure

Cosmetic but confusing: with several accounts per provider you can't tell which account a running subagent is spending.

Version or commit

Reproduced on main c79ef8c454

Environment

Desktop client, two Claude accounts with accent colours; also mobile Agents sheet

Logs or stack traces

n/a (visual)

Screenshots, recordings, or supporting files

Before:
After:

Related: #24896 (the chat subagent card, same gap, PR #24898).