Research tool row grays on the registry's own search signal (Teacher flag)

The teacher registry drops the research tool when no search engine is
live (chat-handler's searchAvailable, 2026-09-20) — the settings row
now grays on the SAME /api/search signal, so the row and the actual
registry availability can never disagree (Teacher-stream flag,
2026-09-22, routed by master).

- researchToolUnavailable (pure): research grays exactly when the live
  status says unavailable; a loading/failed fetch grays nothing; the
  other tools never touch the search port.
- The row: whole-row dim (graph-view graying) + disabled toggle + the
  status reason as the hint — the same line the search-engine row
  under it speaks from.

Two live-caught fixes while here:
- The reason line refreshed IN the pick handler raced the section PUT
  (the hint could contradict the trigger it sits under — caught with a
  Brave pin: trigger 'Brave', hint still 'tavily (auto)'). The refresh
  now keys on the agent prop's engine pin, so it reads post-write
  state.
- The engine menu options render from a map without React keys
  (console warning) — keyed.

Live-verified on 3001: no engine key → research dimmed with reason,
siblings untouched; Tavily key saved + section reopened → row alive
('search engine: tavily (auto)'); Brave pinned without its key → both
rows gray with the honest missing-key reason; Automatic restored.
Gates: typecheck, lint, 803/803, build.

tests/settings-pure.test.ts

@@ -8,7 +8,13 @@
 import assert from "node:assert/strict";
 import { describe, it } from "node:test";
 import { appearanceVars, derivedPalette, resolveAppearance } from "../components/settings/appearance.tsx";
-import { composePromptSwitch, composeSearchEngine, composeSlotModel, composeTool } from "../components/settings/agent-section.tsx";
+import {
+	composePromptSwitch,
+	composeSearchEngine,
+	composeSlotModel,
+	composeTool,
+	researchToolUnavailable,
+} from "../components/settings/agent-section.tsx";
 import { nextOverrides } from "../components/settings/keyboard-section.tsx";
 import { classifyKeys, keyStatusLine, type ProviderKeyInfo } from "../components/settings/keys-section.tsx";
 import type { AgentSettings, AuthKeyStatus, ModelsSettings } from "../src/core";
@@ -282,3 +288,18 @@ describe("first-run gate — slotsConfigured + setupStage (model-slots wave)", (
 		assert.equal(setupStage([{ id: "zai", keyPresent: false }, { id: "openai", keyPresent: true }]), "pickers", "one keyed provider widens the pickers");
 	});
 });
+
+describe("researchToolUnavailable — the row grays on the registry's own signal", () => {
+	it("research grays exactly when the live search status says unavailable", () => {
+		// the SAME /api/search signal chat-handler's searchAvailable uses —
+		// the row and the registry can never disagree (Teacher flag 2026-09-22)
+		assert.equal(researchToolUnavailable("research", { available: false }), true);
+		assert.equal(researchToolUnavailable("research", { available: true }), false);
+		// loading/failed fetch grays nothing (never a false-negative row)
+		assert.equal(researchToolUnavailable("research", null), false);
+		// only research — the other tools never touch the search port
+		for (const id of ["demo", "visual", "pdf"]) {
+			assert.equal(researchToolUnavailable(id, { available: false }), false);
+		}
+	});
+});