fix(tools): wire the searchAvailable gate + research/pdf in the probe phase

Fix wave (2026-09-20), two owner-approved fixes:

Fix 1 — the searchAvailable gate was dead code on the live wiring path:
createTeacherTools RECEIVED ctx.searchAvailable but dropped it when
building the inner JobToolContext, so jobTools' research gate always took
its ?? true default (jobTools-direct tests passed; the production path
never saw the signal). Threaded through — the gate now reads the live
server signal on the teacher registry too.

Fix 2 — the prober registry gains the research + pdf dispatch tools
(owner ruling, 2026-09-20): these two are phase-omnipresent — the prober
can verify facts or read learner PDFs while assessing. demo/visual stay
teach-phase-only (they need the lesson's plan context). Same gating as
the teacher side: toggle-filtered, research subject to the search gate
(honestly absent when no engine is live — the prober's dispatch closes
over the SAME jobTools builder, one gate, one semantics). probe_done
stays always-on-by-construction; the learner-fs set untouched.

- ProberToolContext (extends ProbeDoneContext with the job wiring);
  createProberAgent takes it; chat-handler injects jobStore + spawn +
  searchAvailable into the prober ctx.
- TeacherToolContext.spawn's stale input union fixed (PdfInput was
  missing from the TYPE — runtime already passed it).
- Tests: the threaded-gate describe in web-search.test.ts pins the LIVE
  factory path both ways (teacher + prober); question-tools' registry
  key-sets gain research/pdf; the prober toggle test pins the
  research:false path.

Gates: typecheck, 807/807 tests, lint, build — all green.

src/server/chat-handler.ts

@@ -276,6 +276,12 @@ export async function handleChatRequest(
 						chatId,
 						chatStore: stores.chatStore,
 						deliver: (steer) => void deliverJobResult(steer),
+						// Fix wave (2026-09-20): the prober carries research + pdf
+						// dispatch (owner ruling — phase-omnipresent), so its ctx
+						// needs the same job wiring as the teacher's.
+						jobStore: stores.jobStore,
+						spawn: (job, input) => void jobRunner.runJob(job.id, input),
+						searchAvailable,
 					},
 					agentOpts,
 				);