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/agent/teacher-agent.ts

@@ -10,6 +10,7 @@ import {
 	createProberTools,
 	createTeacherTools,
 	type ProbeDoneContext,
+	type ProberToolContext,
 	type TeacherToolContext,
 } from "./tools.ts";
 import { toPiTools, type LhTool } from "./lh-tool.ts";
@@ -181,7 +182,7 @@ function baseAgentOptions(kind: AgentKind, opts: AgentBuildOptions, tools: Agent
  * handoff wake (see tools/probe.ts) and is the transcript seam the
  * handler keys teacher construction on.
  */
-export function createProberAgent(ctx: ProbeDoneContext, opts: AgentBuildOptions = {}) {
+export function createProberAgent(ctx: ProberToolContext, opts: AgentBuildOptions = {}) {
 	return baseAgentOptions("prober", opts, toPiTools(createProberTools(ctx, opts.toolToggles)));
 }