fix(search): charge native search usage to the active root budget

Problem: an explicitly bounded Agent root omitted known native search usage,
allowing an extra normal tool round after a reported search answer.

Root cause: the search terminal finalized its independent request count and
reported usage without calling the accepted optional root observer.

Fix: observe the finalized record once before optional reporting. Preserve
query isolation, request counts, callback copies, unknown usage, native
completion, paid-backend consent and zero-limit behavior.

Verification: all 239 root packages pass through disjoint commands; four
complete owner race suites and real boot OpenAI/Anthropic HTTP regressions
pass. The compiled preceding owner fails seven events while nine controls
pass. Static gates, six cross-compiles and the literal 16-event Windows
contract pass locally; own native execution remains a hosted gate.

.github/workflows/ci.yml

@@ -421,6 +421,37 @@ jobs:
           retention-days: 7
 
 
+      - name: test (Windows root websearch budget)
+        if: env.RUN_STEPS == 'true' && runner.os == 'Windows' && github.event_name == 'pull_request'
+        timeout-minutes: 5
+        shell: bash
+        run: |
+          set -o pipefail
+          go test -count=1 -timeout=3m -json -run '^Test(RootWebSearch|EffectRootWebSearch)' ./internal/tools/websearch ./internal/assembly/boot | tee "$RUNNER_TEMP/root-websearch-budget.jsonl"
+          jq -se '
+            [.[] | select(.Test == null and .Action == "pass") | .Package] as $packages |
+            [.[] | select(.Test != null and .Action == "pass") | .Test] as $tests |
+            all(.[]; .Action != "skip" and .Action != "fail") and
+            (["reasonix/internal/tools/websearch", "reasonix/internal/assembly/boot"] - $packages | length) == 0 and
+            (["TestRootWebSearchUsageTerminal", "TestRootWebSearchUsageTerminal/reported",
+              "TestRootWebSearchUsageTerminal/no_reporter", "TestRootWebSearchUsageTerminal/missing",
+              "TestRootWebSearchUsageTerminal/no_observer", "TestRootWebSearchUsageTerminal/input_refused",
+              "TestRootWebSearchUsageTerminal/no_factory", "TestRootWebSearchUsageTerminal/factory_error",
+              "TestRootWebSearchUsageTerminal/billed_failure", "TestRootWebSearchUsageTerminal/cancel_after_usage",
+              "TestRootWebSearchUsageTerminal/cancel_before_start", "TestRootWebSearchUsageTerminal/stream_error",
+              "TestEffectRootWebSearchTokenBudget", "TestEffectRootWebSearchTokenBudget/reported",
+              "TestEffectRootWebSearchTokenBudget/disabled", "TestEffectRootWebSearchTokenBudget/missing"] - $tests | length) == 0
+          ' "$RUNNER_TEMP/root-websearch-budget.jsonl"
+
+      - name: upload Windows root websearch budget inventory
+        if: always() && env.RUN_STEPS == 'true' && runner.os == 'Windows' && github.event_name == 'pull_request'
+        uses: actions/upload-artifact@cf430e030ddbb5b0abf93d22962f4752f3646cd9 # v7
+        with:
+          name: root-websearch-budget-windows-${{ github.run_attempt }}
+          path: ${{ runner.temp }}/root-websearch-budget.jsonl
+          if-no-files-found: ignore
+          retention-days: 7
+
       - name: test (Windows root triage budget)
         if: env.RUN_STEPS == 'true' && runner.os == 'Windows' && github.event_name == 'pull_request'
         timeout-minutes: 5