Support async function and custom tool calls across Responses continuations

Problem statement / motivation

Async tools let a model continue independent work after emitting a call and accept its output later on the original call_id. Existing parallel gateway execution waits for a round's tool calls before continuing inference. Function normalization also drops async, and typed function/custom call outputs do not preserve the marker.

Proposed solution

Add typed async declaration/call metadata and a pending-call continuation contract. Keep client-executed function/custom tools application-executed. Preserve the marker and call identity through JSON/SSE output, persistence, and replay. Decide explicitly which upstreams can produce async calls and which gateway-executed tools may opt into nonblocking execution; reject unsupported combinations.

Acceptance criteria

• async: true survives declaration normalization and emitted call serialization for supported function/custom tools.
• A reference scenario starts a slow tool, produces independent output, and submits the eventual tool call output on its original call_id in a later response.
• Streaming exposes a complete call early enough for the application to dispatch it while consuming remaining output.
• Continuations preserve unresolved calls and accept outputs arriving across responses; unknown, duplicate, and conflicting outputs have explicit behavior.
• A synchronous wait-tool example returns completed outputs before wait status; the wait tool remains application-defined.
• Gateway-managed pending work, if enabled, has entry/byte/time limits, cancellation, disconnect policy, and joined task teardown. Acknowledging an async declaration alone must not imply this execution support.
• Record supported combinations with parallel calls and multi-agent mode; preserve ordinary synchronous tool behavior.

Additional context

Related: #181 (completed parallel scheduling), #298 (multi-agent), #241/#244 (streaming boundaries). Extend the shared typed ingestion and ordered delivery path; keep pending-work scheduling in orchestration and ingestion inline.

Use the existing cassette recorder/scenario workflow for OpenAI-reference and gateway recordings; do not hand-author captures.

Source: async tool calling.

Reviewed at cdac36f: implementation. Documentation checked September 18, 2026; pin the implementation-time API/model version when recording conformance.

Start with crates/agentic-server-core/src/tool/function.rs and the shared typed ingestion and ordered delivery path, then review orchestration for continuation and pending-work boundaries. Use the existing cassette recorder/scenario workflow for OpenAI-reference and gateway recordings; done means async metadata and call identity survive serialization, persistence, replay, streaming, and later responses while unsupported combinations and pending-work behavior are explicit.