Define and implement Responses Programmatic Tool Calling support

Problem statement / motivation

The Responses API exposes programmatic_tool_calling plus eligible tool settings such as allowed_callers and output_schema. Agentic API has no tool variant for this capability, and function normalization loses those settings. Existing code-interpreter support does not supply this protocol or runtime contract.

Proposed solution

Start with a documented execution-location decision: an upstream-hosted built-in capability, a gateway-executed isolated runtime, or separately advertised support for both. Define the supported vLLM integration before promising local execution. Implement typed declarations, program calls/outputs, nested client function calls, and continuation for the selected deployment profile.

Acceptance criteria

• A short design records execution ownership, upstream requirements, supported transports, and explicit unsupported-path errors.
• Preserve allowed_callers, output_schema, call IDs, and the supported programmatic item/event contract across storage and continuation.
• Enforce direct/programmatic/either caller policies, including nested client-executed functions and structured outputs.
• An end-to-end scenario calls multiple tools, filters/aggregates intermediate outputs, and returns the final output with correct usage and failure behavior.
• If gateway execution is selected, the runtime is isolated with explicit CPU/memory/output/time/concurrency limits, cancellable execution, and joined teardown. Ambient host filesystem, credentials, subprocesses, and network access are unavailable; external effects go through enabled tools.
• Test nested call failures, malformed outputs, timeout/cancellation, continuation, and JSON/SSE parity using shared ingestion and tool routing.
• Record reference/gateway examples through the existing cassette workflow and document deliberate compatibility limits.

Additional context

Related: #160 (typed tool execution), #308 (serverless code interpreter), #314 (request fidelity). Programmatic Tool Calling is a distinct feature; avoid silently lowering it to an ordinary code-interpreter or function tool.

Source: Programmatic Tool Calling.

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

Start by reading the reviewed implementation at crates/agentic-server-core/src/types/tools/params.rs, then compare the requirements with issues #160, #308, and #314 and the linked Programmatic Tool Calling reference. Establish the execution-location design and supported vLLM integration before implementation. Done means the selected profile satisfies the acceptance criteria, including nested calls, continuation, isolation limits, failure cases, JSON/SSE parity, and cassette examples.