## Why
Tool router construction rebuilds the deferred-tool BM25 index during
session initialization and before each sampling continuation, even when
the searchable tool metadata is unchanged. Local profiling measured
`append_tool_search_executor` at roughly 113 ms per continuation, making
repeated index construction the largest measured router-building cost.
## What changed
- Add a session-scoped `ToolSearchHandlerCache` so continuations and
user turns can reuse the existing handler.
- Key reuse on the complete ordered `Vec<ToolSearchInfo>`, rebuilding
when searchable text, loadable tool specs, source metadata, or ordering
changes.
- Build handlers outside the cache lock and recheck before publishing
them, avoiding holding the mutex during index construction.
## Verification
- `cache_reuses_identical_search_infos_and_rebuilds_changed_inputs`
covers exact cache reuse and invalidation when the ordered search
metadata changes.
- Local rollout profiling showed the initial router build populating the
cache and unchanged later continuations reusing it:
- uncached: 118 ms median across 14 spans from 3 rollouts
- cached: 4 ms median across 12 spans from 3 rollouts
## Why
Default tool search text currently derives identity from both `ToolName`
and `ToolSpec`. For function and namespace specs, this indexes the same
names more than once and also adds a flattened `{namespace}{name}` token
that is not model-visible.
## What changed
- Derive default search text entirely from `ToolSpec` while preserving
names, descriptions, namespace metadata, and recursive schema metadata.
- Keep the default search-text builder private and remove the unused
`ToolName` argument.
- Add coverage for the exact search text generated for a namespaced tool
with nested schema metadata.
## Example
For the `codex_app` namespace and `automation_update` tool (schema terms
omitted):
- Before: `codex_appautomation_update automation update codex_app
codex_app Manage Codex automations. automation_update automation update
...`
- After: `codex_app Manage Codex automations. automation_update
automation update ...`
## Testing
- `just test -p codex-tools`
Deferred tools need to be searchable even when they are not implemented
inside `codex-core`. Extension-provided tools can be registered for
later discovery, but the search metadata path was still owned by
core-specific runtime hooks, which meant the shared `ToolExecutor`
abstraction could not describe how a deferred extension tool should
appear in `tool_search`.
## Changes
- Move `ToolSearchEntry` and `ToolSearchInfo` into `codex-tools` and
re-export them from the shared tools crate.
- Add a default `ToolExecutor::search_info` implementation that derives
loadable tool-search metadata from function and namespace specs.
- Forward search metadata through extension adapters and exposure
overrides while keeping custom search text/source metadata for dynamic,
MCP, and multi-agent tools.
- Remove the old core-local `tool_search_entry` module now that search
metadata lives with the shared executor APIs.
## Testing
- Added `deferred_extension_tools_are_discoverable_with_tool_search`
coverage in `core/src/tools/spec_plan_tests.rs`.
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