feat(relay): hosted-tool conversion fidelity, reasoning normalization, and billing usage integrity (#7137)

* feat(relaykit): preserve hosted tools across conversions

- add protocol-neutral hosted-tool DTOs, conversion metadata, and loss policies
- bridge citations, grounding metadata, and hosted-tool stream lifecycles
- document the public conversion behavior and channel policy controls

* refactor(relaykit): normalize reasoning and thinking intent

- centralize provider-neutral reasoning intent, effort, and budget mappings
- parse model suffixes at the host entry boundary while preserving provider-owned tails
- keep adaptive Claude thinking and explicit zero-token compatibility consistent

* fix(billing): preserve authoritative usage across relay hops

- carry native BillingUsage sidecars through direct and streamed protocol bridges
- merge partial and terminal usage monotonically with safe fallback settlement
- retain cache metadata, penultimate usage, and per-call Gemini tool surcharges

* feat(relay): bridge Responses with Claude and Gemini protocols

- add direct request, response, and stream converters across supported relay formats
- expose Claude count_tokens and Chat-to-Responses compatibility endpoints
- carry conversion diagnostics through the host while retaining the curated public goldens

* fix(relay): wire relaykit conversions into host channels

- connect handlers, adaptors, and channel settings to the standalone conversion layer
- keep model mapping, pricing identity, retries, and provider-specific suffix behavior aligned
- ignore local audit artifacts and retain focused public regression coverage

relaykit/reasonmap/reasonmap.go

@@ -16,6 +16,11 @@ func ClaudeStopReasonToOpenAIFinishReason(stopReason string) string {
 		return "length"
 	case "tool_use":
 		return "tool_calls"
+	case "pause_turn":
+		// Responses has no pause_turn finish reason. Treat the provider's
+		// resumable server-side loop as an incomplete response instead of a
+		// successful stop; the hosted output items preserve continuation state.
+		return "length"
 	case "refusal":
 		return types.FinishReasonContentFilter
 	default: