fix(litellm): use anthropic-messages API for Claude models

- Set api: 'anthropic-messages' for claude-* models through LiteLLM
- Add LiteLLM to resolveCacheControlTtl for cache parameter passthrough
- Enables proper Anthropic cache control headers for cost savings

This fixes the missing cache support by ensuring:
1. Claude models use the correct API format (anthropic-messages)
2. Cache control TTL is resolved and passed through for LiteLLM

src/agents/litellm-models.ts

@@ -22,9 +22,14 @@ export type LitellmModelEntry = {
 };
 
 export function buildLitellmModelDefinition(entry: LitellmModelEntry): ModelDefinitionConfig {
+  // Detect Claude models and use Anthropic Messages API for proper cache control support
+  const isClaude = entry.id.toLowerCase().startsWith("claude-");
+
   return {
     id: entry.id,
     name: entry.name,
+    // Claude models through LiteLLM should use anthropic-messages API for cache control
+    ...(isClaude ? { api: "anthropic-messages" as const } : {}),
     reasoning: entry.reasoning ?? false,
     input: entry.input ? [...entry.input] : ["text"],
     cost: LITELLM_DEFAULT_COST,

src/agents/pi-embedded-runner/extra-params.ts

@@ -32,6 +32,8 @@ function resolveCacheControlTtl(
   if (raw !== "5m" && raw !== "1h") return undefined;
   if (provider === "anthropic") return raw;
   if (provider === "openrouter" && modelId.startsWith("anthropic/")) return raw;
+  // LiteLLM proxying to Anthropic models (claude-*)
+  if (provider === "litellm" && modelId.startsWith("claude-")) return raw;
   return undefined;
 }