chore(engine): log prompt-cache hit/miss per chat call

Add a `tag` option to chat() and have it print one `[cache] <tag>
hit=X miss=Y rate=Z%` line per call. Three Usage-shape variants are
probed in order so the same logger works across providers:

  - DeepSeek (v3+):  usage.prompt_cache_hit_tokens / *_miss_tokens
  - OpenAI / o-series: usage.prompt_tokens_details.cached_tokens
  - Anthropic:        usage.cache_read_input_tokens / *_creation_*

When none of them are present (MiMo / local Ollama / others) we still
print prompt + completion totals so the cost baseline is visible.

Tag every callsite so the log is greppable:
  architect / writer / character-designer / cinematographer / insert-beat

This is the prerequisite for the prefix-cache reordering work that
follows — without per-agent visibility there's no way to tell if a
prompt rearrangement actually moved the needle.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

lib/ai-client/chat.ts

@@ -6,10 +6,65 @@ export type ChatMessage = {
   content: string;
 };
 
+// Different providers expose prompt-cache stats under different keys. We probe
+// for the three forms we've seen in the wild and fall back to total tokens
+// when no cache field exists.
+//
+//   DeepSeek (v3+)    usage.prompt_cache_hit_tokens / prompt_cache_miss_tokens
+//   OpenAI / o-series usage.prompt_tokens_details.cached_tokens
+//   Anthropic / others  usage.cache_read_input_tokens / cache_creation_input_tokens
+//   No-cache (MiMo,
+//     local Ollama, …) only prompt_tokens / completion_tokens — print those
+//                       so we still get a rough cost baseline.
+type Usage = {
+  prompt_tokens?: number;
+  completion_tokens?: number;
+  prompt_cache_hit_tokens?: number;
+  prompt_cache_miss_tokens?: number;
+  prompt_tokens_details?: { cached_tokens?: number };
+  cache_read_input_tokens?: number;
+  cache_creation_input_tokens?: number;
+};
+
+function summarizeUsage(tag: string, usage: Usage | undefined): string {
+  if (!usage) return `[cache] ${tag} no-usage`;
+  const prompt = usage.prompt_tokens ?? 0;
+  const completion = usage.completion_tokens ?? 0;
+  // DeepSeek-style
+  if (typeof usage.prompt_cache_hit_tokens === "number") {
+    const hit = usage.prompt_cache_hit_tokens;
+    const miss = usage.prompt_cache_miss_tokens ?? Math.max(0, prompt - hit);
+    const denom = hit + miss;
+    const rate = denom > 0 ? ((hit / denom) * 100).toFixed(1) : "n/a";
+    return `[cache] ${tag} hit=${hit} miss=${miss} rate=${rate}% completion=${completion}`;
+  }
+  // OpenAI-style
+  const oaiCached = usage.prompt_tokens_details?.cached_tokens;
+  if (typeof oaiCached === "number") {
+    const miss = Math.max(0, prompt - oaiCached);
+    const rate = prompt > 0 ? ((oaiCached / prompt) * 100).toFixed(1) : "n/a";
+    return `[cache] ${tag} hit=${oaiCached} miss=${miss} rate=${rate}% completion=${completion}`;
+  }
+  // Anthropic-style
+  if (typeof usage.cache_read_input_tokens === "number") {
+    const hit = usage.cache_read_input_tokens;
+    const create = usage.cache_creation_input_tokens ?? 0;
+    const denom = hit + create + prompt;
+    const rate = denom > 0 ? ((hit / denom) * 100).toFixed(1) : "n/a";
+    return `[cache] ${tag} hit=${hit} create=${create} miss=${prompt} rate=${rate}% completion=${completion}`;
+  }
+  // No cache field at all
+  return `[cache] ${tag} prompt=${prompt} completion=${completion} (provider didn't report cache stats)`;
+}
+
 export async function chat(
   config: ProviderConfig,
   messages: ChatMessage[],
-  opts?: { temperature?: number; responseFormat?: "json_object" | "text" },
+  opts?: {
+    temperature?: number;
+    responseFormat?: "json_object" | "text";
+    tag?: string;
+  },
 ): Promise<string> {
   const url = `${config.baseUrl.replace(/\/$/, "")}/chat/completions`;
   const body: Record<string, unknown> = {
@@ -35,7 +90,10 @@ export async function chat(
     throw new Error(`Chat API error ${res.status}: ${text}`);
   }
 
-  let json: { choices: { message: { content: string } }[] };
+  let json: {
+    choices: { message: { content: string } }[];
+    usage?: Usage;
+  };
   try {
     json = JSON.parse(text);
   } catch {
@@ -50,5 +108,7 @@ export async function chat(
     );
   }
 
+  console.log(summarizeUsage(opts?.tag ?? "chat", json.usage));
+
   return content;
 }