Refactor document handling to JSON envelope

The `document.md` file has been replaced by `document.json`. This new
format acts as a JSON envelope, containing the original markdown content
along with metadata such as `covered_mtime`, `written_at`, and
`backend_id`.

This change addresses a bug where new recordings arriving during an LLM
analysis call could be ignored. The `covered_mtime` field in the new
JSON envelope ensures that the document's metadata accurately reflects
the state of recordings at the time of analysis, preventing incorrect
"analysis current" states.

Additionally, the `analysis_input.json` file is now removed upon
successful analysis, and a failure marker is used to indicate analysis
failures. This ensures that a case is re-analyzed if necessary.

The `analysis_idle_threshold_secs` configuration option has been
introduced to control the delay before a case in the `Required` state is
automatically promoted to `Idle` and enqueued for LLM analysis.

server/tests/common/config.rs

@@ -66,6 +66,7 @@ pub struct TestConfig {
     whisper_url: Option<String>,
     whisper_timeout_seconds: Option<u64>,
     ollama_url: Option<String>,
+    idle_threshold_secs: Option<u64>,
     label: &'static str,
 }
 
@@ -85,6 +86,7 @@ impl TestConfig {
             whisper_url: None,
             whisper_timeout_seconds: None,
             ollama_url: None,
+            idle_threshold_secs: None,
             label: "common",
         }
     }
@@ -152,6 +154,15 @@ impl TestConfig {
         self
     }
 
+    /// Idle threshold in seconds: how long after the latest recording mtime
+    /// a `Required` case is auto-promoted to `Idle` (and enqueued). Default
+    /// is `Config::test_default()` (= `0`, eager). Tests that exercise the
+    /// "wait for quiet period" branch set a positive value.
+    pub fn with_idle_threshold(mut self, secs: u64) -> Self {
+        self.idle_threshold_secs = Some(secs);
+        self
+    }
+
     /// Materialize the config alone. Bucket-B overrides (`with_llm`,
     /// `with_whisper`, `with_ollama`) are silently dropped — call
     /// [`build_pair`] when those need to take effect.
@@ -170,12 +181,16 @@ impl TestConfig {
             .map(|u| (u.api_key.clone(), u.slug.clone()))
             .collect();
 
-        let config = Arc::new(Config {
+        let mut config = Config {
             data_path,
             users: self.users,
             api_keys,
             ..Config::test_default()
-        });
+        };
+        if let Some(v) = self.idle_threshold_secs {
+            config.analysis_idle_threshold_secs = v;
+        }
+        let config = Arc::new(config);
 
         let mut settings = Settings::default();
         // LLM-related fields used to live on `Settings`; with the multi-backend