docs: correct the false Arc<Mutex> MakeWriter claim

Both the spec and the plan asserted that Arc<Mutex<RotatingLogWriter>>
satisfies tracing-subscriber's MakeWriter via impl MakeWriter for Arc<W>.
That impl requires &'a W: io::Write, and &Mutex<W> does not implement
io::Write. The error reached the implementer and cost a fix round before
being caught; the shipped code uses a bare Mutex.

Corrections are marked inline so the mistake stays visible rather than
being silently erased.

Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01EGntTHCW3sEPy1VBRopNNp

docs/superpowers/plans/2026-07-31-xy-start-on-login.md

@@ -168,9 +168,11 @@ The daemon currently logs only to stderr, which launchd discards. Give it `log_d
 
 **Interfaces:**
 - Consumes: `impl io::Write for RotatingLogWriter` from Task 1.
-- Produces: `pub(crate) fn daemon_writer(log_dir: &Path) -> std::io::Result<Arc<Mutex<RotatingLogWriter>>>`. No later task depends on this.
+- Produces: `pub(crate) fn daemon_writer(log_dir: &Path) -> std::io::Result<Mutex<RotatingLogWriter>>`. No later task depends on this.
 
-`tracing-subscriber` 0.3 already implements `MakeWriter` for `Mutex<W> where W: io::Write` and for `Arc<W>`, so `Arc<Mutex<RotatingLogWriter>>` satisfies `with_writer` directly. No adapter type is needed.
+`tracing-subscriber` 0.3 implements `MakeWriter` for `Mutex<W> where W: io::Write` (`fmt/writer.rs:808`), so a plain `Mutex<RotatingLogWriter>` satisfies `with_writer` directly. No adapter type is needed.
+
+**Corrected 2026-08-01.** This step originally specified `Arc<Mutex<RotatingLogWriter>>`, on the false premise that `impl MakeWriter for Arc<W>` would cover it. That impl requires `&'a W: io::Write`, and `&Mutex<W>` does not implement `io::Write`. The error was caught during implementation and cost one fix round; the shipped code uses a bare `Mutex`.
 
 - [ ] **Step 1: Write the failing test**
 
@@ -215,16 +217,16 @@ Prepend to `crates/xy/src/logging.rs`, above the test module:
 
 ```rust
 use std::path::Path;
-use std::sync::{Arc, Mutex};
+use std::sync::Mutex;
 use xy_supervisor::logs::RotatingLogWriter;
 
 const LOG_FILE_MAX_BYTES: u64 = 10 * 1024 * 1024;
 const LOG_FILE_KEEP: usize = 5;
 
-pub(crate) fn daemon_writer(log_dir: &Path) -> std::io::Result<Arc<Mutex<RotatingLogWriter>>> {
+pub(crate) fn daemon_writer(log_dir: &Path) -> std::io::Result<Mutex<RotatingLogWriter>> {
     let writer = RotatingLogWriter::open(&log_dir.join("xy.log"), LOG_FILE_MAX_BYTES, LOG_FILE_KEEP)?;
 
-    Ok(Arc::new(Mutex::new(writer)))
+    Ok(Mutex::new(writer))
 }
 ```