refactor(src): startup_profiler.py Phase 11.3.2 - extract _log_phase_output

Phase 11.3.2. CONTEXT-MANAGER EXCEPTION.

The plan claimed 'StartupProfiler.phase() is NOT a context manager;
tier-2's claim is factually wrong.' This is incorrect. phase() IS a
context manager:
- Decorated with @contextmanager (src/startup_profiler.py:26)
- Used in 13 'with startup_profiler.phase(...)' call sites in
  src/gui_2.py (lines 308, 311, 327, 338, 343, 627, 629, 631, 669,
  672, 711, 729, 739)

It cannot return Result[None] because:
- @contextmanager requires the function to yield (not return)
- The except body is inside a finally block (which cannot return)

Best partial migration: extract _log_phase_output helper that returns
Result[None]; phase() calls it and ignores the Result (we're in a
finally block).

Audit post-migration:
- _log_phase_output L28 = INTERNAL_COMPLIANT (Heuristic A) ✓
- phase() L54 try/finally = INTERNAL_COMPLIANT (canonical cleanup) ✓

Tests: 12/12 pass (test_audit_allowlist_2d, test_gui_startup_smoke,
test_headless_service, test_startup_profiler, test_warmup_canaries).

This site is documented in the per-site report as a CONTEXT-MANAGER
EXCEPTION. The Heuristic #19 (catch+log) classification remains valid;
the partial migration adds explicit Result-returning helpers where
possible without breaking the context manager pattern.

src/startup_profiler.py

@@ -4,6 +4,8 @@ from contextlib import contextmanager
 from dataclasses import dataclass, field
 from typing import Any, Iterator
 
+from src.result_types import ErrorInfo, ErrorKind, Result
+
 
 @dataclass
 class _Phase:
@@ -12,6 +14,26 @@ class _Phase:
  end_ts:   float = 0.0
 
 
+def _log_phase_output(line: str, phase_name: str) -> Result[None]:
+ """Best-effort stderr write for phase timing output. Returns Result[None].
+
+ Used by phase() (which is a @contextmanager; cannot return Result from
+ its except body because @contextmanager requires yield, not return, and
+ the except is in a finally block).
+ """
+ try:
+  sys.stderr.write(line)
+  sys.stderr.flush()
+  return Result(data=None)
+ except OSError as e:
+  return Result(data=None, errors=[ErrorInfo(
+   kind=ErrorKind.INTERNAL,
+   message=f"phase output failed for {phase_name}: {e}",
+   source="startup_profiler._log_phase_output",
+   original=e,
+  )])
+
+
 @dataclass
 class StartupProfiler:
  _phases:  list[_Phase] = field(default_factory=list)
@@ -34,11 +56,10 @@ class StartupProfiler:
   finally:
    p.end_ts = time.perf_counter()
    self._phases.append(p)
-   try:
-    sys.stderr.write(f"[startup] {name}: {(p.end_ts - p.start_ts) * 1000.0:.1f}ms\n")
-    sys.stderr.flush()
-   except OSError as e:
-    sys.stderr.write(f"[startup] phase output failed for {name}: {e}\n")
+   log_line = f"[startup] {name}: {(p.end_ts - p.start_ts) * 1000.0:.1f}ms\n"
+   log_result = _log_phase_output(log_line, name)
+   if not log_result.ok:
+    _log_phase_output(f"[startup] phase output failed for {name}: {log_result.errors[0].message}\n", name)
 
  def snapshot(self) -> dict[str, Any]:
   phases: dict[str, dict[str, float]] = {}