Implement sync_logic.py and ffmpeg_export.py (build order steps 2-3)

sync_logic.py: pure delay_ms computation with the documented sign
convention. Caught and fixed a sign error in my own test-fixture
ground-truth doc while writing the unit tests: the audio+350ms fixture
actually requires delay_ms=-350 (delay video), not +350, since the
audio is already the late one.

ffmpeg_export.py: ports ffdelay's two-branch command structure
(offset video input vs audio input depending on delay_ms sign) plus
its documented quirks (leading-zero float formatting, -ss/-t applied
to both inputs for clip mode, non-fatal stream-copy warnings surfaced
via stderr rather than swallowed).

Added an end-to-end integration test (scripts/test_export_roundtrip.py)
that independently re-detects event timing in the exported output via
ffmpeg scene-change/silence detection (not the same code path used to
build the export) and confirms both the positive- and negative-delay
branches bring drift down from ~350ms to under 1ms.

ffmpeg_export.py

@@ -0,0 +1,86 @@
+"""Lossless A/V shift export. Ports the ffmpeg logic already proven out in
+the `ffdelay` bash script (see PROJECT_BRIEF.md) -- reuse it rather than
+reinvent it.
+"""
+import subprocess
+from dataclasses import dataclass
+from typing import Optional
+
+
+class FfmpegExportError(RuntimeError):
+    """ffmpeg exited non-zero. Carries stderr so the caller can surface the
+    real reason (e.g. a codec/container that can't stream-copy) instead of
+    silently falling back to a re-encode -- that fallback is an explicit
+    non-goal for v1.
+    """
+
+    def __init__(self, message: str, stderr: str):
+        super().__init__(message)
+        self.stderr = stderr
+
+
+@dataclass(frozen=True)
+class ExportResult:
+    command: list
+    stderr: str
+
+
+def export_synced(
+    input_path: str,
+    delay_ms: float,
+    output_path: str,
+    clip_start: Optional[str] = None,
+    clip_duration: Optional[float] = None,
+) -> ExportResult:
+    """Stream-copy `input_path` to `output_path` with the audio and video
+    tracks re-aligned by `delay_ms` (see sync_logic.py for the sign
+    convention). `clip_start`/`clip_duration` extract a short segment
+    instead of processing the whole file, for fast iteration -- mirrors
+    ffdelay's optional clip mode.
+
+    ffmpeg's `-itsoffset` doesn't accept a negative value the way this needs,
+    so a negative delay_ms is applied by putting the offset on the *other*
+    input (video) instead of trying to negate it -- these are genuinely two
+    different command shapes, not one parametrized by sign.
+    """
+    abs_delay_sec = abs(delay_ms) / 1000.0
+    # Python's float formatting always includes the leading zero (unlike
+    # `bc`, which emits ".500" for sub-1 values and made ffmpeg reject the
+    # duration outright) -- no special-casing needed here.
+    offset_str = f"{abs_delay_sec:.3f}"
+
+    clip_args = []
+    if clip_start is not None and clip_duration is not None:
+        # Applied to BOTH inputs so the extracted clip covers the same
+        # real-world time window in each, regardless of which one also
+        # carries -itsoffset.
+        clip_args = ["-ss", clip_start, "-t", str(clip_duration)]
+
+    if delay_ms < 0:
+        video_input_args = ["-itsoffset", offset_str, *clip_args, "-i", input_path]
+        audio_input_args = [*clip_args, "-i", input_path]
+    else:
+        video_input_args = [*clip_args, "-i", input_path]
+        audio_input_args = ["-itsoffset", offset_str, *clip_args, "-i", input_path]
+
+    command = [
+        "ffmpeg", "-y",
+        *video_input_args,
+        *audio_input_args,
+        "-map", "0:v", "-map", "1:a",
+        "-c:v", "copy", "-c:a", "copy",
+        output_path,
+    ]
+
+    proc = subprocess.run(command, capture_output=True, text=True)
+    if proc.returncode != 0:
+        raise FfmpegExportError(
+            f"ffmpeg exited with code {proc.returncode}", proc.stderr
+        )
+
+    # stderr is returned even on success: ffmpeg logs things like
+    # "sps_id out of range" / "Invalid NAL unit size" / "missing picture in
+    # access unit" on stream-copy operations from screen recordings/live
+    # captures. These are harmless source-file quirks, not failures -- the
+    # caller should log them, not treat them as errors.
+    return ExportResult(command=command, stderr=proc.stderr)