fix(ocr-pipeline): improve page crop spine detection and cell assignment

1. page_crop: Score all dark runs by center-proximity × darkness ×
   narrowness instead of picking the widest. Fixes ad810209 where a
   wide dark area at 35% was chosen over the actual spine at 50%.

2. cv_words_first: Replace x-center-only word→column assignment with
   overlap-based three-pass strategy (overlap → midpoint-range → nearest).
   Fixes truncated German translations like "Schal" instead of
   "Schal - die Schals" in session 079cd0d9.

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

klausur-service/backend/tests/test_page_crop.py

@@ -15,6 +15,7 @@ import pytest
 
 from page_crop import (
     detect_and_crop_page,
+    detect_page_splits,
     _detect_format,
     _detect_edge_projection,
     _detect_left_edge_shadow,
@@ -465,3 +466,101 @@ class TestCropDeterminism:
             assert np.array_equal(ref_crop, crop), (
                 f"Run {i} produced different pixel output"
             )
+
+
+# ---------------------------------------------------------------------------
+# Tests: detect_page_splits — spine scoring logic
+# ---------------------------------------------------------------------------
+
+def _make_book_spread(h: int = 1616, w: int = 2288) -> np.ndarray:
+    """Create a synthetic landscape book spread (two pages side by side).
+
+    Simulates the ad810209 failure case:
+    - A narrow spine shadow near the center (~50% of width)
+    - A wider dark area off-center (~35% of width), simulating a text column
+    - Bright paper flanking the spine on both sides
+    """
+    img = np.full((h, w, 3), 230, dtype=np.uint8)
+
+    # --- Spine shadow: narrow dark valley centered at x = w/2 (1144) ---
+    spine_center = w // 2
+    spine_half_w = 30  # ~60px wide total
+    for x in range(spine_center - spine_half_w, spine_center + spine_half_w + 1):
+        dist = abs(x - spine_center)
+        # Brightness dips from 230 (paper) to 130 (spine)
+        brightness = int(130 + (230 - 130) * min(dist / spine_half_w, 1.0))
+        img[:, x] = brightness
+
+    # --- Off-center dark area at ~35% of width (x=799), wider than spine ---
+    dark_center = int(w * 0.35)
+    dark_half_w = 80  # ~160px wide total (wider than spine)
+    for x in range(dark_center - dark_half_w, dark_center + dark_half_w + 1):
+        dist = abs(x - dark_center)
+        # Brightness dips from 230 to 140 (slightly less dark than spine)
+        brightness = int(140 + (230 - 140) * min(dist / dark_half_w, 1.0))
+        img[:, x] = min(img[0, x, 0], brightness)  # don't overwrite spine if overlapping
+
+    return img
+
+
+class TestDetectPageSplits:
+    def test_portrait_image_returns_empty(self):
+        """Portrait images (width < height * 1.15) should not be split."""
+        img = np.full((1000, 800, 3), 200, dtype=np.uint8)
+        assert detect_page_splits(img) == []
+
+    def test_uniform_image_returns_empty(self):
+        """Uniform brightness image should not detect any spine."""
+        img = np.full((800, 1600, 3), 220, dtype=np.uint8)
+        assert detect_page_splits(img) == []
+
+    def test_prefers_centered_spine_over_wider_offcenter_dark(self):
+        """Scoring should pick the centered narrow spine over a wider off-center dark area.
+
+        This is the regression test for session ad810209 where the old algorithm
+        picked x=799 (35%) instead of x=1144 (50%).
+        """
+        img = _make_book_spread(h=1616, w=2288)
+        pages = detect_page_splits(img)
+
+        assert len(pages) == 2, f"Expected 2 pages, got {len(pages)}"
+
+        # Split point should be near the center (x ~ 1144), not at ~799
+        split_x = pages[0]["width"]  # pages[0] width = split point
+        center = 2288 / 2  # 1144
+
+        assert abs(split_x - center) < 100, (
+            f"Split at x={split_x}, expected near center {center:.0f}. "
+            f"Old bug would have split at ~799."
+        )
+
+    def test_split_produces_two_reasonable_pages(self):
+        """Both pages should be at least 15% of total width."""
+        img = _make_book_spread()
+        pages = detect_page_splits(img)
+
+        if len(pages) == 2:
+            w = img.shape[1]
+            for p in pages:
+                assert p["width"] >= w * 0.15, (
+                    f"Page {p['page_index']} too narrow: {p['width']}px "
+                    f"(< {w * 0.15:.0f}px)"
+                )
+
+    def test_page_indices_sequential(self):
+        """Page indices should be 0, 1, ..."""
+        img = _make_book_spread()
+        pages = detect_page_splits(img)
+        if pages:
+            indices = [p["page_index"] for p in pages]
+            assert indices == list(range(len(pages)))
+
+    def test_pages_cover_full_width(self):
+        """Pages should cover the full image width without gaps or overlaps."""
+        img = _make_book_spread()
+        pages = detect_page_splits(img)
+        if len(pages) >= 2:
+            w = img.shape[1]
+            assert pages[0]["x"] == 0
+            total_w = sum(p["width"] for p in pages)
+            assert total_w == w, f"Total page width {total_w} != image width {w}"