@@ -22,8 +22,9 @@
 //     input"), and, when the summary-capability rule applies, again with
 //     one summary sized for its strictest consumer ("· shared summary"),
 //     and, when queries read windows of one scan on a common grid, again
-//     with one summary per shared segment ("· shared segments"); in
-//     enumeration order, only those with a written physical candidate. A
+//     with one summary per shared segment, once per segmentation, by its
+//     number of segments ("· shared segments ×4"); in enumeration order,
+//     only those with a written physical candidate. A
 //     repeating query's mergeable alternatives also come in tumbling panes
 //     (Pass 2's window-composition rule), e.g. "Q1 Kll · tumbling 1m panes";
 //   - stage2_physical_asap: per logical candidate, its physical candidates
@@ -267,11 +268,11 @@ fn stage_pipeline(
         let inventory = &variant.inventory;
         let index = offset + choice_index(inventory, &candidate.choice) + 1;
         let mut label = label(inventory, &target_owners(inventory), &candidate.choice);
-        label += match candidate.sharing {
-            Sharing::Independent => "",
-            Sharing::IdenticalExpressions => " · shared input",
-            Sharing::SummaryCapability => " · shared summary",
-            Sharing::WindowSegments => " · shared segments",
+        label += &match candidate.sharing {
+            Sharing::Independent => String::new(),
+            Sharing::IdenticalExpressions => " · shared input".into(),
+            Sharing::SummaryCapability => " · shared summary".into(),
+            Sharing::WindowSegments { segments } => format!(" · shared segments ×{segments}"),
         };
         let written = candidate
             .physical
 
@@ -170,13 +170,14 @@ fn example3b_lists_tumbling_candidates() {
     }
 }
 
-/// Example 4, Pattern A repeated monthly: the same 487 logical candidates
-/// as the ad hoc batch (3a), 486 independent or sharing input plus the one
-/// sharing 1-year segments (Q60). The windows (1–5 y) are longer than the
-/// month between runs and no pane width fits, so only the shared segments
-/// can be maintained at ingestion time (Q61): 488 plans. The selected plan
-/// is 3a's, now amortized over monthly runs instead of one run per hour of
-/// horizon.
+/// Example 4, Pattern A repeated monthly: the same 488 logical candidates
+/// as the ad hoc batch (3a), 486 independent or sharing input plus the two
+/// sharing segments (Q60, Q67): four split at the windows' boundaries and
+/// five 1-year ones. The windows (1–5 y) are longer than the month between
+/// runs and no pane width fits, so only the even 1-year segments can be
+/// maintained at ingestion time (Q61); the uneven ones are no stream of
+/// panes: 489 plans. The selected plan is 3a's, now amortized over monthly
+/// runs instead of one run per hour of horizon.
 #[test]
 fn example4a_repeats_monthly_with_only_segments_maintainable() {
     let once = generate(&[
@@ -194,14 +195,14 @@ fn example4a_repeats_monthly_with_only_segments_maintainable() {
     let physical = monthly["stage2_physical_asap"]["candidates"]
         .as_array()
         .unwrap();
-    assert_eq!(physical.len(), 488);
+    assert_eq!(physical.len(), 489);
     let maintained: Vec<_> = physical
         .iter()
         .map(|p| p["label"].as_str().unwrap())
         .filter(|label| label.contains("ingestion time"))
         .collect();
     assert_eq!(maintained.len(), 1, "{maintained:?}");
-    assert!(maintained[0].ends_with("shared segments · ingestion time: Kll ×5 panes"));
+    assert!(maintained[0].ends_with("shared segments ×5 · ingestion time: Kll ×5 panes"));
     let selected = |d: &Value| {
         d["stage3_selection"]["selected"]
             .as_str()
 
@@ -180,51 +180,103 @@ fn stage1_a_identical_expression_rule_shares_only_raw_input() {
     }
 }
 
-/// The windows' boundaries lie on a 1-year grid, so one set of five 1-year
-/// KLL segments over [T − 5y, T] serves all five queries (Q60, in place of
-/// the spec's Exponential Histogram): each query merges the segments its
-/// range covers and estimates p99 from its merge.
+/// The planner chooses the segment grid (Q67): one candidate splits the
+/// windows at their boundaries into four KLL segments over [T − 5y, T], the
+/// oldest two years wide; another uses the even grid of five 1-year
+/// segments. Either serves all five queries (Q60, in place of the spec's
+/// Exponential Histogram): each query merges the segments its range covers
+/// and estimates p99 from its merge.
 #[test]
 fn stage1_a_shared_segments_serve_all_five() {
     let run = run_a();
-    let c = run
+    let candidates: Vec<_> = run
         .logical
         .iter()
-        .find(|c| shares_segments(&c.dag, &c.query_roots))
-        .expect("a candidate sharing window segments");
-    let segments = windowed_builds(c);
-    assert_eq!(segments.len(), 5, "{}", c.id);
-    for (_, algorithm, form, _) in &segments {
-        assert_eq!(*algorithm, SketchAlgorithm::Kll);
-        assert_eq!(*form, WindowForm::Tumbling { length_ms: YEAR_MS });
-    }
-    let read: BTreeSet<usize> = segments.iter().flat_map(|(.., r)| r.clone()).collect();
-    assert_eq!(read, (0..5).collect());
-    // Each query merges as many segments as its range has years.
-    for (q, (_, lookback, _)) in PATTERN_A.iter().enumerate() {
-        let covering = segments.iter().filter(|(.., r)| r.contains(&q)).count() as u64;
-        assert_eq!(covering, lookback / YEAR_MS, "q{}", q + 1);
-    }
-    let estimates: std::collections::BTreeMap<_, _> = segments
+        .filter(|c| shares_segments(&c.dag, &c.query_roots))
+        .collect();
+    let mut widths: Vec<Vec<u64>> = candidates
         .iter()
-        .flat_map(|(build, ..)| estimates_of(&c.dag, *build))
+        .map(|c| {
+            let mut widths: Vec<u64> = windowed_builds(c)
+                .iter()
+                .map(|(_, _, form, _)| match form {
+                    WindowForm::Tumbling { length_ms } => length_ms / YEAR_MS,
+                    other => panic!("{}: {other:?}", c.id),
+                })
+                .collect();
+            widths.sort();
+            widths
+        })
         .collect();
-    assert_eq!(estimates.len(), 5, "one estimate per query");
-    for (estimate, statistic) in estimates {
-        assert_eq!(statistic, SketchStatistic::Quantile { q: 0.99 });
-        let merged = c
-            .dag
-            .producers(estimate)
-            .into_iter()
-            .any(|p| matches!(c.dag.payload(p), LogicalASAPOperatorPayload::SummaryMerge));
-        assert!(
-            merged,
-            "{}: estimate {estimate:?} does not read a merge",
-            c.id
-        );
+    widths.sort();
+    assert_eq!(widths, [vec![1, 1, 1, 1, 1], vec![1, 1, 1, 2]]);
+    for c in candidates {
+        let segments = windowed_builds(c);
+        for (_, algorithm, ..) in &segments {
+            assert_eq!(*algorithm, SketchAlgorithm::Kll);
+        }
+        let read: BTreeSet<usize> = segments.iter().flat_map(|(.., r)| r.clone()).collect();
+        assert_eq!(read, (0..5).collect());
+        // The segments each query merges add up to its range.
+        for (q, (_, lookback, _)) in PATTERN_A.iter().enumerate() {
+            let covered: u64 = segments
+                .iter()
+                .filter(|(.., r)| r.contains(&q))
+                .map(|(_, _, form, _)| match form {
+                    WindowForm::Tumbling { length_ms } => *length_ms,
+                    _ => unreachable!(),
+                })
+                .sum();
+            assert_eq!(covered, *lookback, "{} q{}", c.id, q + 1);
+        }
+        let estimates: std::collections::BTreeMap<_, _> = segments
+            .iter()
+            .flat_map(|(build, ..)| estimates_of(&c.dag, *build))
+            .collect();
+        assert_eq!(estimates.len(), 5, "one estimate per query");
+        for (estimate, statistic) in estimates {
+            assert_eq!(statistic, SketchStatistic::Quantile { q: 0.99 });
+            let merged = c
+                .dag
+                .producers(estimate)
+                .into_iter()
+                .any(|p| matches!(c.dag.payload(p), LogicalASAPOperatorPayload::SummaryMerge));
+            assert!(
+                merged,
+                "{}: estimate {estimate:?} does not read a merge",
+                c.id
+            );
+        }
     }
 }
 
+/// Stage 3 decides the grid: the four boundary segments cost as much to
+/// build as the five 1-year ones (each row lands in one segment) but merge
+/// fewer states, so they are cheaper, and cheapest overall.
+#[test]
+fn stage3_a_selects_the_cheaper_segment_grid() {
+    let run = run_a();
+    let mut costs: Vec<(usize, f64, &str)> = run
+        .logical
+        .iter()
+        .filter(|c| shares_segments(&c.dag, &c.query_roots))
+        .map(|c| {
+            let p = run.physical_of(c).next().unwrap();
+            (
+                windowed_builds(c).len(),
+                run.cost(&p.id).unwrap(),
+                p.id.as_str(),
+            )
+        })
+        .collect();
+    costs.sort_by_key(|(segments, ..)| *segments);
+    let [(4, four, selected), (5, five, _)] = costs[..] else {
+        panic!("{costs:?}");
+    };
+    assert!(four < five, "×4 {four} vs ×5 {five}");
+    assert_eq!(run.selection.selected, selected);
+}
+
 /// The shared-segment candidate is added next to the independent KLL candidates, not instead of them.
 #[test]
 fn stage1_a_keeps_independent_and_shared_window_summaries() {
 
@@ -97,13 +97,17 @@ fn eh_options(run: &Run) -> BTreeMap<Materialization, (String, LogicalASAPNodeId
     options_of(run, None, 5)
 }
 
-/// The physical candidates of the shared-segment candidate (Q60), keyed by
-/// the materialization of its segments, with the newest segment's id.
-fn segment_options(run: &Run) -> BTreeMap<Materialization, (String, LogicalASAPNodeId)> {
+/// The physical candidates of the shared-segment candidate (Q60) with
+/// `count` segments (Q67), keyed by the materialization of its segments,
+/// with the newest segment's id.
+fn segment_options(
+    run: &Run,
+    count: usize,
+) -> BTreeMap<Materialization, (String, LogicalASAPNodeId)> {
     let logical = run
         .logical
         .iter()
-        .find(|c| shares_segments(&c.dag, &c.query_roots))
+        .find(|c| shares_segments(&c.dag, &c.query_roots) && sketch_builds(&c.dag).len() == count)
         .expect("a candidate sharing window segments");
     run.physical_of(logical)
         .map(|p| {
@@ -197,23 +201,31 @@ fn stage2_a_at_rest_drops_the_ingestion_time_option() {
     assert_eq!(found, BTreeSet::from([QueryTimeKept, NotMaterialized]));
 }
 
-/// The shared segments (Q60, in place of the spec's EH) are five builds,
-/// each built once, read by all five queries through their merges and
-/// charged once, in every materialization: at query time for the one-off
-/// batch, and also at ingestion time when it repeats monthly (Q61).
+/// The shared segments (Q60, in place of the spec's EH), four split at the
+/// windows' boundaries or five on the 1-year grid (Q67), are each built
+/// once, read by all five queries through their merges and charged once,
+/// in every materialization: at query time for the one-off batch, and, for
+/// the even grid only, also at ingestion time when it repeats monthly
+/// (Q61). Uneven segments are no stream of panes.
 #[test]
 fn stage2_a_segments_are_built_once_for_all_consumers() {
-    for (workload, expected) in [
-        (once(), BTreeSet::from([NotMaterialized])),
-        (monthly(), BTreeSet::from([IngestionTime, NotMaterialized])),
+    for (workload, count, expected) in [
+        (once(), 4, BTreeSet::from([NotMaterialized])),
+        (once(), 5, BTreeSet::from([NotMaterialized])),
+        (monthly(), 4, BTreeSet::from([NotMaterialized])),
+        (
+            monthly(),
+            5,
+            BTreeSet::from([IngestionTime, NotMaterialized]),
+        ),
     ] {
         let run = run_promql(&workload);
-        let options = segment_options(&run);
+        let options = segment_options(&run, count);
         assert_eq!(options.keys().copied().collect::<BTreeSet<_>>(), expected);
         for (id, _) in options.values() {
             let p = run.physical(id);
             let segments = sketch_builds(&p.dag);
-            assert_eq!(segments.len(), 5, "{id}");
+            assert_eq!(segments.len(), count, "{id}");
             let read: BTreeSet<usize> = segments
                 .iter()
                 .flat_map(|(b, ..)| readers(&p.dag, &p.query_roots, *b))
 
@@ -711,10 +711,9 @@ fn realize(
             };
             build(child, coverage)?
         }
-        WindowForm::Tumbling { pane_ms }
-        | WindowForm::Segments {
-            segment_ms: pane_ms,
-        } => tumbling_state(&child, pane_ms, build)?,
+        WindowForm::Tumbling { .. } | WindowForm::Segments { .. } => {
+            tumbling_state(&child, window, build)?
+        }
     };
     let evaluation = match query {
         Some(query) => ASAPOp::SummaryEstimate {
 
@@ -37,8 +37,9 @@ pub enum Sharing {
     /// The shared-segment rule on top of the identical-expression rule
     /// ([`share_window_segments`]): queries over windows of one scan merge
     /// one shared summary per segment, and the identical segments are merged
-    /// after composition.
-    WindowSegments,
+    /// after composition. One variant per segmentation, by its number of
+    /// distinct segments (Q67).
+    WindowSegments { segments: usize },
 }
 
 impl Sharing {
@@ -62,8 +63,9 @@ pub struct SharingVariant<Id> {
 /// In every variant, the window-composition rule adds tumbling forms of
 /// mergeable alternatives for repeating queries (`demand[i]` is the demand
 /// of `roots[i]`; a root without one gets none). Last, the shared-segment
-/// variant when windows of one scan can share segments: only that form, not
-/// the tumbling forms, for the targets it groups.
+/// variants when windows of one scan can share segments, one per
+/// segmentation (Q67): only that form, not the tumbling forms, for the
+/// targets it groups.
 pub fn stage1_logical_candidates<Id: Clone>(
     roots: Vec<(Id, QueryRoot)>,
     metric_types: &BTreeMap<String, MetricType>,
@@ -93,9 +95,9 @@ pub fn stage1_logical_candidates<Id: Clone>(
     for variant in &mut variants {
         add_window_forms(&mut variant.inventory, demand);
     }
-    if let Some(inventory) = segments {
+    for (segments, inventory) in segments {
         variants.push(SharingVariant {
-            sharing: Sharing::WindowSegments,
+            sharing: Sharing::WindowSegments { segments },
             inventory,
         });
     }