diff --git a/context/code/source.md b/context/code/source.md
index 5eb7d12f..164b3a34 100644
--- a/context/code/source.md
+++ b/context/code/source.md
@@ -129,5 +129,6 @@ backend a Revenue of 128 and the other 38, and the second then disagreed with it
 `Summary::from_parts(Numbers::from_parts(count, exact_sum, least, greatest)?, Agreement)?`. `Agreement`
 is what the non-number values hold: `None`, `One(v)` or `Mixed`. Both constructors return a `Result`
 and **refuse** a state no sequence of values could produce (extremes with no numbers, a NaN extreme,
-least above greatest, one extreme without the other, a number among the "others"), because repairing
+least above greatest, one extreme without the other, a sum with no numbers, no extremes beside a sum
+that is not NaN, a number among the "others"), because repairing
 it would hide an engine's bug.
diff --git a/context/test-evidence/core.md b/context/test-evidence/core.md
index b4969633..d9842ddd 100644
--- a/context/test-evidence/core.md
+++ b/context/test-evidence/core.md
@@ -2317,7 +2317,8 @@ lost). The first two also fail `aggregating_differing_text_values_is_an_explicit
 Review of #356 (Opus): the pool gained `1e16`, `1.0`, `-1e16`, `0.1` and `0.2`. A merge rounding each part's
 sum before pooling, keeping the zero sign, changed 514 production answers while the workspace passed;
 that rival now fails this test, `numbers_pool_what_the_formula_aggregate_pools` (a NaN-free split now,
-compared as bits), `summaries_built_from_parts_merge_as_the_union_does` and the fixture test below, by name.
+compared as bits), `summaries_built_from_parts_merge_as_the_union_does` and the fixture test below, by
+name: four tests, for each of the three round-then-pool variants.
 
 A read of several parts is a merge. The test merges every ordered pair of sequences of up to two
 values and compares with the reference of the concatenation. Rivals: the least not merged, the
@@ -2426,12 +2427,14 @@ Review of #356 (Opus, Codex): `from_parts` repaired some impossible states and a
 The earlier entry's claim that a NaN-extreme rival failed `parts_that_contradict_themselves_...` was
 false: that test passed with the filter removed. Rivals now, each failing this test alone: NaN extreme
 allowed, least above greatest allowed, extremes with no numbers allowed, a number accepted among the
-others.
+others, a sum with no numbers allowed, and no extremes beside a sum that is not NaN allowed.
 
 ### the_exact_sum_of_a_pool_of_cancelling_values_is_not_the_sum_of_rounded_parts
 
 An exactness test needs inputs whose float sum is inexact and no NaN. `1e16 + 1.0 - 1e16` rounded part
-by part is `0.0`; pooled exactly it is `1.0`. This pins the fixture rather than the code.
+by part is `0.0`; pooled exactly it is `1.0`. The merge runs in both orders, so whichever part a rival
+rounds is the inexact one: rounding only the other part, only self, or both each fail it (the first
+version merged one way and a rival rounding only the exact part passed it; Opus caught that).
 
 ### keys_of_a_store_are_its_cells_keys_without_the_values
 
diff --git a/crates/improvise-core/src/model/source.rs b/crates/improvise-core/src/model/source.rs
index b36a8e55..740489b4 100644
--- a/crates/improvise-core/src/model/source.rs
+++ b/crates/improvise-core/src/model/source.rs
@@ -70,7 +70,13 @@ impl Numbers {
         greatest: Option<f64>,
     ) -> Result<Self, BadParts> {
         let bad = |why| Err(BadParts(why));
+        if count == 0 && sum != ExactSum::new() {
+            return bad("a sum with no numbers");
+        }
         match (least, greatest) {
+            (None, None) if count > 0 && !sum.value().is_nan() => {
+                return bad("no extremes but a sum that is not NaN");
+            }
             (None, None) => {}
             (Some(_), Some(_)) if count == 0 => return bad("extremes with no numbers"),
             (Some(l), Some(g)) if l.is_nan() || g.is_nan() => return bad("a NaN extreme"),
@@ -535,13 +541,15 @@ mod tests {
     #[test]
     fn the_exact_sum_of_a_pool_of_cancelling_values_is_not_the_sum_of_rounded_parts() {
         let rounded = [1e16, 1.0].iter().sum::<f64>() + [-1e16].iter().sum::<f64>();
-        let mut pooled = Numbers::of([1e16, 1.0]);
-        pooled.merge(&Numbers::of([-1e16]));
         assert_eq!(
             rounded, 0.0,
-            "the fixture is inexact: summing rounded parts loses the 1"
+            "the fixture is inexact: rounded parts lose the 1"
         );
-        assert_eq!(pooled.total(), 1.0);
+        for (first, second) in [([1e16, 1.0], [-1e16, 0.0]), ([-1e16, 0.0], [1e16, 1.0])] {
+            let mut pooled = Numbers::of(first);
+            pooled.merge(&Numbers::of(second));
+            assert_eq!(pooled.total(), 1.0, "{first:?} then {second:?}");
+        }
     }
 
     fn pair(c: &str, i: &str) -> (String, String) {
@@ -854,7 +862,19 @@ mod tests {
             refused(Numbers::from_parts(2, sum(), Some(1.0), None)),
             "one extreme without the other"
         );
-        let ok = Numbers::from_parts(2, sum(), None, None).unwrap();
+        let mut one = ExactSum::new();
+        one.add(1.0, 1);
+        assert_eq!(
+            refused(Numbers::from_parts(0, one.clone(), None, None)),
+            "a sum with no numbers"
+        );
+        assert_eq!(
+            refused(Numbers::from_parts(2, one, None, None)),
+            "no extremes but a sum that is not NaN"
+        );
+        let mut nan = ExactSum::new();
+        nan.add(f64::NAN, 1);
+        let ok = Numbers::from_parts(2, nan, None, None).unwrap();
         assert!(
             ok.reduce(&AggFunc::Min).unwrap().is_nan(),
             "all NaN: no extremes, a count"