diff --git a/crates/eps/src/ai_editing/tests.rs b/crates/eps/src/ai_editing/tests.rs
index bff3ed60a..529c54a5f 100644
--- a/crates/eps/src/ai_editing/tests.rs
+++ b/crates/eps/src/ai_editing/tests.rs
@@ -2259,6 +2259,58 @@ fn gradient_definitions_skip_the_names_of_swatches_and_patterns() {
     same_lines(&gradient_looks(&read_back(&doc)), &gradient_looks(&doc));
 }
 
+/// Each search for a new gradient name starts after the last name given, so naming many gradients
+/// tries each name once.
+#[test]
+fn gradient_names_are_searched_from_the_last_one_given() {
+    let mut taken: std::collections::HashSet<String> = ["Gradient 2", "Gradient 5"].map(String::from).into();
+    let (mut next, mut tries) = (1, 0);
+    let mut names = vec![];
+    for _ in 0..1000 {
+        let name = writer::free_gradient_name(&mut next, |name| {
+            tries += 1;
+            taken.contains(name)
+        });
+        taken.insert(name.clone());
+        names.push(name);
+    }
+    assert_eq!(names[..4], ["Gradient", "Gradient 3", "Gradient 4", "Gradient 6"]);
+    assert_eq!(names.last().map(String::as_str), Some("Gradient 1002"));
+    assert_eq!(tries, 1002);
+    assert_eq!(next, 1003);
+}
+
+/// Many gradients that no swatch names each get their own name, in order, skipping the names of
+/// swatches and patterns, and the same names when the document is written again.
+#[test]
+fn many_gradient_definitions_take_distinct_names_in_order() {
+    let mut doc = one_object(shapes::rectangle(Rect::new(0.0, 0.0, 10.0, 10.0)), vec![]);
+    let check = square(&mut doc, Rect::new(0.0, 0.0, 5.0, 5.0), Color::BLACK);
+    let mut taken = vectorcraft_doc::PatternDef::new("Gradient 7", vec![Arc::new(check)]);
+    taken.tile = Rect::new(0.0, 0.0, 10.0, 10.0);
+    doc.patterns.push(taken);
+    swatches_plus(&mut doc, vec![Swatch { name: "Gradient 3".into(), paint: solid(Color::rgb(0.0, 0.5, 0.0)), global: false, spot: false }]);
+    let layer = doc.layers[0].id;
+    for i in 0..300u16 {
+        let shade = f32::from(i) / 300.0;
+        let g = ramp(GradientKind::Linear, &[(0.0, Color::rgb(shade, 0.0, 0.0), 1.0, 0.5), (1.0, Color::rgb(0.0, 0.0, 1.0), 1.0, 0.5)]);
+        let x = f64::from(i % 20) * 10.0;
+        let y = f64::from(i / 20) * 6.0;
+        let n = painted(
+            &mut doc,
+            shapes::rectangle(Rect::new(x, y, x + 8.0, y + 5.0)),
+            vec![AppearanceItem::Fill(FillLayer::new(gradient_paint(g, None, None)))],
+        );
+        doc.insert(Some(layer), usize::MAX, n).unwrap();
+    }
+    let text = program(&doc);
+    let names: Vec<&str> = text.lines().filter_map(|l| l.strip_prefix("%AI5_BeginGradient: ")).collect();
+    let expected: Vec<String> =
+        std::iter::once("(Gradient)".to_string()).chain((2..=302).filter(|&i| i != 3 && i != 7).map(|i| format!("(Gradient {i})"))).collect();
+    assert_eq!(names, expected);
+    assert_eq!(text, program(&doc));
+}
+
 /// A hard edge: a linear gradient whose stops all lie at one place is written from its last stop,
 /// and reads back with each stop on its side.
 #[test]
diff --git a/crates/eps/src/ai_editing/writer.rs b/crates/eps/src/ai_editing/writer.rs
index 21c9a7068..78602b939 100644
--- a/crates/eps/src/ai_editing/writer.rs
+++ b/crates/eps/src/ai_editing/writer.rs
@@ -22,7 +22,7 @@ pub(super) use images::PAYLOAD;
 #[cfg(test)]
 pub(super) use images::{IMAGE_DEPTH, IMAGES, LINKS_EMBEDDED};
 #[cfg(test)]
-pub(super) use paints::{GRADIENT_MIX, GRADIENT_SHAPES, GRADIENTS, STOP_LINKS, SWATCH_PLACEMENT};
+pub(super) use paints::{GRADIENT_MIX, GRADIENT_SHAPES, GRADIENTS, STOP_LINKS, SWATCH_PLACEMENT, free_gradient_name};
 
 /// How deep symbols, live objects and pattern tiles may nest in one another (deeper art is left
 /// out). Groups and layers need no limit of their own: a document is never deeper than
@@ -141,6 +141,8 @@ pub(super) struct Writer<'a> {
     canvas: Affine,
     /// The gradient definitions, in the order they are first used.
     gradients: Vec<paints::GradientDef>,
+    /// The number the next new gradient definition's name tries first (1 is `Gradient`).
+    next_gradient: u64,
     /// The pattern definitions written, each after those its art uses.
     patterns: Vec<paints::PatternOut>,
     /// The patterns whose definitions are being written.
@@ -167,6 +169,7 @@ impl<'a> Writer<'a> {
             tile_grad_space: Affine::translate((-origin.0, origin.1)),
             canvas: Affine::new([1.0, 0.0, 0.0, -1.0, origin.0, origin.1]) * to_art,
             gradients: vec![],
+            next_gradient: 1,
             patterns: vec![],
             defining: vec![],
             in_palette: false,
diff --git a/crates/eps/src/ai_editing/writer/paints.rs b/crates/eps/src/ai_editing/writer/paints.rs
index 09af15e18..d805d9d0e 100644
--- a/crates/eps/src/ai_editing/writer/paints.rs
+++ b/crates/eps/src/ai_editing/writer/paints.rs
@@ -49,6 +49,18 @@ fn writable_gradient(g: &Gradient) -> bool {
     g.kind != GradientKind::Freeform && !g.stops.is_empty() && g.stops.len() <= crate::import::MAX_GRADIENT_STOPS
 }
 
+/// The first name of `Gradient`, `Gradient 2`, `Gradient 3`, … numbered `next` or higher for
+/// which `taken` returns false (`Gradient` is number 1); `next` is set past it.
+pub(in crate::ai_editing) fn free_gradient_name(next: &mut u64, mut taken: impl FnMut(&str) -> bool) -> String {
+    loop {
+        let name = if *next <= 1 { String::from("Gradient") } else { format!("Gradient {next}") };
+        *next = next.saturating_add(1);
+        if !taken(&name) {
+            return name;
+        }
+    }
+}
+
 /// The stops of `g` in order of their place on the ramp.
 fn sorted_stops(g: &Gradient) -> Vec<&GradientStop> {
     let mut stops: Vec<&GradientStop> = g.stops.iter().collect();
@@ -72,7 +84,8 @@ impl Writer<'_> {
 
     /// The name of the definition of `g`: the swatch `swatch` names when it is a gradient swatch
     /// of the same gradient, else a definition of the same gradient that no swatch names, else a
-    /// new one (`Gradient`, numbered when taken by another definition, a swatch or a pattern).
+    /// new one ([`free_gradient_name`]: `Gradient`, numbered when taken by another definition, a
+    /// swatch or a pattern).
     fn define_gradient(&mut self, g: &Gradient, swatch: Option<&str>) -> String {
         let used = !self.in_palette;
         let found = swatch
@@ -82,16 +95,10 @@ impl Writer<'_> {
             d.used |= used;
             return d.name.clone();
         }
-        let doc = self.doc;
-        let taken = |gradients: &[GradientDef], name: &str| {
+        let (doc, gradients) = (self.doc, &self.gradients);
+        let name = free_gradient_name(&mut self.next_gradient, |name| {
             gradients.iter().any(|d| d.name == name) || doc.swatch(name).is_some() || doc.pattern(name).is_some()
-        };
-        let mut name = String::from("Gradient");
-        let mut i = 1;
-        while taken(&self.gradients, &name) {
-            i += 1;
-            name = format!("Gradient {i}");
-        }
+        });
         self.gradients.push(GradientDef { name: name.clone(), gradient: g.clone(), swatch: false, used });
         name
     }