diff --git a/CHANGELOG.md b/CHANGELOG.md
index 7591b10..7c95858 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -7,6 +7,11 @@ and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0
 
 ## [Unreleased]
 
+- reject a universe that mixes a literal with a broad type (e.g.
+  `"a" | \`x-${number}\``), which the finite-literal gate let through
+- narrow the tagged-union matcher's handler parameters for a union-valued or
+  optional discriminant, and its fallback to the unhandled tags, instead of
+  passing `never`
 - prune the backlog of completed items and record the TS 7 LSP shutdown
   workaround in development/testing.md
 
diff --git a/backlog.tasks b/backlog.tasks
index d4e1650..33d38a4 100644
--- a/backlog.tasks
+++ b/backlog.tasks
@@ -17,10 +17,10 @@ v1.0:
     ☐ Achieve 100% branch coverage on `src/index.ts`
 
 Matcher:
-☐ when using a union type as a property, the current behavior of tagged union matcher is
+✔ when using a union type as a property, the current behavior of tagged union matcher is @done
     to pass never to handler parameters
     → new matcher function needed or can be fixed in tagged union matcher
-    ☐ optional discriminant (`{ type?: "x" }`) is the same hole: the boolean/nullish change now admits the `undefined` tag, so the factory accepts the key, but `Extract<T, Record<K, V>>` still passes `never` to both the `x` and `undefined` handlers
+    ✔ optional discriminant (`{ type?: "x" }`) is the same hole: the boolean/nullish change now admits the `undefined` tag, so the factory accepts the key, but `Extract<T, Record<K, V>>` still passes `never` to both the `x` and `undefined` handlers @done
 
 Documentation:
 ☐ Bring README.md back to its previous form — synopsis and examples restored, in the correct place
diff --git a/development/library.md b/development/library.md
index b7b6869..3254d4a 100644
--- a/development/library.md
+++ b/development/library.md
@@ -145,6 +145,10 @@ inversion: the handler parameter is the member(s) of `T` whose `PatternKey` is
   object satisfy the exhaustive overload and reaches the `dispatch` throw.
   Rejecting at the boundary avoids threading an open/closed branch through
   `Handlers`, `Fallback` and `MustBePartial`.
+- **The finite-literal predicate is `IsLiteral<PatternKey<T>> extends true`.**
+  `IsLiteral` is `boolean` for a union that mixes a literal with a broad type
+  (`"a" | \`x-${number}\``), so `extends false`would treat the mix as
+supported;`extends true` is the check that rejects it.
 - **Collisions are rejected, not merged.** `Member<T, K>` would be sound (the
   handler gets the union), but the API is one handler per member; rejecting
   keeps `Member` a singleton and the remainder exact.
@@ -208,16 +212,27 @@ The key is a separate call because `K` is inferred from its literal argument and
 
 #### Why
 
-- **Same fallback/remainder machinery as the primitive-union matcher.** `HandledMembers`
-  maps the handled tags to their members and `Exclude<T, …>` is the fallback's
-  parameter; the redundant-fallback guard is the same F-bounded constraint. Only
-  the "universe" changes — `T`'s members instead of primitive values.
+- **Same fallback/remainder machinery as the primitive-union matcher.** `HandledTags`
+  recovers the tag values the map handled (`Member<Tags, keyof Handled>`) and
+  `Narrowed<T, K, Exclude<Tags, …>>` is the fallback's parameter; the
+  redundant-fallback guard is the same F-bounded constraint. Only the "universe"
+  changes — `T`'s members instead of primitive values.
 - **`T extends object`, not `Record<PropertyKey, unknown>`.** An `interface` has
   no implicit index signature, so the `Record` constraint would reject
   interface-based unions. The runtime reads the tag off `object` with one
   assertion, the tagged twin of the primitive-union dispatch's `shape as string | number`.
-- **`MapTaggedUnion` distributes with `Extract`.** A duplicated tag yields a
-  union of members instead of dropping one.
+- **`Narrowed` distributes over `T`, narrowing `K` to the tag.** A member whose
+  `K` cannot take the tag drops out; a duplicated tag yields a union of members
+  instead of dropping one. `T[K] extends V` returns the exact member (a
+  discriminated union's declared interface) untouched, so the mapped form only
+  handles a property that is itself a union.
+- **A union-valued or optional discriminant is supported.** A single shape whose
+  property is a union (`{ color: "red" | "green" | "blue" }`) is narrowed per
+  handler instead of being passed `never`. Matching a defined tag on an optional
+  property (`{ type?: "x" }`) proves the key is present, so it becomes required
+  (`{ type: "x" }`); the `undefined` tag narrows it to `{ type?: never }` under
+  `exactOptionalPropertyTypes` (absence) or `{ type?: undefined }` when the
+  property explicitly admits `undefined`.
 - **A `boolean` / `null` / `undefined` tag goes through the shared
   `PatternKey` / `Member` projection.** `Discriminated` admits those tags
   (they are in `Tag`), but they cannot key a mapped type, so the handler map is
diff --git a/src/matcher-shared.ts b/src/matcher-shared.ts
index fead4cc..ecb1732 100644
--- a/src/matcher-shared.ts
+++ b/src/matcher-shared.ts
@@ -62,11 +62,11 @@ export type Collisions<T> = Extract<T, Stringified<T>>;
 // universe is a finite union of literals with no value/stringification
 // collision: only then can `PatternKey` be inverted unambiguously.
 export type UnsupportedReason<T extends Matchable> =
-    IsLiteral<PatternKey<T>> extends false
-        ? "broad types like string, number and template literals are not supported"
-        : [Collisions<T>] extends [never]
-          ? never
-          : `a value and its stringification collide. Value is "${Collisions<T> & string}"`;
+    IsLiteral<PatternKey<T>> extends true
+        ? [Collisions<T>] extends [never]
+            ? never
+            : `a value and its stringification collide. Value is "${Collisions<T> & string}"`
+        : "broad types like string, number and template literals are not supported";
 
 // The diagnostic for an unsupported universe. Extends `HandlerMap` so the
 // implementation's `handlers: HandlerMap` stays assignable when the gate is
diff --git a/src/primitive-union.test.ts b/src/primitive-union.test.ts
index 17c2ac0..bd307d4 100644
--- a/src/primitive-union.test.ts
+++ b/src/primitive-union.test.ts
@@ -528,6 +528,8 @@ test("getPrimitiveUnionMatcher: a broad universe is rejected", () => {
     getPrimitiveUnionMatcher<number>()({ 1: () => 1 });
     // @ts-expect-error a template literal is open
     getPrimitiveUnionMatcher<`a${string}`>()({ a: () => 1 });
+    // @ts-expect-error a literal mixed with a template literal is open
+    getPrimitiveUnionMatcher<"a" | `x-${number}`>()({ a: () => 1 });
     // @ts-expect-error `string | boolean` is open because of `string`
     getPrimitiveUnionMatcher<string | boolean>()({ true: () => 1 });
     // @ts-expect-error the widening factory rejects broad universes too
diff --git a/src/tagged-union.test.ts b/src/tagged-union.test.ts
index cadb6ed..bd6f4b1 100644
--- a/src/tagged-union.test.ts
+++ b/src/tagged-union.test.ts
@@ -892,6 +892,358 @@ test("getTaggedUnionMatcher: the maximum illegal tag universe is rejected", () =
     });
 });
 
+// ============================================================================
+// Property unions
+// ============================================================================
+
+test("property union: narrows each handler", () => {
+    // Arrange — one shape, not a union of variants; `color` is a union.
+    interface Paint {
+        readonly color: "red" | "green" | "blue";
+        readonly value: number;
+    }
+    const factory = getTaggedUnionMatcher<Paint>()("color");
+
+    // Act
+    const describe = factory({
+        red: (s) => {
+            expectTypeOf(s).toEqualTypeOf<{
+                readonly color: "red";
+                readonly value: number;
+            }>();
+            assert.equal(s.color, "red");
+            return s.value;
+        },
+        green: (s) => {
+            expectTypeOf(s).toEqualTypeOf<{
+                readonly color: "green";
+                readonly value: number;
+            }>();
+            assert.equal(s.color, "green");
+            return s.value;
+        },
+        blue: (s) => {
+            expectTypeOf(s).toEqualTypeOf<{
+                readonly color: "blue";
+                readonly value: number;
+            }>();
+            assert.equal(s.color, "blue");
+            return s.value;
+        },
+    });
+
+    // Assert
+    expectTypeOf(describe).toEqualTypeOf<(shape: Paint) => number>();
+    assert.equal(describe({ color: "red", value: 1 }), 1);
+    assert.equal(describe({ color: "green", value: 2 }), 2);
+    assert.equal(describe({ color: "blue", value: 3 }), 3);
+});
+
+test("property union: a fallback receives the unhandled values", () => {
+    // Arrange
+    interface Paint {
+        color: "red" | "green" | "blue";
+        value: number;
+    }
+    const factory = getTaggedUnionMatcher<Paint>()("color");
+
+    // Act
+    const describe = factory(
+        {
+            red: (s) => {
+                expectTypeOf(s).toEqualTypeOf<{
+                    color: "red";
+                    value: number;
+                }>();
+                assert.equal(s.color, "red");
+                return 1 as const;
+            },
+        },
+        (s) => {
+            expectTypeOf(s).toEqualTypeOf<{
+                color: "green" | "blue";
+                value: number;
+            }>();
+            assert.ok(s.color === "green" || s.color === "blue");
+            return 2 as const;
+        },
+    );
+
+    // Assert
+    expectTypeOf(describe).toEqualTypeOf<(shape: Paint) => 1 | 2>();
+    assert.equal(describe({ color: "red", value: 1 }), 1);
+    assert.equal(describe({ color: "green", value: 2 }), 2);
+    assert.equal(describe({ color: "blue", value: 3 }), 2);
+});
+
+test("property union: a numeric property narrows each handler", () => {
+    // Arrange
+    interface Version {
+        code: 1 | 2 | 3;
+        value: number;
+    }
+    const factory = getTaggedUnionMatcher<Version>()("code");
+
+    // Act
+    const describe = factory({
+        1: (s) => {
+            expectTypeOf(s).toEqualTypeOf<{ code: 1; value: number }>();
+            assert.equal(s.code, 1);
+            return 1;
+        },
+        2: (s) => {
+            expectTypeOf(s).toEqualTypeOf<{ code: 2; value: number }>();
+            assert.equal(s.code, 2);
+            return 2;
+        },
+        3: (s) => {
+            expectTypeOf(s).toEqualTypeOf<{ code: 3; value: number }>();
+            assert.equal(s.code, 3);
+            return 3;
+        },
+    });
+
+    // Assert
+    expectTypeOf(describe).toEqualTypeOf<(shape: Version) => number>();
+    assert.equal(describe({ code: 1, value: 10 }), 1);
+    assert.equal(describe({ code: 2, value: 20 }), 2);
+    assert.equal(describe({ code: 3, value: 30 }), 3);
+});
+
+test("property union: an optional property is required for a defined tag", () => {
+    // Arrange — `type?` is `"x" | undefined`; both handlers must be typed,
+    // and matching `"x"` proves the key is present, so `type` becomes required.
+    interface Maybe {
+        type?: "x";
+        value: number;
+    }
+    const factory = getTaggedUnionMatcher<Maybe>()("type");
+
+    // Act
+    const describe = factory({
+        x: (s) => {
+            expectTypeOf(s).toEqualTypeOf<{ type: "x"; value: number }>();
+            assert.equal(s.type, "x");
+            return 1;
+        },
+        undefined: (s) => {
+            // The input only allows absence (exactOptionalPropertyTypes), so
+            // the `undefined` tag narrows the key to never-present.
+            expectTypeOf(s).toEqualTypeOf<{ type?: never; value: number }>();
+            assert.equal(s.type, undefined);
+            return 2;
+        },
+    });
+
+    // Assert
+    expectTypeOf(describe).toEqualTypeOf<(shape: Maybe) => number>();
+    assert.equal(describe({ type: "x", value: 1 }), 1);
+    assert.equal(describe({ value: 2 }), 2);
+});
+
+test("property union: an optional fallback receives the undefined tag", () => {
+    // Arrange
+    interface Maybe {
+        type?: "x";
+        value: number;
+    }
+    const factory = getTaggedUnionMatcher<Maybe>()("type");
+
+    // Act
+    const describe = factory(
+        {
+            x: (s) => {
+                expectTypeOf(s).toEqualTypeOf<{ type: "x"; value: number }>();
+                assert.equal(s.type, "x");
+                return 1 as const;
+            },
+        },
+        (s) => {
+            expectTypeOf(s).toEqualTypeOf<{
+                type?: never;
+                value: number;
+            }>();
+            assert.equal(s.type, undefined);
+            return 2 as const;
+        },
+    );
+
+    // Assert
+    expectTypeOf(describe).toEqualTypeOf<(shape: Maybe) => 1 | 2>();
+    assert.equal(describe({ type: "x", value: 1 }), 1);
+    assert.equal(describe({ value: 2 }), 2);
+});
+
+test("property union: explicit `undefined` stays distinct from absence", () => {
+    // Arrange — `type?: "x" | undefined` admits an explicit `undefined`, unlike
+    // the bare optional above; the two representations must not be conflated.
+    interface Explicit {
+        type?: "x" | undefined;
+        value: number;
+    }
+    const factory = getTaggedUnionMatcher<Explicit>()("type");
+
+    // Act
+    const describe = factory({
+        x: (s) => {
+            expectTypeOf(s).toEqualTypeOf<{ type: "x"; value: number }>();
+            assert.equal(s.type, "x");
+            return 1;
+        },
+        undefined: (s) => {
+            expectTypeOf(s).toEqualTypeOf<{
+                type?: undefined;
+                value: number;
+            }>();
+            assert.equal(s.type, undefined);
+            return 2;
+        },
+    });
+
+    // Assert
+    expectTypeOf(describe).toEqualTypeOf<(shape: Explicit) => number>();
+    assert.equal(describe({ type: "x", value: 1 }), 1);
+    assert.equal(describe({ type: undefined, value: 2 }), 2);
+    assert.equal(describe({ value: 3 }), 2);
+});
+
+test("property union: a union-valued member is split, not dropped", () => {
+    // A member whose tag is itself a union sits beside a singleton-tag member.
+    // Selecting members with `Extract<T, Record<K, V>>` keeps only the
+    // singleton; the distributive narrowing must keep both sides of the
+    // union-valued member.
+    // Arrange
+    type Mixed = { kind: "a"; a: number } | { kind: "a" | "b"; b: number };
+    const factory = getTaggedUnionMatcher<Mixed>()("kind");
+
+    // Act
+    const describe = factory({
+        a: (s) => {
+            expectTypeOf(s).toEqualTypeOf<
+                { kind: "a"; a: number } | { kind: "a"; b: number }
+            >();
+            assert.equal(s.kind, "a");
+            return 1;
+        },
+        b: (s) => {
+            expectTypeOf(s).toEqualTypeOf<{ kind: "b"; b: number }>();
+            assert.equal(s.kind, "b");
+            return 2;
+        },
+    });
+
+    // Assert
+    expectTypeOf(describe).toEqualTypeOf<(shape: Mixed) => number>();
+    assert.equal(describe({ kind: "a", a: 1 }), 1);
+    assert.equal(describe({ kind: "a", b: 2 }), 1);
+    assert.equal(describe({ kind: "b", b: 3 }), 2);
+});
+
+test("property union: getTaggedUnionMatcherW widens the returns", () => {
+    // Arrange
+    interface Paint {
+        color: "red" | "green" | "blue";
+        value: number;
+    }
+    const factory = getTaggedUnionMatcherW<Paint>()("color");
+
+    // Act
+    const describe = factory({
+        red: (s) => {
+            expectTypeOf(s).toEqualTypeOf<{ color: "red"; value: number }>();
+            assert.equal(s.color, "red");
+            return "r" as const;
+        },
+        green: (s) => {
+            expectTypeOf(s).toEqualTypeOf<{ color: "green"; value: number }>();
+            assert.equal(s.color, "green");
+            return 1 as const;
+        },
+        blue: (s) => {
+            expectTypeOf(s).toEqualTypeOf<{ color: "blue"; value: number }>();
+            assert.equal(s.color, "blue");
+            return true as const;
+        },
+    });
+
+    // Assert
+    expectTypeOf(describe).toEqualTypeOf<(shape: Paint) => "r" | 1 | true>();
+    assert.equal(describe({ color: "red", value: 1 }), "r");
+    assert.equal(describe({ color: "green", value: 2 }), 1);
+    assert.equal(describe({ color: "blue", value: 3 }), true);
+});
+
+test("property union: enforces its contract", () => {
+    // Arrange
+    interface Paint {
+        color: "red" | "green" | "blue";
+        value: number;
+    }
+    const factory = getTaggedUnionMatcher<Paint>()("color");
+
+    // Act / Assert — the calls below must not compile
+    factory({
+        red: () => 1,
+        green: () => 2,
+        blue: () => 3,
+        // @ts-expect-error `yellow` is not a value of `color`
+        yellow: () => 4,
+    });
+    // @ts-expect-error a gap without a fallback is not exhaustive
+    factory({ red: () => 1 });
+    // @ts-expect-error a fallback is redundant once every value is handled
+    factory({ red: () => 1, green: () => 2, blue: () => 3 }, () => 0);
+});
+
+test("property union: a broad value in the union is rejected", () => {
+    // Arrange — the property is a union, but not a finite union of literals: a
+    // template literal cannot be proven exhaustive.
+    interface Template {
+        kind: "a" | `x-${number}`;
+        a: number;
+    }
+
+    // Act / Assert — the calls below must not compile
+    // @ts-expect-error a template literal is not a finite literal
+    getTaggedUnionMatcher<Template>()("kind")({ a: () => 1 });
+    // @ts-expect-error the widening factory rejects broad values too
+    getTaggedUnionMatcherW<Template>()("kind")({ a: () => 1 });
+});
+
+test("property union: a value/stringification collision is rejected", () => {
+    // Arrange — one property holding both a value and its stringification; both
+    // would key the same handler.
+    interface BooleanColliding {
+        kind: "true" | true;
+        a: number;
+    }
+    interface NumericColliding {
+        kind: "1" | 1;
+        a: number;
+    }
+
+    // Act / Assert — the calls below must not compile
+    // @ts-expect-error `true` collides with the string tag `"true"`
+    getTaggedUnionMatcher<BooleanColliding>()("kind")({ true: () => 1 });
+    // @ts-expect-error the number tag `1` collides with the string tag `"1"`
+    getTaggedUnionMatcher<NumericColliding>()("kind")({ "1": () => 1 });
+    // @ts-expect-error the widening factory rejects the collision too
+    getTaggedUnionMatcherW<BooleanColliding>()("kind")({ true: () => 1 });
+});
+
+test("property union: a collision is rejected beside legal members", () => {
+    // Arrange — the map is otherwise complete, so only the collision can make
+    // it invalid.
+    interface Colliding {
+        kind: "true" | true | "a";
+        a: number;
+    }
+
+    // Act / Assert — the call below must not compile
+    // @ts-expect-error `true` collides with the string tag `"true"`
+    getTaggedUnionMatcher<Colliding>()("kind")({ true: () => 1, a: () => 2 });
+});
+
 // ============================================================================
 // Dispatch — runtime behavior
 // ============================================================================
@@ -935,6 +1287,7 @@ interface LabelsProbe {
     readonly tail?: string;
     readonly typeName?: string;
     readonly typeSource?: string;
+    readonly key?: string;
 }
 
 const labelsFor = ({
@@ -944,6 +1297,7 @@ const labelsFor = ({
     tail = "",
     typeName = "Shape",
     typeSource = SHAPE_SOURCE,
+    key = "kind",
 }: LabelsProbe): Promise<readonly string[]> => {
     const session = new LspSession(REPO_ROOT);
     const target: CompletionTarget = {
@@ -951,7 +1305,7 @@ const labelsFor = ({
         source: [
             `import { ${factory} } from "./index.ts";`,
             typeSource,
-            `const m = ${factory}<${typeName}>()("kind")({`,
+            `const m = ${factory}<${typeName}>()("${key}")({`,
             body,
             `}${tail});`,
             "",
@@ -1038,6 +1392,91 @@ test("autocomplete: boolean and nullish tags are offered by name", () => {
     });
 });
 
+// A single shape whose property is a union: the popup is still keyed by the
+// property's values, not by the members of a union type.
+const PALETTE_SOURCE = `interface Palette { color: "red" | "green" | "blue"; value: number }`;
+const FLAG_SOURCE = `interface Flag { kind: true | false | null | undefined; a: number }`;
+
+test("property union: autocomplete offers the property's values", () => {
+    // Arrange
+    const name = "property_union_fresh";
+
+    // Act
+    const labels = labelsFor({
+        name,
+        factory: "getTaggedUnionMatcher",
+        typeName: "Palette",
+        typeSource: PALETTE_SOURCE,
+        key: "color",
+        body: "    /*COMPLETE*/",
+    });
+
+    // Assert
+    return labels.then((result) => {
+        assert.deepEqual([...result], ["blue", "green", "red"]);
+    });
+});
+
+test("property union: autocomplete drops a handled value", () => {
+    // Arrange
+    const name = "property_union_after_key";
+
+    // Act
+    const labels = labelsFor({
+        name,
+        factory: "getTaggedUnionMatcher",
+        typeName: "Palette",
+        typeSource: PALETTE_SOURCE,
+        key: "color",
+        body: "    red: () => 1,\n    /*COMPLETE*/",
+    });
+
+    // Assert
+    return labels.then((result) => {
+        assert.deepEqual([...result], ["blue", "green"]);
+    });
+});
+
+test("property union: autocomplete makes the remaining values optional with a fallback", () => {
+    // Arrange
+    const name = "property_union_with_fallback";
+
+    // Act
+    const labels = labelsFor({
+        name,
+        factory: "getTaggedUnionMatcher",
+        typeName: "Palette",
+        typeSource: PALETTE_SOURCE,
+        key: "color",
+        body: "    /*COMPLETE*/",
+        tail: ", () => 0",
+    });
+
+    // Assert
+    return labels.then((result) => {
+        assert.deepEqual([...result], ["blue?", "green?", "red?"]);
+    });
+});
+
+test("property union: autocomplete offers boolean and nullish values by name", () => {
+    // Arrange
+    const name = "property_union_boolean_nullish";
+
+    // Act
+    const labels = labelsFor({
+        name,
+        factory: "getTaggedUnionMatcher",
+        typeName: "Flag",
+        typeSource: FLAG_SOURCE,
+        body: "    /*COMPLETE*/",
+    });
+
+    // Assert
+    return labels.then((result) => {
+        assert.deepEqual([...result], ["false", "null", "true", "undefined"]);
+    });
+});
+
 // `Mixed` has two common keys, but `id`'s value is not a tag, so only `kind`
 // may serve as the discriminant.
 const MIXED_SOURCE = `type Mixed = { id: Date; kind: "a"; a: number } | { id: Date; kind: "b"; b: number };`;
diff --git a/src/tagged-union.ts b/src/tagged-union.ts
index fabfb07..b43e9ad 100644
--- a/src/tagged-union.ts
+++ b/src/tagged-union.ts
@@ -1,4 +1,4 @@
-import type { Exact, UnknownRecord } from "type-fest";
+import type { Exact, SetRequired, UnknownRecord } from "type-fest";
 
 import type {
     HandlerMap,
@@ -28,34 +28,49 @@ type Discriminated<T extends object> = {
     [K in keyof T]: T[K] extends Matchable ? K : never;
 }[keyof T];
 
-// The member(s) of `T` tagged `V`. `Extract` distributes over the union, so a
+// The member(s) of `T` narrowed to the tag(s) `V`. Distributes over `T`, so a
 // duplicated tag maps to a union of members rather than silently dropping one.
+// A member whose `K` cannot take `V` drops out; the rest have `K` narrowed to
+// `Extract<T[K], V>`. `T[K] extends V` keeps an exact member (a discriminated
+// union's declared interface) untouched, so only a property that is itself a
+// union — or optional — goes through the mapped form. `SetRequired` makes `K`
+// required unless `V` admits `undefined`: a defined tag yields `{ type: "x" }`,
+// while the `undefined` tag keeps `{ type?: never }` (absence) or
+// `{ type?: undefined }` when the property admits an explicit `undefined`.
 // Keyed by `PatternKey`, so `boolean`/`null`/`undefined` tags can key a mapped
-// type; `Member` inverts the projection against the tag set to recover the
-// member.
+// type; `Member` inverts the projection against the tag set to recover the tag.
+type Narrowed<T extends object, K extends keyof T, V> = T extends object
+    ? [Extract<T[K], V>] extends [never]
+        ? never
+        : T[K] extends V
+          ? T
+          : undefined extends V
+            ? { [P in keyof T]: P extends K ? Extract<T[P], V> : T[P] }
+            : SetRequired<
+                  { [P in keyof T]: P extends K ? Extract<T[P], V> : T[P] },
+                  K
+              >
+    : never;
+
 type MapTaggedUnion<T extends object, K extends keyof T> = {
-    [P in PatternKey<Tags<T, K>>]: Extract<T, Record<K, Member<Tags<T, K>, P>>>;
+    [P in PatternKey<Tags<T, K>>]: Narrowed<T, K, Member<Tags<T, K>, P>>;
 };
 
 type Handlers<T extends object, K extends keyof T, R> = {
     [P in PatternKey<Tags<T, K>>]: UnaryFn<MapTaggedUnion<T, K>[P], R>;
 };
 
-// The members `Handled` covers. Mapping over the projected keys keeps every
-// index within `MapTaggedUnion`'s keys, and the conditional drops a stray key
-// outside `T` so it cannot widen the remainder. The remainder is
-// `Exclude<T, …>`, mirroring the primitive-union matcher's
-// `Exclude<T, Member<T, keyof Handled>>`.
-type HandledMembers<T extends object, K extends keyof T, Handled> = {
-    [P in PatternKey<Tags<T, K>>]: P extends keyof Handled
-        ? MapTaggedUnion<T, K>[P]
-        : never;
-}[PatternKey<Tags<T, K>>];
+// The tag values whose `PatternKey` is handled.
+type HandledTags<T extends object, K extends keyof T, Handled> = Member<
+    Tags<T, K>,
+    keyof Handled
+>;
 
 // The fallback is a *second argument*, not a property of the handler map, so
-// its parameter can be the remainder the map left uncovered. See development/library.md.
+// its parameter can be the remainder the map left uncovered: the members
+// narrowed to the tags the map did not handle. See development/library.md.
 type Fallback<T extends object, K extends keyof T, Handled, R> = UnaryFn<
-    Exclude<T, HandledMembers<T, K, Handled>>,
+    Narrowed<T, K, Exclude<Tags<T, K>, HandledTags<T, K, Handled>>>,
     R
 >;