diff --git a/packages/downgrader/tests/v3.1-to-v3.0.test.ts b/packages/downgrader/tests/v3.1-to-v3.0.test.ts
index 9b9d80a..629c267 100644
--- a/packages/downgrader/tests/v3.1-to-v3.0.test.ts
+++ b/packages/downgrader/tests/v3.1-to-v3.0.test.ts
@@ -495,6 +495,7 @@ describe('downgradeSchemaV31ToV30', () => {
       ])
       expect(downgradeSchemaV31ToV30({ type: 'null', enum: [null, 'a'] })).toEqual(onlyNull)
       expect(downgradeSchemaV31ToV30({ enum: [null] })).toEqual(onlyNull)
+      expect(downgradeSchemaV31ToV30({ enum: [null, 'a'] })).toEqual({ enum: [null, 'a'] })
       // Only null may match the type, and the enum rules it out.
       expect(downgradeSchemaV31ToV30({ type: 'null', enum: ['a'] })).toEqual({
         enum: ['a'],
@@ -624,6 +625,11 @@ describe('downgradeSchemaV31ToV30', () => {
       nullable: true,
       enum: [null],
     })
+    // As Zod emits `z.literal(5)`: the written type stays, though 5 is an integer.
+    expect(downgradeSchemaV31ToV30({ type: 'number', const: 5 })).toEqual({
+      type: 'number',
+      enum: [5],
+    })
   })
 
   it('types a const with no type after its value', () => {
@@ -686,6 +692,7 @@ describe('downgradeSchemaV31ToV30', () => {
   it('turns examples into example', () => {
     expect(downgradeSchemaV31ToV30({ examples: ['a', 'b'] })).toEqual({ example: 'a' })
     expect(downgradeSchemaV31ToV30({ example: 'x', examples: ['a'] })).toEqual({ example: 'x' })
+    expect(downgradeSchemaV31ToV30({ examples: [] })).toStrictEqual({})
   })
 
   it('turns tuples into arrays whose items match any item schema', () => {
@@ -784,6 +791,10 @@ describe('downgradeSchemaV31ToV30', () => {
       type: 'string',
       format: 'binary',
     })
+    expect(downgradeSchemaV31ToV30({ type: 'string', contentEncoding: 'binary' })).toEqual({
+      type: 'string',
+      format: 'binary',
+    })
     expect(
       downgradeSchemaV31ToV30({
         type: 'string',
@@ -820,6 +831,11 @@ describe('downgradeSchemaV31ToV30', () => {
         ],
       }),
     ).toEqual({})
+    expect(
+      downgradeSchemaV31ToV30({
+        anyOf: [{ type: 'string', contentMediaType: 'image/png' }, { type: 'integer' }],
+      }),
+    ).toEqual({ anyOf: [{ type: 'string', format: 'binary' }, { type: 'integer' }] })
   })
 
   describe('$ref', () => {
@@ -1145,6 +1161,44 @@ describe('downgradeSchemaV31ToV30', () => {
       ).toEqual({ anyOf: [{ minimum: 0 }], allOf: [{ anyOf: [{ items: {} }, {}] }] })
     })
 
+    it('includes propertyNames beyond a string type, and unevaluatedProperties beside an in-place applicator', () => {
+      // As zod emits z.record with a key pattern.
+      expect(
+        downgradeSchemaV31ToV30({
+          not: { type: 'object', propertyNames: { type: 'string', pattern: '^a' } },
+        }),
+      ).toEqual({})
+      expect(
+        downgradeSchemaV31ToV30({
+          not: { allOf: [{ properties: { a: {} } }], unevaluatedProperties: false },
+        }),
+      ).toEqual({})
+    })
+
+    it('excludes propertyNames: true, and unevaluatedProperties beside additionalProperties', () => {
+      expect(downgradeSchemaV31ToV30({ not: { type: 'object', propertyNames: true } })).toEqual({
+        not: { type: 'object' },
+      })
+      expect(
+        downgradeSchemaV31ToV30({
+          not: { additionalProperties: { type: 'string' }, unevaluatedProperties: false },
+        }),
+      ).toEqual({ not: { additionalProperties: { type: 'string' } } })
+    })
+
+    it('carries over to a schema around it, through allOf, anyOf, properties, not, and oneOf', () => {
+      const lost = { type: 'object', propertyNames: { pattern: '^a' } } as const
+      expect(downgradeSchemaV31ToV30({ not: { allOf: [lost, { required: ['a'] }] } })).toEqual({})
+      expect(downgradeSchemaV31ToV30({ not: { anyOf: [lost, { type: 'string' }] } })).toEqual({})
+      expect(
+        downgradeSchemaV31ToV30({ not: { properties: { a: lost, b: { type: 'string' } } } }),
+      ).toEqual({})
+      expect(downgradeSchemaV31ToV30({ not: { not: lost } })).toEqual({})
+      expect(
+        downgradeSchemaV31ToV30({ not: { anyOf: [{ minimum: 0 }], oneOf: [lost, {}] } }),
+      ).toEqual({})
+    })
+
     it('excludes keywords that restrict nothing, and tuples converted exactly', () => {
       // As zod emits a discriminated union whose member holds a record or a tuple.
       const record: OpenAPIV3_1.SchemaObject = {
@@ -1483,6 +1537,9 @@ describe('unusual input', () => {
         ],
       }),
     ).toEqual({ oneOf: [{ type: 'object' }, { type: 'string' }] })
+    expect(
+      downgradeSchemaV31ToV30({ oneOf: [{ type: 'object', if: undefined }, { type: 'string' }] }),
+    ).toEqual({ oneOf: [{ type: 'object' }, { type: 'string' }] })
   })
 
   it('tolerates malformed input without throwing', () => {
diff --git a/packages/downgrader/tests/v3.2-to-v3.1.test.ts b/packages/downgrader/tests/v3.2-to-v3.1.test.ts
index 2021069..34ac525 100644
--- a/packages/downgrader/tests/v3.2-to-v3.1.test.ts
+++ b/packages/downgrader/tests/v3.2-to-v3.1.test.ts
@@ -359,12 +359,14 @@ describe('downgradeSpecV32ToV31', () => {
         200: { summary: 'OK', description: 'Everything went fine' },
         201: { summary: 'Created' },
         204: {},
+        404: { $ref: '#/components/responses/NotFound' },
       },
     })
     expect(out?.responses).toEqual({
       200: { description: 'Everything went fine' },
       201: { description: 'Created' },
       204: { description: '' },
+      404: { $ref: '#/components/responses/NotFound' },
     })
   })
 
diff --git a/packages/downgrader/tests/validate.ts b/packages/downgrader/tests/validate.ts
index 510c345..3d1d717 100644
--- a/packages/downgrader/tests/validate.ts
+++ b/packages/downgrader/tests/validate.ts
@@ -92,6 +92,8 @@ export async function expectValidDowngrade<I extends object, O extends object>(
   const before = structuredClone(doc)
   const output = downgrade(doc)
   await expectValidAs(output, to)
+  // It holds only what JSON keeps, with no key holding `undefined`.
+  expect(output).toStrictEqual(JSON.parse(JSON.stringify(output)))
   expectNoNewDanglingRefs(doc, output)
   expect(doc).toEqual(before)
   return output