diff --git a/docs/current-handoff.md b/docs/current-handoff.md
index a01fe08..ef564e7 100644
--- a/docs/current-handoff.md
+++ b/docs/current-handoff.md
@@ -36,7 +36,8 @@ If YES, the next live experiment is one timed/costed OpenAI UI investigation mea
 ## OpenAI alternate structured output
 
 - The OpenAI provider now honors a caller-supplied structured-output schema, applying its existing strict-schema transport projection; absent an alternate schema, initial reconstruction retains its existing strict schema and transport normalization.
-- This deterministically fixes the focused-deconstruction 502 root cause: its six-field schema now reaches OpenAI rather than being replaced by the initial-reconstruction schema. Zero live calls occurred.
+- The next live run exposed incomplete recursive strict projection: OpenAI rejected `relationships.items` because it lacked `additionalProperties: false`. The projector now recognizes every `type: "object"` node, including property-less objects in array items, and recursively enforces strict object schemas while preserving initial-reconstruction optionality/nullability behavior.
+- This deterministically fixes the focused-deconstruction transport failures. Provider and focused-route suites pass with zero live calls.
 - Next boundary: retry one fresh real Playwright-driven OpenAI/Terra investigation; do not reuse the failed focused-deconstruction attempt.
 
 ## Repository checkpoint
diff --git a/lib/llm/provider.js b/lib/llm/provider.js
index 1ca6762..9fa2182 100644
--- a/lib/llm/provider.js
+++ b/lib/llm/provider.js
@@ -112,7 +112,7 @@ const openAIReconstructionJsonSchema = createOpenAIStrictSchema(
 /** @internal OpenAI Structured Outputs requires every object property. */
 export function createOpenAIStrictSchema(schema, zodSchema = reconstructionV2Schema) {
   const projected = structuredClone(schema);
-  projectOpenAIStrictSchema(projected, zodSchema, projected);
+  projectOpenAIStrictSchema(projected, zodSchema, projected, new WeakSet());
   return projected;
 }
 
@@ -157,10 +157,12 @@ function zodAcceptsNull(schema) {
   return schema?.isNullable?.() === true;
 }
 
-function projectOpenAIStrictSchema(schema, zodSchema, rootSchema) {
+function projectOpenAIStrictSchema(schema, zodSchema, rootSchema, visited) {
   const resolved = resolveSchema(schema, rootSchema);
+  if (!resolved || typeof resolved !== "object" || visited.has(resolved)) return;
+  visited.add(resolved);
   const shape = zodObjectShape(zodSchema);
-  if (resolved?.properties) {
+  if (resolved?.type === "object" || resolved?.properties) {
     if (shape) {
     for (const [key, property] of Object.entries(resolved.properties)) {
       const propertyZodSchema = shape[key];
@@ -169,17 +171,28 @@ function projectOpenAIStrictSchema(schema, zodSchema, rootSchema) {
       }
     }
     }
-    resolved.required = Object.keys(resolved.properties);
+    if (resolved.properties) resolved.required = Object.keys(resolved.properties);
+    resolved.additionalProperties = false;
   }
 
   if (resolved?.items) {
-    projectOpenAIStrictSchema(resolved.items, zodArrayItem(zodSchema), rootSchema);
+    projectOpenAIStrictSchema(resolved.items, zodArrayItem(zodSchema), rootSchema, visited);
   }
   if (resolved?.properties) {
     for (const [key, property] of Object.entries(resolved.properties)) {
-      projectOpenAIStrictSchema(property, shape?.[key], rootSchema);
+      projectOpenAIStrictSchema(property, shape?.[key], rootSchema, visited);
     }
   }
+  for (const branch of [
+    ...(resolved.anyOf ?? []),
+    ...(resolved.oneOf ?? []),
+    ...(resolved.allOf ?? []),
+  ]) {
+    projectOpenAIStrictSchema(branch, null, rootSchema, visited);
+  }
+  for (const definition of Object.values(resolved.$defs ?? resolved.definitions ?? {})) {
+    projectOpenAIStrictSchema(definition, null, rootSchema, visited);
+  }
 }
 
 function schemaAllowsNull(schema, rootSchema) {
diff --git a/tests/llm/provider.test.js b/tests/llm/provider.test.js
index 3ee063c..adcf95c 100644
--- a/tests/llm/provider.test.js
+++ b/tests/llm/provider.test.js
@@ -220,6 +220,7 @@ describe("OpenAI reconstruction provider experiment seam", () => {
       const resolved = resolveLocalRef(schema, root);
       if (resolved?.properties) {
         expect(resolved.required).toEqual(Object.keys(resolved.properties));
+        expect(resolved.additionalProperties).toBe(false);
         Object.values(resolved.properties).forEach((property) => assertAllPropertiesRequired(property, root));
       }
       (resolved?.anyOf ?? []).forEach((branch) => assertAllPropertiesRequired(branch, root));
@@ -337,7 +338,9 @@ describe("OpenAI reconstruction provider experiment seam", () => {
       "possibleFollowUpQuestions",
     ]);
     expect(Object.keys(schema.properties)).toEqual(schema.required);
+    expect(schema.additionalProperties).toBe(false);
     expect(schema.properties).toHaveProperty("targetNodeId");
+    expect(schema.properties.relationships.items.additionalProperties).toBe(false);
     expect(schema.properties).not.toHaveProperty("reconstruction");
     expect(schema.properties).not.toHaveProperty("inputClassification");
     expect(result.response).toEqual({
@@ -350,6 +353,46 @@ describe("OpenAI reconstruction provider experiment seam", () => {
     });
   });
 
+  it("recursively projects arbitrary nested object schemas as strict", () => {
+    const alternateSchema = {
+      type: "object",
+      properties: {
+        groups: {
+          type: "array",
+          items: {
+            type: "object",
+            properties: {
+              details: {
+                type: "object",
+                properties: { value: { type: "string" } },
+              },
+            },
+          },
+        },
+      },
+    };
+
+    const projected = createOpenAIStrictSchema(alternateSchema, null);
+
+    expect(projected.additionalProperties).toBe(false);
+    expect(projected.properties.groups.items.additionalProperties).toBe(false);
+    expect(projected.properties.groups.items.properties.details.additionalProperties).toBe(false);
+  });
+
+  it("projects property-less objects nested in arrays as strict", () => {
+    const projected = createOpenAIStrictSchema({
+      type: "object",
+      properties: {
+        relationships: { type: "array", items: { type: "object" } },
+      },
+    }, null);
+
+    expect(projected.properties.relationships.items).toEqual({
+      type: "object",
+      additionalProperties: false,
+    });
+  });
+
   it("extracts ordered output_text parts from raw Responses output", async () => {
     const provider = createOpenAIReconstructionProvider({
       apiKey: "test-key",