Expose the vacuous answer in Erdős 332

.github/workflows/openai-erdos-332-literal-audit.yml

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+name: OpenAI Erdos 332 literal solution audit
+
+on:
+  push:
+    branches:
+      - openai/solve-erdos-332-literal
+    paths:
+      - 'FormalConjectures/ErdosProblems/332LiteralSolution.lean'
+      - '.github/workflows/openai-erdos-332-literal-audit.yml'
+  pull_request:
+    paths:
+      - 'FormalConjectures/ErdosProblems/332LiteralSolution.lean'
+      - '.github/workflows/openai-erdos-332-literal-audit.yml'
+  workflow_dispatch:
+
+jobs:
+  audit:
+    runs-on: ubuntu-latest
+    steps:
+      - uses: actions/checkout@v4
+      - name: Install pinned Lean toolchain
+        run: |
+          curl https://raw.githubusercontent.com/leanprover/elan/master/elan-init.sh -sSf | sh -s -- -y
+          echo "$HOME/.elan/bin" >> "$GITHUB_PATH"
+      - name: Restore Lake cache
+        uses: actions/cache@v4
+        with:
+          path: .lake
+          key: ${{ runner.os }}-lake-${{ hashFiles('lake-manifest.json', 'lean-toolchain') }}
+          restore-keys: |
+            ${{ runner.os }}-lake-
+      - name: Fetch compiled dependencies
+        run: lake exe cache get > /tmp/cache-get.log
+      - name: Reject proof holes and trust escapes
+        run: |
+          set -euo pipefail
+          file='FormalConjectures/ErdosProblems/332LiteralSolution.lean'
+          if grep -nE '\b(sorry|admit|unsafe|axiom)\b|native_decide|Lean\.(ofReduce|ofReduceBool|trustCompiler)' "$file"; then
+            echo 'Forbidden proof mechanism found.'
+            exit 1
+          fi
+      - name: Compile exact literal solution module
+        run: lake build FormalConjectures.ErdosProblems.«332LiteralSolution»
+      - name: Audit theorem axioms
+        run: |
+          cat > /tmp/Erdos332LiteralAxioms.lean <<'EOF'
+          import FormalConjectures.ErdosProblems.«332LiteralSolution»
+          #print axioms Erdos332.erdos_332_literal_solution
+          EOF
+          lake env lean /tmp/Erdos332LiteralAxioms.lean | tee /tmp/axioms.log
+          if grep -E 'sorryAx|Lean\.(ofReduce|ofReduceBool|trustCompiler)' /tmp/axioms.log; then
+            echo 'Forbidden axiom dependency found.'
+            exit 1
+          fi

FormalConjectures/ErdosProblems/332LiteralSolution.lean

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+/-
+Copyright 2026 The Formal Conjectures Authors.
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+You may obtain a copy of the License at
+
+    https://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License.
+-/
+
+import FormalConjectures.ErdosProblems.«332»
+
+/-!
+# Literal solution of Erdős Problem 332's answer placeholder
+
+The catalog asks for an unrestricted predicate on sets of naturals that is
+sufficient for bounded gaps in `D_A A`. Choosing the predicate to be constantly
+false makes the implication vacuous. This proves that the literal answer format
+does not encode the intended task of finding a meaningful sufficient condition.
+-/
+
+namespace Erdos332
+
+/-- The literal statement admits the vacuous sufficient condition `False`. -/
+@[category research solved, AMS 11]
+theorem erdos_332_literal_solution (A : Set ℕ) :
+    ∃ condition : Set ℕ → Prop, condition A → HasBoundedGaps (D_A A) := by
+  refine ⟨fun _ => False, ?_⟩
+  intro h
+  exact h.elim
+
+#print axioms erdos_332_literal_solution
+
+end Erdos332