From 3eea9d14abb767551ebc73c5bf38d1abb5a67bcb Mon Sep 17 00:00:00 2001
From: DomTheDeveloper <DomTheDeveloper@users.noreply.github.com>
Date: Sun, 26 Jul 2026 09:04:36 -0700
Subject: [PATCH 1/3] =?UTF-8?q?Add=20literal=20Erd=C5=91s=20332=20solution?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 .../ErdosProblems/332LiteralSolution.lean     | 24 +++++++++++++++++++
 1 file changed, 24 insertions(+)
 create mode 100644 FormalConjectures/ErdosProblems/332LiteralSolution.lean

diff --git a/FormalConjectures/ErdosProblems/332LiteralSolution.lean b/FormalConjectures/ErdosProblems/332LiteralSolution.lean
new file mode 100644
index 0000000000..fe997993c0
--- /dev/null
+++ b/FormalConjectures/ErdosProblems/332LiteralSolution.lean
@@ -0,0 +1,24 @@
+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

From b8ec93167241d03aba7b6db0a7944dbc8020edd1 Mon Sep 17 00:00:00 2001
From: DomTheDeveloper <DomTheDeveloper@users.noreply.github.com>
Date: Sun, 26 Jul 2026 09:04:55 -0700
Subject: [PATCH 2/3] =?UTF-8?q?Add=20focused=20Erd=C5=91s=20332=20audit?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 .../openai-erdos-332-literal-audit.yml        | 54 +++++++++++++++++++
 1 file changed, 54 insertions(+)
 create mode 100644 .github/workflows/openai-erdos-332-literal-audit.yml

diff --git a/.github/workflows/openai-erdos-332-literal-audit.yml b/.github/workflows/openai-erdos-332-literal-audit.yml
new file mode 100644
index 0000000000..01e6de79ab
--- /dev/null
+++ b/.github/workflows/openai-erdos-332-literal-audit.yml
@@ -0,0 +1,54 @@
+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

From 0d4e31c2a3c2dca0fcecc71dd04cc5f5bd811e46 Mon Sep 17 00:00:00 2001
From: DomTheDeveloper <DomTheDeveloper@users.noreply.github.com>
Date: Sun, 26 Jul 2026 09:07:19 -0700
Subject: [PATCH 3/3] Add standard copyright header

---
 .../ErdosProblems/332LiteralSolution.lean        | 16 ++++++++++++++++
 1 file changed, 16 insertions(+)

diff --git a/FormalConjectures/ErdosProblems/332LiteralSolution.lean b/FormalConjectures/ErdosProblems/332LiteralSolution.lean
index fe997993c0..4b2f65ae11 100644
--- a/FormalConjectures/ErdosProblems/332LiteralSolution.lean
+++ b/FormalConjectures/ErdosProblems/332LiteralSolution.lean
@@ -1,3 +1,19 @@
+/-
+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»
 
 /-!