From b8b49653b09cde1f1a0fc3c360d438fc466bc8be Mon Sep 17 00:00:00 2001
From: DomTheDeveloper <DomTheDeveloper@users.noreply.github.com>
Date: Sun, 26 Jul 2026 10:03:37 -0700
Subject: [PATCH 1/3] feat: prove two nontrivial unitary-perfect examples

---
 .../ErdosProblems/1052Proof.lean              | 305 ++++++++++++++++++
 1 file changed, 305 insertions(+)
 create mode 100644 FormalConjectures/ErdosProblems/1052Proof.lean

diff --git a/FormalConjectures/ErdosProblems/1052Proof.lean b/FormalConjectures/ErdosProblems/1052Proof.lean
new file mode 100644
index 0000000000..fddc22d07b
--- /dev/null
+++ b/FormalConjectures/ErdosProblems/1052Proof.lean
@@ -0,0 +1,305 @@
+/-
+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.«1052»
+
+/-!
+# Kernel proofs for the nontrivial unitary-perfect examples
+
+This module verifies the two larger examples through the multiplicativity of the
+sum of unitary divisors. It avoids enumerating every integer below the examples.
+-/
+
+namespace Erdos1052
+
+/-- All unitary divisors of `n`. -/
+def unitaryDivisors (n : ℕ) : Finset ℕ :=
+  {d ∈ Finset.Ico 1 (n + 1) | d ∣ n ∧ d.Coprime (n / d)}
+
+/-- The sum of all unitary divisors of `n`. -/
+def sigmaStar (n : ℕ) : ℕ := ∑ d ∈ unitaryDivisors n, d
+
+private lemma properUnitaryDivisors_prime_pow (p k : ℕ) (hp : p.Prime) (hk : 1 ≤ k) :
+    properUnitaryDivisors (p ^ k) = {1} := by
+  ext d
+  simp only [properUnitaryDivisors, Finset.mem_filter, Finset.mem_Ico, Finset.mem_singleton]
+  constructor
+  · rintro ⟨⟨h1, hlt⟩, hdvd, hcop⟩
+    obtain ⟨j, hjk, rfl⟩ := (Nat.dvd_prime_pow hp).mp hdvd
+    have hp1 : 1 < p := hp.one_lt
+    have hjlt : j < k := (Nat.pow_lt_pow_iff_right hp1).mp hlt
+    rcases Nat.eq_zero_or_pos j with hj0 | hjpos
+    · simp [hj0]
+    · exfalso
+      have hdiv : p ^ k / p ^ j = p ^ (k - j) := by
+        rw [Nat.pow_div hjk (by omega)]
+      rw [hdiv] at hcop
+      have hp_dvd_j : p ∣ p ^ j := dvd_pow_self p (by omega)
+      have hp_dvd_kj : p ∣ p ^ (k - j) := dvd_pow_self p (by omega)
+      have : p ∣ Nat.gcd (p ^ j) (p ^ (k - j)) := Nat.dvd_gcd hp_dvd_j hp_dvd_kj
+      rw [Nat.Coprime] at hcop
+      rw [hcop] at this
+      have := Nat.eq_one_of_dvd_one this
+      omega
+  · rintro rfl
+    refine ⟨⟨le_refl 1, ?_⟩, one_dvd _, ?_⟩
+    · have : 1 < p := hp.one_lt
+      calc
+        1 < p := this
+        _ = p ^ 1 := (pow_one p).symm
+        _ ≤ p ^ k := Nat.pow_le_pow_right (le_of_lt this) hk
+    · simp
+
+private lemma self_mem_unitaryDivisors (n : ℕ) (hn : 1 ≤ n) : n ∈ unitaryDivisors n := by
+  simp only [unitaryDivisors, Finset.mem_filter, Finset.mem_Ico]
+  refine ⟨⟨hn, by omega⟩, dvd_refl n, ?_⟩
+  rw [Nat.div_self hn]
+  exact Nat.coprime_one_right n
+
+private lemma properUnitaryDivisors_eq (n : ℕ) :
+    properUnitaryDivisors n = (unitaryDivisors n).erase n := by
+  ext d
+  simp only [properUnitaryDivisors, unitaryDivisors, Finset.mem_filter, Finset.mem_Ico,
+    Finset.mem_erase]
+  constructor
+  · rintro ⟨⟨h1, hlt⟩, hdvd, hcop⟩
+    exact ⟨by omega, ⟨h1, by omega⟩, hdvd, hcop⟩
+  · rintro ⟨hne, ⟨h1, hle⟩, hdvd, hcop⟩
+    have : d < n := lt_of_le_of_ne (by omega) hne
+    exact ⟨⟨h1, this⟩, hdvd, hcop⟩
+
+private lemma isUnitaryPerfect_iff_sum_unitaryDivisors (n : ℕ) :
+    IsUnitaryPerfect n ↔ (∑ d ∈ unitaryDivisors n, d = 2 * n ∧ 0 < n) := by
+  constructor
+  · rintro ⟨hsum, hpos⟩
+    refine ⟨?_, hpos⟩
+    rw [properUnitaryDivisors_eq n] at hsum
+    have hmem := self_mem_unitaryDivisors n hpos
+    have htotal := Finset.add_sum_erase (unitaryDivisors n) id hmem
+    simp only [id] at htotal
+    omega
+  · rintro ⟨hsum, hpos⟩
+    refine ⟨?_, hpos⟩
+    rw [properUnitaryDivisors_eq n]
+    have hmem := self_mem_unitaryDivisors n hpos
+    have htotal := Finset.add_sum_erase (unitaryDivisors n) id hmem
+    simp only [id] at htotal
+    omega
+
+private lemma unitaryDivisors_prime_pow (p k : ℕ) (hp : p.Prime) (hk : 1 ≤ k) :
+    unitaryDivisors (p ^ k) = {1, p ^ k} := by
+  have hpos : 1 ≤ p ^ k := Nat.one_le_iff_ne_zero.mpr (pow_ne_zero k hp.pos.ne')
+  have hstep : unitaryDivisors (p ^ k) = insert (p ^ k) (properUnitaryDivisors (p ^ k)) := by
+    rw [properUnitaryDivisors_eq]
+    rw [Finset.insert_erase (self_mem_unitaryDivisors _ hpos)]
+  rw [hstep, properUnitaryDivisors_prime_pow p k hp hk, Finset.pair_comm]
+
+private lemma sigmaStar_prime_pow (p k : ℕ) (hp : p.Prime) (hk : 1 ≤ k) :
+    sigmaStar (p ^ k) = 1 + p ^ k := by
+  rw [sigmaStar, unitaryDivisors_prime_pow p k hp hk]
+  rw [Finset.sum_pair]
+  have hp1 : 1 < p := hp.one_lt
+  have : 1 < p ^ k := by
+    calc
+      (1 : ℕ) < p := hp1
+      _ = p ^ 1 := (pow_one p).symm
+      _ ≤ p ^ k := Nat.pow_le_pow_right (by omega) hk
+  omega
+
+private lemma mem_unitaryDivisors {n d : ℕ} (hn : 0 < n) :
+    d ∈ unitaryDivisors n ↔ d ∣ n ∧ d.Coprime (n / d) := by
+  simp only [unitaryDivisors, Finset.mem_filter, Finset.mem_Ico]
+  constructor
+  · rintro ⟨_, hdvd, hcop⟩
+    exact ⟨hdvd, hcop⟩
+  · rintro ⟨hdvd, hcop⟩
+    have hdpos : 1 ≤ d := Nat.one_le_iff_ne_zero.mpr (by rintro rfl; simp at hdvd; omega)
+    have hdle : d ≤ n := Nat.le_of_dvd hn hdvd
+    exact ⟨⟨hdpos, by omega⟩, hdvd, hcop⟩
+
+private lemma sigmaStar_mul_coprime {m n : ℕ} (hm : 0 < m) (hn : 0 < n)
+    (hmn : m.Coprime n) : sigmaStar (m * n) = sigmaStar m * sigmaStar n := by
+  rw [sigmaStar, sigmaStar, sigmaStar, Finset.sum_mul_sum]
+  rw [← Finset.sum_product']
+  have key : ∀ d, d ∣ m * n → d.gcd m * d.gcd n = d := by
+    intro d hddvd
+    have hgm : d.gcd m ∣ d := Nat.gcd_dvd_left d m
+    have hgn : d.gcd n ∣ d := Nat.gcd_dvd_left d n
+    have hcop_g : (d.gcd m).Coprime (d.gcd n) :=
+      Nat.Coprime.coprime_dvd_left (Nat.gcd_dvd_right d m)
+        (Nat.Coprime.coprime_dvd_right (Nat.gcd_dvd_right d n) hmn)
+    have hprod_dvd : d.gcd m * d.gcd n ∣ d := hcop_g.mul_dvd_of_dvd_of_dvd hgm hgn
+    have hd_dvd : d ∣ d.gcd m * d.gcd n := by
+      have h1 : d ∣ d.gcd m * n := dvd_gcd_mul_of_dvd_mul hddvd
+      have hgm_pos : 0 < d.gcd m := Nat.gcd_pos_of_pos_right d hm
+      obtain ⟨e, he⟩ := hgm
+      have he_dvd_n : e ∣ n := by
+        have h1' : d.gcd m * e ∣ d.gcd m * n := by rw [← he]; exact h1
+        exact (Nat.mul_dvd_mul_iff_left hgm_pos).mp h1'
+      have he_dvd_d : e ∣ d := ⟨d.gcd m, by rw [mul_comm]; exact he⟩
+      have he_gcd : e ∣ d.gcd n := Nat.dvd_gcd he_dvd_d he_dvd_n
+      calc
+        d = d.gcd m * e := he
+        _ ∣ d.gcd m * d.gcd n := Nat.mul_dvd_mul_left _ he_gcd
+    exact Nat.dvd_antisymm hprod_dvd hd_dvd
+  apply Finset.sum_nbij' (i := fun d => (d.gcd m, d.gcd n)) (j := fun p => p.1 * p.2)
+  · intro d hd
+    rw [mem_unitaryDivisors (Nat.mul_pos hm hn)] at hd
+    obtain ⟨hddvd, hdcop⟩ := hd
+    have hgm_dvd_m : d.gcd m ∣ m := Nat.gcd_dvd_right d m
+    have hgn_dvd_n : d.gcd n ∣ n := Nat.gcd_dvd_right d n
+    have hgm_dvd_d : d.gcd m ∣ d := Nat.gcd_dvd_left d m
+    have hgn_dvd_d : d.gcd n ∣ d := Nat.gcd_dvd_left d n
+    have hkey := key d hddvd
+    have hquot : (m * n) / d = (m / d.gcd m) * (n / d.gcd n) := by
+      conv_lhs => rw [← hkey]
+      rw [Nat.div_mul_div_comm hgm_dvd_m hgn_dvd_n]
+    simp only [Finset.mem_product]
+    refine ⟨?_, ?_⟩
+    · rw [mem_unitaryDivisors hm]
+      refine ⟨hgm_dvd_m, ?_⟩
+      have h1 : (m / d.gcd m) ∣ (m * n) / d := by rw [hquot]; exact Dvd.intro _ rfl
+      have hcop1 : d.Coprime ((m * n) / d) := hdcop
+      have : d.Coprime (m / d.gcd m) := hcop1.coprime_dvd_right h1
+      exact this.coprime_dvd_left hgm_dvd_d
+    · rw [mem_unitaryDivisors hn]
+      refine ⟨hgn_dvd_n, ?_⟩
+      have h1 : (n / d.gcd n) ∣ (m * n) / d := by rw [hquot]; exact Dvd.intro_left _ rfl
+      have hcop1 : d.Coprime ((m * n) / d) := hdcop
+      have : d.Coprime (n / d.gcd n) := hcop1.coprime_dvd_right h1
+      exact this.coprime_dvd_left hgn_dvd_d
+  · rintro ⟨a, b⟩ hab
+    simp only [Finset.mem_product] at hab
+    obtain ⟨ha, hb⟩ := hab
+    rw [mem_unitaryDivisors hm] at ha
+    rw [mem_unitaryDivisors hn] at hb
+    obtain ⟨hadvd, hacop⟩ := ha
+    obtain ⟨hbdvd, hbcop⟩ := hb
+    rw [mem_unitaryDivisors (Nat.mul_pos hm hn)]
+    have hab_dvd : a * b ∣ m * n := Nat.mul_dvd_mul hadvd hbdvd
+    refine ⟨hab_dvd, ?_⟩
+    have hdiv : m * n / (a * b) = (m / a) * (n / b) := by
+      rw [Nat.div_mul_div_comm hadvd hbdvd]
+    rw [hdiv]
+    have ha_n : a.Coprime n := Nat.Coprime.coprime_dvd_left hadvd hmn
+    have hb_m : b.Coprime m := Nat.Coprime.coprime_dvd_left hbdvd hmn.symm
+    have ha_nb : a.Coprime (n / b) := ha_n.coprime_dvd_right (Nat.div_dvd_of_dvd hbdvd)
+    have hb_ma : b.Coprime (m / a) := hb_m.coprime_dvd_right (Nat.div_dvd_of_dvd hadvd)
+    exact Nat.Coprime.mul_left (Nat.Coprime.mul_right hacop ha_nb)
+      (Nat.Coprime.mul_right hb_ma hbcop)
+  · intro d hd
+    rw [mem_unitaryDivisors (Nat.mul_pos hm hn)] at hd
+    exact key d hd.1
+  · rintro ⟨a, b⟩ hab
+    simp only [Finset.mem_product] at hab
+    obtain ⟨ha, hb⟩ := hab
+    rw [mem_unitaryDivisors hm] at ha
+    rw [mem_unitaryDivisors hn] at hb
+    obtain ⟨hadvd, _⟩ := ha
+    obtain ⟨hbdvd, _⟩ := hb
+    have ha_n : a.Coprime n := Nat.Coprime.coprime_dvd_left hadvd hmn
+    have hb_m : b.Coprime m := Nat.Coprime.coprime_dvd_left hbdvd hmn.symm
+    have hgm : (a * b).gcd m = a := by
+      rw [mul_comm a b]
+      exact Nat.gcd_mul_of_coprime_of_dvd hb_m hadvd
+    have hgn : (a * b).gcd n = b := Nat.gcd_mul_of_coprime_of_dvd ha_n hbdvd
+    rw [Prod.ext_iff]
+    exact ⟨hgm, hgn⟩
+  · intro d hd
+    rw [mem_unitaryDivisors (Nat.mul_pos hm hn)] at hd
+    exact (key d hd.1).symm
+
+private lemma sigmaStar_one : sigmaStar 1 = 1 := by
+  have : unitaryDivisors 1 = {1} := by
+    ext d
+    simp only [unitaryDivisors, Finset.mem_filter, Finset.mem_Ico, Finset.mem_singleton]
+    constructor
+    · rintro ⟨⟨h1, h2⟩, _, _⟩
+      omega
+    · rintro rfl
+      exact ⟨⟨le_refl 1, by omega⟩, dvd_refl 1, by simp⟩
+  rw [sigmaStar, this, Finset.sum_singleton]
+
+private lemma sigmaStar_mul {m n : ℕ} (hmn : m.Coprime n) :
+    sigmaStar (m * n) = sigmaStar m * sigmaStar n := by
+  rcases Nat.eq_zero_or_pos m with hm0 | hm
+  · subst hm0
+    rw [Nat.coprime_zero_left] at hmn
+    subst hmn
+    simp [sigmaStar_one]
+  rcases Nat.eq_zero_or_pos n with hn0 | hn
+  · subst hn0
+    rw [Nat.coprime_zero_right] at hmn
+    subst hmn
+    simp [sigmaStar_one]
+  exact sigmaStar_mul_coprime hm hn hmn
+
+private lemma isUnitaryPerfect_iff_sigmaStar (n : ℕ) :
+    IsUnitaryPerfect n ↔ (sigmaStar n = 2 * n ∧ 0 < n) := by
+  rw [isUnitaryPerfect_iff_sum_unitaryDivisors n, sigmaStar]
+
+private lemma sigmaStar_prime {p : ℕ} (hp : p.Prime) : sigmaStar p = 1 + p := by
+  have h := sigmaStar_prime_pow p 1 hp (le_refl 1)
+  simpa using h
+
+/-- The factorization proof that `87360` is unitary perfect. -/
+@[category test, AMS 11]
+theorem isUnitaryPerfect_87360_kernel : IsUnitaryPerfect 87360 := by
+  rw [isUnitaryPerfect_iff_sigmaStar]
+  refine ⟨?_, by norm_num⟩
+  have hfac : (87360 : ℕ) = 2 ^ 6 * (3 * (5 * (7 * 13))) := by norm_num
+  rw [hfac]
+  rw [sigmaStar_mul (by norm_num), sigmaStar_mul (by norm_num),
+      sigmaStar_mul (by norm_num), sigmaStar_mul (by norm_num)]
+  rw [sigmaStar_prime_pow 2 6 (by norm_num) (by norm_num),
+      sigmaStar_prime (p := 3) (by norm_num), sigmaStar_prime (p := 5) (by norm_num),
+      sigmaStar_prime (p := 7) (by norm_num), sigmaStar_prime (p := 13) (by norm_num)]
+  norm_num
+
+/-- The factorization proof of the fifth known unitary-perfect example. -/
+@[category test, AMS 11]
+theorem isUnitaryPerfect_146361946186458562560000_kernel :
+    IsUnitaryPerfect 146361946186458562560000 := by
+  rw [isUnitaryPerfect_iff_sigmaStar]
+  refine ⟨?_, by norm_num⟩
+  have hfac : (146361946186458562560000 : ℕ) =
+      2 ^ 18 * (3 * (5 ^ 4 * (7 * (11 * (13 * (19 * (37 * (79 * (109 * (157 * 313)))))))))) := by
+    norm_num
+  rw [hfac]
+  rw [sigmaStar_mul (by norm_num), sigmaStar_mul (by norm_num),
+      sigmaStar_mul (by norm_num), sigmaStar_mul (by norm_num),
+      sigmaStar_mul (by norm_num), sigmaStar_mul (by norm_num),
+      sigmaStar_mul (by norm_num), sigmaStar_mul (by norm_num),
+      sigmaStar_mul (by norm_num), sigmaStar_mul (by norm_num),
+      sigmaStar_mul (by norm_num)]
+  rw [sigmaStar_prime_pow 2 18 (by norm_num) (by norm_num),
+      sigmaStar_prime (p := 3) (by norm_num),
+      sigmaStar_prime_pow 5 4 (by norm_num) (by norm_num),
+      sigmaStar_prime (p := 7) (by norm_num),
+      sigmaStar_prime (p := 11) (by norm_num),
+      sigmaStar_prime (p := 13) (by norm_num),
+      sigmaStar_prime (p := 19) (by norm_num),
+      sigmaStar_prime (p := 37) (by norm_num),
+      sigmaStar_prime (p := 79) (by norm_num),
+      sigmaStar_prime (p := 109) (by norm_num),
+      sigmaStar_prime (p := 157) (by norm_num),
+      sigmaStar_prime (p := 313) (by norm_num)]
+  norm_num
+
+#print axioms isUnitaryPerfect_87360_kernel
+#print axioms isUnitaryPerfect_146361946186458562560000_kernel
+
+end Erdos1052

From cd084320af253eb37cb62a0384f36e52c88f3a14 Mon Sep 17 00:00:00 2001
From: DomTheDeveloper <DomTheDeveloper@users.noreply.github.com>
Date: Sun, 26 Jul 2026 10:03:52 -0700
Subject: [PATCH 2/3] ci: audit unitary-perfect example proofs

---
 .../openai-unitary-perfect-audit.yml          | 57 +++++++++++++++++++
 1 file changed, 57 insertions(+)
 create mode 100644 .github/workflows/openai-unitary-perfect-audit.yml

diff --git a/.github/workflows/openai-unitary-perfect-audit.yml b/.github/workflows/openai-unitary-perfect-audit.yml
new file mode 100644
index 0000000000..83a18534ba
--- /dev/null
+++ b/.github/workflows/openai-unitary-perfect-audit.yml
@@ -0,0 +1,57 @@
+name: OpenAI unitary-perfect example audit
+
+on:
+  pull_request:
+    branches: [main]
+    paths:
+      - '.github/workflows/openai-unitary-perfect-audit.yml'
+      - 'FormalConjectures/ErdosProblems/1052Proof.lean'
+
+permissions:
+  contents: read
+
+jobs:
+  audit:
+    runs-on: ubuntu-latest
+    timeout-minutes: 45
+    steps:
+      - uses: actions/checkout@v6
+      - name: Install pinned Lean toolchain
+        run: |
+          curl -fsSL https://raw.githubusercontent.com/leanprover/elan/master/elan-init.sh | sh -s -- -y --default-toolchain none
+          echo "$HOME/.elan/bin" >> "$GITHUB_PATH"
+      - uses: actions/cache@v4
+        with:
+          path: |
+            .lake/packages
+            .lake/build
+          key: unitary-perfect-${{ runner.os }}-${{ runner.arch }}-${{ hashFiles('lean-toolchain', 'lake-manifest.json', 'lakefile.toml') }}
+          restore-keys: unitary-perfect-${{ runner.os }}-${{ runner.arch }}-
+      - name: Fetch compiled dependencies
+        run: lake exe cache get
+      - name: Reject proof holes and trust escapes
+        run: |
+          set -euo pipefail
+          f=FormalConjectures/ErdosProblems/1052Proof.lean
+          ! grep -nE '\b(sorry|admit)\b|native_decide|decide \+native|unsafe|axiom |Lean\.ofReduce|Lean\.ofReduceBool|Lean\.trustCompiler' "$f"
+      - name: Compile exact proof module
+        run: |
+          set -o pipefail
+          lake env lean FormalConjectures/ErdosProblems/1052Proof.lean 2>&1 | tee /tmp/unitary-perfect-build.log
+      - name: Audit exact theorem axioms
+        run: |
+          cat > /tmp/UnitaryPerfectAxioms.lean <<'EOF'
+          import FormalConjectures.ErdosProblems.«1052Proof»
+          #print axioms Erdos1052.isUnitaryPerfect_87360_kernel
+          #print axioms Erdos1052.isUnitaryPerfect_146361946186458562560000_kernel
+          EOF
+          lake env lean /tmp/UnitaryPerfectAxioms.lean 2>&1 | tee /tmp/unitary-perfect-axioms.log
+          ! grep -E 'sorryAx|Lean\.ofReduce|Lean\.ofReduceBool|Lean\.trustCompiler' /tmp/unitary-perfect-axioms.log
+      - uses: actions/upload-artifact@v4
+        if: always()
+        with:
+          name: unitary-perfect-${{ github.run_id }}
+          path: |
+            /tmp/unitary-perfect-build.log
+            /tmp/unitary-perfect-axioms.log
+          if-no-files-found: warn

From e2b0f213c1afd4451bdc4f5be9ee0265e2c16870 Mon Sep 17 00:00:00 2001
From: DomTheDeveloper <DomTheDeveloper@users.noreply.github.com>
Date: Sun, 26 Jul 2026 10:09:49 -0700
Subject: [PATCH 3/3] ci: build the registered unitary-perfect module

---
 .github/workflows/openai-unitary-perfect-audit.yml | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/.github/workflows/openai-unitary-perfect-audit.yml b/.github/workflows/openai-unitary-perfect-audit.yml
index 83a18534ba..d8535f8a31 100644
--- a/.github/workflows/openai-unitary-perfect-audit.yml
+++ b/.github/workflows/openai-unitary-perfect-audit.yml
@@ -37,7 +37,7 @@ jobs:
       - name: Compile exact proof module
         run: |
           set -o pipefail
-          lake env lean FormalConjectures/ErdosProblems/1052Proof.lean 2>&1 | tee /tmp/unitary-perfect-build.log
+          lake build FormalConjectures.ErdosProblems.«1052Proof» 2>&1 | tee /tmp/unitary-perfect-build.log
       - name: Audit exact theorem axioms
         run: |
           cat > /tmp/UnitaryPerfectAxioms.lean <<'EOF'