[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"ZkhvRg8Dr9":3},"# Erdős Problem #870\n\n> **Additive bases with many representations and no minimal subbasis**\n>\n> David Corneliu Turturean, [Fulcrum](https://fulcrum.inc), `davidct@mit.edu`\n\n[![Build Lean formalization](https://github.com/davidturturean/erdos-870/actions/workflows/build-lean.yml/badge.svg)](https://github.com/davidturturean/erdos-870/actions/workflows/build-lean.yml)\n[![Lean 4.30.0-rc2](https://img.shields.io/badge/Lean-4.30.0--rc2-blue)](lean-toolchain)\n[![License: Apache-2.0](https://img.shields.io/badge/License-Apache--2.0-blue.svg)](LICENSE)\n\nThis repository contains the paper and Lean 4 formalization of a negative answer to [Erdős Problem #870](https://www.erdosproblems.com/870).\n\n## Result\n\nFor every integer `k >= 3` and every real constant `C > 0`, there is a set `E` of positive integers such that:\n\n1. every sufficiently large integer is a sum of at most `k` elements of `E`;\n2. every sufficiently large `n` has at least `C log n` nondecreasing representations using at most `k` elements of `E`; and\n3. no order-`k` subbasis of `E` is minimal.\n\nThus no constant `c(k)` can force a minimal subbasis from a logarithmic representation lower bound.\n\nThe public Lean declarations are:\n\n```lean\nErdos870.main_theorem : Erdos870Target\nErdos870.erdos_870    : Erdos870Target\n```\n\n## Start here\n\n- **For the mathematics:** read [`paper/erdos_870_paper.pdf`](paper/erdos_870_paper.pdf).\n- **For the final theorem:** open [`Erdos870/Main.lean`](Erdos870/Main.lean), then [`Erdos870/MainTheorem.lean`](Erdos870/MainTheorem.lean).\n- **For a guided tour:** use [`notes/FORMALIZATION_GUIDE.md`](notes/FORMALIZATION_GUIDE.md) and [`notes/PAPER_TO_LEAN.md`](notes/PAPER_TO_LEAN.md).\n- **For verification details:** read [`notes/AXIOM_AUDIT.md`](notes/AXIOM_AUDIT.md) and inspect the GitHub Actions run.\n\n## Proof at a glance\n\nThe proof has one order-two probabilistic source and two deterministic endings.\n\n- The Larsen-Larsen order-two construction supplies a sparse basis with logarithmically many exact pair representations and a deletable element in every order-two subbasis.\n- For `k >= 4`, a finite residue-class filler gadget reserves two summands for the order-two core, amplifies the representation count, and preserves non-minimality.\n- For `k = 3`, a parity reduction leaves only one filler. A clustered canary construction controls all finite shifts created by that reduction simultaneously.\n\nThe formalization expands the probabilistic estimates, finite schedules, exceptional sets, Borel-Cantelli arguments, and interface conversions consumed by the final theorem.\n\n## Repository contents\n\n```text\n.\n├── Erdos870.lean                    # Lean root importing the complete project\n├── Erdos870/\n│   ├── Main.lean                    # Public theorem `erdos_870`\n│   ├── MainTheorem.lean             # Core theorem `main_theorem`\n│   ├── Defs.lean                    # Representations, bases, minimality, target\n│   ├── Filler.lean                  # Deterministic k >= 4 and k = 3 reductions\n│   ├── Order2Input.lean             # Repaired Larsen-Larsen order-two input\n│   ├── ClusteredInput.lean          # Finite-shift clustered order-three input\n│   ├── ProbabilityTools.lean        # Probability and concentration utilities\n│   ├── ClaudeProbCore.lean          # Core probabilistic estimates\n│   ├── LLProp5AntiClustering.lean   # Anti-clustering argument\n│   └── LarsenLarsenL7BC.lean        # Lemma 7 and Borel-Cantelli support\n├── paper/\n│   ├── erdos_870_paper.pdf          # Paper\n│   ├── erdos_870_paper.tex          # LaTeX source\n│   └── README.md\n├── notes/\n│   ├── FORMALIZATION_GUIDE.md\n│   ├── PAPER_TO_LEAN.md\n│   ├── AXIOM_AUDIT.md\n│   └── SOURCE_PROVENANCE.md\n├── scripts/\n│   ├── check_no_placeholders.py\n│   ├── check_local_imports.py\n│   ├── check_axioms.py\n│   └── print_axioms.sh\n├── problem_statement.md\n├── Makefile\n├── lakefile.toml\n├── lake-manifest.json\n├── lean-toolchain                   # Lean 4.30.0-rc2\n├── CITATION.cff\n├── NOTICE\n└── LICENSE                          # Apache-2.0 for code and repository text\n```\n\n## Build and verify\n\nInstall [`elan`](https://github.com/leanprover/elan), then run:\n\n```bash\nlake exe cache get\nmake check\n```\n\nThe individual commands are:\n\n```bash\nmake imports   # all project-local imports resolve\nmake audit     # no active sorry/admit or project assumption declarations\nmake build     # compile the project\nmake axioms    # print and check final kernel dependencies\n```\n\nLean and Mathlib are pinned to `v4.30.0-rc2`, including transitive revisions in `lake-manifest.json`.\n\n## Build the paper\n\n```bash\nmake paper\n```\n\nA pre-built PDF is committed at [`paper/erdos_870_paper.pdf`](paper/erdos_870_paper.pdf), so reading the paper does not require a TeX installation.\n\n## Trust boundary\n\nThe packaged active source passes the included comment-aware static audit with no admitted proof and no project-specific assumption declaration. The final files run `#print axioms`. CI performs a clean build and checks that the reported dependencies are limited to:\n\n```text\n[propext, Classical.choice, Quot.sound]\n```\n\nThese are standard Lean principles used through Mathlib. See [`notes/AXIOM_AUDIT.md`](notes/AXIOM_AUDIT.md).\n\n## Relationship to Problem #868\n\nThe order-two input builds on Daniel Larsen and Michael Larsen's negative answer to Erdős Problem #868. A separate standalone repository will package that theorem in its exact two-summand form and will include their paper with explicit attribution.\n\n## Citation\n\n```bibtex\n@misc{turturean2026erdos870,\n  author       = {David Turturean},\n  title        = {Additive bases with many representations and no minimal subbasis: A negative answer to Erdős Problem #870},\n  year         = {2026},\n  howpublished = {GitHub repository},\n  url          = {https://github.com/davidturturean/erdos-870}\n}\n```\n\nSee [`CITATION.cff`](CITATION.cff) for machine-readable metadata.\n\n## License and disclosure\n\nThe Lean code and repository documentation are licensed under Apache-2.0. The paper remains copyright David Turturean. The natural-language proof used GPT-5.4-Pro then GPT-5.5-Pro; the Lean formalization used GPT-5.5-Pro, Codex (GPT-5.5-xhigh), and Claude Code (Opus 4.8, maximum thinking). Compute was provided by [Fulcrum](https://fulcrum.inc). David Turturean is responsible for the final mathematical claims and exposition. See [`LICENSE`](LICENSE) and [`NOTICE`](NOTICE).\n",1782661959475]