[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"vCUoqoxYl1":3},"# The uniform-constant Erdős unit-distance conjecture is false — formalized\n\nA complete, machine-checked Lean 4 formalization of L. Alpöge's one-page\ndisproof of the uniform-constant form of Erdős's unit-distance conjecture:\n\n> **Theorem** (`Erdos.erdos_unit_distance_uniform_constant_false`).\n> For every `C > 0` and every `N` there exist `n ≥ N` and an `n`-point set\n> `P ⊆ ℝ²` with more than `n^(1 + C / log log n)` unit-distance pairs.\n\nThis is the literal negation of Erdős's 1946 conjecture\n`ν(n) ≤ n^(1 + C/log log n)`.  The final axiom audit:\n\n```\n'Erdos.erdos_unit_distance_uniform_constant_false' depends on axioms:\n[propext, Classical.choice, Quot.sound]\n```\n\nNo `sorry`, no extra axioms.  (Note this is *weaker* than the\n[lean-eval problem `erdos_unit_distance_conjecture_false`](https://lean-lang.org/eval/problems/erdos_unit_distance_conjecture_false/),\nwhich asks for a fixed power gain `ν(n) ≥ n^(1+δ)` — the OpenAI 2026\nconstruction via class field towers.  Alpöge's one-pager deliberately\ntrades the power gain for elementary inputs, and that is what is\nformalized here.)\n\n## Layout\n\nA single Lake project on **Lean v4.31.0-rc2 / Mathlib master**, with three git\ndependencies:\n\n- **[Mathlib](https://github.com/leanprover-community/mathlib4)**;\n- **[PrimeNumberTheoremAnd](https://github.com/AlexKontorovich/PrimeNumberTheoremAnd)**,\n  whose sorry-free `chebyshev_asymptotic_pnt` is the only *analytic* input;\n- **[TauCeti](https://github.com/FormalFrontier/TauCeti)**, which supplies the\n  reusable *algebraic* number theory (see below).\n\nThe library is organised by subject:\n\n| module | contents |\n|---|---|\n| `Counting` | unit-distance pair counts; transport `ℂ ≃ ℝ²` |\n| `GeometricCore` | grid-pigeonhole packing/doubling in polydiscs; the translation argument |\n| `PrimesMod4` | the `j`-th primes `≡ 1, 3 (mod 4)`; polynomial growth from PNT-in-AP; the modulus `m t` |\n| `MultiquadraticField` | `K_g = ℚ(i, √q₀, …, √q_{g-1})`: degree `2^(g+1)` via TauCeti's square-class descent; CM structure |\n| `Discriminant` | the concrete diagonal trace form for `K_g`; `log h_{K_g} = O(2^g·g log g)` via TauCeti's effective class-number and discriminant bounds |\n| `IdealFamily` | unramifiedness, inertia `≤ 2`, conjugation freeness for `K_g`; the `2^(t·2^(g-1))` ideals over `m` via TauCeti's conjugate-transversal family |\n| `NormFibre` | the two pigeonholes; the fibre `Z = {z ∈ 𝔟 : z·z∗ = μ}`; unit-square index from TauCeti |\n| `PointCount` | Minkowski embedding, separation, the planar point set |\n| `Main` | the counting assembly and the theorem |\n\n### What comes from TauCeti\n\nThe reusable number theory this proof needs has been upstreamed into\n[TauCeti](https://github.com/FormalFrontier/TauCeti), so this repository now holds only\nthe concrete `K_g` construction and the disproof assembly.  The general results imported\nand applied here are:\n\n- effective ideal-count, class-number, and unit-square-index bounds\n  (`TauCeti.NumberField.classNumber_le_bound`, `units_sq_index_le`) and the discriminant\n  comparison `abs_discr_le_of_basis_isIntegral`;\n- the conjugate-transversal ideal family\n  (`TauCeti.DedekindDomain.exists_transversal_family`);\n- multiquadratic square-class descent (`TauCeti.Multiquadratic.sqrtTower`,\n  `squareClass_of_sqrt_mem`) and the quadratic intermediate-field lemma\n  `TauCeti.IntermediateField.mem_sup_adjoin_sq`.\n\nVerify with:\n\n```\nlake exe cache get && lake build\n```\n\nand then\n\n```lean\nimport ErdosUnitDistance\n#print axioms Erdos.erdos_unit_distance_uniform_constant_false\n-- [propext, Classical.choice, Quot.sound]\n```\n\n- `informal-proof.md` — a faithful transcription of the informal proof.\n- `formalization.yaml` — provenance and resource-usage metadata.\n\n## Independent verification\n\n[kim-em/erdos-unit-distance-comparator](https://github.com/kim-em/erdos-unit-distance-comparator)\nchecks this library with\n[leanprover/comparator](https://github.com/leanprover/comparator): a\nMathlib-only `Challenge.lean` states the theorem, and comparator\ncertifies — without trusting any proof code here — that the library\nproves exactly that statement from the standard axioms, replaying the\nproof through the Lean kernel.\n\n## Proof structure\n\n1. **Geometric core** — grid-pigeonhole packing and doubling bounds for\n   lattice points in polydiscs of `ℂ^d` (no measure theory), and the\n   translation argument producing unit distances after projection to one\n   complex coordinate.\n2. **Arithmetic construction** — the multiquadratic CM field\n   `K = ℚ(i, √q₀, …, √q_{g-1})` (`q_j` the `j`-th prime `≡ 3 mod 4`) of\n   degree `2^(g+1)`; for `m` the product of the first `t` primes\n   `≡ 1 mod 4`, at least `2^(t·2^(g-1))` ideals `𝔄` with `𝔄𝔄∗ = (m)`\n   (unramifiedness, inertia degree `≤ 2` via the exponent-2 Galois group,\n   conjugation acting freely on primes, and a transversal count);\n   class-group and unit-square-class pigeonholes then produce a norm\n   fibre `Z = {z ∈ 𝔟 : z z̄ = μ}` of exponential size, all of one\n   archimedean modulus.  The general tools here — square-class descent,\n   the effective ideal-count / class-number / discriminant / unit-square\n   bounds, and the conjugate-transversal count — come from TauCeti and are\n   instantiated at the concrete `K_g` and `m`.\n3. **Assembly** — explicit-constant bookkeeping: `log(ν/n) ≫ t·2^g`,\n   `log n ≪ 2^g·t log t`, `log log n ≍ g`; choosing `g ≈ C log t` and `t`\n   large refutes the bound for the given `C`.\n\n## Provenance\n\nFormalized 2026-06-11 (one working day) by an orchestrated ensemble —\nClaude (Anthropic), Aristotle (Harmonic), and Codex (OpenAI) — directed\nfrom a single Claude Code session.  See `formalization.yaml` for details\nand resource usage.  Migrated onto TauCeti and bumped to Lean v4.31.0-rc2\n/ Mathlib master on 2026-06-17.\n",1788025895666]