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The higher-dimensional Erdos distinct-distances conjecture

Theorem 1.1: for every integer $d\ge3$ there is $c_d>0$ such that every set of $n\ge2$ distinct points in $\mathbb R^d$ determines at least $c_dn^{2/d}$ distinct distances, matching the integer grid up to the constant. No dependence of $c_d$ on $d$ is made explicit, nothing is claimed for pinned distances (many distances from one point), and the planar problem, whose conjectured order is $n/\sqrt{\log n}$ , is untouched beyond reusing the Guth-Katz bound. Priority: Aksoy Yazici earlier claimed the same constant-factor bound in all dimensions $d\ge3$ , but withdrew the manuscript because a step was wrong; it is not an opposite claim.

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The higher-dimensional Erdos distinct-distances conjecture · Higher-dimensional distinct distances

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Registry verification: unreviewed · announcement · candidate

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VibeMathed record: The higher-dimensional Erdos distinct-distances conjecture

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture; it is the full constant-factor statement $|\Delta(P)|\ge c_dn^{2/d}$ for every $d\ge3$ with no position assumption, not an $n^{2/d-o(1)}$ bound. The proof runs a minimal-counterexample induction on dimension through rigid-motion flats, Hilbert function estimates and approximate complete intersections; it was not refereed here. The paper itself notes an earlier claim of the same bound (Aksoy Yazici) that was withdrawn as incorrect. No Lean formalization exists for this family.