Dead end: U60 (from-best)

Settle Lemma.erdos_1082.parts.i: prove it or disprove it Method: Prove Lemma.erdos_1082.parts.i or its negation: ∀ (A : Finset ℝ²) (hA_n3c : NonTrilinear (A : Set ℝ²)), A.card / 2 ≤ distinctDistances A. Question from Erdős problem 1082 (https://www.erdosproblems.com/1082): Let $A\subset {ℝ}^2$ be a set of $n$ points with no three on a line. Does $A$ determine at least $\lfloor n/2\rfloor$ distinct distances? Status on erdosproblems.com: open, falsifiable. Formal statement: formal-conjectures FormalConjectures/ErdosProblems/1082.lean. Start from proofs/Lemma.erdos_1082.parts.i.lean. A counterexample that only shows the formal statement is wrong (an edge case the problem didn't mean) still counts, and the lab's maintainers will report it upstream; say so in your notes. If neither is in reach, prove a useful special case or lemma, post what blocks you to the chat, and submit the files you started from unchanged, with notes on what you tried and what blocked you: the notebook keeps them as a dead end so others don't repeat it. Outcome: found nothing better than the starting point. Threshold was 0. Submitter notes (data): U60 dead end for Lemma.erdos_1082.parts.i. I printed the definitions with #guard_msgs. distinctDistances A = (distanceSet A).card : ℕ, NonTrilinear A = A.Triplewise (¬Collinear ℝ {x,y,z}), and A.card/2 is ℕ floor division, which matches ⌊n/2⌋. The formalization is faithful and I found no edge case. Erdős's question is open; the best known lower bound is ⌈(n-1)/3⌉ (Szemerédi). I attempted neither a proof nor a disproof. I am submitting the starting files unchanged.

{"proofs":{}}