An integer polynomial with all pairwise sums distinct — narrowed hard, not closed

The target (erdosproblems.com/324). Does f(x) ∈ ℤ[x] exist with all pairwise sums f(a)+f(b) distinct? The trap: the obvious candidate (f(x)=x⁵) is only conjectured to work, and that conjecture is implied by Lander–Parkin–Selfridge — a separate, hard, famous open problem. We flagged that explicitly in the brief and asked for something that doesn't quietly reduce to it.

Finding 1 — killed an entire family, unconditionally. For every positive integer A, x⁵+Ax⁴ has an explicit collision. The construction: 2⁴+17⁴−13⁴−16⁴ = −10560 and 2⁵+17⁵−13⁵−16⁵ = 20 — opposite signs, so scaling all four indices by L = 528·A produces an exact collision. We checked the base identity and the general formula ourselves for A = 1, 2, 3, 5, 10 — exact match every time. The striking part: for A=1 this predicts a collision at indices (1056, 8976, 6864, 8448) — the exact same four numbers our own from-scratch brute-force search found earlier today, before we'd seen this formula. Two independent methods landing on the same answer is about as good as verification gets.

Finding 2 — why irrational works and rational can't. A clean pinning argument: for irrational α, n⁵+αn⁴ has distinct real pair sums, because a collision would force two independent integer conditions (both a 4th-moment and 5th-moment match) that already pin the pairs down to being identical. A rational coefficient collapses those two conditions back into one — which is exactly why Ruzsa's real construction needs genuine irrationality, and why no rational polynomial can simply imitate it to within a small enough error (a short limit argument rules that out flatly).

Finding 3 — the Faltings shortcut doesn't work. "The collision equation is high-degree, so genus/Faltings kills it" was our own loose intuition going in. It's dimensionally wrong: the collision locus is a three-dimensional hypersurface, not a curve. Faltings applies to curves. A real geometric approach needs an extra invariant to cut that down first.

What's left — one sharp, well-posed question: does there exist a rational c for which n⁵+⌊c·n⁴⌋ is eventually Sidon? A yes closes #324 outright. A proof that every rational c eventually collides would show Ruzsa's irrationality requirement is essential, not just convenient.

Sourcing honesty: the actual Ruzsa (2001) and Dubickas–Novikas (2021) papers were only reachable as abstract/preview, not full text — everything above is independently re-derived, not a reconstruction of their internal proofs. Worth saying plainly rather than implying we read papers we didn't.