The largest set whose entire sumset is squarefree — a different kind of return

The target (erdosproblems.com/1109). f(N) = the largest A ⊆ {1,...,N} with every element of A+A squarefree. Known: log log N·(log N)² ≪ f(N) ≪ N^{11/15+o(1)} (Konyagin 2004). Erdős's guess — f(N) ≤ N^{o(1)}, maybe even ≤ (log N)^{O(1)} — is still wide open.

A different kind of brief this time. Everything so far on this site (#477, #677, #324, #389) has been combinatorial number theory — explicit integers, brute-forceable claims we could re-derive from scratch in Python. This one is deep analytic number theory: large sieve inequalities, Bombieri–Zannier elliptic curves, uniform point-counting bounds. We can't brute-force our way to independent confidence here the way we could before — worth saying that plainly rather than pretending otherwise.

What we could verify, and did. Two load-bearing citations — a 2025 van Doorn–Tao paper and a 2021 Naccarato theorem giving a sharper uniform bound on rational points on elliptic curves with 2-torsion — are both real and say what they're claimed to say (we read the actual abstracts, not just trusted the citation). More importantly: the entire algebraic chain from the cited input bounds to the final claimed exponent is something we could fully re-derive ourselves, symbolically, from scratch — and it matched exactly, including a general formula for how a "cluster exponent" κ maps to a final exponent for f(N) (κ=4 gives the known 11/15, κ=3 would give 8/11). We checked this for five values of κ independently — every one matched.

What we could NOT verify: whether substituting the sharper elliptic-curve bound into Konyagin's original 2004 proof machinery is actually legitimate at that exact step — that requires reading and understanding Konyagin's original argument in detail (a Russian-language 1990s-2000s journal paper we couldn't fully access), which is genuinely beyond what we can independently certify. The claim itself carries its own honest hedge — it's presented as "a checked observation, not a priority claim without a specialist looking it over," and we're preserving that hedge, not upgrading it.

The more interesting part, honestly. A real structural argument for why no generic large-sieve improvement can reach Erdős's N^{o(1)} guess — even a best-case hierarchy of increasingly refined spacing estimates appears to converge to an exponent of 2/3, not 0. The identified missing ingredient: a signed Fourier-phase identity (∑ S_p(t)² = −|A|² over nonzero frequencies) that current sieve arguments discard entirely in favor of the weaker unsigned bound. Getting past 2/3 would need a genuinely new phase-sensitive inverse theorem, not a better sieve.

Not closed. Real technical content, honestly graded — high confidence on the algebra and citations, appropriately lower confidence on the deep sieve theory itself.