Erdos's question on the maximum modulus of real Littlewood polynomials: is it at least (1 + c) sqrt(N)?

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Comparator statement: LittlewoodFiniteFlatness.lean evidence for this event

This event attributed to Unreleased internal OpenAI model

Companion: Ultraflat real Littlewood polynomials (two-sided, Oct 5) evidence for this event

Erdos's question on the maximum modulus of real Littlewood polynomials: is it at least (1 + c) sqrt(N)? parent of this event

Lean proof (OAI.AsymptoticallyMinimalLittlewood.main) evidence for this event

Lean proof of finite-exponent flatness (OAI.AsymptoticallyMinimalLittlewoodFiniteFlatness.main) evidence for this event

Theorem 1.1: $\min_{\epsilon}\max_{|z|=1}|\sum_{k<N}\epsilon_k z^k|=(1+o(1))\sqrt N$ as $N\to\infty$ through all integers, so no constant $c>0$ as in Erdos's question exists. The construction relaxes to real coefficients in $[-1,1]$ with small defect, using sampled quadratic-phase modes and a hypergraph interval packing, then rounds to signs by a Lovett-Meka partial-coloring argument. It is existential and gives no rate. The principal paper gives no uniform lower bound; the October 5 companions add $|P|\ge\sqrt N/16$ and then full two-sided ultraflatness $(1\pm\epsilon)\sqrt N$ at every large length. parent of this event

VibeMathed record: Erdos's question on the maximum modulus of real Littlewood polynomials: is it at least (1 + c) sqrt(N)? evidence for this event

Comparator statement: AsymptoticallyMinimalLittlewood.lean evidence for this event

Companion: Nearly minimal maxima and positive minima of Littlewood polynomials evidence for this event

Turyn's conjecture that binary merit factors are bounded evidence for this event