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Statement

Conventions (Section 2, p. 3): $\mathbb T=\mathbb R/\mathbb Z$ with Haar measure of mass one, $\mathrm e(t)=\exp(2\pi it)$, and $\widehat f(k)=\int_{\mathbb T}f(t)\mathrm e(-kt)\,dt$.

Proposition 5.1. Two absolute constants $C>0$ and $\delta_0>0$ serve every $\delta$ in $(0,\delta_0)$. Each such $\delta$ fixes a threshold $N_0(\delta)$ and a finite tail constant $K_\delta$, and every integer $N\ge N_0(\delta)$ then admits a continuous $B_N:\mathbb T\to\mathbb C$ obeying

$$B_N(-t)=\overline{B_N(t)},\qquad 1\le|B_N(t)|\le1+C\delta\quad(t\in\mathbb T),$$ $$\sqrt N\,|\widehat B_N(k)|\le1+C\sqrt\delta\quad(0\le k<N),\qquad \sum_{k<0\ \text{or}\ k\ge N}|\widehat B_N(k)|\le K_\delta N^{-1}.$$

The coefficients $\widehat B_N(k)$ are all real, and the Fourier series converges absolutely to $B_N$.

The constant $C$ multiplying $\delta$ and $\sqrt\delta$ is absolute; the threshold $N_0(\delta)$ and the tail constant $K_\delta$ depend on all the auxiliary data chosen from $\delta$ and are not quantified.

Source. OpenAI, Ultraflat real Littlewood polynomials, release folder preprints/Ultraflat-real-Littlewood-polynomials-October-5-2026; TeX sections/waves.tex lines 10--26 (label prop:waves), PDF p. 9; proof pp. 10--14 (sections/waves.tex lines 41--392, Figure 1 on p. 13). Read 2026-10-07.

Read depth. Claims checked: the statement and the conventions it relies on were read clause by clause in the TeX source, together with the statement of Lemma 5.2 and Proposition 4.1 which the proof invokes. The proof was read for its structure (below) and no step was checked. Nothing here is independently reviewed.

Proof pointer

The proof (pp. 10--14) has four parts. First, the data of Proposition 4.1 (Section 4, pp. 6--9) are fixed: a real trigonometric polynomial $F(y)=\sum_{a\in\mathcal A}c_a\mathrm e(a\cdot y)$ on $\mathbb T^m$ with $\|F\|_\infty\le1+\delta$, weights $w_a=|a\cdot v|$ summing to a number in $[1-C\delta,1)$, and moduli $m_a=|c_a|/\sqrt{w_a}\in[1,1+C\delta]$. Lemma 5.2 (signed interval packing, quoted from the companion) places disjoint arcs of length $w_a/H$ centered at $-a\cdot\theta_h$ on the circle, one for each frequency $a$ and each $h\le H$. On each arc the manuscript puts a wave of constant modulus $m_a$ whose phase $N\psi_{a,h}$ has derivative running over a subinterval $J_h$ of $((h-1)/H,h/H)$ in the normalized index $x=k/N$; the inverse curvature is $w_a\chi_h(x)^2$ for a taper $\chi_h$ equal to a small $\sigma$ near the ends of $J_h$ and to $1$ on most of it, so stationary phase assigns the wave a leading scaled coefficient $|c_a|\chi_h(x)$ while the modulus stays $m_a$. Second, with a cutoff $G_h$ the endpoint pieces are bounded by Lemma 2.3 (large curvature $1/(w_a\sigma^2)$) and the interior leading terms, by Lemma 2.2, sum over $a$ to $G_h(x)\chi_h(x)F(k\theta_h+NvQ_h(x))$, so the norm bound on $F$ controls the scaled coefficient by $1+2\delta+o(1)$ on the union of the main arcs. Third, the gaps, of total length $L\le C\delta$, are filled by waves whose leading phase derivative is piecewise linear through disjoint derivative intervals $[P_j,R_j]\subset(1/4,3/4)$ (Figure 1), with short steep outer pieces costing $C\sqrt\delta$ by Lemma 2.3 and at most one middle piece near a given $x$ costing $C\sqrt L$; moduli interpolate linearly and an affine phase $\gamma_{j,N}$ matches the adjoining values, with $B_N(0)=B_N(1/2)=1$ and conjugate reflection to the other half-circle. This gives the modulus bounds and the coefficient cap $1+C\sqrt\delta$ after $N_0(\delta)$ absorbs the $o(1)$. Fourth, for $k<0$ or $k\ge N$ two integrations by parts on each piece, with the first boundary terms canceling at every join, where $B_N$ has no jump and adjacent pieces share their leading phase derivative (and at $\pm1/2$ because $B_N=1$, $\phi'=1/8$ and $k$ is an integer), give $|\widehat B_N(k)|\le C_{\mathrm{data}}(N+|k|)^{-2}$, whose sum is $O(N^{-1})$; absolute convergence, representation by continuity and reality of the coefficients from conjugate symmetry follow.

Dependencies

Internal: Proposition 4.1 (proved in Section 4 from Lemma 2.1 and Lemma 3.2, adapting the companion's Section 2 design) and Lemmas 2.2 and 2.3 (proved in Section 2). Imported without proof: Lemma 5.2, the companion Nearly minimal maxima and positive minima of Littlewood polynomials, Lemma 3.1, whose proof there rests on Pippenger--Spencer 1989 in the form of Alon--Yuster 2005, Lemma 2.1. External premises are taken at statement level; none was checked here.

Bears on

• Problem 1150: reaches the problem only through Theorem 1, which rounds this function to signs; it is the analytic half of the claimed negative answer. Unverified here; the page's status rests on acceptance evidence.