# General moment/SOS analysis of Chowla's cosine problem Working mathematical notes, 18 October 2026 UTC. This is **not a proof or a counterexample to Chowla's conjecture**. The results below identify rigorous limitations of proposed general proof shortcuts. No novelty claim is made. Let A be a nonempty finite subset of the positive integers, N=|A|, f_A(theta) = sum_{a in A} cos(a theta), K(A) = -min_theta f_A(theta), m_r = (1/(2pi)) integral_0^{2pi} f_A(theta)^r dtheta. ## 1. Exact identities and the quadratic moment bound For S=A union (-A), Fourier orthogonality gives m_r = 2^{-r} #{(s_1,...,s_r) in S^r : s_1+...+s_r=0}. In particular m_0=1, m_1=0, m_2=N/2, and all m_r>=0. With normalized convolution on the circle, f_A * f_A = f_A/2. For every real polynomial p and every K>=K(A), E[(f_A+K) p(f_A)^2] >= 0. Writing H_d=(m_{i+j}) and J_d=(m_{i+j+1}), 0<=i,j<=d, gives J_d + K H_d positive semidefinite. For d=1, optimizing p(x)=x-c gives the exact lower bound K(A) >= [sqrt(m_3^2+4m_2^3)-m_3]/(2m_2). It implies a square-root bound when m_3=O(N^{3/2}), but not uniformly when m_3 is of order N^2. This is closely related to Roth's weighted Cauchy-Schwarz argument, reproduced in Bedert, Lemma 4.1, rather than a new solution: https://arxiv.org/html/2509.05260v3 ## 2. Why nonnegative Fourier coefficients and convolution idempotence alone cannot work On G=Z/(2N+1), use normalized counting measure and define F_N(j)=sum_{a=1}^N cos(2pi a j/(2N+1)). The complete geometric sum gives exactly F_N(0)=N, F_N(j)=-1/2 for j!=0. Nevertheless: - E F_N=0 and E F_N^2=N/2; - the nonzero Fourier coefficients equal 1/2; - F_N*F_N=F_N/2; - every moment is a nonnegative exact signed-relation count in G; - F_N+1/2 is nonnegative; - every positive-polynomial/SOS inequality and every Laplace-transform inequality implied by support in [-1/2,N] is satisfied, at every order. Explicitly, E F_N^r = [N^r + 2N(-1/2)^r]/(2N+1). Therefore the group-independent package of these identities cannot imply K>=c sqrt(N), even if all moments and all positivity constraints are used. A successful argument must use a property distinguishing the torsion-free integer frequency group from finite cyclic groups. This finite-group example is **not** a counterexample on the continuous circle: its zero-sum relations are modulo 2N+1, and negative excursions between the grid points have been discarded. ## 3. A stronger obstruction on actual integer-frequency cosine sums ### Claim: bounded-degree whole-sum moment tests are insufficient For each fixed positive integer d there is a finite constant L_d and a sequence of genuine sets A_M subset {1,...,M-1}, with |A_M|~M/2, such that E[(f_{A_M}+L_d) p(f_{A_M})^2] >= 0 for every polynomial p of degree at most d, once M is sufficiently large. Coefficients of p may depend arbitrarily on A_M and M. Thus no method consisting solely of these bounded-degree whole-sum polynomial tests can obtain a lower bound growing with N for all sets A. In particular it cannot prove the square-root conjecture. The degree must grow, or additional tests must use the frequencies/positions rather than just a fixed number of moments of the entire sum. The argument was independently checked by the root/chowla_lower_bound agent, including the rescaling, the variance constant, strict positive definiteness, and the scope of the conclusion. This remains working research, without an external novelty or publication review. ### Step 1: a probabilistic existence construction Independently choose X_a in {0,1}, 1<=at) <= 2 exp(-2t^2/(M-1)). Taking t=2sqrt(M log M) makes the union-bound probability tend to zero. Grid interpolation contributes O(1/M). Evaluating at theta=0 also gives N_M/M -> 1/2. These are actual sets with coefficients exactly zero or one. No repeated frequencies or weights occur in f_{A_M}. ### Step 2: moment asymptotics Put e_M=f_{A_M}-g_M. Orthogonality gives the deterministic bound ||e_M||_2^2 = (1/2) sum_{a 1/4. For every fixed integer r>=3, m_r / M^{r-1} -> c_r, where psi(u) = (1/2) [sin(u/2)/(u/2)]^2, c_r = (1/(2pi)) integral_R psi(u)^r du > 0. To justify the perturbation step, use ||g_M||_1=O(1) and, for s>=1, integral |g_M|^s = O(M^{s-1}). In the binomial expansion of (g_M+e_M)^r, every mixed term is o(M^{r-1}) by the uniform error bound. The pure error term satisfies integral |e_M|^r <= ||e_M||_infinity^{r-2} ||e_M||_2^2 = O(M^{r/2} (log M)^{(r-2)/2}) = o(M^{r-1}) for r>=3. The Fejer-kernel limit follows by changing variables u=Mtheta, since g_M(u/M)/M -> psi(u), and F_M(u/M)/M is bounded by min(1,pi^2/u^2). The subtracted constant contributes negligibly to the moments of order at least two. **The variance constant is 1/4 for the actual random set, not the Fejer-mean value c_2=1/6.** Ignoring this distinction would give a wrong matrix limit. For reference, c_r = [sum_{k=0}^{r-1} (-1)^k binom(2r,k) (r-k)^{2r-1}] / [2^r (2r-1)!], so c_2=1/6 and c_3=11/160. These evaluations are optional; the positive integral representation is sufficient for the proof. ### Step 3: the limiting moment matrix is positive for a constant shift Set v(y)=(1,y,...,y^d)^T and consider Q_{M,d}(L) = E[(f_{A_M}+L) v(f_{A_M}/M) v(f_{A_M}/M)^T]. This is a congruence transformation of J_d+L H_d by diag(1,M^{-1},...,M^{-d}). Its entries are Q_{ij} = m_{i+j+1}/M^{i+j} + L m_{i+j}/M^{i+j}. For fixed d and fixed L, the moment limits give Q_{M,d}(L) -> [ L b^T ] [ b C ], where indices in b and C range from 1 to d and b_1=1/4, b_j=c_{j+1} for j>=2, C_{ij}=c_{i+j+1}. For any nonzero real vector alpha, alpha^T C alpha = (1/(2pi)) integral_R psi(u) [sum_{i=1}^d alpha_i psi(u)^i]^2 du >0. The strict inequality holds because psi takes a nontrivial interval of positive values, and a nonzero polynomial cannot vanish throughout that interval. Hence C is positive definite. Choose, for example, L_d = b^T C^{-1} b + 1. The limiting block matrix is strictly positive definite by its Schur complement. Entrywise convergence in this fixed dimension then makes Q_{M,d}(L_d) positive definite for all sufficiently large M. Congruence preserves positivity, proving the claim for every polynomial of degree at most d, including adaptively chosen coefficients. For d=1, b^T C^{-1} b=(1/4)^2/(11/160)=10/11. This illustrates the bounded nature of the moment estimate; it is not a claimed value of the actual minimum. ### Step 4: a distribution indistinguishable by any fixed number of moments The obstruction can be stated beyond a single SOS test. Let V_d be the polynomials in f of degree at most d with their actual L2 inner product, and let T be the orthogonal compression to V_d of multiplication by f. It is self-adjoint. Its Rayleigh quotients are / = E[f p(f)^2]/E[p(f)^2]. The proved positivity gives every eigenvalue of T at least -L_d. Since f<=N_M, every eigenvalue is also at most N_M. Take the spectral measure of T at the constant vector 1. It is a positive probability measure nu_M supported in [-L_d,N_M]. It has exactly the same moments as f through order 2d+1. Indeed T^j1=f^j for j<=d. For i,j<=d, = E[f^{i+j+1}], because orthogonal projection may be removed against f^i in V_d. This covers every order 1 through 2d+1; order zero is one. Therefore the first 2d+1 exact moments of a genuine large cosine sum can also belong to a probability distribution whose negative support stays bounded. Merely knowing those finitely many moments, their positivity constraints, and the upper bound f<=N cannot force a growing negative minimum. This is the Gaussian-quadrature viewpoint, derived here directly by finite-dimensional spectral theory. ### What this result does not say - It does not obstruct moment order growing with N. - It does not obstruct tests involving several shifted sums, individual frequencies, Fourier coefficients of nonlinear expressions, or arithmetic order. - It does not say the true minimum of the constructed f_A stays bounded. Indeed known general theorems imply it diverges; Hoeffding only gives the upper estimate K(A_M)=O(sqrt(M log M)). - It does not refute Chowla's conjecture. ## 4. Context from current primary literature Bedert's argument uses frequency-sensitive auxiliary functions and asymmetric additive boundaries, not only scalar moments of the whole sum. See especially Lemma 4.1, the auxiliary functions in Sections 5-6, and Proposition 7.3: https://arxiv.org/html/2509.05260v3 Shankar's September 2026 preprint combines the Roth-type count with an exact boundary-averaging identity and the asymmetric-boundary estimate to obtain a uniform n^{1/5} lower bound: https://arxiv.org/html/2609.05338v1 These papers supply relevant primary-source context, not a square-root proof. Their frequency-sensitive structure is precisely the information discarded by the scalar fixed-degree moment relaxation above.