Partial sums of the Legendre symbol at rational points: an exact finite identity, the rank φ(M)/2, and the complete list of vanishing initial sums

For a prime \( p ≡ 3 4 \), the partial sums of the Legendre symbol at rational points \( a/M \) are governed by a single exact, finite identity: the Fourier transform of \( a ↦ S_p(a/M) \) over \( (Z/M)^× \) equals an explicit cyclotomic coefficient \( γ_ψ \), given by a short Euler product over the primes dividing \( M \) but not the conductor of \( ψ \), times a generalised Bernoulli number \( B1,χ_pψ \). From this one identity follow the rank \( φ(M)/2 \) of the block vector within a fixed signature, a closed form for the coefficient of \( h(-p) \) together with an exact mean-value law, the fact that a mesh closes over class numbers of imaginary quadratic fields precisely when \( M ∈ \{4,6,8,10,12,24\} \), and — the main new result — the proof that the century-old list of rational points where \( S_p \) vanishes identically is complete, consisting only of \( 1/4, 3/4 \) and \( 1/10, 3/10, 7/10, 9/10 \). Outside the closing meshes nothing actually fails: the invariants simply have higher degree, so that \( M = 14, 18 \) obey an explicit quadratic law tied to relative class numbers of sextic CM fields and \( M = 30, 60 \) to cyclic quartic CM fields over \( Q(√5) \). Every claim was verified numerically to \( 10⁻¹⁴ \), and the only non-elementary step remaining is a non-degeneracy statement for the joint values \( L(1,χ_pψ) \).