A Theta-Kernel Reformulation of Riemann-\ (\) Growth and the Obstruction to Blockwise Positivity

Let Φ be the Riemann theta kernel and D(z)=∫0∞Φ(u)cos(zu) du, so that Ξ(z)=ξ(1/2+iz)=4D(z). The Riemann hypothesis is equivalent to the global growth criterion ∂y|D(x+iy)|2>0 for all real x and all y>0; in the form ∂y|D(x+iy)|2=18ℜ(ξ′(s)ξ(s)¯) with s=1/2+y+ix, this criterion is an identity between entire quantities and remains meaningful at the zeros of ξ. Using the symmetric Hadamard product we close the half-plane σ>1 unconditionally, localising the obstruction to the strip 1/20, the globally summed longitudinal and transverse sectors cancel to all algebraic orders as x→∞ (each is of size x−5, their sum is exponentially small, and their ratio tends to −1), and deduces that for every y>0 and all sufficiently large x at least one aligned block is negative. Universal phase-aligned blockwise positivity is therefore impossible, and independent algebraic-order estimates of the two sectors, followed by addition, cannot determine the sign, which survives only in the exponentially small remainder left after the cancellation. Global positivity remains equivalent to the Riemann hypothesis and remains open; what the theta-kernel representation shows is that it cannot be reconstructed from independent local positivity. Positivity here is intrinsically nonlocal. Correlated groupings of blocks, exact resummation, and globally coupled identities are not excluded. No proof of the Riemann hypothesis is claimed.