Paper 134DX: Positive Contact Responses and the Riemann Theta Kernel A Constructive Bridge and Its Obstructions

This paper connects the real occupied-contact calculations of Papers 49J–49L and T9 to thetheta-response criteria of Papers 134AP, 134CO and 134CQ. For a positive self-adjoint operator anda normalized receiver, an explicit common real feature map represents the divided-difference kernelof the negative logarithm of its resolvent response. Frequency-dependent elimination preserves thatresponse, but the logarithmic derivative of the full determinant includes an additional eliminated-state trace. The exact four-mode sector of the 92-site cuboctahedral shell provides a completepositive interlacing example. Its pole-residue ratios nevertheless exclude identification with thelogarithmic derivative of any entire function under any common positive amplitude and spectralrescaling. A second obstruction is specific to the theta integral: averaging the individual componentresponses omits a strictly positive variance term and produces a function whose Taylor series atzero has zero radius of convergence. A third obstruction shows that trace-normalized bounded-degree FCC stiffness determinants converge to an exponential, not the normalized Riemann symbol.Trace-norm estimates and an independently specified infinite positive control show what a controlledreplacement must accomplish. Eleven executable jobs reconstruct the native reduction and test thealgebra, finite theta matrices, elimination identities and limits. The positive contact-side kernel isconstructed; its identification with the Riemann theta kernel remains the unresolved arithmetic step.No Riemann Hypothesis proof is claimed.