Paper 134ED: Theta Source Operators and Quantitative Cancellation Bounds Complete quadrature errors, finite Hankel detection, and arithmetic obstructions

We study the logarithmic response of the function constructed from the theta function in Paper 134EC, keeping track of both positive and negative contributions. We show that the associated Hankel form has a self adjoint Markov representation built from the original source. Its normalized kernel is smooth and rapidly decaying. The number of independent negative directions in the resulting operator equals the number of distinct conjugate pairs of nonreal zeros of the original function. For the basic Gaussian quadrature approximation, we derive an exact formula for the approximation error. This formula separates the error into explicit factors, including a positive transform of the source, and establishes a precise limiting rate for the error coefficient. However, the exact Schur correction contains an additional coupled term whose sign cannot be determined from the basic approximation error alone. We also establish explicit upper and lower bounds connecting the size of the operator’s negative part, its total negative contribution, and a reciprocal measure of zero height. These results allow finite Hankel positivity tests and carefully controlled companion approximations to place quantitative restrictions on where nonreal zeros could occur. Under the assumption that the reciprocal zero height measure vanishes, we determine the precise logarithmic growth scale of the scalar response. This scale is the cube root of the ratio between the input and its logarithm. Companion operators built from bounded weighted shifts have a larger growth scale. Comparison examples show why the estimates must preserve cancellation between positive and negative contributions. Finite theta sums and nonconstant monotone tapers have both positive and negative directions, even though they converge in operator norm. Smooth changes to the source can preserve any chosen finite section of the response, leave the original tail unchanged, and retain the critical strip while introducing nonreal zeros. In these examples, the modular traces lose the positive weights required for an ordinary positive spectral representation. Finally, we derive an exact signed energy formula using translations associated with prime numbers. This formula reproduces the same decisive Schur quantity and provides bounds for finite cutoffs, together with classes of test functions for which positivity can be established. The central unresolved question is whether the integrated theta form is nonnegative for every admissible test function. That statement is equivalent to the vanishing of the reciprocal zero height measure and to the Riemann hypothesis. This paper does not prove it.