Paper 134EC: Signed Theta Corrections and the Remaining Cancellation Bound Exact quadrature repairs, a smaller operator core, and a nearby-source obstruction

Paper 134EB left open a particular estimate involving the theta source. It requires the logarithm of a specified source quantity to become negligible compared with the square of the logarithm of the approximation order. This continuation establishes three partial results. First, the first two errors in canonical Gaussian quadrature determine an exact correction of rank two to the response Hankel form at every approximation order. For the actual theta source, the canonical approximation using two quadrature nodes produces both a negative test value and a negative Schur complement. The omitted higher moments repair both failures, and these repairs are verified by rigorous rational bounds. However, the general correction has both positive and negative directions. Positivity of the raw moment data therefore does not automatically transfer to the response matrix. Second, we construct an explicit feedback truncation that reduces the Gram estimate to a much smaller core. The size of this core grows no faster than a constant times the logarithm of the approximation order divided by the logarithm of that logarithm. The normalized contribution from transient effects tends to zero. This reduction does not require control of the conditioning of an eigenvector basis. Third, we use the Rodgers and Tao theorem to identify deformations in the negative heat direction that satisfy all the source upper bounds used previously while still having nonreal zeros. Keeping the original normalization scale fixed, we show that the change in the corresponding companion operator has an explicit trace norm bound proportional to the deformation size. At the same time, every feedback magnitude decreases. These examples show that smaller feedback and the previously established upper bounds are insufficient to rule out nonreal zeros. Together, these results establish a specific finite correction for the theta source and identify a limitation of arguments based only on the sizes of individual contributions. The required cancellation estimate across all approximation orders remains unproved, and the paper does not establish the Riemann hypothesis.