Paper 134EB: Finite-Window Energies and the Remaining Theta Growth Bound An explicit family with polynomial conditioning and a complete infinite-tail estimate

Paper 134EA reduced an adaptive energy approach to the Riemann hypothesis to two main requirements: limiting growth in the finite part of the problem and controlling its interaction with the infinite remainder. This paper constructs an explicit family of correlated energies for which both the conditioning and the full contribution of the infinite tail are controlled at every approximation order. Using a finite time window that grows logarithmically with the approximation order, we obtain positive real matrices whose condition numbers grow no faster than a constant times the square of the order multiplied by its logarithm. We also establish an explicit bound on the complete response, with a controlled prefactor and a certified exponential growth rate. Without assuming the Riemann hypothesis, we prove that these certified rates converge to the imaginary spectral height of the exact theta operator. This quantity measures how far the operator’s spectrum extends away from the real axis. The energy construction therefore introduces no additional gap in the limit. Its limiting growth rate is zero exactly when the Riemann hypothesis holds. We reformulate the endpoint matrix norm estimate using a finite Taylor polynomial and a positive quantity obtained from a Gram matrix. This leaves one specific source estimate unproved: the logarithm of the remaining source quantity must become negligible compared with the square of the logarithm of the approximation order as that order increases. A separate estimate involving theta moments shows that the feedback contribution decreases faster than any fixed geometric decay rate. For the prescribed family of energies, the boundary conditioning and infinite tail requirements are therefore resolved. The remaining zero growth estimate has not been proved, so the argument does not yet establish the Riemann hypothesis.