Paper 134EA: Adaptive Energy Bounds for the Theta Evolution An exact real operator, correlated energies and a finite-block certificate with an infinite tail

actly DZ’s exponential transform. The free weighted shift has a provedsubexponential bound; the theta dependence enters through one rank-one feedback term. The sourceinequality 8Mm+2 ≤ (2m + 81/80)Mm gives an explicit geometric bound for the entire feedbacktail. A time-dependent energy identity includes metric variation and the cost of recovering theoriginal norm. Every diagonal adaptive metric has an unavoidable positive exponential allowance inthis realization, so correlations between coordinates are essential. A full matrix energy on a finiteblock, joined to the complete infinite tail, gives a global bound whose rate depends on boundaryconditioning. Directed interval verification of a twelve-dimensional correlated energy establishes|P (iy)| ≤ 1.09 × 1011e0.29|y| for every real y. The remaining task is to make the certified ratesapproach zero. That convergence, and hence the Riemann Hypothesis, is not established here.