Quantitative bounds for the three-block Aluthge transition

Three fixed real Jordan blocks are joined by two positive links vanishing as powers of a small parameter. A candidate quantitative estimate for ordinary half Aluthge iteration bounds the moving-gauge error against a calibrated positive bidiagonal reference by a conservative logarithmic modulus, uniformly over all sufficiently late observation times and gaps in an arbitrary shrinking envelope. Constants and sufficient finite thresholds have terminating mathematical constructions; general practical numerical values are not evaluated. The proof combines relative discrete-prefix entry, a two-cut filtration, a transverse tube and forced logarithmic contraction. The scope is fixed positive data in three blocks; the endpoint rate, complex phases, generic perturbations and additional blocks are outside the result.