A Parameterized Cluster–Multiplicity Sieve for the Zeros of the Riemann Zeta Function

Building on Lamzouri’s Hilbert-space proof (arXiv:2609.02882), this preliminary note retains multiplicity information discarded in the final simple/distinct-zero estimates. It proves a one-parameter unconditional cluster–multiplicity sieve separating multiple critical-line zeros from off-line functional-equation pairs, derives optimized upper bounds for zeros of multiplicity at least r, and records an entropy bound for the multiplicity distribution. In particular, the number of distinct zeros of multiplicity at least 3 is at most 0.054584 N(T) + o(N(T)); the corresponding bounds for multiplicity at least 4 and 5 are 0.028095 N(T) + o(N(T)) and 0.018730 N(T) + o(N(T)). The note does not improve the 67.25% simple-critical-line proportion; it refines the distribution of its complement. This is a preliminary research note, and no claim of established literature priority is made.