Closed certificates for fixed mirror-descent convergence (C73-15)

Closed-certificate trees and countable-ordinal rankings characterize convergence of fixed mirror-descent rules for the canonical regularizer and step-schedule class. The stated criterion is infinitary. Contents. Original proof PDF and LaTeX source, a complete research archive with pinned GitHub source snapshots and existing release assets, and a provenance manifest. The archive retains available verification records, source audits, reproducibility instructions and Lean source files. Original files are preserved byte-for-byte. Status. Author-submitted preprint and accompanying research materials. GitHub submission and Zenodo deposit do not constitute independent peer review or catalogue acceptance. This deposit records the existing results and their stated scope; archival checks do not certify mathematical correctness or replace a Lean build. AI use. The hand-proof manuscript and its supporting research materials were developed with GPT Codex assistance, including proof development and AI-assisted checks, as documented in the original repository. These checks are not represented as independent human peer review. Source versions. fixed-mirror-convergence-characterization — 000960f40cf5f7565032a9ba5a2986c93b5e0c5b Rights. CC BY 4.0 applies to the author-controlled materials that did not already carry a license. Existing file-level licenses, copyright notices and third-party attributions remain in force. Original manuscript and repository files have not been edited for this deposit.