Quellen und Lizenzen

VIES, European Commission: structure table in its FAQ, in our own notation. VIES is never queried, by the site or for it. Its notice forbids retransmitting or reproducing its contents, and its FAQ says the Commission does not disclose the algorithms.
IBAN structure by country: ISO 13616 registry, version 101, read from the table in the open-source python-stdnum package (LGPL). The registrar's own download answered 403 to our request, so this table has not been compared with its latest version.
Decreto-Lei 14/2013 (PT). Ministerio del Interior and Orden EHA/451/2008 (ES). Service-Public.fr (FR). Bundeszentralamt für Steuern (DE). Agenzia delle Entrate (IT). Belastingdienst (NL). GOV.UK (UK).
Check digit rules for VAT numbers are not official: they come from public descriptions, tested on examples from the python-stdnum documentation. That code was read, not run.