en.wikipedia.org/wiki/Pope_Paul_V

The wording is anachronistic. The Accademia dei Lincei’s own history says Federico Cesi founded it in Rome in 1603 with three young colleagues, and that Galileo was among its members from 1611. The same official history says it was only later, in 1847, that Pius IX re-established it as the "Pontificia Accademia dei Nuovi Lincei". So describing the 1611 academy as "papal" is incorrect.

This cannot be correct as written. A Catholic-Hierarchy entry for St. Albert Cardinal de Louvain lists his canonization date as 9 Aug 1613. Separately, the Holy See’s list of popes shows that Paul V’s pontificate ended on 28 Jan 1621. Since Paul V was already dead by January 1621, he could not have canonized Albert on 9 August 1621.

The surviving 1616 certificate does not say Galileo could use the geocentric model as a theoretical device. Bellarmine’s certificate says Galileo had been notified that the doctrine attributed to Copernicus—that the earth moves around the sun and the sun stands at the center—"cannot be defended or held." A scholarly summary by Maurice Finocchiaro explains Bellarmine’s clarification this way: Galileo was warned not to believe, support, or defend Copernicanism as true, but only as a hypothesis. So if one speaks of a model being usable hypothetically, it is the Copernican/heliocentric model, not the geocentric one.

This overstates Paul V’s role. Official St. Peter’s Basilica materials say construction of the current basilica began in 1506 and was completed in 1626. Paul V died in 1621. A Vatican page on the facade explains that Carlo Maderno worked on the facade from 1607 to 1614 on commission from Paul V, which matches the inscription mentioned in the article. So Paul V did sponsor major phases of the project, but he did not finance the final completion of the entire basilica as such.