Every Patek carries two numbers: a case number engraved on the case, and a movement number engraved on the movement. On a genuine, unmolested watch they correspond to each other in the maison's records, and the Extract from the Archives states both.
The extract is ordered from Patek Philippe for a fee and confirms the production date, the sale date, the calibre, the case metal and the original dial configuration. A watch whose extract describes a different dial from the one fitted has been redialled or reassembled — which is not counterfeiting but is a material fact.
The maison also stamps the Patek Philippe Seal on modern movements, replacing the Geneva Seal it used previously. Both are quality marks applied to the movement, and both are struck rather than printed.
No public database exists. The extract is the mechanism, and it is worth the fee on any watch above a modest value.