The large American factories numbered movements sequentially and, in several cases, left production records that were later compiled into published tables. This is why an American movement can often be dated within a year or two while a contemporary Swiss watch cannot.
The number that matters is on the movement, not the case. American cases were bought separately by jewellers, so case and movement numbers are unrelated and a case serial dates only the case.
Published tables are usually interpolations. A table listing a range for a given year is asserting that production passed through that range in that period, which is an approximation built from surviving ledgers, shipping records and observed examples.
Grade names and run numbers add resolution. Many factories recorded movements in runs of a given grade, so the combination of serial, grade marking and jewel count can confirm or contradict a table reading.
State the basis of your date. 'Circa 1912, from published Hamilton serial tables' is a research claim a later reader can check; a bare date is not.