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It's pretty easy to disprove.
Jan 1 can be one of 7 days.
It is either a leap year or not.
Hence there are only 14 patterns of dates for a calendar year.
One year in 4 is a leap year, so the pattern of days in the year would repeat after 28 years, except for the Gregorian century rules which make it repeat after 400 years.
In 400 years there are 401 months with 31 days where the 31st is a Tuesday. For July (and January, March and May) there are 58 such months. That is once every 6.897 years on average. August and October only have 56. December has 57.
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