Goines On: Before 1901, after 2099?

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Goines looked at his archives and discovered that he had years ago pieced together more about calendar recycling than he was currently re-piecing together. He had tackled the subject in some depth eight years earlier, even written a sestina about it for the Goineses’ 47^th wedding anniversary.
    While he didn’t think he still had the brain power those earlier investigations required, he was pleased that he was managing to re-tool a thing or two for his “parlor trick.”
    The earlier Goines had avoided thinking about what to do to determine directly the day of the week that a year preceding 1901 or following 2099 began on, rather than just to total the day-shifts for regular and non-leap years and then divide by 7. He had written, “In order to use the template for periods involving a 00 year that is not a leap year [1900 & 2100], the appropriate regular-year calendar must be used for the 00 year itself, and the template must be shifted to apply to subsequent years.” The phrase “appropriate…calendar” signaled that the younger Goines had not developed the mnemonics for exploring the regions before or after the period 1901–2099.

And today’s Goines hadn’t either…until the very night he specified “taking shortcuts” within the original 1901–2099 region. (That was the night he was going to securely memorize the 28-year cycles for those years and practice their use; 1909-1936, 1937-64, etc.)
    As usual, writing another note to himself before going to bed prompted him to start musing about bridging the 1900 & 2100 bumps.
    The key, he thought, was identifying precisely where his current 28-year cycle (2021-2048) could take up again, back into the 19^th and forward into the 22^nd Centuries by Pope Gregory XIII’s calendar, after repeating over and over again in 1909–1936, 1937–1964, 1965–1992, 1993–2020, 2021–2048, 2049–2076 (and, partially, in 2077–2099, with a 28-year cycle’s first 23 years; in fact, the years 1901–1908 were the last 8 years of a cycle).
    But he thought he needed first to work out in more detail the 6₁-11-6₂-5 pattern of sub-cycles within the 28-year cycle, both for his researches and for “oiling” the machinery of his parlor trick so that he could navigate quickly inside a 28-year cycle after he had associated a given date with it:

1. The first 6-year sub-cycle (“6₁”) begins immediately following a leap year and includes 1 leap year:
   |_|_|_|L|_|_|
   
2. The 11-year sub-cycle begins immediately preceding a leap year and includes 3 leap years:
         |_|L|_|_|_|L|_|_|_|L|_| 
3. The second 6-year sub-cycle (“6₂”) begins two years after a leap year and includes 1 leap year:
               |_|_|L|_|_|_|
   
4. The 5-year sub-cycle begins and ends with a leap year and includes 2 leap years:
                     |L|_|_|_|L|.
   The four sub-cycles accounted for all 7 leap years, the number of unique calendars for any 28-year cycle.

    Right off, Goines observed that the first three sub-cycles would furnish the 23 years 2077-2099 (6₁ + 11 + 6₂). The 1901-1908 years, however, needed furnishing for 8 years, which would take half of a 6₂ sub-cycle plus a 5-year sub-cycle. Now, that was interesting! Assuming that the interrupted cycle would continue backwards into the 19^th Century after the bump over the 1900 non-leap-year, what were the bump’s and the cycle’s characteristics?
    But he needed to focus on one thing at a time, and that would be the 2100, 2200, & 2300 bumps….

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