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Every Hindu lunar month (masa) is really the same underlying cycle of the Moon; the difference is only where you draw the line between one month and the next. The Amanta system (followed across South India, Maharashtra, and Gujarat) ends a month at Amavasya (new moon) and starts the next at the following Shukla Pratipada. The Purnimanta system (followed across most Hindi-speaking North Indian states) instead ends a month at Purnima (full moon).
Because of this offset, a Purnimanta month's Krishna Paksha (dark fortnight) always falls inside what the Amanta system still calls the previous month, while its Shukla Paksha (bright fortnight) lines up with the Amanta month of the same name. Both systems track the identical Moon; they simply name the fortnights differently around the seam.
An Adhika (leap/extra) or Kshaya (rare, compressed) month is a whole intercalary month inserted (or, for Kshaya, contracted) to keep the lunar calendar aligned with the solar year; it isn't one of the ordinary Krishna/Shukla halves either system's naming logic is built around. So for the stretch of days an Adhika or Kshaya month covers, the Purnimanta system doesn't apply its usual name-shift at all: it simply carries the same Amanta label for that month. Once the calendar returns to an ordinary (Nija) month, the two systems resume differing by their usual half-month offset.