What does Median Tithi (मध्यमा तिथी) signify?
Median Tithi (मध्यमा तिथी) signifies a tithi, in the vicinity of which (+/- 15 Tithis), a specific solar day would occur (e.g. day of summer solstice, winter solstice, spring equinox, fall equinox, etc.) within a period of about ~72 years.
Path of the sun (ecliptic) is the reference frame either for a purely solar calendar or Luni-solar calendar. While indian calendar is Luni-solar calendar, there is no ‘purely solar’ calendar in existence that is capable of incorporating astronomy nuances such as precession of equinoxes. As a result we will employ otherwise prevalent ‘Gregorian (and Julian) calendar’ as the closest approximation for ‘purely’ solar calendar.
For a given ~72 years period, the range of Tithis over which (say) a day of winter solstice would occur is fixed.
in 2650 CE (~600 years from now), Median tithi for the day of winter solstice would be Margashirsha purnima (Margashirsha Shukla 15). This also means, Tithis for the day of winter solstice would range from Shukla Pratipada to Amawasya. Of course this is a trivially true statement, as range for tithis for a specific day of solar calendar (e.g. day of winter solstice) would encompass ‘all 30 tithis (15 of Shukla paksha and 15 of Vadya/Krishna paksha).
This means if a historical evidence demands matching of certain lunar tithi with that of a specific day of solar calendar (e.g. day of winter solstice); it is always possible to find a combination that would satisfy this requirement.
Having said that, once a year (per solar calendar) is determined (or fixed), the lunar tithi is also automatically fixed for (say) day of winter solstice, for that specific year and thus non-negotiable.
On the other hand, how a month, specifically a lunar month, is named depends on multiple factors.
These multiple factors include ‘purnamanta’ vs. ‘amanta’ methods/systems of lunar months and corresponding nomenclature procedures, methods to determine which month is to be interpreted as ‘Adhika masa’ and nomenclature of these ‘adhika masa’, but even more critical, how the nomenclature of adjacent lunar months is affected, due to insertion of these ‘adhika masa’.
Recognition of rarely occurring ‘Kshaya masa’ adds its own complications to naming of lunar months.
This is the background against which problem(s) of identification of lunar months during the Mahabharata times (based on references to lunar months of Mahabharata text) should be analyzed.
And that is exactly what we will do. Stakes are high, for our understanding of ancient Indian history, and so are the rewards.