Shri Phadnis writes…
As far as I know, today’s system depends on Sun’s transit through Rashis. It is prevailing for the last 500 years at least. Is there a recognized cycle of years after which the Date of Chaitra Shuddha Pratipada repeats? If it does then only one can say that an exact match between lunar and solar year is achieved. Shri. Oak may like to verify.
I repeat that with ‘one lunar month’ as a correction unit, it is doubtful whether any system of Adhika Masas, will give an exact match between Lunar and solar years after a specific cycle of even several years.
Vedanga Jyotisha (Laghadha) employs 19 year cycle with use of ‘one lunar month’ as a correction unit that achieves exact match between ‘lunar’ and ‘solar’ year every 19 years.
The time of Vedanga Jyotisha, based on internal (and very intricate and thus precise) astronomy references can be estimated.
Do keep in mind, that Vedanga Jyotisha is a practical manual (calendar making) and thus updates are expected. Consequently, time estimated (by Shri P V Holay) is to be understood as that of the latest update, rather than its original composition).
Shri P V Holay estimated time of latest correction/update to Yajus Path (Vedanga Jyotisha) to be that of 2352 BCE and that of Rik path (Vedanga Jyotisha) to be much prior to (but definitely not later than) 2976 BCE.
One may refer to excellent and groundbreaking work ‘Vedic Astronomy (Vedanga Jyotisha)’ by Shri Prabhakar Vyankatesh Holay (published 1988?).
In Yajus cycle (Yajus-path of Vedanga Jyotisha), adhika masa appear at the interval of 2.5, 2.5, 3. 2.5, 3. 2.5, 3 years in a 19 year Yuga cycle.
In Rik cycle (Rik-path of Vedanga Jyotisha) adhik masa appear at he interval of 3, 2.5, 3, 2.5, 3, 2.5, 2.5 years in a 19 year Yuga cycle.
One may do a quick test of this exact match between Lunar and Solar calendar, for example, by noting down the Tithi/nakshatra of one’s birth date (per Julian calendar, e.g. 11 January 2016) and again, note down the Tithi/nakshatra for the same, 19 years afterwards. Tithi and nakshatra (and lunar month) would match ‘exactly’ for this day.
Effects due to precession of equinoxes would enter the picture over a long period of time (~1000-3000 years) and appropriate corrections, especially in determining the beginning of new Luni-solar year, would need to be made.