The Merge Families of Greedy Mersenne Decomposition Occur at the Base-4 Digits of 1/π

Greedy Mersenne decomposition prices every position of an admitted window in closed form: its cost table is the bit length of a linear function of 2^m modulo 2^p - 1, with the straggler positions read off Pascal's triangle. This note prices the one phenomenon that closure leaves out — the merge, a predicted straggler whose cost lands exactly on the generic cost. We derive the merge inequality from the cost model, show that the modulus and window parameters clear out of it, force its offset to the top bit of the central binomial coefficient, and reduce its solvability at row j to an integer-exact window test on frac(log_2 binom(j, floor(j/2))). The window's crossings sit at n* = 4^m/pi, shifted by exactly one quarter at first order in Stirling's series, inside intervals whose length is exactly 2/pi at every lap; the resulting hit/miss forecast has scored four consecutive laps committed-first, the last a two-row prediction across a thirty-thousand-row void, landed to the row. The main result is the reduction behind that scorecard: the merge family recurs infinitely often if and only if the x4 orbit of 1/pi — equivalently, the base-4 digit sequence of 1/pi — falls in quarter-shifted intervals of length 2/pi infinitely often. Infinitude is therefore open exactly as far as the digits of pi are open, and no farther.