Closed-Form Plancherel Mass of the Syracuse Stationary Measure: The Constant 7/45

Let πk denote the stationary distribution of Tao’s Syracuse Markov chain on the multiplica-
tive group (Z/3kZ)∗ and let Sk = P
3∤ξ |ˆμk(ξ)|2 be the associated high-frequency Plancherel
mass. We identify the leading constant in the level-deviation norm |dk|2 ·3k−1 → c in closed form
as c = 7/45, or equivalently S∞ = 7/15. The identification rests on a chain of rigorous structural
identities: a Plancherel decomposition of Sk; a conservation law and leading-mode identity for
the bilinear pair-form moments; and the diagonal recursion operator Tdiag = (1/5) · [[1, 1], [4, 4]],
whose eigenstructure forces the asymptotic mass onto the (1, 4)-eigendirection. The value 7/15
is certified by exact rational computation of Sk through k = 6 together with the closed forms
S1 = 2/3 and S2 = 10/21. The remaining question (the rate at which Sk approaches 7/15)
is shown, through an exhaustive operator-theoretic search, to admit no finite-rank discrete-
eigenvalue closure; we state it as an explicit open problem. The constant enters Tao’s measure-
theoretic Collatz program as the leading coefficient in the L2 bound on Syracuse trajectory-
measure increments; this paper closes its value and isolates exactly what remains.