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

Let πk be the stationary distribution of Tao’s Syracuse Markov chain on (Z/3kZ)∗ andSk = 3∤ξ |ˆµk(ξ)|2 its high-frequency Plancherel mass. We identify the leading constant in|dk|2 · 3k−1 → c in closed form as c = 7/45, equivalently S∞ = 7/15, via a Plancherel decomposition of Sk, a conservation law for the bilinear pair-form moments, and a diagonal recursionoperator Tdiag = (1/5)[[1,1],[4,4]] whose eigenstructure forces the asymptotic mass onto the(1, 4)-eigendirection. The value is certified by exact rational computation of Sk through k = 6,with S1 = 2/3 and S2 = 10/21. The algebraic decay rate of Sk → 7/15 is 1/3; the rigorous √Ncancellation underlying the associated exponential-sum bound [1, Remark 27] remains open, butwe narrow it, ruling out the Cochrane and van der Corput routes and showing any pointwiseN estimate insufficient, which leaves sum–product, band-ℓ1, and smooth-completion methods.This constant is the leading L2 coefficient for Syracuse trajectory-measure increments in Tao’sprogram; the paper closes its value and isolates what remains.