Square Root Coupling Between Structure and Entropy: A Perspective from Riemann Phase Modulation

We report a stable structural relationship emerging from Riemann phase modulations and

entropy growth: the structural function ϕ(n) maintains a nearly constant ratio with H(n),

where H(n) = log(1+ϕ(n)2) denotes entropy. This coupling shows high stability across the first

200 Riemann zeros, consistent with a square-root scaling. Additionally, the structural derivative

ratio K(n) = dlog |ϕ(n)|

dlog H(n) stabilizes around 0.5, suggesting a scaling regime complementary to the

entropy attractor at K = 1 in prior work.