A Family of Cotangent-Polylogarithm Integrals, Harmonic Sums, and Level-Four Cyclotomy

For integers v greater-than 2 and parameters 0 less than lambdaw less than 1, we study the family Iv (lambda): = integral cot pi/2 0 x Liv (lambda 1+cotx ) dx. We prove absolute convergence, reduce the integral to a Dirichlet series with positive coefficients, and evaluate it in terms of a depth-two multiple polylogarithm, with w=(1+i)/2:Iv (lambda)=pi 2 ImLiv(lambdaw)-ReLiv,1 (lambdaw,w-1). We then derive an exact second-order recurrence for the Dirichlet coefficients, an Euler differential descent in the deformation parameter, a functional difference equation for the weight generating function, and an exponentially convergent expansion in generalised harmonic numbers and zeta values. As consequences, the sequence Iv:=Iv(1)is strictly decreasing, Iv = pi4+(pi4-12) 2-v+0(3-v),v aero lazy eight, and no nontrivial linear recurrence with constant coefficients in v can hold. At lambda=1,the family gives a concrete one-dimensional model for level-four cyclotomic periods built from harmonic sums.