Orthogonal Polynomials of the Semilocal Euler–Gamma Measure: Effective Reconstruction of the Weil Form and an Arcsine Limit Theorem at the Prime Two

For a finite set P of primes and σ>0 let wP,σ(t)= LP(σ+it)², where LP(s)=π-s/2Γ(s/2)∏ₚ(1-p⁻ˢ)⁻¹. At σ=1/2 this is the spectral measure of the semilocal scaling operator, and its logarithmic σ-derivative produces the archimedean and prime terms of the semilocal Weil quadratic form. We study the orthogonal polynomials of these measures from this viewpoint. First, for every λ>1 and every P containing the primes p≤λ², truncating the complete Weil form in the polynomial grading reconstructs it with an explicit error O(log N/N), uniformly on the unit ball of H² functions supported in [-logλ,logλ]. Second, for P=\2\ and σ∈[1/4,3/4], we prove that the normalized inverse Gram corner dₙ(σ), which governs the ratios of the Jacobi coefficients of w_\2\,σ and w,σ, converges in C¹ as n→∞. The limit is a series of ordered words in the coordinate projection onto ²(Z≤0) and spectral projections of the operators (2b)-(2b)X, b≠0, on ²(Z), whose spectral measures are arcsine laws; already its second-order term differs from the value predicted by Toeplitz theory. Consequently the ratios of the Jacobi coefficients and of the leading coefficients converge, and the σ-derivative of log aₙ(\2\,σ)-log aₙ(,σ) tends to zero uniformly in σ. The proofs exploit that multiplication by eⁱᵘᵗ acts in the Meixner–Pollaczek basis as a hyperbolic element of $SU(1,1)$. Time flies, and I want to explore more of the unknown.