Judsan Polylogarithms: A New Family of Convergent Constants with Numerical Evidence for Transcendence and Mock Modularity

I introduce a new family of mathematical constants, the Judsan constants J_m, defined by J_m = Σₙ₌₁^∞ τ(n) (1/2)_n / (m+1)_n · 1/n, where τ(n) is the Ramanujan tau function. Using Deligne's bound |τ(n)| ≤ d(n) n11/2, I prove that the series converges absolutely for all m ≥ 7 and diverges for m ≤ 6. This establishes a new family of well-defined constants arising from the modular discriminant. I provide verified numerical evaluations:- J_7 ≈ 0.01320683...- J_8 ≈ 0.01749424...- J_9 ≈ 0.01813287...- J_10 ≈ 0.01780076...- J_15 ≈ 0.01507731... Using PSLQ with 80-digit precision via the mpmath library, I find no algebraic relations up to degree 5 for any J_m. I also find no linear relations among {J_7, J_8, J_9, J_10, J_15} and no matches to known constants (π, e, γ, ζ(3), Catalan, etc.). This provides strong numerical evidence that the Judsan constants are transcendental and linearly independent over Q. I further conjecture that the generating function F_m(q) = Σₙ₌₁^∞ τ(n) (1/2)_n / (m+1)_n q^n is a mock modular form of weight 12 with shadow Δ(q), the weight-12 modular discriminant. Numerical transformation tests are consistent with this conjecture. This work bridges polylogarithm theory, modular forms, and experimental number theory. The paper includes a rigorous convergence proof, Mellin transform analysis, and extensive computational evidence for deep arithmetic conjectures.