Exponential Sums of the Titan Polynomial: Numerical Evidence for Non-Generic Monodromy

We compute the additive exponential sums S_p = Σ exp(2πi Q(n)/p) for the Titan polynomial Q(n) = n⁴⁷ − (n−1)⁴⁷, a degree-46 cyclotomic norm form satisfying the palindromic symmetry Q(n) = Q(1−n). For the 111 primes p ≡ 1 (mod 47) up to 50,000, the normalized magnitude |S_p|/√p has observed mean ≈ 1.50 and maximum ≈ 8.65 (at p = 283). This mean lies far below both the Gaussian random walk prediction (≈ 5.97 for 45 independent unit vectors) and the generic compact symplectic group USp(44) prediction (≈ 3.74). The palindromic symmetry selects symplectic over orthogonal monodromy; by Katz's theorem, USp(44) is the expected generic group. The dramatic shortfall suggests the geometric monodromy group is a proper subgroup of USp(44), constrained by the cyclotomic origin Q(n) = Φ₄₇(n/(n−1)). We conjecture that this anomalous cancellation arises from a Jacobian splitting: the associated algebraic variety decomposes into lower-dimensional CM abelian sub-varieties indexed by the characters of (ℤ/47ℤ)×. The repository includes the paper (5 pages), two data CSVs (111 data points and summary statistics), a 4-panel PDF figure, and three scripts (Python computation, Python figure generation, SageMath original).