Paper 31 — The Euler–Gauss Obstruction for Black Circle Quasi-Normal Mode Coefficients

We determine the analytic structure of A_part(ω; 2, 0), the Wronskian projection coefficient entering the Black Circle quasi-normal mode condition for l=2, m²=0. Applying the Euler integral formula to the source integrals of Paper 17, we obtain A_part as a combination of ₃F₂(1) hypergeometric series. The central result is a universal identity: the balance parameter of these series is S=−iΩ, where Ω is the spectral parameter itself. This is a structural consequence of the Fuchsian exponent sum at the outer boundary and holds independently of the Frobenius branch and of l. Since S∉ℤ_{>0} at any physical QNM frequency, all standard summation theorems (Gauss, Pfaff–Saalschütz, Watson, Whipple) fail, and A_part admits no representation as a finite Gamma-function ratio (OP 30.1, Derived-Negative). We further show that the dominant near-z=1 approximation has O(1) gap at physical QNM frequencies, and that the Weniger u-transform fails structurally for every physical QNM, with the discriminant δ₂/2=√23/4 — an algebraic invariant of the l=2 Black Circle ODE — controlling the obstruction. New algebraic identities are derived: p₁p₂=l(l+1)/2 and cross-Wronskian Δ=δ_l/(2Ω).

Part of the series Zero-Divisors in Cayley–Dickson Algebras (Papers 1–32, Zenodo).

Papers referenced in this work:

Paper 10 — Loop, Threshold, and Residue (Conjecture 6.3 origin):
https://doi.org/10.5281/zenodo.20507204

Paper 15 — The Kummer Connection Matrix for the Black Circle:
https://doi.org/10.5281/zenodo.20554392

Paper 16 — The Limit-Circle Obstacle for Black Circle QNMs:
https://doi.org/10.5281/zenodo.20554795

Paper 17 — The Physical Boundary Condition at r=0 (source of A_part definition):
https://doi.org/10.5281/zenodo.20555042

Paper 30 — High-Precision QNM Spectrum of the Black Circle Interior (numerical checks):
https://doi.org/10.5281/zenodo.20733658