When 3 Horizons meet...

When a charged de Sitter black hole is pushed to its ultracold limit, all three of its horizons — Cauchy, event, and cosmological — coincide at a triple root of the lapse. At the well-studied double merger (the Nariai limit), the quasinormal spectrum develops an exceptional point: modes coalesce and the potential becomes Pöschl–Teller. It is natural to expect the triple merger to do the same, only more violently. This paper proves it does the opposite.

Resolving the horizon cluster before taking the limit, the surface gravity vanishes as ε² while the barrier height vanishes as ε³, so the control ratio √V₀/α diverges as ε^(−1/2) and the system is driven away from the branch-point condition. Both exponents follow from a single merger-multiplicity law fixed by root counting alone: at a merger of multiplicity q the near-horizon equation is Bessel of order (q−1)/(q−2) — integral only for the triple root — and the exceptional-point mechanism operates only at q = 2. Direct spectral computation confirms the mechanism: eigenvalue condition numbers decrease monotonically along the approach, the opposite of defective coalescence.

The slowly rotating polar sector then develops a genuine first-order obstruction: a trace pole δ(tr V) ∼ −x⁻³ at the triple horizon, certified here against both classes of reformulation that could have dissolved it — it transforms by pure conjugation under every horizon-regular GL(2) change of master variables, and it is stable under isospectral Darboux deformation. The obstruction localises to the electromagnetic channel, is proportional to the azimuthal number m — axisymmetric modes are unobstructed, making it a Zeeman-type corotation effect — and forces, for m ≠ 0, a boundary layer governed by a single universal inner equation of Heun class.

This paper condenses and extends the monograph Reciprocal Horizons and the Horizon Octave (v23), where the full symbolic verifications, certificate scripts, and solver validation are recorded.

quasinormal modes, black hole perturbation theory, Reissner–Nordström–de Sitter, ultracold black hole, triple horizon, exceptional point, Nariai limit, surface gravity, slow rotation, Einstein–Maxwell, Darboux transformation, Heun equation, Kodama–Ishibashi, non-Hermitian spectral theory, de Sitter space pseudospectrum, strong cosmic censorship, Kerr–Newman–de Sitter, boundary layer analysis, WKB approximation, spectral methods, Chebyshev collocation, symbolic computation, matched asymptotic, degenerate horizon, root multiplicity, isospectrality, Regge–Wheeler, Zerilli