CARD-VI: A Zero-Parameter Derivation of h and Bandwidth-Limited Deviations in Gravitational Ringdown

CARD‑VI extends the curvature‑based framework developed in earlier volumes into the strong‑field, rapidly evolving regime of black‑hole ringdown. The model introduces no new free parameters in the gravitational sector. Instead, all medium constants are derived from the curvature structure of the Hydrogen ground state through the scale‑invariant curvature‑cost law expressed in plain text as “s equals r times theta.” This geometric relation, combined with the causal ceiling imposed by the speed of light, fixes a minimum angular update and a finite refresh interval for curvature evolution in the underlying medium. When curvature changes faster than this refresh interval, the medium cannot update curvature continuously, producing bandwidth‑limited deviations from the smooth exponential ringdown predicted by General Relativity. CARD‑VI formalizes this behavior by constructing a unified taxonomy of perturbation families, mode‑mixing pathways, and curvature‑driven update rules that govern the transition from smooth Kerr ringdown to bandwidth‑limited behavior. The framework identifies the specific geometric conditions under which curvature evolution becomes under‑resolved, leading to discrete‑step curvature updates, mode‑family cross‑talk, and departures from the standard exponential decay structure. These effects are not introduced as new physics but arise naturally from the finite refresh interval implied by the curvature‑cost law and the causal speed limit. The resulting model provides a unified organizational structure for all ringdown modes, including their mixing behavior, hierarchy, and deviation channels. It also clarifies how bandwidth‑limited curvature evolution produces predictable, structured departures from the Kerr spectrum without violating the underlying geometric constraints of General Relativity. CARD‑VI therefore serves as the conceptual and mathematical backbone for interpreting the deviation patterns explored in CARD‑IV and the multi‑mode inference strategies developed in CARD‑V, completing the trilogy’s unified curvature‑based description of black‑hole ringdown.