HDC-CBC/Θ Asymptotic Limit of Projectability and Return to the Basal Domain

HDC–CBC/Θ — Asymptotic Limit of Projectability and Return to the Basal Domain The HDC–CBC framework — the Correlational Disequilibrium Hypothesis and Correlated Bubble-Cosmos — establishes that geometry does not constitute the fundamental ontological level of reality, but rather an effective projection of an underlying correlational structure. In the previous volumes of the corpus, this architecture has been progressively developed: Δ formalized the necessary emergence of the projected regime from the instability of the basal state of maximal coherence; ΔΩcₜ articulated the effective dynamical closure between basal rupture and historical coherence; and Ωcₜ described the evolution of the observable universe as a dynamical trajectory of scale-dependent correlational coherence. The present work addresses the remaining question of the program: whether the projected geometric regime can persist indefinitely under prolonged correlational relaxation, or whether the dynamics of the framework itself imposes an asymptotic limit on projectability. Starting from the dissipative gradient flow associated with the basal energy functional, it is shown, under minimal structural hypotheses, that an admissible trajectory tends asymptotically toward a critical set where the correlational gradient vanishes. Assuming structural coherence between the projectability map and the correlational disequilibrium that sustains it, it is argued that the derivative of degenerates at that limit. Consequently, the geometric regime — defined by the non-degeneration of projectability — possesses an internal asymptotic limit. The extinction of geometry is not interpreted as dynamical collapse, singular transition or thermodynamic catastrophe, but as a loss of descriptive validity. When projectability degenerates, metric structure, cosmological scale and emergent time cease to be physically meaningful. This loss of projectability finds its operational correspondence in the loss of executability of the ΔΩcₜN domain: the transition toward marginal or broken states does not represent a technical failure of the model, but the computational manifestation of the exhaustion of the projected regime. The system thus returns to the non-geometric basal domain — the Greater Cosmos — not as a temporal reversal or as a cosmological cycle, but as the asymptotic disappearance of the projected phase. There is no inter-bubble memory or causal inheritance between successive projections, because time, causality and geometry belong exclusively to the projected regime. With this, Θ completes the structural architecture of the HDC–CBC corpus through the sequence Δ → ΔΩcₜ → Ωcₜ → Θ: emergence of projection, effective dynamical closure, historical evolution of coherence and asymptotic extinction of projectability. The projected universe is neither eternal nor ontologically fundamental, but a transient phase of correlational disequilibrium. This volume formally closes the emergence–articulation–evolution–extinction axis without introducing new fields, constants or external mechanisms, establishing the asymptotic limit of projectability as an intrinsic consequence of the correlational principle.