The biological cosmological constant Λ_B: Exploratory propensity, dynamical habitability and the geometric origin of...

The emergence of life represents a fundamental geometric phase transition in the state space of complex systems, rather than a mere sequence of chemical syntheses. Building upon the framework of Biorelativity, we introduce Λ_B, the biological cosmological constant, which quantifies the intrinsic exploratory propensity of a system, also conceived as the basal pressure driving the expansion of the eco-evolutionary manifold into the adjacent possible. By mathematically formalising Λ_B as proportional to the squared fractional expansion rate of the Fisher Information Volume, we provide a rigorous criterion to distinguish between chemical persistence (subcriticality) and genuine open-ended evolution. Furthermore, we demonstrate that macroevolutionary trajectories in the Phanerozoic record, traditionally debated through logistic and dynamic models, represent a multi-stage geometric evolution. We argue that the biosphere transitioned from a high-volatility inflationary regime in the Cambrian to a geometrically stabilised, high-curvature state in the Meso-Cenozoic. In this mature regime, Λ_B is counterbalanced by relational interaction curvature κ_bio to form an asymptotic Ricci soliton, maintained through continuous thermodynamic dissipation. This geometric perspective offers a novel theoretical distinction between structural and dynamical habitability, reconceptualises lithopanspermia as the robust transport of dynamically subcritical systems and recasts the early Earth as a geometric amplifier where Gaian feedbacks tune geometric parameters to secure long-term dynamical habitability. Ultimately, this synthesis suggests that extraterrestrial life-detection strategies should prioritise the search for thermodynamic and atmospheric signatures characteristic of dynamically expanding information spaces, including intermittent or patchy chemical anomalies indicative of aborted geometric expansions, providing a unifying mathematical foundation for understanding the evolutionary potential of planetary biospheres.