Occupancy in a two-coordinate absorbing Markov model of sequential development: an exact renewal first-passage law and its domain of validity

Description When we model the long-term survival and growth of complex systems, such as technological civilizations or macro-evolutionary lineages, we often rely on single-track metrics like energy consumption. However, real development is multi-dimensional. This work steps away from oversimplified, one-dimensional progress tracking by introducing a strict mathematical framework that evaluates development across two independent axes simultaneously: an energy axis (throughput or scale) and a substrate axis (such as the transition from biological to post-biological frameworks). By framing development as a path finding its way through a two-dimensional grid, the model tracks how systems move forward before hit by an absorbing "hazard", representing existential termination or exit from observability. Written in a accessible yet mathematically precise style, the text breaks down two core concepts: The Survival Horizon ($A_k$): The probability that a system can scale up its energy capacity multiple times while remaining completely anchored to its initial baseline substrate. The State Occupancy ($\Omega$): A method for calculating where a scattered population of systems is most likely to be found at any given moment when observing a cross-section of development. Implied Meanings for Evolutionary Possibility Spaces The true weight of this model lies in what it suggests about the distribution of long-lived systems within the cosmic possibility space. It demonstrates that the balance between the speed of raw energetic expansion ($r_E$) and structural substrate transitions ($r_S$) acts as an invisible, deterministic regulator. If structural or technological transitions follow standard, memoryless timelines (exponential tails), the presence of high-energy systems on ancestral substrates becomes exponentially rare. The possibility space is violently pinched, funneling surviving systems into highly specific, predictable configurations. Conversely, if structural transitions are bottlenecks governed by complex, heavy-tailed waiting distributions, the possibility space blooms: high-energy, early-substrate systems transition from being impossibly rare anomalies to polynomially persistent features of the landscape. Ultimately, this paper shifts the conversation from asking if complex systems survive, to mapping exactly where they must inevitably pool across the dimensional grid of development if they do.