Viability Boundaries of Living Systems: A Phenomenological Theory

The boundary between living and nonliving matter has resisted rigorous formalization despite substantial progress in characterizing the molecular and thermodynamic mechanisms of life. Under what minimal conditions can an open system sustain a viable trajectory, and why does every such trajectory necessarily end in finite time? These questions form the subject of the present work. The Viability Boundaries of Living Systems (VBLS) theory addresses this problem not through an additional checklist of defining features such as metabolism, self-replication, or evolvability, but through an explicit dynamical construction. It is built from a system of four coupled nonlinear ordinary differential equations, each derived from an independent phenomenological principle and governing one of four state variables: the repair reserve R (t), the assimilation efficiency ηabs (t), the integrity potential E (t), and the structural entropy Σ (t). The theory formalizes the viability boundary as a critical surface ℳmin in parameter space, namely the locus at which the attractor of R (t) coincides with its bifurcation threshold R*, separating configurations capable of sustaining life from those in which life cannot occur. The finite lifetime of any living system is shown to be a mathematical theorem within the proposed framework rather than introduced as a postulate: any trajectory originating above the viability boundary reaches it within finite time. The integrity-potential channel (E-channel) guarantees this finiteness under the model assumptions. The biologically dominant repair-reserve channel (R-channel) offers a second, later endpoint: as structural entropy accumulates, it progressively depresses the repair-reserve attractor below its death threshold. The two channels are functionally irreducible, neither threshold following from the other. The mechanism that turns the E-channel's finiteness guarantee into a theorem is history-dependent memory. The system's capacity to absorb resources degrades under the cumulative load of past damage, not its instantaneous state. In this nonlocal, integral mechanism, assimilation efficiency decays with accumulated structural entropy — a formulation absent from conventional thermodynamic descriptions. Because these two channels operate on E (t) and R (t) alone, the framework can be described concisely as an (E, R) -system with memory, distinguishing VBLS from prior thermodynamic theories that lacked a second, distinct channel of degradation. Three functionally irreducible viability barriers are identified, whose joint satisfaction is necessary for a viable trajectory to exist: an autocatalytic amplification barrier (R-channel), a timescale-separation barrier (E-channel), and a memory barrier governing assimilation efficiency ηabs (E-channel). Each barrier guards a distinct state variable of the system; violation of any single one rules out a viable trajectory, and this violation cannot generally be compensated by external conditions. This yields a structural explanation for the rarity of abiogenesis, suggesting that the probability of a spontaneous transition may scale as the product of three independently small barrier-crossing probabilities. Together with this insight, the theory may be useful for experimental programs, diagnosing for each candidate system which barrier remains unsatisfied and what minimal conditions are required to satisfy it. The parameters of ℳmin are informed by multiple empirical anchors, including the minimal synthetic cell syn3. 0 — the experimentally characterized organism with the smallest known viable genome — yielding a concrete, quantitative estimate of the viability boundary. When the governing equations are integrated numerically, the three phases of the life cycle — growth, plateau, and collapse — follow directly from the structure of these equations, not from an added assumption. A scaling invariant ℬ ≡ (E0 − E*) · τenergy, with the dimension of action (J·s), combines the structural energy budget with a characteristic thermodynamic timescale. Its mass-normalized form, ℬ* = ℬ/M^ (5/4), is empirically observed to be approximately conserved across the 306 mammalian species examined in the verification analysis. The theory further generates a systematic table of living and sub-biological structures. Rows are organized by memory regime, columns by position relative to ℳmin. Most cells are populated by known organisms. Some cells are excluded outright by the finiteness theorem. The remaining empty cells constitute falsifiable predictions: classes of structures the theory permits but that have not yet been observed. Examples include syn3. 0, sitting essentially on ℳmin; obligate endosymbionts and organelles, lying below it; and spores and tardigrades in anabiosis, occupying a separate class of pre-boundary states. Each is also classified by death mechanism and evolutionary fate. The work continues the tradition of Ervin Bauer (1935) and Ilya Prigogine (1977): Bauer's principle of steady disequilibrium and Prigogine's theory of dissipative structures are unified and extended here by a nonlocal memory mechanism, providing a rigorous dynamical framework and an explicit mathematical criterion distinguishing living from nonliving matter. https: //github. com/vaavdeev/VBLS