This work introduces the Λ (Lambda) Index, a dimensionless metric describing the structural balance between the resources required to sustain a system’s operating conditions and the useful output produced by the system. The Λ Index is defined as: Λ = Rconditions / Rₒut where Rconditions represents the resources required to maintain the operating conditions of the system and Rₒut represents the useful output produced by the system. Building on this metric, the paper formulates the Λ Law of Architectural Scalability, which states that the long-term viability of a system is governed by the geometric scaling relationship between sustaining conditions and useful function. In geometric form, the index evolves with system scale S as: Λ (S) ∝ Sᵃ⁻ᵇ where a describes the scaling geometry of sustaining conditions and b describes the scaling geometry of useful output. The framework identifies three fundamental architectural regimes: Architectural Leverage (a b) — sustaining conditions grow faster than useful output, leading to structural limits When sustaining conditions dominate system growth, technological optimization cannot compensate for the underlying scaling imbalance. This regime produces what we define as the Architectural Sunk Cost Trap, where increasing technical sophistication primarily sustains the system itself rather than producing external value. To complement the Λ Law, the paper introduces the Diagnostic Triangle of System Viability, which states that a system remains structurally viable only if three conditions are satisfied simultaneously: Λ < 1System StabilityArchitectural Autonomy The framework provides a general diagnostic tool for evaluating the scalability and long-term structural viability of technological, infrastructural, and socio-technical systems. Representative examples include communication networks, transportation infrastructures, energy systems, hydrogen architectures, data centers, and launch systems.
Roman Goncharenko (Sat,) studied this question.