Hierarchical Structures Composed of Humans Inevitably Decay: A Formal Framework, Models, and a Proof (Part I)

Part I of a two-part series presenting a formal framework for analysing the long-run stability of hierarchical social structures composed of humans. The paper proves that every such structure decays in finite time almost surely, with the source of decay being the combination of entities' finite lifespan and their bounded rationality. Key contributions: a formal statement and proof of inevitable decay (Theorem T) via two independent channels — internal (bounded rationality plus finite lifespan, via a generalised Borel–Cantelli argument) and external (environmental shocks modelled as a Poisson process) ; a minimal stochastic model M1 treating decay as a continuous-time Markov chain of qualitative damage accumulation, with an explicit upper bound on the expected lifetime Eτ ≤ D*/ (ε Λ Δ̄) and an exponential (Kramers-type) bound under healing; extension M2 adding network tie dynamics and coalition formation, yielding a minimax characterisation of stability across multiple decay channels; extension M3 incorporating asymmetric retaliation (organiser vs. follower) and informational cascades of betrayal, explaining why codified bureaucratic empires die of stagnation rather than palace coups; a decomposition of lifetime into entity parameters ΘE and mechanism parameters ΘM, coupled through implementation fidelity ηᵢmpl; an argument that bounded rationality simultaneously makes hierarchy the only scalable form of large-scale human organisation (cognitive cost Θ (N) vs. Θ (N²) for flat networks) and condemns every instance to eventual collapse; a testable prediction (contrapositive proposition) that the lifetime of a hierarchical structure grows with the degree to which decision-making is externalised from individual human judgement into codified systems (law, ritual, collective bodies, algorithms). Companion Part II addresses external influence, localisation of power, the point of irreversibility, and mechanisms of silent institutional adaptation. Keywords: hierarchical organisations, bounded rationality, organisational decay, stochastic models, continuous-time Markov chains, coalition dynamics, institutional stability, iron law of oligarchy, codification, formal social theory