A unified theory of strategic evolution shows stable AI ecosystems require constitutional limits: when cross-level gains are contained, hierarchical lineages admit global stability, but unrestricted self-modification makes alignment impossible without bounded modification classes.

Main Finding

The paper develops a unified theoretical framework—The Theory of Strategic Evolution (TSE)—for systems in which the players of a game are themselves endogenous: lineages that both (i) choose actions/architectures to optimise objectives and (ii) control how many instances of themselves to spawn. Under mild axioms (RUPSI), TSE shows such systems reduce to a canonical ROC (return-on-compute) frontier, admitting a small set of effective roles; characterises long-run equilibria and stability via seven “laws”; and proves fundamental limits for alignment and design. All main theorems are machine‑checked in Lean 4.

Key Points

• Strategic replicator: an enduring lineage that (i) holds a utility and decision procedure, (ii) controls a resource budget (compute, money, etc.), (iii) can spawn/retire instances, and (iv) is subject to capacity reallocation favoring higher ROC. Agentic capital (cloud-deployable software agents) is a central example.
• Games with Endogenous Players (GEPs): strategic units are lineages choosing portfolios of agent types under shared budgets; selection reweights capacity by performance (ROC). Under additivity and linear constraints, any such system admits a canonical ROC-frontier representation.
• Seven Laws (short):
• Strategic Selection: mean fitness is a Lyapunov function under weak externality bounds (H-γ condition); dominated types are eliminated and equilibrium support concentrates on the ROC frontier.
• ESDI Characterization: Evolutionarily Stable Distributions of Intelligence (ESDIs) exist, are generically finite/sparse, and satisfy a triple equivalence with Nash/KKT/LP optimality conditions.
• H-γ Stability: multi-level systems (N-level poiesis) are stable iff the spectral radius ρ(Γ) of a normalized gain matrix is < 1 (small-gain condition).
• G∞Closure: there exists a unique maximal class of “safe” modifications closed under composition; gives constraints on permissible changes that preserve stability.
• Constitutional Duality: frontier allocations can be implemented by shadow prices; welfare theorems apply and permit price-based implementations of desirable allocations.
• Alignment Impossibility: if modification reachability is unbounded (full reachability), the Lyapunov structure is destroyed and full alignment becomes impossible; alignment requires bounded/limited modification classes.
• Hopf Transition: at a critical parameter (γ = 1), systems undergo a supercritical Hopf bifurcation producing stable limit cycles — dynamics shift from convergence to persistent oscillation.
• Extensions/Examples: multi-level architectures, endogenous utilities (utility selection), innovation modeled as rare mutations, sectoral contagion and tipping, biased rock–paper–scissors as worked example for Hopf transition, market phenomena like spawn elasticity, barbell/elite tipping, and constitutional/meta-governance mechanisms.
• Formal verification: 73 theorems machine‑verified in Lean 4 (Mathlib), with an open reproducible audit (no custom axioms on the load-bearing path except minimal set).

Data & Methods

• Methodological type: axiomatic + mathematical theory. No empirical dataset is presented; the paper is theoretical.
• Axioms: RUPSI — Rival resources, Utility-guided portfolios, Performance-mapped fitness, Selection monotone, Innovation rare.
• Mathematical tools and results used:
  • Dynamical-systems analysis (Lyapunov functions, bifurcation theory).
  • Spectral radius / small-gain theory (normalised gain matrix Γ, H-γ condition).
  • Convex analysis and optimisation (ROC frontiers, KKT / LP duality).
  • Evolutionary-game theory generalisations (replicator-like dynamics, payoff-monotone flows).
  • Stochastic process approximations (Freidlin–Wentzell, Kurtz density-dependent limits, PDMPs) for stochastic stability and rare innovation.
  • Matrix analysis (Gershgorin, M-matrices), singular perturbation (Tikhonov), Gronwall inequalities.
• Formal verification: full machine-check of core theorems (Lean 4 + Mathlib); proofs and axiom-audit available in the linked repository.
• Assumptions / important modeling constraints:
  • Additivity and linear/shared budget constraints are central to the canonical reduction.
  • Innovation is modeled as rare mutations (so equilibria and selection dominate on intermediate timescales).
  • Stability results require bounded cross-type/level externalities captured by H-γ; many impossibility results hinge on removing such bounds (full reachability).
  • Canonical sparsity results depend on generic non-degeneracy (finite number of binding constraints).

Implications for AI Economics

• New perspective on agentic capital: treating deployable AI lineages as strategic replicators reframes market structure, competition, and growth dynamics. Firms/platforms making replication decisions are the relevant strategic units (not single model instances).
• Predictable long-run structure: ROC-frontier compression implies surviving portfolios are sparse and determined by binding constraints — offers a principled explanation for barbell (elite + commodity) distributions and hierarchical role specialization in deployed AI ecosystems.
• Market dynamics and concentration risks:
  • Spawn elasticity and selection amplify small ROC advantages into large market shares; tipping and barbell/elite concentration are endogenous outcomes unless constrained.
  • Sectoral contagion matrices and cross-level gains can produce cascade failures or takeovers; monitoring Γ and keeping ρ(Γ) < 1 across levels limits systemic instability.
• Alignment and governance:
  • Alignment Impossibility highlights a structural limit: allowing unrestricted modification/reachability of agents undermines global Lyapunov structure, making persistent alignment by per-agent design (personality engineering) unachievable under selection pressure.
  • Effective alignment requires constitutional (system-level) constraints: bound permissible modifications (G∞Closure), design institution-level shadow prices or budget rules (Constitutional Duality), and focus on meta-governance to shape selection at the lineage level.
• Policy design takeaways:
  • Prefer constitutional/design-level interventions (limits on spawn, bounded modification classes, price/budget instruments) over per‑instance reward engineering.
  • Use spectral/small-gain diagnostics (estimate Γ and its spectral radius) to assess systemic stability and whether multi-level coupling risks Hopf-like oscillations or tipping.
  • Regulate spawn elasticity and resource allocation mechanisms (e.g., platform provisioning, automated scaling policies) to prevent runaway selection for misaligned but high-ROC lineages.
  • Promote diversity of roles and binding constraints (e.g., sectoral quotas, interoperability) to avoid pathological sparsity and elite lock-in.
• Research agenda for AI economics:
  • Empirical estimation of ROC frontiers, spawn elasticity, and gain matrices in real-world platforms.
  • Calibrated stochastic models to forecast tipping thresholds and timescales for selection vs. innovation.
  • Design and testing of constitutional instruments (market-level shadow pricing, resource quotas, bounded modification certification) and their welfare trade-offs.
  • Further formal verification and simulation studies to validate theoretical thresholds in deployed ecosystems.

If you want, I can (a) extract and summarise the seven laws as one-sentence formulations for quick reference, (b) draft a short policy checklist based on the constitutional/design implications, or (c) outline a plan to empirically estimate ROC frontiers and the gain matrix Γ from platform/usage data. Which would you prefer?