Collective Intelligence Phase Engineering

Collective Intelligence Phase Engineering: Audited Retention and Achieved Growth How can we determine whether a group of AI agents has developed a genuinely useful collective capability—and whether that capability remains after coordination, additional computational resources, or temporary assistance is removed? This paper develops a mathematical framework for collective intelligence phase engineering: observing, controlling, and sustaining beneficial collective regimes under explicit evidence, verification, and resource constraints. Rather than equating collective intelligence with agent count, consensus, or short-term benchmark performance, the framework defines an operational collective-intelligence phase through a set of measurable, protocol-dependent conditions. These conditions can include verified task performance, interaction-specific advantages, verification reliability, resource availability, and retained capability growth. A normalized phase margin describes whether the declared conditions are satisfied without implying a physical or thermodynamic phase transition. A central contribution is the distinction between temporary collective performance and durable capability growth. The paper evaluates retained capabilities using held-out tasks and complete evolving comparator trajectories with matched information and resource entitlements. This separates immediate task improvement, benefits attributable to a specified interaction channel, absolute retained growth, and acceleration relative to an autonomous baseline. To make collective-intelligence phases operationally controllable, the framework combines partially observed robust control, sequential evidence, and bounded interventions. It addresses phase entry and persistence, adaptive allocation of production and verification resources, unresolved verification work, informative but costly observations, and fully funded continuation policies. It also develops methods for selecting reusable artifacts and workflows from finite catalogues through adaptive, partially observed, and explicitly paid audits. The mathematical results connect audited success labels, actual phase attainment, and retained mean growth through calibrated error budgets, convex optimization, conditional moment bounds, and receipt-based resource accounting. Time-uniform statistical certificates and predictable clipping methods support the assessment of achieved capability growth under dependent observations and potentially heavy-tailed outcomes, including mathematical examples with infinite variance. The paper also proposes a reproducible research program involving social learning, complex search, and verification-constrained multi-agent workflows. These settings can be used to investigate when interventions improve collective performance, when verification becomes a bottleneck, and when apparent progress fails to produce lasting capability. Scope and limitations: This is a theoretical and methodological contribution, not an empirical demonstration of collective-intelligence acceleration. Its guarantees depend on explicitly declared models, justified verification and statistical assumptions, matched comparators, and funded execution. The framework does not establish universal collective-intelligence gains, physical criticality, or general artificial superintelligence. Instead, it provides a structured foundation for measuring, testing, and potentially promoting verifiable and persistent collective capabilities in human and artificial multi-agent systems.