Observing and Accelerating Collective Capability Growth

This paper develops a finite-time mathematical framework for observing, attributing, and accelerating collective capability growth under explicit resource, verification, evidence, and continuation constraints. The central problem is not merely whether a collective system can generate more output, but whether a specified interaction contributes to sustained growth of separately measured task and research capacities while preserving verification capacity, protected service floors, resource feasibility, and unresolved-obligation control. The framework combines joint service-capacity measurement, interaction ablation, supporting-price inequalities, information-theoretic recognition bounds, adaptive confidence sequences, finite-horizon control, and executable continuation. Collective interaction is evaluated against resource-, information-, and time-matched comparison policies that are reoptimized after the interaction is removed. This prevents simple communication volume, agent count, replicated compute, or unverified artifact production from being credited as collective capability growth. A finite-horizon theorem jointly bounds terminal capability, statistical recognition against multiple negative models, and deferred repair obligations. Time-dependent supporting prices capture the timing of evidence acquisition and debt. Complementary sufficient certificates use bounded adaptive observations with explicit allowances for calibration bias and drift. The paper also formulates finite backward-induction policies that first seek statistically supported entry into a growth regime with funded continuation and then maximize the minimum attainment of task and research capacity targets. Additional results analyze interaction-specific growth residuals, verification and repair queues, structural support and seed requirements, finite resource limits, dependence diagnostics, and a three-capacity serial model coupling task, research, and verification capacity. Synthetic integer executions demonstrate how verification, repair, latency, calibration, evidence acquisition, and communication compete for the same finite budget, and how excessive communication can eliminate otherwise feasible growth. The framework is relevant to collective intelligence, multi-agent AI systems, AI research automation, recursive capability growth, AGI/ASI research, verified autonomous research systems, and resource-constrained AI governance. Its claims are protocol-relative and finite-horizon: the paper does not claim observation of artificial general intelligence, artificial superintelligence, a physical phase transition, or intelligence growth in any deployed system.