MEASURING INTELLIGENCE QUOTIENTS OF HYBRID HUMAN–MACHINE–AI SYSTEMS A Categorical and Measurement-Theoretic Framework

We develop a mathematically explicit framework for measuring the intelligence of hybrid systemscomposed of human, conventional-machine, and artificial-intelligence (AI) agents. Agents and environmentsare modeled as morphisms in a Markov category, so that the law of an interaction history, thenormalized value of a policy, and the aggregate test score are all defined without ambiguity. We separatethree objects that earlier treatments conflated: a raw performance functional (discounted reward), anaggregate score (a weighted average over a battery of environments), and a quotient (a standardized, normreferencedtransform of the score). For complexity-weighted batteries we prove a sharp normalizabilitytheorem for weighted Levin complexity, showing that the convex weighting proposed in 2013 does not ingeneral define a probability distribution, and we give the one-parameter correction. Hybrid networks aremodeled as typed open Petri nets; using the freeness of the symmetric monoidal category generated by anet, we show that classical, stochastic, fuzzy, timed, quantum, and “generalized-uncertainty” Petri netsare all symmetric monoidal functors out of the same syntax into different semantic categories, relatedby change-of-base functors. Rewards are themselves functors; we show that undiscounted reward isinterleaving-invariant whereas step-discounted reward is not, which forces a choice of time semanticsfor concurrent systems. Collective intelligence is analyzed through the Möbius transform of a coalitionalcharacteristic function: we prove that any pairwise (2-additive) model is blind to genuinely triadichuman–machine–AI synergy, that the difference-form MIQ used in the earlier work is maximized byeliminating all human agents, and we prove a routing-threshold theorem characterizing when a hybridteam outperforms its best member. For quantum Petri nets we give a correct instrument-based firing rule,prove that complex arc weights alone produce no observable quantum effects, and prove that, againstclassical environments and without resource bounds, quantum agents attain exactly the same scores asclassical ones. A synthetic worked example illustrates the theory.