Generative Calculus II: Capability Attribution Under Augmentation

Abstract

Modern AI capability is increasingly a property of an augmented system rather than a frozen model: test-time computation, retrieved evidence, tools, memory, interfaces, and control rules can all change what is feasible. Generative Calculus II (GC-II) asks where such gains come from. It lifts factorial attribution from scalar scores to task–scale–error–budget capability envelopes and defines an Envelope Attribution Spectrum using Boolean-lattice Möbius inversion. We prove exact attribution, additive-accounting characterization, endpoint non-identifiability, factorial identifiability, and strict refinement of scalar accuracy. We then prove a matched-summary whole-envelope separation: for every fixed projection order, there are balanced envelope families with identical low-order coordinate projections but exact membership-circuit complexities O(n) and Ω(2ⁿ/n), using classical Shannon circuit counting as the lower-bound ingredient. The accompanying public implementation provides reproducible computational results. On handwritten digits, third-order hold-out prediction reduces full-system error from 0.0795 to 0.0162. In a fixed-weight 16-condition FLAN-T5-small pilot comprising 240 evaluations, accuracy rises from 0.20 to 0.60, higher-order interaction mass is 0.4667, and third-order terms predict the held-out full system to machine precision. GC-II therefore separates model capability from system capability while making both attribution and reconstruction complexity auditable.