Prolegomena to any future Aufbau: Rethinking Constitutive Metaphysics

We develop a structural critique of purely local, closure-based "Aufbau-style" metaphysics---bottom-up constitutive approaches that seek to reconstruct the world from a sparse local basis and a stock of rules. Our central claim is that such projects are conditionally correct but deeply limited: they presuppose that the foundational theory is C-classical, in the sense that its local structure and fully unfolded structural content uniquely determine the global world (up to the relevant notion of equivalence). This assumption holds, at best, in regimes like Newtonian and Maxwellian physics, where the relevant higher cohomological and higher-categorical structure is trivial. In genuinely modern regimes such as general relativity, gauge theories, and effective field theories with nontrivial topology and anomalies, C-classicality fails. Local closure yields at most a stabilized structural "shadow"; global cohomological invariants and extension data are indispensable. We use spectral sequences and cohomology to articulate the distinction between C-classical and non-C-classical theories, and we show how non-categoricity results in logic and concrete examples from physics instantiate the same extension phenomenon: a fully stabilized shadow that admits multiple non-isomorphic global realizations. We argue that this does not kill the generative impulse, but forces a neo-Aufbau: an ontology of structure plus cohomological and higher-categorical extension, grounded in frameworks such as topos theory and homotopy type theory.