Philosophy’s Vacant Chair in Science: A Formal Theory of the Constitutional Layer

This paper addresses a structural question at the intersection of philosophy and science: why has the constitutional layer of theoretical systems remained institutionally vacant since the late 19th century, and what are the formal consequences of this vacancy? We introduce a functional distinction between two layers present in any theoretical construction. The operational layer executes inference, computation, and expansion. The constitutional layer specifies what a theory is, within what scope it is valid, and what identity criteria it adopts. Historical analysis from 1870 to 2000 demonstrates that the divergence of mathematical physics from philosophical methodology produced an institutional separation in which the constitutional layer was systematically left without an explicit custodian. The central formal result, the Internal Closure Impossibility Theorem, proves that no operational layer can generate its own constitutive conditions as internal operations. The proof proceeds by reductio ad absurdum: assuming such internal generation exists produces an infinite regress that cannot terminate within finite computational resources. An external fixation point, termed Arbitrium, is therefore structurally necessary. This result is not a reformulation of Gödel incompleteness or Tarski's truth hierarchy. Gödel operates at the syntactic level of formal systems; Tarski at the level of linguistic hierarchies. The present theorem operates at the constitutive level of theoretical construction itself. The framework employed is Noology, a formal system developed within Cognitional Mechanics, which provides axiomatic foundations for the functional separation of constitutional and operational layers. Re-implementing the audit function in a form compatible with contemporary mathematical physics constitutes the logical responsibility identified by this paper.