Identity Persistence after Brouwer and Hilbert_ Formal Admissibility, Constructive Witness, and Invariant Continuity

Abstract The historical dispute between Hilbertian formalism and Brouwerian intuitionism is often remembered as a conflict over logic, proof, and mathematical existence. This paper argues that the deeper structural division is better understood as a distinction between formal admissibility and constructive existence. Hilbert asks whether a symbolic move is lawful inside a formal system; Brouwer asks whether the asserted object or proposition has been constructed or witnessed. Modern mathematics did not resolve this dispute by fully choosing one side, but produced a layered settlement in which classical formal practice, constructive methods, type theory, homotopy type theory, and formal verification each occupy distinct roles. This paper positions identity persistence under transformation as a structurally distinct layer within that settlement. Identity persistence is not reducible to formal admissibility, constructive existence, homotopy type theory, or formal verification. Its primitive is regime-bound persistence across admissible transformation: the question of what makes a transformed object, process, or decision the same one through change. The paper develops the argument through four counterfactual tests: formal legality without invariant preservation, construction without persistence criteria, verification without replay-stable identity, and probability without a stable object of measurement. It then translates the framework into consequential computation, where a decision must become a replayable identity-bearing artifact rather than merely an output. Finally, the paper applies the framework downstream to Earth as an explicitly interpretive example of a bounded, gravitationally coherent, density-stratified, multi-rate system whose identity persists through coupled transformation. The paper is a philosophical companion to the Universal Identity and Persistence forcing theorem. It provides orientation, counterfactual tests, and applied translation; it does not prove the forcing theorem or extend the formal stack.