Substrate Undecidability: Why No Internal Observer Can Decide Whether Reality Is Discrete or Continuous, and What Can Adjudicate Instead

Is physical reality discrete or continuous? This question is usually treated as one question with a definite, if currently unknown, answer — but it is really at least two independent questions: whether a bounded region admits only finitely many distinguishable states (information density), and whether the time-evolution law is computable or accesses non-computable structure (dynamics). This paper proves, and machine-checks in Lean 4 against Mathlib, that neither question can be decided by any observer that is itself part of the system being observed and that has only finite resources. Two independent barriers are established: a topological barrier (substrate discreteness is not a finitely determined, clopen property of observable transcripts, so no finite body of evidence decides it) and a recursion-theoretic barrier (no candidate physical law, even when supplied to the observer, can be internally certified as correct, by reduction to the halting problem). Under Perfect Self-Containment (PSC), substrate discreteness becomes an outsourced property whose only admissible adjudicator — by a uniqueness argument this paper cites from the companion NEMS programme rather than re-derives — is minimum-description-length (MDL) selection over complete law-sets. That selection favors the finite-description class: computable transcripts together with an adjudicative extension (a "glimmer") that resolves an apparent tension with the programme's own account of physical randomness, since the purely computable class alone would contradict its determinism no-go result. The paper explains why Planck-scale grid searches are structurally unable to settle the question, identifies asymptotic scaling exponents as the only finite observables carrying substrate information, and states its results' limits explicitly: the glimmer extension's classification is fully machine-checked, but two physical premises underlying it — that the actual Born measure constitutes complete internal semantics, and that a degree-0' adjudication bound proved elsewhere for a specific physical framework transfers to this paper's framework-agnostic model — are left open, together with the question of which substrate actually obtains.