Branch Indexical Theorem: A No-Go for Observable Single-History Selection

A single-history selection law is branch indexical when it marks one branch by an inaccessible label while leaving the unitary dynamics unchanged. The Branch Indexical Theorem identifies a quiescent class of such laws and shows every member is operationally silent on the post-decoherence pointer algebra, a conditional no-go for observable selection within that class. Silence is membership in an equivalence class induced by restriction to that algebra, so the strength of the claim is fixed by the choice of algebra. The boundary carries the contribution. One-way diagnostics on measurable functionals detect a population shift, generated inter-block coherence, an affinity defect, and the failure of local realizability, each implying a named restriction fails; no converse holds, and a null reading certifies nothing. Mechanistic amplitude blindness admits no output-level diagnostic. And leaving the class does not purchase observability: a state-dependent law can exit quiescence and remain silent, so restriction violation is necessary for empirical content and does not secure it. Three results carry the classification beyond exact idealizations, bounding the deviation under finite calibration error and finite decoherence, and closing the multishot channel through history-wise conditional fidelity. The result does not derive the Born rule, solve the measurement problem, choose an ontology, or introduce a new empirical prediction; Born universality is derived from calibration on a spanning family with affinity rather than assumed. The framework separates branch selection, branch actuality, and the truthmaker of actuality, and names which commitments a putatively empirical single-history proposal has already made.