The International System of Units recognises seven base dimensions, but nothing in dimensional analysis fixes that number, and the system’s own history shows it moving. This paper asks what constrains the decision to found a new one. Its principal result is negative. A dimensional formalism can establish that the records of a domain admit an independently re-scalable coordinate; it cannot establish what quantity that coordinate is the dimension of. This is proved rather than diagnosed: readings that mark the same quantities and predict the same motion in the records present a procedure with identical input, so no procedure whose input is the exponent matrix and the observed action of the available interventions can separate them. The gap is therefore not an obstruction inside the algebra, which a better theorem might remove, but an absence in what the algebra is given. Three consequences follow: the standard independence condition on a new dimension claims more than it establishes, a rank test cannot distinguish an economic column from a phenomenal one, and a dimensionally admissible group can carry an estimator that recovers nothing in particular. What governs founding is therefore not supplied by the formalism alone, and the rest of the paper asks what does. Seven conditions are proposed as necessary, sorted by the kind of necessity each carries: three are constitutive, in that failing one yields no founding rather than a defective one; two hold within a declared scope; one is a truth condition on an accompanying claim; and the seventh divides in two, a derived requirement that the founding refine the dimensional partition and an undertaken criterion that what it excludes was independently meaningful. Six candidates are worked against the conditions: plane angle as a control, thermodynamic temperature as a consistency check rather than corroboration, economic value as a case the conditions convict, information and qualitative kind because they return verdicts of other kinds, and sensory intensity because it is the only one on the restricted branch the formal results are stated for, admissible there on premises the paper names and does not defend. The estimator result is stated for readers outside this programme, because the conflation of dimensional admissibility with estimator identifiability does not announce itself: the offending construction satisfies every check the framework can run.