Underdetermination of Finite Sequences and Construct Validity in Intelligence Testing

Numerical sequence completion items are among the most familiar tasks in reasoning tests: a respondent isshown a finite set of terms and asked to state the next one. This paper defends a narrower claim than thesurface format of the task suggests: given only the observed terms, the next term is not uniquely determined asa matter of logic unless additional assumptions are imposed on the class of admissible rules or on the criterionused to select a rule. The point is formalized through Lagrange interpolation and the family P(x)+W(x)H(x),and is then extended, in a form that is more than a structural analogy, to Raven matrices, treated as finitecompletion problems subject to a privileged transformation grammar whose role can be made explicit.The psychometric consequence of this argument is limited but important. Logical underdetermination doesnot imply that sequence items or matrix items are useless or psychometrically invalid. It does imply that theirinterpretation cannot rest solely on the idea that the correct answer is already entailed by the data. The papershows, in particular, that uniqueness in Raven-type items follows conditionally from an assumed grammarof transformations, and that for every distractor in a multiple-choice format there exists a rule, outside thatgrammar, under which that distractor is the logically correct answer. A brief discussion of the g factor clarifieswhy an item may correlate with other tests and contribute to the estimate of a general factor without thatfact, by itself, settling the question of what cognitive content the item rewards.The paper closes by sketching a minimal empirical follow-up that could estimate how often alternative butinternally coherent continuations arise in practice. The conclusion is cautious but firm: such items maymeasure real abilities, but their construct validity requires more explicit constraints on the space of admissiblesolutions and greater caution when a conventional scoring rule is used to support a broad interpretation of“intelligence”. This work originated outside academic psychology. The author is an independent researcher with a background in CNC machining and GPU programming, with no prior publications in psychometrics. The motivation was straightforward: encountering numerical sequence and matrix items in a standard intelligence test, and noticing that the uniqueness of the expected answer was assumed rather than demonstrated. The observation that Lagrange interpolation makes this non-uniqueness formally explicit, and that the same argument extends to Raven matrices through a grammar-based conditional uniqueness result, does not appear to have been made in this specific form in the existing literature, despite the mathematical tools having been available for centuries. This paper is an attempt to make that argument precise.