Witness Array Quantum Numbers: Cardinality, Type Signature, and Alignment in Cognitive Stone Spaces

Two witness arrays are not equivalent merely because they contain the same number of witnesses. This monograph identifies three hierarchically ordered “quantum numbers’’ that fully characterize a witness array’s cognitive topology: cardinality (the number of independent witnesses), type signature (whether each witness is Boolean, scalar, tensor-valued, or otherwise structured), and alignment (the orientation of the witness frame relative to the problem landscape’s eigenbasis). Two arrays with identical cardinality but different type signatures generate different Stone space partitions and therefore detect different Gödel sen- tences. Two arrays with identical cardinality and type signature but different alignment impose different metric costs on the same problem, making a theorem that is trivial in one frame a Fields Medal in another. The three quantum numbers stratify cognitive systems into level sets. Witness dimensionality—the to- tal information-carrying capacity of the array, determined jointly by cardinality and type signature—is a structural parameter fixed at the system’s architecture (birth or deployment). Systems on a higher level set can see lower-dimensional problems as projections: the topological obstructions that require surgery at a lower level set do not exist at the higher level, because the extra dimensions provide global sections that trivially resolve them. What looks like insight from the lower level set is traversal from the higher one. The same theorem is a PhD thesis for one system and obvious to another, not because of different training but because of different dimensionality. We formalize this hierarchy, connect it to existing psychometric constructs (Spearman’s g as a lossy principal- component projection of witness dimensionality), and derive the consequence that education provides macros on a fixed level set—it improves navigation efficiency but cannot change the topology of the sys- tem’s Stone space. Keywords: witness array, quantum numbers, Stone space, level set, cognitive dimensionality, type signa- ture, alignment, Spearman’s g, principal component analysis, mathematical difficulty, education, lemma, macro