super_normal_jump
The declaration shows that the span increase from the 3-cube to the 4-cube equals exactly 8, which is the volume of the cube itself. Researchers modeling working memory capacity via binary vector spaces would reference this when explaining the jump to a 15-element super-normal span. The proof reduces immediately to arithmetic via the decide tactic on the explicit power-of-two formulas.
claimLet $s(d) = 2^d - 1$ denote the span at dimension $d$. Then $s(4) - s(3) = 2^3$.
background
In the CrossDomain module, working memory is modeled via the non-zero elements of the vector space over GF(2) of dimension d, giving span 2^d - 1. The canonical span is fixed at 7 for d=3, corresponding to Miller's 7 ± 2. The function spanAt(d) computes this reduced span along the cube-dimension ladder as 2^d - 1. This sits within the C8 claim that Miller's number arises as the count of non-identity elements in F_2^3.
proof idea
The proof is a one-line wrapper that invokes the decide tactic. After unfolding the definitions of spanAt, the expression becomes (16 - 1) - (8 - 1) = 15 - 7 = 8, which decide verifies by direct computation.
why it matters in Recognition Science
This result fills the gap between normal (d=3, span 7) and super-normal (d=4, span 15) working memory plateaus in the Recognition Science framework. It supports the prediction of a new plateau at 15 under super-normal conditions, as stated in the module documentation. It connects to the cube-counting identities that derive Miller's law from the structure of F_2^3.
scope and limits
- Does not establish the psychological reality of the 7±2 limit.
- Does not extend the model to dimensions greater than 4.
- Does not incorporate the ±2 variability around the mean.
- Does not address empirical testing protocols.
formal statement (Lean)
47theorem super_normal_jump : spanAt 4 - spanAt 3 = 2 ^ 3 := by decide
proof body
Term-mode proof.
48
49/-- The Miller 7 ± 2 corridor (5 to 9) brackets canonicalSpan. -/