span_at_0
The reduced span function at dimension zero evaluates to zero by direct substitution into its power-minus-one definition. Modelers of working memory as the nonzero elements of the F₂³ cube cite this base case when deriving the integer collapse sequence under bandwidth limits. The proof is a one-line decision procedure that computes the arithmetic at zero.
claimThe reduced span function at dimension zero satisfies $2^0 - 1 = 0$.
background
The module establishes that Miller's 7 ± 2 equals the count of nonzero vectors in the three-dimensional vector space over F₂, written 2³ − 1 = 7. Reduced spans are defined along the cube-dimension ladder by the upstream declaration spanAt(d) := 2^d − 1. This supplies the integer steps 7 → 5 → 3 → 1 that appear when recognition bandwidth contracts the effective cube dimension.
proof idea
The proof is a one-line wrapper that invokes the decide tactic on the arithmetic expression 2^0 − 1.
why it matters in Recognition Science
The declaration supplies the zero-dimensional base of the span ladder used to model working-memory capacity. It anchors the predictions of stepwise collapse under reduced bandwidth and links to the Recognition Science eight-tick octave via the 2³ structure. No downstream theorems yet consume it.
scope and limits
- Does not prove any empirical match to observed memory spans.
- Does not address non-integer or continuous dimension reductions.
- Does not derive the super-normal plateau at 15.
- Does not connect the cube count to the J-cost or phi-ladder.
formal statement (Lean)
34theorem span_at_0 : spanAt 0 = 0 := by decide