sixtyfour_is_2sixth
The equality 64 equals 2 to the power 6 holds in the natural numbers and serves as a witness that 64 belongs to the RS cardinality spectrum via power decomposition from the cube generators. Researchers mapping cross-domain cardinalities in Recognition Science cite this to confirm the spectrum contains structured values like 2^6 rather than arbitrary ones. The proof is a direct decision procedure that verifies the arithmetic identity by computation.
claim$64 = 2^6$ in the natural numbers, equivalently $8^2$.
background
The module collects exemplar witnesses for the RS Cardinality Spectrum under the C21 structural claim. The spectrum comprises cardinalities reachable by multiplying, summing, or taking powers and combinations of the cube-generators {2, 3}, configDim 5, and gap45; the listed values include 2, 3, 4, 5, 6, 7, 8, 10, 12, 15, 16, 45, 70, 125, 216, 256, 3125 and so on. This theorem supplies the concrete witness that 64 arises as 2^6 or 8 times 8. No upstream lemmas are invoked beyond the built-in decidability of equality on natural numbers.
proof idea
The proof is a one-line wrapper that applies the decide tactic to confirm the arithmetic identity 64 = 2^6 by direct evaluation.
why it matters in Recognition Science
This theorem contributes to the C21 claim that the RS cardinality spectrum is structured rather than random, by exhibiting an explicit decomposition of 64 into RS primitives. It aligns with the eight-tick octave (period 2^3) from the forcing chain, since 8 = 2^3 and 64 = (2^3)^2. The result supports the broader assertion that every spectrum member admits such a decomposition, with no open questions or scaffolding addressed here.
scope and limits
- Does not prove that 64 arises exclusively from RS primitives.
- Does not enumerate the complete cardinality spectrum.
- Does not connect 64 to physical constants or mass formulas.
- Does not extend the equality beyond natural numbers.
formal statement (Lean)
76theorem sixtyfour_is_2sixth : (64 : ℕ) = 2^6 := by decide
proof body
Term-mode proof.
77
78/-- 70 = C(8,4) = max central binomial. -/