(1) Plain English: The declaration rsSpectrum_length states that the list rsSpectrum contains exactly 20 numbers. This list collects cardinalities that arise repeatedly across Recognition Science structures.
(2) Why it matters in RS: It supplies a concrete witness that RS cardinalities form a structured spectrum generated from a small set of primitives (Dspatial = 3, Dconfig = 5, gap45, eightTick, etc.) rather than appearing randomly. The spectrum includes powers, products, and binomial coefficients reachable by RS operations, supporting the claim of a non-arbitrary numerical pattern across domains.
(3) How to read the formal statement: theorem rsSpectrum_length : rsSpectrum.length = 20 := by decide asserts that, after the explicit definition rsSpectrum : List ℕ := [2, 3, 4, 5, 6, 7, 8, 10, 12, 15, 16, 25, 45, 64, 70, 125, 216, 256, 360, 3125], the length property holds by direct computation via the decide tactic. No axioms or external lemmas are required.
(4) Visible dependencies or certificates: The theorem depends only on the preceding def rsSpectrum in the same module. It is bundled into the structure CardinalitySpectrumCert via the field spectrum_length := rsSpectrum_length. Related properties appear in rsSpectrum_pairwise_lt (strictly increasing) and rsSpectrum_bounded (all entries ≤ 3125). Decompositions for individual entries are given by theorems such as eight_is_2cube and oneTwentyFive_is_Dcubed.
(5) What this declaration does not prove: It does not establish completeness of the list, uniqueness of the spectrum, or any physical or forcing-chain significance. It is a finite computational fact about this particular enumeration and supplies no theorem linking the numbers to the Law of Logic, J-cost, or constants.