Explain the Lean def `cardinalitySpectrumCert` in module `IndisputableMonolith.CrossDomain.CardinalitySpectrum`. Write for an educated reader who knows science and programming but may not know Lean. Cover: (1) what the declaration says in plain English, (2) why it matters in Recognition Science, (3) how to read the formal statement, (4) visible dependencies or certificates in the supplied source, and (5) what this declaration does not prove. Cite only declarations present in the supplied Recognition source.

The declaration cardinalitySpectrumCert in module IndisputableMonolith.CrossDomain.CardinalitySpectrum is a concrete instance of the CardinalitySpectrumCert structure.

(1) In plain English: it bundles explicit proofs that the RS spectrum numbers (2, 3, 4, 5, 6, 7, 8, 10, 12, 15, 16, 25, 45, 64, 70, 125, 216, 256, 360, 3125) arise directly from the generators Dspatial = 3, Dconfig = 5, gap45 = 45, eightTick = 8, and cubeFaces = 6, together with the facts that the spectrum list has length 20, is strictly increasing, and is bounded above by 3125.

(2) Why it matters in Recognition Science: the module states that RS produces a structured numerical spectrum from a small set of primitives (powers and products of 2, 3, 5 and the gap), rather than arbitrary integers; the certificate records that every listed cardinality admits an RS decomposition.

(3) How to read the formal statement: CardinalitySpectrumCert is a structure whose fields are equalities such as Dspatial_is_3 : Dspatial = 3, gap_as_D : gap45 = Dspatial^2 * Dconfig, eightTick_as_D : eightTick = 2 ^ Dspatial, full_turn : 360 = eightTick * gap45, choose_central : 70 = Nat.choose 8 4, D_cubed : 125 = Dconfig^3, D_fifth : 3125 = Dconfig^5, spectrum_length : rsSpectrum.length = 20, spectrum_pairwise : rsSpectrum.Pairwise (· < ·), and spectrum_bounded : ∀ n ∈ rsSpectrum, n ≤ 3125. The definition cardinalitySpectrumCert supplies the proofs for each field (rfl or references to the module's decide lemmas).

(4) Visible dependencies or certificates in the supplied source: it directly uses the generator equalities eightTick_eq, gap45_eq, cubeFaces_eq and the spectrum theorems rsSpectrum_length, rsSpectrum_pairwise_lt, rsSpectrum_bounded, plus individual decompositions such as three_is_Dspatial, seventy_is_choose_8_4, oneTwentyFive_is_Dcubed and threeOne25_is_D_fifth.

(5) What this declaration does not prove: it does not establish that the listed numbers are the only cardinalities appearing in RS, does not derive any physical predictions or constants, does not connect to the forcing chain or J-cost uniqueness, and supplies no uniqueness or completeness claim for the spectrum.