Explain the Lean theorem `rsSpectrum_length` 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.

Here is an explanation of the rsSpectrum_length theorem, broken down into the five requested areas.

1. Plain English Meaning

The theorem states that the list of canonical numbers, rsSpectrum, contains exactly 20 elements. This list enumerates the first 20 integer cardinalities (sizes of sets or spaces) that appear repeatedly across the framework.

2. Why it Matters in Recognition Science

Recognition Science (RS) asserts that physical and cognitive structures are governed by a shared, structured mathematical foundation rather than arbitrary parameters. The module demonstrates that domain sizes across the RS stack fall into a specific numerical spectrum built from small primitive generators (like $D_{spatial} = 3$, $D_{config} = 5$, the 8-tick cadence, and the gap 45). The rsSpectrum_length theorem acts as a structural sanity check, formalizing that exactly 20 such exemplar values have been aggregated to form the first cross-domain wave.

3. Reading the Formal Statement

The theorem in Lean is written as:

theorem rsSpectrum_length : rsSpectrum.length = 20 := by decide

• rsSpectrum.length applies the list length function to the predefined list of numbers.
• = 20 asserts that the resulting count is the natural number 20.
• := by decide is the proof tactic. It tells the Lean compiler to simply evaluate the list computation and verify that $20 = 20$. No complex logical deduction is required because it is a finite, computable property.

4. Visible Dependencies and Certificates

The theorem depends directly on the definition of rsSpectrum, which explicitly lists the integers [2, 3, 4, 5, 6, 7, 8, 10, 12, 15, 16, 25, 45, 64, 70, 125, 216, 256, 360, 3125].

Furthermore, this theorem is used as a formal witness. It is bundled into the overarching CardinalitySpectrumCert certificate (specifically in the spectrum_length field). This certificate mathematically guarantees that the RS spectrum possesses specific properties, alongside being strictly bounded by 3125 and strictly increasing.

5. What this Declaration Does Not Prove

This theorem is purely a computational property of a specific Lean list. It does not prove that these are the only valid cardinalities in Recognition Science, nor does it mathematically bind every physical RS domain to map injectively into this list. It simply asserts the size of an explicitly enumerated set of exemplar domain sizes.