Explanation of rsSpectrum_pairwise_lt

(1) What the declaration says in plain English

The theorem asserts that the list rsSpectrum (containing 20 canonical RS cardinalities) is strictly increasing: every element is smaller than all later elements.

(2) Why it matters in Recognition Science

RS claims that cardinalities of domain types across the framework form a structured spectrum generated from primitives {2, 3, 5, gap45}. This theorem verifies the list is ordered and duplicate-free, confirming the spectrum is a deliberate, non-random numerical structure rather than an arbitrary collection.

(3) How to read the formal statement

theorem rsSpectrum_pairwise_lt : rsSpectrum.Pairwise (· < ·) := by decide

In standard math: Let rsSpectrum = [2, 3, 4, 5, 6, 7, 8, 10, 12, 15, 16, 25, 45, 64, 70, 125, 216, 256, 360, 3125]. The Pairwise predicate (from Mathlib) requires that for every pair of indices i < j, rsSpectrum[i] < rsSpectrum[j]. The proof by decide evaluates the concrete list at compile time.

(4) Visible dependencies or certificates in the supplied source

• Defined in the same module as rsSpectrum and supported by rsSpectrum_length and rsSpectrum_bounded.
• Included in the certificate structure CardinalitySpectrumCert via the field spectrum_pairwise.
• Relies on generator equalities such as eightTick_eq, gap45_eq, and individual spectrum-member theorems (e.g., three_is_Dspatial).
• The full certificate is witnessed by cardinalitySpectrumCert.

(5) What this declaration does not prove

It only confirms ordering of this fixed list via decidable evaluation. It proves nothing about physical interpretations, uniqueness of the spectrum, connections to forcing chains, constant derivations, or any theorem outside this module.