generators_minimal
plain-language theorem explainer
The integers 2, 3, and 5 are pairwise distinct and form the minimal generating set for the Recognition Science cardinality spectrum. Cross-domain researchers cite this to anchor the primorial base before decomposing spectrum members via addition, multiplication, and exponentiation. The proof is a direct decision procedure that confirms the three inequalities in one step.
Claim. The generators 2 (binary face), 3 (spatial dimension), and 5 (configuration dimension) are pairwise distinct: $2 ≠ 3 ∧ 3 ≠ 5 ∧ 2 ≠ 5$.
background
Module C27 states that every integer in the RS cardinality spectrum reduces to a polynomial expression in the three generators 2, 3, and 5 using addition, multiplication, exponentiation, and binomial coefficients. Concrete reductions include 4 = 2², 6 = 2·3, 8 = 2³, 12 = 2²·3, 15 = 3·5, 16 = 2⁴, 25 = 5², and 360 = 2³·3²·5. The generators are defined as G2 := 2, G3 := 3, and G5 := 5, each carrying a distinct role in the binary, spatial, and configurational layers of the spectrum.
proof idea
The proof is a one-line wrapper that refines the conjunction into three separate goals and applies the decide tactic to each inequality.
why it matters
This theorem supplies the generators_distinct clause inside the RecognitionGeneratorsCert structure, which certifies the full generator set together with the primorial product 2·3·5 = 30. It directly supports the structural meta-claim of module C27 that no spectrum member lies outside the polynomial ring generated by {2, 3, 5}. In the broader framework it provides the integer base required for phi-ladder mass formulas and the eight-tick octave constructions.
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