G5
G5 assigns the integer constant 5 as the configuration dimension generator within the Recognition Science spectrum. Cross-domain proofs that decompose spectrum members such as 15 and 125 cite this value when applying multiplication and exponentiation to the base set. The definition is a direct constant assignment requiring no further reduction steps.
claim$G_5 = 5$, where $G_5$ denotes the configuration dimension generator in the set of primitive generators for the RS cardinality spectrum.
background
The Recognition Generators module shows that every integer in the RS spectrum reduces to polynomials in the generators G = {2, 3, 5} under addition, multiplication, exponentiation, and binomial coefficients. G5 supplies the third generator, fixed at 5 to match the configuration dimension. Upstream results define configDim as d + 2 for spatial dimension d, yielding 5 when d = 3, and similarly set the constant to 5 in cosmology and game-theory contexts.
proof idea
G5 is introduced by the direct definition assigning the natural number 5. Downstream theorems then invoke this constant inside equalities that are discharged by decide.
why it matters in Recognition Science
G5 completes the minimal generator triple whose products recover spectrum members and the third primorial 30. It is used by fifteen_decomp, oneTwentyFive_decomp, primorial_product, and the RecognitionGeneratorsCert structure. This placement supports the module meta-claim that the spectrum is generated from {2, 3, 5} and aligns with the framework forcing of three spatial dimensions plus two ledger degrees of freedom.
scope and limits
- Does not derive the value of configDim from the forcing chain.
- Does not prove minimality of the generator set.
- Does not enumerate or decompose any spectrum integer.
- Does not supply physical units or scaling for the integer 5.
formal statement (Lean)
43def G5 : ℕ := 5 -- configDim
proof body
Definition body.
44
45/-! ## Spectrum decompositions. -/
46