z8Size
The constant for the cardinality of the cyclic group ℤ/8ℤ is set to 8, which equals 2 to the third power, in the Recognition Science treatment of abstract harmonic analysis. Harmonic analysts working with discrete Fourier transforms on groups of order 8 cite this when verifying the five locally compact groups. The definition is direct and requires no further computation.
claimLet $n$ be the number of elements in the cyclic group $ℤ/8ℤ$. Then $n = 8$.
background
The Recognition Science framework derives abstract harmonic analysis from five canonical locally compact groups: the real numbers, the integers, the circle group, the p-adic numbers, and the general linear group over the rationals. These groups correspond to a configuration dimension of 5. Within this setting, the discrete Fourier transform of order 8 operates on the cyclic group $ℤ/8ℤ$, whose order is 8, equal to $2^3$, aligning with the eight-tick octave of period $2^3$ in the framework.
proof idea
The declaration is a direct definition assigning the literal natural number 8. It forms the basis for the theorem that equates it to $2^3$ using the decide tactic, with no additional lemmas invoked.
why it matters in Recognition Science
This supplies the cardinality needed for the AbstractHarmonicAnalysisCert structure, which records both the five groups and the equality of the size to $2^3$. It anchors the connection to the eight-tick octave in the Recognition Science chain for discrete Fourier transforms. The value supports the certification that there are zero sorries in the module.
scope and limits
- Does not derive the cardinality from deeper Recognition Science axioms such as the J-uniqueness or forcing chain.
- Does not prove any algebraic properties of $ℤ/8ℤ$ other than its order.
- Does not address the other four locally compact groups or Pontryagin duality.
- Does not relate to physical constants like the fine structure constant or mass ladder.
Lean usage
example : z8Size = 8 := rflformal statement (Lean)
28def z8Size : ℕ := 8