AbstractHarmonicAnalysisCert
AbstractHarmonicAnalysisCert packages the assertions that five locally compact groups are recognized and that the cyclic group of order eight has cardinality exactly two cubed. Researchers formalizing harmonic analysis inside the Recognition Science framework cite the certificate to fix the configuration dimension and the DFT-8 octave size. The declaration is a bare structure whose two fields are direct cardinality and equality statements with no computational content.
claimA structure asserting that the number of canonical locally compact groups equals five and that the cardinality of the cyclic group of order eight equals eight (hence two cubed).
background
The module develops abstract harmonic analysis from Recognition Science by enumerating five canonical locally compact groups: the real line, the integers, the circle group, the p-adic numbers, and the general linear group over the rationals. These are collected in an inductive type that derives decidable equality and finite type, so its cardinality is immediately computable as five. The auxiliary definition z8Size fixes the size of the cyclic group of order eight at eight, which equals two cubed and matches the eight-tick octave required for discrete Fourier analysis on that group. Pontryagin duality is noted as linking the dual of the integers to the circle group.
proof idea
The declaration is a structure definition. It introduces two fields, one recording the cardinality of the locally compact groups and one recording the size equality for the order-eight cyclic group. No lemmas are applied and no tactics appear; the fields stand as direct assertions.
why it matters in Recognition Science
This structure serves as the type of the concrete certificate constructed by the sibling definition that supplies the group count and octave size. It anchors the setup for abstract harmonic analysis and aligns with the eight-tick octave in the forcing chain. The construction prepares duality arguments that connect the integer lattice to the circle while remaining independent of the spatial dimension three and the fine-structure constant bounds.
scope and limits
- Does not derive the five groups from the J-uniqueness functional equation.
- Does not prove any Fourier transform identities or character sums.
- Does not relate the group count to spatial dimension three.
- Does not address physical constants or mass ladders.
formal statement (Lean)
31structure AbstractHarmonicAnalysisCert where
32 five_groups : Fintype.card LCGroup = 5
33 z8_size : z8Size = 2 ^ 3
34