AbstractHarmonicAnalysisCert
AbstractHarmonicAnalysisCert packages the assertion that five locally compact groups are enumerated and that the cyclic group of order eight has cardinality exactly eight. Researchers formalizing harmonic analysis on the phi-ladder would cite the structure when setting up DFT-8 or invoking Pontryagin duality. The definition assembles these facts directly from the inductive enumeration of the group type and the constant definition of the group order.
claimThe structure consists of the two assertions that the finite cardinality of the set of locally compact groups equals five and that the size of the cyclic group of order eight equals $2^3$.
background
The module identifies five canonical locally compact groups (real line, integers, circle, p-adics, general linear group over the rationals) whose count equals the configuration dimension five. The cyclic group of order eight is introduced as the domain for DFT-8, with its order fixed at eight to match the eight-tick octave of period $2^3$ in the Recognition Science framework. Pontryagin duality is noted as the correspondence sending the integers to the circle group.
proof idea
The declaration is a structure definition whose first field records the cardinality of the inductively defined locally compact groups type (five constructors) and whose second field records the direct equality of the size constant to eight. No tactics or lemmas beyond the inductive definition and the constant definition are required.
why it matters in Recognition Science
The structure supplies the data package used by the downstream definition that constructs a concrete certificate instance. It anchors the abstract harmonic analysis layer to the eight-tick octave and the five-group configuration, thereby licensing DFT-8 within the Recognition Science derivation of harmonic analysis from the forcing chain.
scope and limits
- Does not derive the count of five groups from the T0-T8 forcing chain.
- Does not assign physical interpretations or dynamics to the five groups.
- Does not prove Pontryagin duality; it only records the group enumeration.
- Does not compute the group order; it only records the constant definition.
formal statement (Lean)
31structure AbstractHarmonicAnalysisCert where
32 five_groups : Fintype.card LCGroup = 5
33 z8_size : z8Size = 2 ^ 3
34