abstractHarmonicAnalysisCert
abstractHarmonicAnalysisCert assembles a certificate confirming five locally compact groups and the order-8 cyclic group size of exactly 8 in the Recognition Science harmonic analysis setup. Mathematicians formalizing abstract harmonic analysis from RS principles would cite this to anchor the DFT-8 construction on ℤ/8ℤ. The definition is a direct structure literal that invokes two decide lemmas for the cardinality facts.
claimLet LCGroup be the finite type of the five canonical locally compact groups. The certificate asserts that the cardinality of LCGroup equals 5 and that the size of the cyclic group of order 8 satisfies $z8Size = 2^3$.
background
The module develops abstract harmonic analysis from Recognition Science by identifying five canonical locally compact groups (ℝ, ℤ, S¹, ℚₚ, GL_n(ℚ)) whose count equals configuration dimension D = 5. It encodes the discrete Fourier transform on the cyclic group ℤ/8ℤ of order 8 = 2^3, consistent with the eight-tick octave. Pontryagin duality appears as the dual of ℤ being S¹.
proof idea
The definition is a one-line structure constructor that supplies the five_groups field from lcGroupCount and the z8_size field from z8Size_2cubed.
why it matters in Recognition Science
This definition completes the Lean encoding of the abstract harmonic analysis certificate in the Recognition Science mathematics module. It anchors the five-group count and 2^3 size for the cyclic group, linking directly to the eight-tick octave (T7) and the derivation of three spatial dimensions (T8). The structure closes the module's formalization of harmonic analysis on these groups.
scope and limits
- Does not list the explicit five groups or derive their count from RS axioms.
- Does not prove Pontryagin duality or any duality theorems.
- Does not connect the certificate to physical constants or mass formulas.
- Does not address the Recognition Composition Law or forcing chain steps beyond T7.
formal statement (Lean)
35def abstractHarmonicAnalysisCert : AbstractHarmonicAnalysisCert where
36 five_groups := lcGroupCount
proof body
Definition body.
37 z8_size := z8Size_2cubed
38
39end IndisputableMonolith.Mathematics.AbstractHarmonicAnalysisFromRS