DFTMode
The eight-element finite type indexes the DFT harmonic modes of the RS frequency comb. Cross-domain spectrum work cites it to label the eight modes k from 0 to 7 that carry the theta-band frequencies. The declaration is realized as a direct type abbreviation to the standard finite set of cardinality 8.
claimThe mode index set is the finite type with eight elements, denoted $M := Fin 8 = {0,1,2,3,4,5,6,7}$.
background
The C24 module treats the eight DFT modes as the vertices of the Q3 cube and equips them with the RS frequency comb ν_k = k · 5φ / 8 Hz, where the carrier 5φ ≈ 8.09 Hz lies in the theta band. All frequencies are non-negative and the highest mode remains below 9 Hz. DFTMode supplies the discrete index domain for this comb.
proof idea
One-line abbreviation that sets the type equal to the standard finite type Fin 8.
why it matters in Recognition Science
The type supplies the domain for the harmonic frequency definition and the spectrum certificate structure. It implements the eight-tick octave (T7) of the forcing chain by fixing cardinality exactly 2^3. Downstream theorems on non-negativity, factoring, and strict monotonicity of the frequencies all quantify over this index set.
scope and limits
- Does not assign physical units or frequency values to individual modes.
- Does not prove the 2^3 cardinality; that is supplied by the upstream count theorem.
- Does not restrict the type to any particular physical interpretation beyond indexing.
formal statement (Lean)
29abbrev DFTMode : Type := Fin 8
proof body
Definition body.
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