Constants Gap Weight Projection Dft8 Mode Norm Sq Sum

The unit-norm DFT mode

A discrete Fourier transform (DFT) breaks a sampled signal into a set of frequency modes. The lemma dft8_mode_normSq_sum states that for the framework's eight-tick cycle, each mode's squared magnitude sums to exactly 1 across all eight time steps. In plainer terms, every frequency mode carries precisely one unit of total energy, no more and no less. This is a normalization fact: it fixes the scale of each mode so that none is amplified or damped by the transform itself.

The result is a recognition (a discrete record of events) framework detail. The framework models an eight-tick clock, and the DFT's eight modes are its frequency components. The lemma proves that the transform's columns are orthonormal, meaning each mode is independent and carries equal weight. This matters because later definitions, such as the projection weight w8_projected, divide by total energy to convert fractions into per-cell weights. Without the unit-norm fact, that normalization would be arbitrary.

The lemma does not claim that the DFT modes are physically real, nor that the eight-tick cycle itself is derived from experiment. It is a statement about the framework's chosen mathematical structure. The framework's library proves the lemma from the definition of the DFT mode; it does not prove that this DFT is the only possible one, nor that the normalization constant 64 is forced by physics. Those remain definitional choices within the framework.

What the lemma changes is simple: it removes a hidden degree of freedom. Before it, the normalization of the DFT modes could have been anything. After it, the scale is fixed, and subsequent calculations have a clear, auditable starting point. This is claim hygiene: making the operator and measure explicit so no silent assumption slips in.

/-- DFT-8 mode `k` has unit norm (columns are orthonormal). -/
lemma dft8_mode_normSq_sum (k : Fin 8) :
    (∑ t : Fin 8, Complex.normSq (dft8_mode k t)) = 1 := by
  -- ⟨column k, column k⟩ = 1, and ⟨v,v⟩ = ∑ conj(v_t) * v_t = ∑ normSq(v_t).
  have hcol := dft8_column_orthonormal k k
  have h1 : (∑ t : Fin 8, star (dft8_entry t k) * dft8_entry t k) = (1 : ℂ) := by
    simpa using (by simpa using hcol)
  -- Rewrite each term `star z * z` as `(normSq z : ℂ)`.
  have h1' : (∑ t : Fin 8, ((Complex.normSq (dft8_entry t k) : ℝ) : ℂ)) = (1 : ℂ) := by
    simpa [Complex.normSq_eq_conj_mul_self] using h1
  -- Pull the cast out of the sum.
  have h1'' : ((∑ t : Fin 8, (Complex.normSq (dft8_entry t k) : ℝ)) : ℂ) = (1 : ℂ) := by
    simpa [Complex.ofReal_sum] using h1'
  -- Back to ℝ, and rewrite `dft8_mode`.
  have hreal : (∑ t : Fin 8, (Complex.normSq (dft8_entry t k) : ℝ)) = (1 : ℝ) :=
    Complex.ofReal_injective (by simpa using h1'')
  simpa [dft8_mode] using hreal

What this page does not claim

The DFT modes are physically real or experimentally observed. The eight-tick cycle itself is derived from first principles rather than chosen as a model. The normalization constant 64 is forced by physics rather than by definitional choice.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Constants/GapWeight/Projection.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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