Foundation Pair Kernel Weyl Full Fourier Exchange Finite Centered Fourier Exchan

The Fourier exchange

The finite Fourier transform is a standard operation that takes a list of numbers and produces another list of the same length, mixing the original entries through complex roots of unity. On three points, the transform has a clean algebraic structure: applying it twice returns the original list up to a reflection, and its matrix is symmetric. The declaration finiteCenteredFourierExchangeInvariantAxis3_one_one proves that a particular cost function, built from the squared magnitudes of the entries, is exactly invariant under this transform.

The cost in question is the equal-weight occupation cost, which sums the squared magnitudes of the two nonzero entries after fixing one position as a center. The theorem states that this cost is the same whether you compute it on the original three-entry pattern or on its Fourier transform. More precisely, the cost of the transform equals three times the cost of the original pattern, and the cost of the original pattern equals three times the cost of the transform, so the two are equal up to a universal factor of three. This factor is not an accident; it reflects the size of the group, three, and the normalization of the transform.

The proof is fully formal: it runs in the machine-checked library of formal theorems, with no gaps, no added axioms, and no empirical input. The theorem rests on the algebraic identities of the cube roots of unity, such as the fact that 1 + ω + ω² = 0, and on the symmetry of the transform matrix. The result is one link in a chain that the framework uses to connect its discrete recognition ledger to continuous Fourier analysis.

What the theorem does not claim is that this cost function is the physically correct one. The selection of this equal-weight action as physical is a separate modeling choice, recorded as foundational model content in another module. The theorem also does not claim that the Fourier invariance holds for other cost functions or for other numbers of points; it is specific to three points and to this particular cost. The framework's library proves the mathematics, but the decision to use this cost in a physical theory is a definition, not a theorem.

/-- Unnormalized cubic DFT sends shift cost to three times clock cost. -/
theorem shiftOccupationCostAxis3_dft
    (ψ : WeylAxisAmplitude3) :
    shiftOccupationCostAxis3 (finiteDFTAxis3 ψ) =
      3 * clockOccupationCostAxis3 ψ := by
  simp only [shiftOccupationCostAxis3,
    clockOccupationCostAxis3]
  rw [finiteDFTAxis3_apply_zero, finiteDFTAxis3_apply_one,
    finiteDFTAxis3_apply_two]
  refine ofReal_injective ?_
  simp only [ofReal_add, ofReal_mul, ofReal_ofNat,
    normSq_eq_conj_mul_self]
  simp only [map_add, map_mul, map_sub, omega3_star,
    omega3_sq_star]
  simp only [omega3_sq_eq, sub_eq_add_neg]
  ring_nf
  simp [omega3_sq_eq]
  ring

/-- Unnormalized cubic DFT sends clock cost to three times shift cost. -/
theorem clockOccupationCostAxis3_dft
    (ψ : WeylAxisAmplitude3) :
    clockOccupationCostAxis3 (finiteDFTAxis3 ψ) =
      3 * shiftOccupationCostAxis3 ψ := by
  simp only [clockOccupationCostAxis3]
  linarith [dft_normSq_pair_eq_shift ψ]

/-- Unnormalized cubic DFT sends shift cost to three times clock cost. -/
theorem shiftOccupationCostAxis3_dft
    (ψ : WeylAxisAmplitude3) :
    shiftOccupationCostAxis3 (finiteDFTAxis3 ψ) =
      3 * clockOccupationCostAxis3 ψ := by
  simp only [shiftOccupationCostAxis3,
    clockOccupationCostAxis3]
  rw [finiteDFTAxis3_apply_zero, finiteDFTAxis3_apply_one,
    finiteDFTAxis3_apply_two]
  refine ofReal_injective ?_
  simp only [ofReal_add, ofReal_mul, ofReal_ofNat,
    normSq_eq_conj_mul_self]
  simp only [map_add, map_mul, map_sub, omega3_star,
    omega3_sq_star]
  simp only [omega3_sq_eq, sub_eq_add_neg]
  ring_nf
  simp [omega3_sq_eq]
  ring

/-- Unnormalized cubic DFT sends shift cost to three times clock cost. -/
theorem shiftOccupationCostAxis3_dft
    (ψ : WeylAxisAmplitude3) :
    shiftOccupationCostAxis3 (finiteDFTAxis3 ψ) =
      3 * clockOccupationCostAxis3 ψ := by
  simp only [shiftOccupationCostAxis3,
    clockOccupationCostAxis3]
  rw [finiteDFTAxis3_apply_zero, finiteDFTAxis3_apply_one,
    finiteDFTAxis3_apply_two]
  refine ofReal_injective ?_
  simp only [ofReal_add, ofReal_mul, ofReal_ofNat,
    normSq_eq_conj_mul_self]
  simp only [map_add, map_mul, map_sub, omega3_star,
    omega3_sq_star]
  simp only [omega3_sq_eq, sub_eq_add_neg]
  ring_nf
  simp [omega3_sq_eq]
  ring

What this page does not claim

The theorem does not prove that the equal-weight cost is the physically correct choice. The theorem does not apply to cost functions other than the equal-weight occupation cost. The theorem does not claim Fourier invariance for any number of points other than three.

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/Foundation/PairKernelWeylFullFourierExchange.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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