Foundation Pair Kernel Weyl Full Fourier Exchange Clock Occupation Cost Axis3 At

The attached transform identity

The discrete Fourier transform (DFT) is a standard tool that takes a list of numbers and rewrites it as a list of frequency components. For a signal with three entries, the transform is a 3 by 3 matrix of complex exponentials. A central fact about the DFT is that it preserves total energy: the sum of squared magnitudes of the input equals the sum of squared magnitudes of the output, up to a scale factor. This theorem concerns a specific cost function built from those squared magnitudes.

The declaration clockOccupationCostAxis3At_centerAttached_eq_dft proves an identity about two ways to measure a signal's cost. The first way, clockOccupationCostAxis3At, picks one of the three positions as a center and sums the squared magnitudes of the other two positions, multiplied by three. The second way, shiftOccupationCostAxis3, sums the squared magnitudes of all three positions, also multiplied by three. The theorem states that if you first rotate the signal so that a chosen position becomes the center, then apply the DFT, the clock cost of that rotated, transformed signal equals the shift cost of the plain DFT of the original signal.

In plainer terms, the identity says that the cost you compute after a rotation and a Fourier transform is the same as the cost you would compute from the transform alone, as long as you use the right cost function for each stage. This is a symmetry property: the order of operations, rotate then transform, does not change the final cost when measured this way. The theorem is proved in the framework's machine-checked library of formal theorems, with no unproved assumptions and no empirical input.

What the theorem does not claim is that this cost function is the physically correct one. The selection of the equal-weight action as physical remains a foundational model, a definitional choice, not a proved consequence. The identity is a mathematical fact about a particular cost function, not a statement about which cost function nature uses. It also does not claim that the cost is invariant under all transformations, only under the specific combination of rotation and DFT described.

theorem clockOccupationCostAxis3At_centerAttached_eq_dft
    (center : ZMod 3)
    (ψ : WeylAxisAmplitude3) :
    clockOccupationCostAxis3At center
        (centerAttachedFiniteDFTAxis3 center ψ) =
      clockOccupationCostAxis3
        (finiteDFTAxis3 ψ) := by
  have h1 :
      (center + 1) - center =
        (1 : ZMod 3) := by
    ring
  have h2 :
      (center + 2) - center =
        (2 : ZMod 3) := by
    ring
  simp only [clockOccupationCostAxis3At,
    centerAttachedFiniteDFTAxis3,
    clockOccupationCostAxis3, h1, h2,
    omega3_pow_mul_normSq]

theorem clockOccupationCostAxis3At_centerAttached_eq_dft
    (center : ZMod 3)
    (ψ : WeylAxisAmplitude3) :
    clockOccupationCostAxis3At center
        (centerAttachedFiniteDFTAxis3 center ψ) =
      clockOccupationCostAxis3
        (finiteDFTAxis3 ψ) := by
  have h1 :
      (center + 1) - center =
        (1 : ZMod 3) := by
    ring
  have h2 :
      (center + 2) - center =
        (2 : ZMod 3) := by
    ring
  simp only [clockOccupationCostAxis3At,
    centerAttachedFiniteDFTAxis3,
    clockOccupationCostAxis3, h1, h2,
    omega3_pow_mul_normSq]

What this page does not claim

The theorem does not claim that this cost function is the physically correct one; that selection remains a foundational model. The theorem does not claim invariance under all transformations, only under the specific rotation and DFT combination.

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)

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