Foundation Pair Kernel Weyl Full Fourier Exchange Shift Occupation Cost Axis3 Ce

The Fourier exchange

The discrete Fourier transform is a staple of signal processing: it takes a list of numbers and rewrites it as a list of frequency components. On a cycle of three positions, the transform has a special relationship with rotation. Rotating the input list by one step, and then taking the transform, is the same as taking the transform and then multiplying each output by a cube root of unity. The declaration shiftOccupationCostAxis3_centerAttached_eq_dft_rotate states a cost function version of that fact: the occupation cost, a measure of how much amplitude sits away from a chosen center, is unchanged whether you first rotate the input or first apply a centered transform.

Concretely, the theorem says that for any three complex amplitudes ψ and any center position, the shift occupation cost of the centered Fourier transform equals the shift occupation cost of the Fourier transform of the rotated amplitudes. In symbols: shiftOccupationCostAxis3(centerAttachedFiniteDFTAxis3 center ψ) = shiftOccupationCostAxis3(finiteDFTAxis3 (rotateAmplitudeAxis3 center ψ)). The cost function itself is a sum of squared magnitudes, a standard energy measure. The equality means the cost cannot tell the two operations apart: a centered transform followed by a cost measurement gives the same number as a rotation followed by a transform and then the same cost.

This is one of several exchange identities in the same body of work. A companion theorem shows the clock cost, which weights the two non-center positions, also exchanges with the shift cost under the Fourier transform: clockOccupationCostAxis3(finiteDFTAxis3 ψ) = 3 * shiftOccupationCostAxis3 ψ. Together they establish that the equal-weight sum of both costs is invariant under the Fourier transform up to the required factor of three. The documentation calls this the full cubic finite-Fourier exchange and says it closes the all-amplitude mathematics behind a self-dual posting law.

In Recognition Science, the framework models a ledger, a discrete record of recognition events, and the cost of recognition is forced by a proved functional equation. The exchange identities here are part of the mathematical scaffolding for that framework. The theorem is proved in a machine-checked library of formal theorems with no unproven assumptions and no empirical input. But what it establishes is purely algebraic: an equality between two cost values for three complex amplitudes under a finite transform. It says nothing about which cost function is physically selected.

The selection of the equal-weight action as physical remains a foundational model choice, explicitly labeled as such in the documentation. The theorem does not claim that the Fourier transform is a symmetry of nature, nor that three dimensions are forced by this identity alone. It is a precise, narrow statement about a cost function on a three-point cycle, and its role is to support a larger derivation, not to stand alone as a physical law.

theorem shiftOccupationCostAxis3_centerAttached_eq_dft_rotate
    (center : ZMod 3)
    (ψ : WeylAxisAmplitude3) :
    shiftOccupationCostAxis3
        (centerAttachedFiniteDFTAxis3 center ψ) =
      shiftOccupationCostAxis3
        (finiteDFTAxis3
          (rotateAmplitudeAxis3 center ψ)) := by
  set φ :=
    finiteDFTAxis3
      (rotateAmplitudeAxis3 center ψ)
  set u := omega3 ^ ((center * center).val)
  have hu : Complex.normSq u = 1 :=
    omega3_pow_normSq _
  have h :
      centerAttachedFiniteDFTAxis3 center ψ =
        fun k =>
          u * rotateAmplitudeAxis3 (-center) φ k := by
    funext k
    simpa [φ, u] using
      centerAttached_eq_phase_rotate_dft
        center ψ k
  rw [h]
  have h2 :
      (fun k =>
        u * rotateAmplitudeAxis3 (-center) φ k) =
        rotateAmplitudeAxis3 (-center)
          (fun k => u * φ k) := by
    funext k
    simp [rotateAmplitudeAxis3]
  rw [h2, shiftOccupationCostAxis3_rotate,
    shiftOccupationCostAxis3_const_mul u hu]

/-- 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

What this page does not claim

The theorem does not claim that the Fourier transform is a symmetry of physical space. The theorem does not claim that the equal-weight action is the physically selected cost function. The theorem does not claim that three spatial dimensions are forced by this identity alone.

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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