Foundation Pair Kernel Weyl Full Fourier Exchange Finite Fourier Exchange Invari

The Fourier exchange

The discrete Fourier transform is a standard tool that rewrites a sequence of numbers into a sequence of frequency components. On a system with three slots, the transform maps one set of three complex amplitudes to another. The Recognition Science framework defines two cost functions on these amplitudes: a shift cost, which measures how much energy sits in adjacent pairs, and a clock cost, which measures how much energy sits away from a chosen center. The theorem finiteFourierExchangeInvariantAxis3_one_one establishes that applying the Fourier transform swaps these two costs exactly, multiplying each by three. In symbols, if F is the transform and ψ is an amplitude, then the shift cost of Fψ equals three times the clock cost of ψ, and the clock cost of Fψ equals three times the shift cost of ψ.

The factor of three is not an accident of normalization. It reflects the size of the system: with three slots, the Fourier transform spreads a unit of energy across all three frequencies, so the cost measured in one domain appears three times larger in the other. The theorem is proved in a machine-checked library of formal theorems, meaning every step of the argument is verified by a computer kernel with no gaps and no unproved assumptions. The proof uses only the standard properties of the Fourier transform on the cyclic group of order three, together with the explicit definitions of the two costs.

In Recognition Science, this exchange is part of a larger structure. The framework models physical states as amplitudes on a discrete set of slots, and it treats the cost of a state as a fundamental quantity. The Fourier exchange shows that the two natural costs are dual to each other: what looks like a shift cost in one domain is exactly a clock cost in the other, up to the factor of three. This duality is what makes the equal-weight combination of the two costs invariant under the Fourier transform, a property the framework uses in its self-dual posting law. The selection of that particular combination as physical, however, is a modeling choice, not a consequence of the theorem.

The theorem does not claim that the Fourier transform preserves either cost individually. It claims only that the transform exchanges them, with the factor of three. It also does not claim that the equal-weight combination is the unique physical cost, or that the factor of three has any meaning beyond the size of the system. The theorem is a precise statement about a finite three-slot system, and its content is exactly the exchange property, nothing more.

/-- 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

What this page does not claim

The theorem does not prove that either cost is individually preserved by the Fourier transform. The theorem does not establish that the equal-weight combination is the unique or correct physical cost. The theorem does not assign any physical meaning to the factor of three beyond the size of the system.

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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