Foundation Pair Kernel Weyl Full Fourier Exchange Realized Posting Center Fourie

Fourier symmetry of the posting law

The discrete Fourier transform (DFT) is a standard tool that takes a list of numbers and rewrites it as a list of wave-like components. For a system with three points, the transform is a 3 by 3 matrix of complex exponentials. The Recognition Science declaration realizedPostingCenter_fourierExchange27 establishes a precise symmetry: for a certain cost function defined on such three-point configurations, applying the DFT leaves the cost unchanged up to a fixed factor of three. In plainer terms, the cost of a configuration equals three times the cost of its Fourier-transformed counterpart, and vice versa, so the two are interchangeable up to that scale.

The cost function in question is built from two pieces. The first, called the shift occupation cost, measures how much amplitude sits away from a chosen center point. The second, the clock occupation cost, measures how much amplitude sits at the two non-center points. The theorem shows that the Fourier transform swaps these two roles: the shift cost of a transformed configuration equals three times the clock cost of the original, and the clock cost of the transformed equals three times the shift cost of the original. This is the exchange property the declaration names.

The proof is fully machine-checked in the framework's library of formal theorems. It uses no unproved assumptions and no empirical input. The result is a theorem in the strict sense: it is derived from definitions and earlier lemmas by a proof checker. The mathematics itself is classical Fourier analysis on a finite group, applied to a specific cost functional. The framework's contribution is the selection of that cost functional as physically meaningful, a choice that the declaration does not make.

What the declaration does not claim is equally important. It does not assert that this cost function is the one realized in nature. That selection is a separate modeling step, explicitly marked as foundational model content in another module. The theorem also does not say that the Fourier symmetry holds for any cost function, only for the particular equal-weight combination it defines. And it does not imply anything about systems with more than three points; the result is specific to the three-point case.

For a reader meeting Recognition Science for the first time, the practical upshot is this: the framework has a rigorously proved symmetry that connects its discrete ledger of recognition events to the standard mathematics of Fourier analysis. The symmetry is a structural fact, not an empirical observation. It is one of the pieces that the framework uses to build its account of physical law, but it is not itself a claim about measurement or about which physical system realizes the cost function.

/-- 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

def twoWeightOccupationCostAxis3At
    (center : ZMod 3)
    (shiftWeight clockWeight : ℝ)
    (ψ : WeylAxisAmplitude3) : ℝ :=
  shiftWeight * shiftOccupationCostAxis3 ψ +
    clockWeight * clockOccupationCostAxis3At center ψ

What this page does not claim

The declaration does not claim that this cost function is the one realized in nature. The theorem does not hold for arbitrary cost functions, only for the specific equal-weight combination it defines. The result does not extend to systems with more than three points.

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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