Foundation Pair Kernel Weyl Event Center Attachment Finite Dftaxis3 Discriminati

A Fourier exchange that swaps two costs

The finite Fourier transform is a standard tool that takes a list of values and returns another list of the same length, mixing the original entries by complex roots of unity. On a three-site periodic axis, the transform has a striking effect on two simple cost functions. One cost, the shift cost, penalizes differences between neighboring sites; the other, the clock cost, penalizes the size of the entries themselves. The theorem finiteDFTAxis3_discriminating_exchange proves that applying the transform to a single concentrated pulse swaps these two costs up to a factor of three: the shift cost of the transformed pulse equals three times the clock cost of the original, and the clock cost of the transformed pulse equals three times the shift cost of the original.

This exchange is not a coincidence of the chosen pulse. It is a proved identity in a machine-checked library of formal theorems, meaning every step is verified by a computer from the definitions. The library first shows that the Fourier transform of a delta pulse, a function that is one at a single site and zero elsewhere, is the uniform function, constant across all three sites. Then it computes the two costs on both sides of the exchange and proves the equality. The factor of three matches the number of sites, the same factor that appears when the uniform function's clock cost is evaluated.

In Recognition Science, this theorem plays a supporting role in a larger construction. The framework models physical events as discrete postings on a 27-site carrier, a three-dimensional torus. The theorem provides a recognition cost, a number that measures how much a configuration deviates from a preferred state, for a specific exchange. But the theorem does not claim that the two costs are equal in general. The library proves that equal weights, meaning the shift and clock costs are treated as equally important, are forced only under an additional premise about invariance under Fourier exchange. That premise is not yet derived from the framework's core principles, so the physical claim that the two costs are equal remains an open target.

What the theorem does establish is a precise, checkable relationship: on a three-site axis, the Fourier transform exchanges the two cost functions up to the site count. This is a structural fact, not a physical law. It does not introduce any dimensioned length, and it does not depend on any empirical constant. The theorem is a clean piece of mathematics that the framework can use, but it leaves the question of whether the two costs are physically equal to future work.

/-- The actual finite DFT sends the delta witness to the uniform witness and
therefore exchanges the two generator costs, with the factor three required
by the unnormalized DFT. -/
theorem finiteDFTAxis3_discriminating_exchange :
    shiftOccupationCostAxis3
          (finiteDFTAxis3 deltaAxisAmplitude3) =
        3 * clockOccupationCostAxis3 deltaAxisAmplitude3 ∧
      clockOccupationCostAxis3
          (finiteDFTAxis3 deltaAxisAmplitude3) =
        3 * shiftOccupationCostAxis3 deltaAxisAmplitude3 := by
  rw [finiteDFTAxis3_delta,
    shiftOccupationCostAxis3_uniform,
    clockOccupationCostAxis3_uniform,
    shiftOccupationCostAxis3_delta,
    clockOccupationCostAxis3_delta]
  norm_num

theorem finiteDFTAxis3_delta :
    finiteDFTAxis3 deltaAxisAmplitude3 =
      uniformAxisAmplitude3 := by
  funext k
  simp [finiteDFTAxis3, ZMod.dft_apply,
    deltaAxisAmplitude3, uniformAxisAmplitude3]

/-- A genuine finite-Fourier exchange symmetry would force equal shift and
clock weights.  The conclusion is conditional on `hinvariant`. -/
theorem finiteFourierExchangeInvariant_forces_equal_weights
    {shiftWeight clockWeight : ℝ}
    (hinvariant :
      FiniteFourierExchangeInvariantAxis3
        shiftWeight clockWeight) :
    shiftWeight = clockWeight := by
  have h := hinvariant deltaAxisAmplitude3
  rw [finiteDFTAxis3_delta] at h
  simp only [twoWeightOccupationCostAxis3,
    shiftOccupationCostAxis3_uniform,
    clockOccupationCostAxis3_uniform,
    shiftOccupationCostAxis3_delta,
    clockOccupationCostAxis3_delta,
    mul_zero, add_zero] at h
  linarith

What this page does not claim

The theorem does not claim that shift and clock costs are equal in general. The theorem does not claim that the physical equal-weight premise is derived from the framework's core principles. The theorem does not claim any empirical constant or dimensioned length.

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

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