Foundation Pair Kernel Weyl Self Dual Continuum Scale Self Dual Weyl Mesh Scale

The self-dual mesh scale

In signal processing, a finite set of N evenly spaced samples and its Fourier transform form a reciprocal pair: if the samples are spaced by a distance s in one domain, the transform's samples are spaced by 1/(N s) in the other. The product of the two spacings is always 1/N, a fixed constraint. A coordinate system is called self-dual when the spacing in both domains is the same number, so the two domains become interchangeable. Solving the constraint with equal spacings gives s = 1/√N, the unique positive value that balances the pair.

This classical fact has a long history. The discrete Fourier transform, in its modern form, was popularized by James Cooley and John Tukey in 1965, though the underlying harmonic analysis dates to Joseph Fourier's 1822 Théorie analytique de la chaleur. The reciprocal relation between time and frequency is the heart of the uncertainty principle in physics: a signal confined in one domain spreads in the other. The self-dual point, where the two spreads match, is a natural reference for comparing a function with its transform.

In Recognition Science, the framework's machine-checked library of formal theorems takes this classical setup as a model. The declaration selfDualWeylMeshScale defines, for a positive integer N, the number 1/√N as the mesh scale, the spacing of a discrete coordinate grid. The framework proves three things about it: the scale is positive, it satisfies the balance equation N·s² = 1, and it is the unique positive solution. The uniqueness theorem is exact within the model: any positive spacing that balances the finite Fourier pair must equal 1/√N.

What the declaration does not claim is physical length. The framework explicitly states that its use as a physical length is not established: the Fourier exchange-invariance premise is not proved, and no dimensioned event-length unit is supplied. The module constructs a canonical relative continuum chart, a way to compare scales across different N, not an atomic radius or a measured constant. The proof is axiom-clean, with no empirical constants, but the step from a mathematical self-duality to a physical space remains open.

The consequence is a clean separation. The mathematics gives a unique, provable answer to a well-posed question: what spacing makes a finite Fourier pair self-dual? The physics, if any, must come from additional premises that the framework does not yet provide. This is a model of disciplined derivation: the theorem is solid, and the limits are named.

/-- The coefficient-free relative chart selected by finite Fourier balance. -/
def selfDualWeylMeshScale (N : ℕ) : ℝ :=
  1 / Real.sqrt N

theorem selfDualWeylMeshScale_pos
    (N : ℕ) (hN : 0 < N) :
    0 < selfDualWeylMeshScale N := by
  unfold selfDualWeylMeshScale
  positivity

theorem selfDualWeylMeshScale_balance
    (N : ℕ) (hN : 0 < N) :
    SelfDualWeylMeshBalance N
      (selfDualWeylMeshScale N) := by
  constructor
  · exact selfDualWeylMeshScale_pos N hN
  · unfold selfDualWeylMeshScale
    have hN0 : (N : ℝ) ≠ 0 := by positivity
    have hsqrt0 : Real.sqrt (N : ℝ) ≠ 0 := by positivity
    rw [one_div, inv_pow]
    field_simp
    exact (Real.sq_sqrt
      (show (0 : ℝ) ≤ (N : ℝ) by
        exact_mod_cast Nat.zero_le N)).symm

/-- Finite-Fourier self-duality uniquely fixes the relative mesh exponent and
normalization. -/
theorem selfDualWeylMeshBalance_unique
    (N : ℕ) (hN : 0 < N)
    (meshScale : ℝ)
    (hbalance : SelfDualWeylMeshBalance N meshScale) :
    meshScale = selfDualWeylMeshScale N := by
  rcases hbalance with ⟨hmesh, hsq⟩
  have hcanonical :=
    selfDualWeylMeshScale_balance N hN
  rcases hcanonical with ⟨hcanonical_pos, hcanonical_sq⟩
  have hN0 : (N : ℝ) ≠ 0 := by positivity
  have hsquares :
      meshScale ^ 2 =
        selfDualWeylMeshScale N ^ 2 := by
    apply (mul_left_cancel₀ hN0)
    rw [hsq, hcanonical_sq]
  nlinarith

/-- Scope certificate for the relative chart. -/
structure WeylSelfDualContinuumScaleCert (N : ℕ) (hN : 0 < N) : Prop where
  scale_positive :
    0 < selfDualWeylMeshScale N
  scale_balanced :
    SelfDualWeylMeshBalance N (selfDualWeylMeshScale N)
  scale_unique :
    ∀ meshScale,
      SelfDualWeylMeshBalance N meshScale →
        meshScale = selfDualWeylMeshScale N
  -- Absent by design (were typed `True`, so disciplined nothing):
  -- status_fourier_exchange_from_recognition_open (OPEN ledger status);
  -- status_relative_not_dimensioned (unit interpretation of the relative
  --   chart, not a kernel equation).

What this page does not claim

The declaration does not prove that physical space is discrete or that the mesh scale is a measured constant. The declaration does not derive the fine-structure constant or any other dimensioned physical quantity. The declaration does not claim that the Fourier exchange-invariance premise holds in reality; it only constructs a model.

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/PairKernelWeylSelfDualContinuumScale.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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