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

The self-dual mesh scale

In signal analysis, a finite Fourier pair links N samples in one domain to N samples in another. When the two domains are position and frequency, their coordinates become reciprocal: a finer mesh in one means a coarser mesh in the other. The Recognition Science declaration selfDualWeylMeshScale_balance establishes the single positive scale at which this reciprocity is exact. For any positive integer N, the balance condition is N times the square of the mesh scale equals 1, written N * meshScale^2 = 1. Solving gives meshScale = 1 / sqrt N, the unique positive solution.

The declaration proves two things as a theorem in the machine-checked library of formal theorems. First, the proposed scale 1 / sqrt N is positive and satisfies the balance equation. Second, it is the only positive real number that does: any meshScale meeting the balance condition must equal 1 / sqrt N. This uniqueness is exact, not approximate. The proof uses no sorry (an admission of an unfinished proof) and no new axiom; it relies only on the standard logical postulates of the ambient type theory.

What the declaration does not claim is physical length. The Recognition surface does not prove the premise that position and frequency coordinates actually exchange in the physical world, and it supplies no dimensioned unit for event length. The declaration therefore constructs a canonical relative continuum chart, a dimensionless coordinate system for comparing scales, not an atomic radius or any measured physical quantity. It is a statement about a mathematical balance condition, not about the size of a particle.

In the framework's account, this balance is a building block: it fixes the relative mesh exponent and normalization for a finite Fourier carrier. The practical consequence is that when the framework models a finite N-phase system, the self-dual scale is not a free parameter. It is forced by the reciprocity condition alone. The framework proves this forcing; it does not prove that the condition applies to any particular physical system.

/-- Positive coordinate scale balancing a finite Fourier carrier with its
reciprocal carrier. -/
def SelfDualWeylMeshBalance (N : ℕ) (meshScale : ℝ) : Prop :=
  0 < meshScale ∧ (N : ℝ) * meshScale ^ 2 = 1

/-- 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

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

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

What this page does not claim

The declaration does not prove that position and frequency coordinates actually exchange in any physical system. The declaration does not provide a dimensioned unit for event length or any measured physical quantity. The declaration does not establish that the self-dual scale applies to particles or atoms.

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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