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

The self-dual mesh balance

In signal analysis, a finite Fourier pair relates a sequence of N samples to N frequency components. Position and frequency coordinates become reciprocal: a fine mesh in one domain means a coarse mesh in the other. The Recognition Science declaration selfDualWeylMeshBalance_unique establishes that for any positive integer N, there is exactly one positive scale that balances the two domains perfectly. That scale is 1 divided by the square root of N, written mathematically as 1/√N.

The theorem is exact and unconditional within its setting. It states: if a positive mesh coordinate satisfies N times the square of that coordinate equals 1, then the coordinate must equal 1/√N. The proof is machine-checked in the framework's library of formal theorems, with no gaps and no added assumptions. The balance condition itself is simple: the product of the number of phases and the squared scale must equal unity.

What the declaration does not claim is equally important. The theorem says nothing about physical length. The framework does not prove the premise that Fourier exchange-invariance holds in nature, and it supplies no dimensioned unit for event-length. The result constructs a canonical relative continuum chart, a mathematical reference frame, not an atomic radius or any measured physical quantity. No empirical constant enters the statement.

In Recognition Science, this uniqueness result is a structural anchor. The framework models a discrete record of events, and the balance condition selects the natural scale for comparing a finite carrier with its reciprocal. The payoff is a fixed relative coordinate system that any N-phase system can adopt, without fitting parameters. The classical lesson stands independently: in finite Fourier analysis, the self-dual scale is not a choice but a consequence of the balance equation.

/-- 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 weylSelfDualContinuumScale_cert
    (N : ℕ) (hN : 0 < N) :
    WeylSelfDualContinuumScaleCert N hN where
  scale_positive := selfDualWeylMeshScale_pos N hN
  scale_balanced := selfDualWeylMeshScale_balance N hN
  scale_unique :=
    fun meshScale =>
      selfDualWeylMeshBalance_unique N hN meshScale

/-- 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

What this page does not claim

The theorem does not assign any physical length or atomic radius to the self-dual scale. The result does not prove that Fourier exchange-invariance holds in nature. The declaration introduces no empirical constant and makes no measured prediction.

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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