Foundation Pair Kernel Periodic3 Torus Source Transform Zero Eq Zero Of Neutral

The neutral source

The declaration concerns a finite periodic three-dimensional grid, a torus of N sites in each direction. A source, a complex number assigned to each site, is called neutral when its values sum to zero. The theorem states that a neutral source has a zero Fourier coefficient at the zero frequency: its total contribution to the constant mode vanishes.

In the framework's discrete Fourier analysis, each frequency mode is a complex exponential on the torus. The zero-frequency mode is the constant function, and its Fourier coefficient is exactly the sum of the source values. The theorem, torusSourceTransform_zero_eq_zero_of_neutral, proves this identity by rewriting the zero-frequency transform as the total source and then applying the neutrality hypothesis.

This result is a technical lemma within the framework's construction of a periodic carrier for its recognition dynamics. It ensures that a neutral source does not excite the constant mode, a necessary condition for the spectral decomposition to separate the zero-frequency component from the rest. The theorem is proved in the machine-checked library of formal theorems, with no unverified assumptions beyond the standard axioms of the underlying type theory.

The declaration does not claim that every neutral source is physically realizable, nor that the torus is the correct model for physical space. It does not assert a finite-volume convergence theorem, nor does it identify the periodic carrier with the infinite Fourier readout. The shift-eigenmode equations remain hypotheses, and the bridge from this discrete construction to the physical three-dimensional space is an open target.

/-- Neutrality forces the zero source-transform coefficient to vanish. -/
theorem torusSourceTransform_zero_eq_zero_of_neutral
    {N : ℕ} [NeZero N] (rho : TorusSite3 N → ℂ)
    (hneutral : torusSourceNeutral rho) :
    torusSourceTransform rho (0 : Fin 3 → Fin N) = 0 := by
  rw [torusSourceTransform_zero_eq_totalSource]
  exact hneutral

What this page does not claim

The theorem does not assert that the torus is the physical model of three-dimensional space. It does not prove that every neutral source is physically realizable. It does not establish a finite-volume convergence theorem.

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/PairKernelPeriodic3.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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