Gravity Analysis Bloch Cell Sum Eventually Nonaliased

The eventual non-aliasing theorem

The declaration eventually_nonaliased is a theorem in the framework's machine-checked library of formal theorems. It concerns a sum over a three-dimensional grid of N cells along each axis. The summand is a product of two cosine functions, each with a phase that depends on the cell position and a fixed frequency vector m. The theorem states that for any fixed nonzero frequency vector m, there exists a threshold N₀ such that for every grid size N larger than N₀, the doubled frequency 2m is non-aliased on at least one axis. In plain terms, this means that the grid is fine enough that the frequency does not wrap around and coincide with itself, a condition that makes the sum collapse to a simple closed form.

The practical consequence is a clean identity. When the doubled frequency is non-aliased, the sum over all N³ cells of the product of two phase-shifted cosines equals N³ times the cosine of the phase difference, divided by two. The theorem guarantees that this simplification is not a rare accident but an eventual certainty: for any fixed nonzero frequency, refining the grid always reaches a regime where the identity applies. This is a purely classical discrete Fourier orthogonality result, proved without any physics assumptions or extra axioms, using only standard analysis and algebra.

The theorem does not claim anything about the continuum limit or about the value −1/4 that appears in the downstream physics program. That target remains open. The theorem also does not claim that the non-aliasing condition holds for every grid size; it only guarantees it for all sufficiently large N. Finally, it does not assert anything about what happens when the frequency is zero, since the hypothesis explicitly requires a nonzero component.

/-- For a fixed nonzero frequency vector `m`, the non-aliasing hypothesis of
`cellSum_cos_mul_cos` holds for all sufficiently large `N`: once
`N > 2 * |m i|` on a nonzero axis, `N` cannot divide `2 * m i`. -/
theorem eventually_nonaliased (m : Fin 3 → ℤ) (hm : ∃ i : Fin 3, m i ≠ 0) :
    ∀ᶠ N : ℕ in Filter.atTop, ∃ i : Fin 3, ¬ (N : ℤ) ∣ 2 * m i := by
  obtain ⟨i, hi⟩ := hm
  rw [Filter.eventually_atTop]
  refine ⟨2 * (m i).natAbs + 1, fun N hN => ⟨i, fun hdvd => ?_⟩⟩
  have hne : 2 * m i ≠ 0 := mul_ne_zero two_ne_zero hi
  have hle : (N : ℤ) ≤ |2 * m i| :=
    Int.le_of_dvd (abs_pos.mpr hne) ((dvd_abs _ _).mpr hdvd)
  rw [Int.abs_eq_natAbs] at hle
  omega

/-- For a fixed nonzero frequency vector `m`, the non-aliasing hypothesis of
`cellSum_cos_mul_cos` holds for all sufficiently large `N`: once
`N > 2 * |m i|` on a nonzero axis, `N` cannot divide `2 * m i`. -/
theorem eventually_nonaliased (m : Fin 3 → ℤ) (hm : ∃ i : Fin 3, m i ≠ 0) :
    ∀ᶠ N : ℕ in Filter.atTop, ∃ i : Fin 3, ¬ (N : ℤ) ∣ 2 * m i := by
  obtain ⟨i, hi⟩ := hm
  rw [Filter.eventually_atTop]
  refine ⟨2 * (m i).natAbs + 1, fun N hN => ⟨i, fun hdvd => ?_⟩⟩
  have hne : 2 * m i ≠ 0 := mul_ne_zero two_ne_zero hi
  have hle : (N : ℤ) ≤ |2 * m i| :=
    Int.le_of_dvd (abs_pos.mpr hne) ((dvd_abs _ _).mpr hdvd)
  rw [Int.abs_eq_natAbs] at hle
  omega

/-- For a fixed nonzero frequency vector `m`, the non-aliasing hypothesis of
`cellSum_cos_mul_cos` holds for all sufficiently large `N`: once
`N > 2 * |m i|` on a nonzero axis, `N` cannot divide `2 * m i`. -/
theorem eventually_nonaliased (m : Fin 3 → ℤ) (hm : ∃ i : Fin 3, m i ≠ 0) :
    ∀ᶠ N : ℕ in Filter.atTop, ∃ i : Fin 3, ¬ (N : ℤ) ∣ 2 * m i := by
  obtain ⟨i, hi⟩ := hm
  rw [Filter.eventually_atTop]
  refine ⟨2 * (m i).natAbs + 1, fun N hN => ⟨i, fun hdvd => ?_⟩⟩
  have hne : 2 * m i ≠ 0 := mul_ne_zero two_ne_zero hi
  have hle : (N : ℤ) ≤ |2 * m i| :=
    Int.le_of_dvd (abs_pos.mpr hne) ((dvd_abs _ _).mpr hdvd)
  rw [Int.abs_eq_natAbs] at hle
  omega

What this page does not claim

The theorem does not establish the continuum target value −1/4, which remains open. The theorem does not claim the non-aliasing condition holds for every grid size, only for all sufficiently large ones. The theorem does not address the case of a zero frequency vector.

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

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Derived articles

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