φ-Dilation Operator

noncomputable def FUST.PhiBloch.phiDilate (f : ℂ → ℂ) : ℂ → ℂ

φ-dilation operator: U_φf = f(φz)

Equations

• FUST.PhiBloch.phiDilate f z = f (↑FUST.φ * z)

theorem FUST.PhiBloch.phiDilate_comp (f g : ℂ → ℂ) : phiDilate (f ∘ g) = f ∘ phiDilate g

φ-dilation is an algebra homomorphism on function composition

theorem FUST.PhiBloch.phiDilate_iter (f : ℂ → ℂ) (k : ℕ) (z : ℂ) : phiDilate^[k] f z = f (↑φ ^ k * z)

φ-dilation iterated k times

Diff6 Commutation with φ-Dilation

theorem FUST.PhiBloch.Diff6_phiDilate_comm (f : ℂ → ℂ) (z : ℂ) : DζOperator.Diff6 (phiDilate f) z = phiDilate (DζOperator.Diff6 f) z

Diff6 commutes exactly with φ-dilation: Diff6 ∘ U_φ = U_φ ∘ Diff6

φ-Bloch Eigenfunction Characterization

def FUST.PhiBloch.IsPhiBlochEigenfunction (f : ℂ → ℂ) (c : ℂ) : Prop

A function is a φ-Bloch eigenfunction if U_φ[f] = c · f

Equations

• FUST.PhiBloch.IsPhiBlochEigenfunction f c = ∀ (z : ℂ), FUST.PhiBloch.phiDilate f z = c * f z

theorem FUST.PhiBloch.Diff6_preserves_Bloch (f : ℂ → ℂ) (c : ℂ) (hf : IsPhiBlochEigenfunction f c) : IsPhiBlochEigenfunction (DζOperator.Diff6 f) c

Diff6 preserves Bloch eigenfunctions: if U_φ f = c·f, then U_φ(Diff6 f) = c·(Diff6 f)

Unified Spectral Order Classification

The Diff6 commutation theorem holds for ALL functions (no periodicity assumption). Matter order types are classified by the Mellin spectral support of f, providing a unified framework for crystals, quasicrystals, and amorphous materials.

def FUST.PhiBloch.HasCrystallineOrder (f : ℂ → ℂ) : Prop

Crystalline order: f decomposes into finitely many φ-Bloch eigenmodes

Equations

• One or more equations did not get rendered due to their size.

theorem FUST.PhiBloch.Diff6_commutation_unconditional (f : ℂ → ℂ) (z : ℂ) : DζOperator.Diff6 (phiDilate f) z = phiDilate (DζOperator.Diff6 f) z

Diff6 commutation is unconditional: holds for crystal, quasicrystal, AND amorphous

theorem FUST.PhiBloch.Diff6_finset_sum {ι : Type u_1} (s : Finset ι) (cs : ι → ℂ) (fs : ι → ℂ → ℂ) (z : ℂ) : DζOperator.Diff6 (fun (w : ℂ) => ∑ i ∈ s, cs i * fs i w) z = ∑ i ∈ s, cs i * DζOperator.Diff6 (fs i) z

Diff6 distributes over finite sums

theorem FUST.PhiBloch.crystalline_Diff6_preserves (f : ℂ → ℂ) (hf : HasCrystallineOrder f) : HasCrystallineOrder (DζOperator.Diff6 f)

Crystalline order implies Diff6 f inherits the Bloch structure

φ-Lattice and φ-Dilation Connection

noncomputable def FUST.PhiBloch.phiLatticeSample (f : ℂ → ℂ) (z₀ : ℂ) (n : ℤ) : ℂ

Equations

• FUST.PhiBloch.phiLatticeSample f z₀ n = f (↑FUST.φ ^ n * z₀)

theorem FUST.PhiBloch.phiLatticeSample_eq_phiDilate (f : ℂ → ℂ) (z₀ : ℂ) (n : ℕ) : phiLatticeSample f z₀ ↑n = phiDilate^[n] f z₀

φ-lattice sampling is iterated φ-dilation