FUST Schrödinger Equation: Fζ f = λf

The eigenvalue equation Fζ f = λf corresponds to the Schrödinger equation Ĥψ = Eψ:

• Fζ acts as the Hamiltonian operator
• ker(Fζ) = ground state (zero eigenvalue, photon/light-cone state)
• Active modes n ≡ 1,5 (mod 6): Fζ(zⁿ) = λ(n)·zⁿ with λ(n) ≠ 0
• Kernel modes n ≡ 0,2,3,4 (mod 6): Fζ(zⁿ) = 0
• Variational principle: δ|Fζf|² = 0 ⟹ Fζ f = 0 (Euler-Lagrange)

Eigenvalue λ(n) decomposes into space and time channels:

• Re(λ) = 6·c_S(n): temporal channel (Φ_S)
• Im(λ) = 10√15·c_A(n): spatial channel (Φ_A, via AF_coeff = 2i√3)

Eigenvalue Equation

Fζ acts on monomials zⁿ as an eigenvalue operator: Fζ(zⁿ) = λ(n)·zⁿ. Active modes: n ≡ 1,5 (mod 6). Kernel modes: n ≡ 0,2,3,4 (mod 6).

theorem FUST.SchrodingerEquation.eigenvalue_mod1 (k : ℕ) (z : ℂ) : ∃ (ev : ℂ), FζOperator.Fζ (fun (w : ℂ) => w ^ (6 * k + 1)) z = ev * z ^ (6 * k + 1)

Active mode n ≡ 1 (mod 6): eigenvalue exists

theorem FUST.SchrodingerEquation.eigenvalue_mod5 (k : ℕ) (z : ℂ) : ∃ (ev : ℂ), FζOperator.Fζ (fun (w : ℂ) => w ^ (6 * k + 5)) z = ev * z ^ (6 * k + 5)

Active mode n ≡ 5 (mod 6): eigenvalue exists

theorem FUST.SchrodingerEquation.ground_state_mod0 (k : ℕ) (z : ℂ) : FζOperator.Fζ (fun (w : ℂ) => w ^ (6 * k)) z = 0

Kernel mode n ≡ 0 (mod 6): zero eigenvalue

theorem FUST.SchrodingerEquation.ground_state_mod2 (k : ℕ) (z : ℂ) : FζOperator.Fζ (fun (w : ℂ) => w ^ (6 * k + 2)) z = 0

Kernel mode n ≡ 2 (mod 6): zero eigenvalue

theorem FUST.SchrodingerEquation.ground_state_mod3 (k : ℕ) (z : ℂ) : FζOperator.Fζ (fun (w : ℂ) => w ^ (6 * k + 3)) z = 0

Kernel mode n ≡ 3 (mod 6): zero eigenvalue

theorem FUST.SchrodingerEquation.ground_state_mod4 (k : ℕ) (z : ℂ) : FζOperator.Fζ (fun (w : ℂ) => w ^ (6 * k + 4)) z = 0

Kernel mode n ≡ 4 (mod 6): zero eigenvalue

Superposition Principle

Fζ is ℂ-linear: Fζ(af + bg) = aFζf + bFζg. This is the quantum superposition principle.

theorem FUST.SchrodingerEquation.superposition (f g : ℂ → ℂ) (a b z : ℂ) : FζOperator.Fζ (fun (t : ℂ) => a * f t + b * g t) z = a * FζOperator.Fζ f z + b * FζOperator.Fζ g z

Superposition: Fζ is linear over ℂ

theorem FUST.SchrodingerEquation.scalar_eigenvalue (c : ℂ) (f : ℂ → ℂ) (z : ℂ) : FζOperator.Fζ (fun (t : ℂ) => c * f t) z = c * FζOperator.Fζ f z

Scalar multiplication: Fζ(c·f) = c·Fζ(f)

Critical Point: Variational Principle

The Euler-Lagrange equation for the action |Fζf|²: δL = 0 for all test variations η ⟹ Fζ f = 0.

Fζ(id)(z) = eigenCoeff·z with eigenCoeff.re > 0 provides a nontrivial test variation.

theorem FUST.SchrodingerEquation.Fζ_detects_active (z : ℂ) (hz : z ≠ 0) : FζOperator.Fζ (fun (t : ℂ) => t) z ≠ 0

Fζ detects active states: Fζ(id)(z) ≠ 0 for z ≠ 0

theorem FUST.SchrodingerEquation.critical_point_condition (f : ℂ → ℂ) (z : ℂ) (hz : z ≠ 0) : (∀ (η : ℂ → ℂ), FζOperator.Fζ f z * FζOperator.Fζ η z = 0) → FζOperator.Fζ f z = 0

δL = 0 for all η ⟹ Fζ f = 0 (Euler-Lagrange equation)

Spacetime Channel Decomposition

Eigenvalue λ = 5·c_A·AF_coeff + 6·c_S decomposes into:

• Re(λ) = 6·c_S (temporal/symmetric channel Φ_S)
• Im(λ) = 10√15·c_A (spatial/antisymmetric channel Φ_A)

theorem FUST.SchrodingerEquation.eigenvalue_re_is_time (c_A c_S : ℝ) : (5 * ↑(√5 * c_A) * DζOperator.AF_coeff + 6 * ↑c_S).re = 6 * c_S

Time channel: Re(eigenvalue) = 6·c_S

theorem FUST.SchrodingerEquation.eigenvalue_im_is_space (c_A c_S : ℝ) : (5 * ↑(√5 * c_A) * DζOperator.AF_coeff + 6 * ↑c_S).im = 10 * √15 * c_A

Space channel: Im(eigenvalue) = 10√15·c_A

theorem FUST.SchrodingerEquation.tau_trace (c_A c_S : ℂ) : 5 * c_A * DζOperator.AF_coeff + 6 * c_S + (-5 * c_A * DζOperator.AF_coeff + 6 * c_S) = 12 * c_S

τ-conjugation trace: λ₁ + λ₅ = 12·c_S

theorem FUST.SchrodingerEquation.tau_product (c_A c_S : ℂ) : (5 * c_A * DζOperator.AF_coeff + 6 * c_S) * (-5 * c_A * DζOperator.AF_coeff + 6 * c_S) = 36 * c_S ^ 2 + 300 * c_A ^ 2

τ-conjugation product: λ₁·λ₅ = 36·c_S² + 300·c_A²