SuzukiSkeleton.lean

Declarations

Source

/-
NguyenRH1/SuzukiSkeleton.lean — [SKELETON] statement-level formalization
of paper draft.tex §3.2 (R-003 exact factorization on Suzuki's operator
line + the shape of the Corollary-1.6 reduction).

All proofs are `sorry` BY DESIGN (task 5.1 first pass). `#print axioms`
shows `sorryAx` until proofs are filled in; no new axioms are introduced.

Archive references: R-003, D-026, D-027, D-041, D-051, D-059.
-/
import Mathlib

open Complex

noncomputable section

namespace NguyenRH1

/-! ## 1. The limit factor F -/

/-- The completed ξ-function from Mathlib's completedRiemannZeta
(Λ(s) = π^(−s/2) Γ(s/2) ζ(s)): ξ(s) = ½ s(s−1) Λ(s). -/
def xiFun (s : ℂ) : ℂ := s * (s - 1) / 2 * completedRiemannZeta s

/-- The R-003 limit factor F(z) = z² ξ(½−iz) / ξ′(½−iz) (draft §3.2). -/
def Ffactor (z : ℂ) : ℂ :=
  z ^ 2 * xiFun (1 / 2 - I * z) / deriv xiFun (1 / 2 - I * z)

/-! ## 2. Suzuki's characteristic function -/

/-- Suzuki's characteristic function (draft §3.2, D-019):
W(a,θ;z) = (z−i) V₊(z) + e^{iθ} (z+i) V₋(z) with V± the Fourier
transforms of the deficiency vectors v±(a,·) on [−a,a]. The deficiency
vectors themselves (Suzuki's construction, ℓ² solutions of his
deficiency equations) are formalization content and enter as
parameters. -/
def suzukiW (a θ : ℝ) (vplus vminus : ℝ → ℂ) (z : ℂ) : ℂ :=
  (z - I) * (∫ x in -a..a, vplus x * cexp (I * z * x)) +
    cexp (I * θ) * (z + I) * (∫ x in -a..a, vminus x * cexp (I * z * x))

/-- The explicit sine skeleton (R-003): the leading term of
R₀ = 2 e^{a+iθ/2} sin(az − θ/2) + o(eᵃ). -/
def sineSkeleton (a θ : ℝ) (z : ℂ) : ℂ :=
  2 * cexp (a + I * θ / 2) * Complex.sin (a * z - θ / 2)

/-- [SKELETON] R-003 exact factorization (draft §3.2, Theorem 3.2):
W = F·(R₀ + R_τ) with R₀ the sine skeleton up to o(eᵃ) (the o(eᵃ)
correction is folded into R_τ here; the precise asymptotic is the
D-026/D-059 content, including the punctured-compact reading of the
limit formula). -/
theorem R003_exact_factorization (a θ : ℝ) (vplus vminus : ℝ → ℂ) :
    ∃ Rτ : ℂ → ℂ, ∀ z : ℂ,
      suzukiW a θ vplus vminus z =
        Ffactor z * (sineSkeleton a θ z + Rτ z) := by
  sorry

/-! ## 3. The Corollary-1.6 reduction — deliberately NOT stated

The D-041 reduction "(1.12) only if spec(D) ⊂ ℝ" [proved] /
"converse" [skeleton L3/L4] cannot be stated non-vacuously without a
faithful formalization of the limit formula (1.12) itself (uniform
convergence on punctured compacts, with the gauge φ and boundary flow
θ(a) existentially quantified). A placeholder Prop would be either
vacuously true (making a `sorry`-backed "theorem" dishonestly strong)
or unfaithful. Per the project's honesty rules this is deferred: it is
the first item to state once the factorization proof (above) is
formalized, since (1.12)'s left-hand side is exactly `suzukiW`. -/

end NguyenRH1

end