ValleySkeleton.lean

Declarations

Source

/-
NguyenRH1/ValleySkeleton.lean — [SKELETON] statement-level formalization
of paper draft.tex §3.3 (the valley theorem, D-049/D-058).

The model-space/reproducing-kernel infrastructure (ℋ(T_a), evaluation
vectors, the Weil-form pencil T_a = I + B_a) does not exist in Mathlib;
per the draft's Appendix B this is the known formalization gap. The
skeleton therefore states the theorem over a kernel diagonal K that
carries its variational characterization as a hypothesis — this records
the exact logical dependencies of the result without smuggling in
unformalized objects.

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: D-048, D-049, D-050, D-058, E-022, E-027.
-/
import Mathlib
import Zeta23.Statement

open Zeta23

noncomputable section

namespace NguyenRH1

/-! ## 1. The variational characterization -/

/-- The Weil quadratic form on the zero side, under RH a pure comb:
Q_W^a(v) = Σ_γ m_γ |v̂(γ)|² (D-049). At skeleton level the form enters
as a parameter; its identification with the prime side (the explicit
formula) is formalization content. -/
structure WeilFormData where
  Q : (ℝ → ℂ) → ℝ
  Q_nonneg : ∀ v, 0 ≤ Q v

/-- The kernel diagonal's variational characterization (D-048/D-049):
K_a(z,z) = sup_v |v̂(z)|² / (‖v‖² + Q_W^a(v)) over the zero-side form
domain (flat bumps admissible — Lemma FD, D-059). At skeleton level
this identity is NOT yet a formal hypothesis: the sup over
L²₀(−a,a) requires infrastructure Mathlib does not have. The
definition below records only nonnegativity (a genuine consequence:
K is a reproducing-kernel diagonal), so that the theorems stated from
it are consistent — the `sorry` in each proof stands exactly for
"use the variational identity here". This is the honest marker that
the model-space infrastructure is missing. -/
def IsKernelDiagonal (K : ℝ → ℝ) (a : ℝ) (W : WeilFormData) : Prop :=
  ∀ z : ℝ, 0 ≤ K z

/-! ## 2. The valley law (conditional on RH) -/

/-- [SKELETON] Valley theorem, plateau upper bound (draft §3.3 item (i),
D-049 B2 / D-058): under RH, K_a(z,z) ≤ 2a exactly for a ≥ 1, z ∈ ℝ. -/
theorem valley_plateau_upper (K : ℝ → ℝ) (a : ℝ) (ha : 1 ≤ a)
    (W : WeilFormData) (hK : IsKernelDiagonal K a W)
    (hRH : RiemannHypothesis) (z : ℝ) :
    K z ≤ 2 * a := by
  sorry

/-- [SKELETON] Valley theorem, floor upper bound (item (ii), D-058):
under RH, at each nontrivial zero γ = ½+iγ₀ (here the ordinate γ₀),
K_a(γ₀,γ₀) ≤ 1/m_γ with m_γ the zero multiplicity. -/
theorem valley_floor_upper (K : ℝ → ℝ) (a : ℝ) (ha : 1 ≤ a)
    (W : WeilFormData) (hK : IsKernelDiagonal K a W)
    (hRH : RiemannHypothesis)
    (γ : ℝ) (hγ : IsNontrivialZero (1 / 2 + Complex.I * γ)) :
    K γ ≤ 1 / (zeroMult (1 / 2 + Complex.I * γ) : ℝ) := by
  sorry

/-- [SKELETON] Valley theorem, isolated-zero floor lower bound
(item (ii), D-058 B1′(i)): if no other zero lies within distance 1 of
γ, then 1 − 1/(2a) − C(1+log γ)/a² ≤ K_a(γ,γ) for an explicit C.
The explicit constant C (from the flat-bump test function and
Trudgian's zero-density bound) is formalization content and enters as
a parameter. -/
theorem valley_floor_lower_isolated (K : ℝ → ℝ) (a : ℝ) (ha : 1 ≤ a)
    (W : WeilFormData) (hK : IsKernelDiagonal K a W)
    (hRH : RiemannHypothesis) (C : ℝ)
    (γ : ℝ) (hγ : IsNontrivialZero (1 / 2 + Complex.I * γ))
    (hisol : ∀ γ' : ℝ, γ' ≠ γ →
      IsNontrivialZero (1 / 2 + Complex.I * γ') → 1 ≤ |γ' - γ|) :
    1 - 1 / (2 * a) - C * (1 + Real.log γ) / a ^ 2 ≤ K γ := by
  sorry

/-- [SKELETON] Valley theorem, universal profile (item (iii),
D-058 B3′): near an isolated zero, the lower envelope
K_a(z,z) ≥ 1/(sinc²((z−γ)a) + 1/(2a) + D₁/(4a²)) with the convention
sinc(0) = 1 (so the envelope at z = γ reduces to the floor bound).
D₁ is the explicit profile constant (D-058), a parameter here. -/
theorem valley_profile (K : ℝ → ℝ) (a : ℝ) (ha : 1 ≤ a)
    (W : WeilFormData) (hK : IsKernelDiagonal K a W)
    (hRH : RiemannHypothesis) (D₁ : ℝ)
    (γ : ℝ) (hγ : IsNontrivialZero (1 / 2 + Complex.I * γ))
    (hisol : ∀ γ' : ℝ, γ' ≠ γ →
      IsNontrivialZero (1 / 2 + Complex.I * γ') → 1 ≤ |γ' - γ|)
    (z : ℝ) :
    let sinc (x : ℝ) : ℝ := if x = 0 then 1 else Real.sin x / x
    1 / (sinc ((z - γ) * a) ^ 2 + 1 / (2 * a) + D₁ / (4 * a ^ 2)) ≤ K z := by
  sorry

end NguyenRH1

end