/- GID: D5/X_Frontier/Hearts
   generality: E
   mirror-B: none(waiver:frozen-heart-statements)
   mirror-E: none(waiver:open-hearts-are-not-experimental-evidence)
   anchors: []
   digest: Freeze O-5 zero localization and the classical O-6 Weil positivity statement. -/

import D5.S3.Weil.WeilIdentity
import Mathlib.Algebra.Order.Archimedean.Real.Basic
import Mathlib.Analysis.Meromorphic.Basic
import Mathlib.Analysis.Real.Sqrt
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Topology.Algebra.InfiniteSum.Defs

namespace D5.X_Frontier.Hearts

open D5.S3.Weil.TestFunctions
open D5.S3.Weil.ZeroSum

/-!
O-5 is the independent-control heart for the canonical golden Euler germ.
Its unresolved proof body is intentional: the declaration
records the open claim without asserting that the claim has been proved.

O-6 is the classical positivity heart associated with Weil's 1952 criterion.
D5-T0018 binds its test class, involution, convolution square,
multiplicity-aware zero sum, and symmetric convergence convention to concrete
definitions. The closed proposition is declared below without a proof: the
statement is in place, while its truth remains open.
-/

/-- The expanding golden eigenvalue. -/
noncomputable def phi : ℝ := (1 + Real.sqrt 5) / 2

/-- The contracting conjugate used by the Beatty/Witt exponent. -/
noncomputable def psi : ℝ := 1 - phi

/-- The canonical Beatty shift `S(v)`. -/
noncomputable def beatty (v : ℕ) : ℤ := ⌊((v : ℝ) + 1) * phi⌋ - 1

/-- The concrete expanding exponent `beta(v) = S(v) - v * psi`. -/
noncomputable def beta (v : ℕ) : ℝ := (beatty v : ℝ) - (v : ℝ) * psi

-- Preserve the frozen O-5 instance path after importing the O-6 analytic stack.
attribute [-instance] instCommCStarAlgebraComplex in
/-- The concrete Euler germ for `Z_qc` and its Witt cascade. -/
noncomputable def eulerGerm (s : ℂ) : ℂ :=
  ∏' p : ℕ, if p.Prime then
    ∑' v : ℕ, Complex.cpow (p : ℂ) (-s * (beta v : ℂ))
  else 1

/-- The structural pole contributed by `zeta (phi^3 * s)`. -/
noncomputable def structuralPole : ℝ := 1 / phi ^ 3

/-- The structural zero and pulled-back critical line. -/
noncomputable def structuralZero : ℝ := 1 / (2 * phi ^ 2)

attribute [-instance] instCommCStarAlgebraComplex Complex.instRCLike in
/--
O-5 (independence): the canonical Euler germ has a meromorphic continuation
whose analytic zeros in the declared open band lie on the structural line.
The `AnalyticAt` guard makes the zero test independent of arbitrary values a
total function may carry at its poles.
-/
theorem o5_independence :
    ∃ Zqc : ℂ → ℂ,
      MeromorphicOn Zqc {s | 0 < s.re} ∧
      (∀ s : ℂ, 1 / phi ^ 2 < s.re → Zqc s = eulerGerm s) ∧
      ∀ s : ℂ,
        1 / (2 * phi ^ 3) < s.re →
        s.re < 1 / phi ^ 2 →
        AnalyticAt ℂ Zqc s →
        Zqc s = 0 →
        s.re = structuralZero := by
  sorry

/--
O-6 (Weil positivity), monumented on 2026-07-16. This is the syntactically
closed, non-vacuous proposition audited in `HeartsDraft`: every value is bound,
and no Weil functional or zero set is a free parameter. It is not proved here.

Classically, Weil's criterion is bidirectional: positivity on this full test
class holds if and only if RH. Consequently, if a zero exists off the `1 / 2`
line, some test function in the class violates positivity. The same framework
is therefore also a language for exhibiting a counterexample; whether RH is
true or false, work on this statement produces truth.

The system owner's prior is recorded without entering the proposition: on
2026-07-16 the owner expected zeros off the `1 / 2` line to exist. The motive
for authorizing this monument was the truth produced by the process, not an
expectation of reaching the summit. Exact authorization:
"虽然我认为1/2离线零点存在,但我们去证明黎曼猜想没问题,因为这个过程会产生大量truth"

Audit and authorization provenance: the three-seat audit accepted the literal
draft definition after faithfulness, non-vacuity, and no-RH-theft checks; the
owner's 2026-07-16 authorization above admits that audited statement into the
protected Hearts module. This definition names the `Prop` to be proved; it
does not assert a proof, theorem, or axiom.
-/
def o6WeilPositivityStatement : Prop :=
  ∀ (Z : ZeroData) (g : WeilTestFunction)
      (hZero : SymmetricConvergent Z (convolutionSquare g)),
    0 ≤ (zeroSum Z (convolutionSquare g) hZero).re

end D5.X_Frontier.Hearts