RecognitionThetaModularIdentity

formal source

 274
 275    We package this as a Prop on the existence of a continuous
 276    prefactor `ρ : ℝ → ℝ` realizing the identity. -/
 277structure RecognitionThetaModularIdentity : Prop where
 278  prefactor : ∃ ρ : ℝ → ℝ, Continuous ρ ∧
 279    ∀ t : ℝ, 0 < t →
 280      recognitionTheta (1 / t) = ρ t * recognitionTheta t
 281
 282/-- Sub-conjecture A.3: the Mellin transform of `Θ̃_RS` factors as
 283    `ζ(s) · G_RS(s)` where `G_RS` is a meromorphic function inheriting
 284    the reflection symmetry from the modular identity.
 285
 286    We package the existence of the factorization as a Prop. -/
 287structure RecognitionThetaMellinFactor : Prop where
 288  factorization : ∃ G : ℂ → ℂ,
 289    -- (Stand-in for: G is meromorphic and has reflection symmetry.)
 290    G ≠ 0 ∧
 291    -- The actual Mellin identity requires defining the Mellin
 292    -- transform of recognitionTheta (which depends on convergence)
 293    -- and stating its factorization.  We leave the Prop open at
 294    -- this abstract level.
 295    True
 296
 297/-! ## Master certificate -/
 298
 299/-- The structural facts about the Recognition Theta function
 300    established in this module. -/
 301theorem recognition_theta_certificate :
 302    -- (1) phi-rung is completely additive on positive integers.
 303    (∀ {m n : ℕ}, m ≠ 0 → n ≠ 0 → phiRung (m * n) = phiRung m + phiRung n) ∧
 304    -- (2) phi-rung at 1 is zero.
 305    phiRung 1 = 0 ∧
 306    -- (3) chi8 is bounded by 1.
 307    (∀ n : ℕ, |chi8 n| ≤ 1) ∧