LogicComplexAnalyticSubstrateCert
The LogicComplexAnalyticSubstrateCert structure records an equivalence between recovered LogicComplex numbers and standard ℂ together with transport of the Riemann zeta, completed zeta, Euler product limit, and functional equation. Number theorists building Mellin and theta bridges in the Recognition framework cite it to justify using Mathlib analytic tools on logic-derived inputs. The structure is a record type whose concrete instance is assembled from transport maps and two existing limit and reflection lemmas.
claimLet $L$ be the type of complex numbers whose real and imaginary parts lie in the recovered reals. A certificate consists of an equivalence $L ≃ ℂ$, the identities that the logic Riemann zeta equals the standard zeta after transport to ℂ and likewise for the completed zeta, the assertion that the partial Euler products over the first $n$ primes tend to the logic zeta whenever the real part exceeds 1, and the functional equation relating the completed zeta at $s$ and at $1-s$ under the transport maps.
background
The module supplies a compatibility layer that lets recovered complex numbers (LogicComplex, whose components are LogicReal) interact with Mathlib's standard complex analysis. Transport is performed by toComplex, which applies toReal to each component, and the inverse fromComplex. The auxiliary definitions logicRiemannZeta and logicCompletedRiemannZeta are obtained simply by composing the Mathlib functions riemannZeta and completedRiemannZeta with toComplex. The module doc states the Phase 2 decision: retain Mathlib ℂ as the analytic substrate and make every recovered-complex statement explicit via this transport.
proof idea
The declaration is a structure definition. Its concrete instance is built by supplying the equivalence equivComplex, reflexivity for both zeta transport fields, the lemma logicRiemannZeta_eulerProduct_tendsto for the Euler-product limit, and the lemma logicCompletedRiemannZeta_one_sub for the completed functional equation.
why it matters in Recognition Science
The certificate is required by the analytic_substrate field of RecoveredComplexMellinBridge and of LogicThetaZetaBridge. It implements the Phase 2 commitment to perform all zeta operations inside Mathlib ℂ, thereby allowing the Recognition framework to inherit the standard functional equation and Euler product without redeveloping contour integration on the recovered line. It closes the interface between the logic-derived numbers and classical analytic number theory.
scope and limits
- Does not define holomorphy or contour integration on LogicComplex.
- Does not prove any specific zeta values or zero locations.
- Does not address the Riemann hypothesis.
- Does not extend the certificate to other L-functions.
formal statement (Lean)
67structure LogicComplexAnalyticSubstrateCert where
68 carrier_equiv : LogicComplex ≃ ℂ
69 zeta_transport : ∀ s : LogicComplex,
70 logicRiemannZeta s = riemannZeta (toComplex s)
71 completed_zeta_transport : ∀ s : LogicComplex,
72 logicCompletedRiemannZeta s = completedRiemannZeta (toComplex s)
73 euler_product :
74 ∀ s : LogicComplex, 1 < (toComplex s).re →
75 Tendsto (fun n : ℕ ↦
76 NumberTheory.finitePrimeLedgerPartition (toComplex s) (Nat.primesBelow n))
77 atTop (𝓝 (logicRiemannZeta s))
78 completed_functional_equation :
79 ∀ s : LogicComplex,
80 logicCompletedRiemannZeta (fromComplex (1 - toComplex s)) =
81 logicCompletedRiemannZeta s
82
83/-- The analytic substrate compatibility certificate for Phase 2. -/