euler_rh_conditional
An explicit cost-covering package for the Euler product carrier, together with topological floor coverage for every defect sensor, produces a contradiction if any sensor has nonzero charge. Researchers deriving conditional forms of the Riemann hypothesis from recognition science cost structures would cite this as the concrete instantiation step. The argument is a direct one-line application of the cost-covering bridge lemma.
claimLet $P$ be a budgeted carrier package for the Euler product. Suppose that for every defect sensor $s$ the topological floor of $s$ is covered by $P$. Then for any defect sensor $s$ whose charge (zero multiplicity) is nonzero, a contradiction follows.
background
The module instantiates the abstract RS carrier/sensor framework with the Euler product of the zeta function. Core objects are the prime sum $∑_p p^{-σ}$, the Hilbert-Schmidt norm squared $∑_p p^{-2σ}$, the carrier log series, and the logarithmic derivative bound. The architecture stacks the annular cost framework, the cost-covering bridge, and this concrete Euler layer, yielding a holomorphic nonvanishing carrier $C(s)$ on Re$(s)>1/2$ that satisfies the budgeted carrier interface. Upstream results supply the CostCoveringPackage structure (an explicit BudgetedCarrier witness) and the DefectSensor structure (charge, real part, and location in the critical strip).
proof idea
The proof is a one-line wrapper that applies the rh_from_cost_covering lemma from CostCoveringBridge, supplying the package, the given sensor, the nonzero-charge hypothesis, and the coverage predicate instantiated at that sensor.
why it matters in Recognition Science
This theorem completes the concrete-to-abstract link in the Euler instantiation chain and is invoked directly by the full riemann_hypothesis_euler_conditional result. It realizes the cost-covering axiom for the Euler carrier, advancing the Recognition Science forcing chain (T0-T8) toward a conditional Riemann hypothesis. The parent theorem sharpens the claim by quantifying over all sensors with real part strictly between 1/2 and 1.
scope and limits
- Does not prove the Riemann hypothesis unconditionally.
- Does not address zeros with real part exactly 1/2.
- Does not supply numerical bounds on the carrier derivative.
- Does not treat the left half of the critical strip.
Lean usage
theorem riemann_hypothesis_euler_conditional (pkg : CostCoveringPackage) (hcover : ∀ sensor : DefectSensor, DefectTopologicalFloorCovered pkg sensor) : ∀ (σ₀ : ℝ) (hσ : 1/2 < σ₀) (hσ1 : σ₀ < 1) (m : ℤ) (hm : m ≠ 0), let sensor : DefectSensor := ⟨m, σ₀, ⟨hσ, hσ1⟩⟩; False := fun σ₀ hσ hσ1 m hm => euler_rh_conditional pkg hcover ⟨m, σ₀, ⟨hσ, hσ1⟩⟩ hmformal statement (Lean)
936theorem euler_rh_conditional (pkg : CostCoveringPackage)
937 (hcover : ∀ sensor : DefectSensor, DefectTopologicalFloorCovered pkg sensor)
938 (sensor : DefectSensor) (hm : sensor.charge ≠ 0) : False :=
proof body
Term-mode proof.
939 rh_from_cost_covering pkg sensor hm (hcover sensor)
940
941/-- The complete RH conditional on RS, given an explicit cost-covering package
942 and topological-floor coverage. -/