DeprecatedDefectAnnularCostBounded
DeprecatedDefectAnnularCostBounded packages the proposition that annular costs stay bounded for the canonical defect-sampled family of any nonzero-charge sensor. Legacy analytic proofs of the Riemann hypothesis via the cost-covering bridge cited it as the key boundedness assumption. The definition is a direct existential statement over a uniform real bound K for the sequence of annular costs indexed by mesh refinement N.
claimFor a defect sensor $s$ with nonzero charge, the proposition asserts that there exists $K > 0$ such that for every natural number $N$, the annular cost of the $N$-th mesh of the canonical defect-sampled family generated by $s$ is at most $K$.
background
In the Euler product instantiation module the abstract RS carrier/sensor framework is realized via the Euler product of zeta. DefectSensor carries a nonzero charge; canonicalDefectSampledFamily produces the sequence of meshes whose annular costs are measured by the annularCost function imported from AnnularCost. Upstream cost primitives appear in MultiplicativeRecognizerL4.cost (derivedCost on positive ratios) and ObserverForcing.cost (J-cost of a recognition event), which supply the non-negative cost measure underlying the annular construction. The module doc states the instantiation chain: primes to Hilbert-Schmidt norm to det2 convergence to holomorphic nonvanishing C(s) on Re(s) > 1/2, after which the cost-covering axiom becomes applicable.
proof idea
This is a direct definition whose body is the existential statement itself. No lemmas are applied and no tactics are used; the declaration simply names the boundedness Prop for consumption by downstream inconsistency theorems.
why it matters in Recognition Science
The definition supplies the (now deprecated) bounded annular cost hypothesis for the legacy analytic route to RH. It is invoked directly by analytic_rh, rh_chain_summary_legacy, rh_from_complex_analysis_axioms and rh_no_zeros_in_strip, each of which derives False via defect_annular_cost_bounded_inconsistent. In the Recognition Science framework it corresponds to the cost-covering step that follows the EulerCarrier interface (T5 J-uniqueness through T8 D=3) but is retired in favor of the unified ontology route. It touches the question of which cost axioms remain realizable without contradiction once the Euler product is substituted for the abstract carrier.
scope and limits
- Does not assert that any concrete sensor satisfies the boundedness condition.
- Does not derive the existence of the bound K from other axioms.
- Does not apply to sensors with zero charge.
- Does not replace the preferred unified_rh theorem.
formal statement (Lean)
815def DeprecatedDefectAnnularCostBounded (sensor : DefectSensor)
816 (hm : sensor.charge ≠ 0) : Prop :=
proof body
Definition body.
817 ∃ K : ℝ, ∀ N : ℕ,
818 annularCost ((canonicalDefectSampledFamily sensor hm).mesh N) ≤ K
819
820/-- **Formal inconsistency proof.**
821
822The axiom `defect_annular_cost_bounded` is logically inconsistent with
823`not_realizedDefectAnnularCostBounded`, which proves that bounded cost is
824impossible for any sampled family with nonzero charge. This theorem
825makes the inconsistency machine-checkable so it cannot be overlooked. -/