DefectTopologicalFloorCovered
DefectTopologicalFloorCovered encodes the predicate that a budgeted carrier bounds the annular topological floor for a defect sensor's charge at every mesh depth N. Researchers deriving the Riemann Hypothesis conditionally from Recognition Science cost-covering assumptions cite it as the remaining topological requirement. The predicate is introduced by direct definition as the universal inequality between the summed topological floors and the scaled carrier budget.
claimFor a cost-covering package $pkg$ with budgeted carrier $C$ and a defect sensor $s$ with charge $m$, the topological floor is covered when the annular topological floor for $N$ rings and charge $m$ is at most the carrier budget scale of $C$, for every natural number $N$.
background
The CostCoveringBridge module packages the RS cost-covering architecture for the Riemann Hypothesis after the budget interface is realized. CostCoveringPackage is the explicit structure containing a BudgetedCarrier witness for the realized carrier trace and its annular excess budget. DefectSensor is the structure recording the multiplicity $m$ (charge) of a zero of zeta together with its real part in the open strip $(1/2,1)$.
proof idea
This is a definition, not a derived theorem. It directly encodes the covering condition as the universal statement that annularTopologicalFloor N (sensor.charge) is bounded above by carrierBudgetScale (pkg.carrier) for all natural numbers N. No lemmas or tactics are invoked; the predicate is the abbreviation of this inequality.
why it matters in Recognition Science
The predicate supplies the central hypothesis for the main theorem rh_from_cost_covering, which obtains a contradiction from any nonzero-charge sensor under the covering assumption and thereby yields the conditional Riemann Hypothesis. It is required in the CostCoveringCert structure that packages the full result. In the Recognition Science framework it closes the topological step of the RH bridge after the unconditional annular bounds of AnnularCost, forcing multiplicity zero for any zero with real part exceeding 1/2.
scope and limits
- Does not prove existence of any carrier package that satisfies the bound.
- Does not construct an explicit budgeted carrier or compute its scale.
- Does not address zeros with real part exactly 1/2.
- Does not quantify growth of the topological floor beyond the stated inequality.
- Does not connect to the phi-ladder or J-cost from the forcing chain.
formal statement (Lean)
123def DefectTopologicalFloorCovered (pkg : CostCoveringPackage) (sensor : DefectSensor) : Prop :=
proof body
Definition body.
124 ∀ N : ℕ, annularTopologicalFloor N sensor.charge ≤ carrierBudgetScale pkg.carrier
125
126/-! ### §4. The conditional Riemann Hypothesis -/
127
128/-- The uniform charge ring sample exactly saturates the topological floor. -/