cost_finiteness_of_EBBA
Under the Euler boundary bridge assumption, every defect sensor has finite annular cost. Researchers deriving the Riemann hypothesis from Recognition Science would cite this as the closing step of the direct T1 realizability argument. The proof is a one-line term application of the cost-finiteness equivalence to the unified RH theorem under the bridge hypothesis.
claimIf the remaining ontology bridge hypothesis holds, then for every defect sensor the annular cost remains bounded.
background
The Unified RH module replaces earlier direct bounded-cost assertions with a three-component architecture. CostDivergent for a defect sensor means its annular cost exceeds any finite bound under uniform-charge sampling, growing as Theta(m squared log N) for nonzero charge m. EulerBoundaryBridgeAssumption is the remaining hypothesis that collapse of the realized defect family transports to boundary approach for the Euler realizability proxy.
proof idea
The proof is a one-line term wrapper: the right-to-left direction of the cost-finiteness equivalence applied to the unified RH theorem instantiated on the supplied bridge hypothesis.
why it matters in Recognition Science
This theorem completes the direct T1 defect route in the ontological argument for the Riemann hypothesis. The module architecture shows that T1 forbids boundary approach for realizable ledgers, so the bridge forces finite cost and excludes nonzero-charge sensors. It touches the open question of discharging the external bridge hypothesis itself.
scope and limits
- Does not prove the Euler boundary bridge assumption.
- Does not supply explicit numerical cost bounds.
- Does not apply to carriers other than the Euler trace.
- Does not address divergence without the bridge hypothesis.
formal statement (Lean)
826theorem cost_finiteness_of_EBBA
827 (bridge : EulerBoundaryBridgeAssumption) :
828 ∀ sensor : DefectSensor, ¬CostDivergent sensor :=
proof body
Term-mode proof.
829 cost_finiteness_iff_rh.mpr (unified_rh bridge)
830
831/-! ### §11b. Direct T1 Defect Route — One Scalar
832
833The strongest version of the ontological argument uses a single scalar
834and no proxy at all. The cost scalar `1/(1 + annularCost)` is both:
835
836* **The physical ledger entry** (in RS, the ledger records `1/(1 + cost)`)
837* **The T1-testable observable** (its defect IS the cost)
838
839For charge `0`: cost is bounded, cost scalar stays away from `0`, T1 defect
840is bounded. The sensor has a physically realizable ledger.
841
842For charge `≠ 0`: cost diverges, cost scalar → `0`, T1 defect → `∞`.
843The sensor **cannot** have a physically realizable ledger.
844
845T1 says: only sensors with physically realizable ledgers exist.
846Therefore only charge-`0` sensors exist. That is RH. -/
847
848/-- **Direct T1 exclusion**: if the cost scalar's T1 defect were bounded for
849a nonzero-charge sensor, that contradicts the proved divergence. This is the
850one-scalar core of the RS ontological argument. -/