EBBA_of_rh
The theorem shows that absence of any defect sensor with nonzero charge implies the Euler boundary bridge assumption. Researchers linking the Riemann hypothesis to T1-bounded physical realizability in Recognition Science would cite it to equate the bridge hypothesis with RH. The term-mode proof introduces a sensor, instantiates the Euler physically realizable ledger, and obtains the required boundary transport by absurdum on the charge hypothesis.
claimAssume that every defect sensor has zero charge, i.e., there is no sensor with nonzero multiplicity at a zero of the zeta function. Then the Euler boundary bridge assumption holds: for every defect sensor the collapse boundary transport is satisfied once the physically realizable Euler ledger is instantiated.
background
The Unified RH module implements a T1-bounded realizability architecture with three components: CostDivergent (nonzero charge forces unbounded annular cost), EulerTraceAdmissible (convergent Euler carrier with bounded logarithmic derivative), and PhysicallyRealizableLedger (scalar proxy whose T1 defect remains uniformly bounded). A DefectSensor is the structure recording the multiplicity (charge) and real part of a zero of zeta, so that the inverse zeta has a pole of that order at the sensor location. EulerBoundaryBridgeAssumption is the residual hypothesis that realized-defect collapse transports to boundary approach for the bounded Euler proxy.
proof idea
The proof is a direct term construction. It intros an arbitrary sensor, instantiates the PhysicallyRealizableLedger via the upstream lemma euler_physically_realizable, and then supplies the exact structure whose toLedgerBoundary field maps any collapse hypothesis to a contradiction by applying the supplied charge-vanishing assumption.
why it matters in Recognition Science
The result supplies the right-to-left direction of the equivalence EBBA_iff_rh, establishing that EulerBoundaryBridgeAssumption is logically identical to the Riemann hypothesis under the encoding that forbids nonzero-charge sensors. It thereby confirms the module claim that the bridge hypothesis is RH expressed through the T1-bounded realizability architecture, connecting the number-theoretic statement to the Recognition Science forcing chain and the prohibition on T1 boundary approach for realizable ledgers.
scope and limits
- Does not prove the Riemann hypothesis.
- Does not locate any zeros or constrain their real parts.
- Does not apply when nonzero-charge sensors exist.
- Does not extend beyond the Euler carrier.
formal statement (Lean)
724theorem EBBA_of_rh
725 (hrh : ∀ sensor : DefectSensor, sensor.charge ≠ 0 → False) :
726 EulerBoundaryBridgeAssumption := by
proof body
Term-mode proof.
727 intro sensor
728 letI : PhysicallyRealizableLedger sensor := euler_physically_realizable sensor
729 exact { toLedgerBoundary := fun hm _ => absurd hm (fun h => hrh sensor h) }
730
731/-- `EulerBoundaryBridgeAssumption` is logically equivalent to RH.
732
733This theorem makes machine-checkable the observation documented in §9:
734the bridge hypothesis is not weaker than RH — it IS RH expressed through
735the T1-bounded realizability architecture. -/