zeroInducedBridge_iff_rsPhysicalThesis
The equivalence shows that the zero-induced proxy physicalization bridge holds exactly when the RS physical thesis is true for zeta zeros. Researchers reducing the Riemann hypothesis inside Recognition Science cite this to link the directed Euler ledger to the physical existence predicate. The proof is a direct biconditional split that applies the proxy-to-physical-existence equivalence in both directions at each zeta-zero sensor.
claimThe statement that for every strip zero ρ of ζ with ½ < Re(ρ) < 1 the proxy physicalization bridge holds at the associated zeta defect sensor of charge 1 is equivalent to the Recognition Science physical thesis that zeta zeros are physical recognition events.
background
The Proxy Physicalization Bridge module works after the directed Euler ledger has produced admissible traces and T1-bounded realizability proxies for every DefectSensor. The remaining transport is from PhysicallyRealizableLedger to the PhysicallyExists predicate defined via eulerLedgerScalarState. ZeroInducedProxyPhysicalizationBridge quantifies over all strip zeros ρ of ζ and requires that ProxyPhysicalizationBridge holds for the zetaDefectSensor at Re(ρ) with charge 1. RSPhysicalThesis is the claim that these zeros correspond to physical recognition events. Upstream ledger factorization and phi-forcing structures supply the J-cost and defect-distance calibration used in the sensors.
proof idea
The term proof opens with constructor to split the biconditional. The forward direction assumes the zero-induced bridge, then applies proxyPhysicalizationBridge_iff_physicallyExists to obtain the RS physical thesis. The reverse direction assumes the RS thesis and applies the same equivalence in the opposite direction.
why it matters in Recognition Science
This result feeds directly into zeroInducedBridge_iff_rh, which equates the zero-induced bridge to the Riemann hypothesis. It closes the explicit gap noted in the module documentation between the realizability proxy and physical existence. In the Recognition Science framework it connects the T1-bounded ledger construction to the claim that zeta zeros are physical, advancing the reduction from the forcing chain through the recognition composition law to RH.
scope and limits
- Does not prove that the proxy physicalization bridge holds for any sensor.
- Does not address zeros of ζ outside the critical strip.
- Does not treat sensors with charge other than 1.
- Does not resolve the Riemann hypothesis.
Lean usage
example (h : ZeroInducedProxyPhysicalizationBridge) : RSPhysicalThesis := (zeroInducedBridge_iff_rsPhysicalThesis).mp hformal statement (Lean)
126theorem zeroInducedBridge_iff_rsPhysicalThesis :
127 ZeroInducedProxyPhysicalizationBridge ↔ RSPhysicalThesis := by
proof body
Term-mode proof.
128 constructor
129 · intro hzipb ρ hzero hlo hhi
130 exact (proxyPhysicalizationBridge_iff_physicallyExists _).mp (hzipb ρ hzero hlo hhi)
131 · intro hrs ρ hzero hlo hhi
132 exact (proxyPhysicalizationBridge_iff_physicallyExists _).mpr (hrs ρ hzero hlo hhi)
133
134/-- `ZeroInducedProxyPhysicalizationBridge` is equivalent to the absence of
135strip zeros of ζ: for charge-1 sensors, the bridge evaluates to `False`,
136so quantifying over strip zeros asserts their non-existence. -/