all_physicallyExist_of_rh
The theorem shows that the Riemann hypothesis, formulated as the absence of any nonzero-charge defect sensor, implies every defect sensor has bounded T1 defect and therefore physically exists. Researchers unifying number theory with Recognition Science physical realizability would cite this as the converse half of the RH-existence equivalence. The term proof introduces the sensor and applies the forward direction of the ontological dichotomy after a contradiction step that invokes the hypothesis.
claimIf every defect sensor has charge zero, then for every defect sensor there exists a real bound $K$ such that the T1 defect of its Euler ledger scalar state is at most $K$ for all natural numbers $N$.
background
The Unified RH module replaces an earlier total-cost ledger with a three-part architecture: CostDivergent (nonzero charge forces unbounded annular cost), EulerTraceAdmissible (convergent carrier with bounded log derivative), and PhysicallyRealizableLedger (T1 defect uniformly bounded). A DefectSensor is a structure carrying an integer charge (multiplicity of a zeta zero) and real part locating the zero in the critical strip. PhysicallyExists sensor holds precisely when the Euler ledger scalar state at that sensor satisfies the bounded T1 defect condition: there exists $K$ such that defect(eulerLedgerScalarState sensor N) ≤ K for all N.
proof idea
The term proof introduces an arbitrary sensor, then feeds the hypothesis into a contradiction subproof that assumes the sensor fails to physically exist. The subproof applies the hypothesis to derive a nonzero charge, contradicting the global assumption that all charges are zero; the resulting falsehood is discharged by the forward direction of the ontological dichotomy lemma.
why it matters in Recognition Science
This theorem supplies the left-to-right direction of the equivalence rh_iff_all_physical, which equates the number-theoretic statement (no nonzero-charge sensors) with the Recognition Science claim that every sensor is physically realizable. It closes the T1-bounded realizability chain inside the Unified RH module and connects directly to the Euler carrier admissibility results. The equivalence raises the open question whether the Riemann hypothesis is literally identical to a physical existence principle under the RS forcing chain.
scope and limits
- Does not prove the Riemann hypothesis itself.
- Does not treat the cost-divergent case for nonzero-charge sensors.
- Does not invoke the phi-ladder or the alpha band.
- Does not extend the T1 bound to annular cost or boundary transport.
Lean usage
theorem rh_implies_physical (hrh : ∀ s : DefectSensor, s.charge ≠ 0 → False) : ∀ s : DefectSensor, PhysicallyExists s := all_physicallyExist_of_rh hrhformal statement (Lean)
961theorem all_physicallyExist_of_rh
962 (hrh : ∀ sensor : DefectSensor, sensor.charge ≠ 0 → False) :
963 ∀ sensor : DefectSensor, PhysicallyExists sensor := by
proof body
Term-mode proof.
964 intro sensor
965 exact (ontological_dichotomy sensor).mp (by
966 by_contra hm
967 exact hrh sensor hm)
968
969/-- **RH ↔ all sensors physically exist.** Machine-verified equivalence
970between the number-theoretic statement and the RS physical principle. -/