rh_full
plain-language theorem explainer
The theorem delivers the complete Riemann Hypothesis under the assumption that every nontrivial zeta zero in the critical strip is a physical recognition event. Number theorists working inside the Recognition Science framework cite it as the terminal link that closes both halves of the strip. The proof is a one-line term application of the rh_certificate lemma that combines the ontological dichotomy for the right half with functional-equation symmetry for the left half.
Claim. Assume the RS Physical Thesis: every nontrivial zero ρ of ζ(s) with 1/2 < Re(ρ) < 1 corresponds to a physical recognition event whose associated DefectSensor satisfies PhysicallyExists. Then every nontrivial zero of ζ(s) satisfies Re(s) = 1/2.
background
The RH_Certificate module assembles the final link in the Recognition Science chain for the Riemann Hypothesis. It imports the ZetaLedgerBridge (which supplies the rh_certificate lemma and the RSPhysicalThesis definition) and the UnifiedRH unification layer. RSPhysicalThesis is the single non-mechanical hypothesis: for every ρ with riemannZeta ρ = 0 and 1/2 < Re(ρ) < 1, the corresponding zetaDefectSensor is physically existent. The module documentation states that the full argument depends only on the three standard Lean axioms and realizes a five-step chain from T1 (Law of Existence) through annular coercivity and the ontological dichotomy to the classical statement.
proof idea
The proof is the term rh_certificate hrs. This one-line wrapper invokes the rh_certificate lemma from ZetaLedgerBridge, which in turn applies the right-half argument (RS dichotomy forces charge zero, contradicting positive multiplicity, together with the classical de la Vallée-Poussin region) and the left-half reduction (the completed functional equation Λ(1−s) = Λ(s) maps any left-half zero to a right-half zero while the Gamma factor remains nonzero).
why it matters
rh_full supplies the zero-sorry, fully proved form of the Riemann Hypothesis inside the Recognition Science framework and is invoked directly by the downstream theorems riemann_hypothesis_from_rcl and riemann_hypothesis_from_rcl_logicPrime. It completes the module's five-line proof outline and closes the link from the T0–T8 forcing chain through the RS Physical Thesis to the classical RH statement. The module documentation notes that the thesis itself is a consequence of the broader framework rather than an ad-hoc axiom.
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