RH_Statement
plain-language theorem explainer
The declaration defines the standard mathematical statement of the Riemann hypothesis for the completed zeta function. Number theorists exploring conditional routes under geometric or physical hypotheses would cite this packaging. It is introduced as a plain definition with no proof obligations, serving as the target payload for a downstream conditional theorem.
Claim. The Riemann hypothesis asserts that every non-trivial zero $ρ$ of the completed Riemann zeta function satisfies $Re(ρ) = 1/2$. Formally: for all $ρ ∈ ℂ$, if the completed zeta vanishes at $ρ$ and $Im(ρ) ≠ 0$, then $Re(ρ) = 1/2$.
background
In Recognition Science the completed Riemann zeta function is the standard analytic continuation imported from Mathlib, with non-vanishing properties recorded in sibling lemmas such as completedRiemannZeta_ne_zero_of_re_gt_one. The local module ports number-theoretic statements into the Recognition framework by importing structures for J-cost, ledger factorization, and spectral emergence from upstream modules. The doc-comment explicitly flags the statement as conditional on ClassicalAnalysis, RecognitionGeometry, and OscillationBounds, placing it outside the core T0-T8 forcing chain.
proof idea
This declaration is a direct definition that encodes the universal quantification over complex zeros of the completed zeta function. No lemmas or tactics are applied; the body is simply the predicate itself.
why it matters
The definition supplies the payload for the downstream theorem RH_conditional, which applies an explicit RHConditionalPackage of assumptions. It addresses the Millennium Prize problem conditionally rather than as part of the T0-T8 chain or the Recognition Composition Law. The package structure replaces earlier unconditional axiom approaches, allowing callers to supply the conditional proof payload.
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