RH_conditional
plain-language theorem explainer
The theorem extracts the Riemann Hypothesis statement from an explicit package bundling classical assumptions, recognition geometry assumptions, and oscillation bounds. A researcher working on conditional proofs of the Riemann Hypothesis under Recognition Geometry would cite this to isolate the payload without embedding axioms. The proof is a direct field projection from the supplied package structure.
Claim. Given a package containing classical assumptions, recognition geometry assumptions, oscillation bounds, and the Riemann Hypothesis statement, the theorem asserts the Riemann Hypothesis: for all complex numbers $ρ$, if the completed Riemann zeta function vanishes at $ρ$ and $ρ$ has nonzero imaginary part, then the real part of $ρ$ equals $1/2$.
background
The module defines RH_Statement as the proposition that all non-trivial zeros of the completed Riemann zeta function lie on the critical line Re(s) = 1/2. The RHConditionalPackage structure bundles the necessary assumptions (classical_assumptions, recognition_geometry_assumptions, oscillation_bounds) together with the rh field holding the conditional payload. This replaces earlier unconditional axiom-valued functions with an explicit package that callers must supply.
proof idea
The proof is a direct extraction of the rh field from the input package via the term pkg.rh. This is a one-line term-mode wrapper that applies the structure field accessor to deliver RH_Statement.
why it matters
This declaration supplies the interface for conditional proofs of the Riemann Hypothesis in the Recognition Science framework. It lies outside the core T0-T8 forcing chain and serves as a port for legacy conditional routes under oscillation hypotheses. The parent structure RHConditionalPackage organizes the assumptions, while the statement itself remains an open Millennium Prize problem.
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