hodge_implies_bsd
plain-language theorem explainer
The theorem shows that the Hodge conjecture scaffold directly yields the Birch-Swinnerton-Dyer structural placeholder in the Recognition Science ledger. Number theorists exploring RS-derived connections between algebraic cycles and elliptic curve ranks would cite this link. The proof reduces to a single application of the input hypothesis because the Hodge placeholder is defined identically to the BSD placeholder.
Claim. If the Hodge conjecture scaffold holds, then the Birch-Swinnerton-Dyer structural input holds, where both placeholders reduce to the irrationality of $phi$.
background
In module M-006 the Recognition Science framework supplies a structural scaffold for Hodge-type algebraicity statements. The upstream definition bsd_from_ledger is the structural placeholder for the RS route connecting elliptic-curve rank to L-function vanishing order and is set to Irrational phi. The sibling definition hodge_from_ledger is the structural placeholder for the RS Hodge-conjecture program and is defined to be identical to bsd_from_ledger.
proof idea
The proof is a term-mode identity that returns the hypothesis h directly. Because hodge_from_ledger is definitionally equal to bsd_from_ledger, the implication is immediate by substitution.
why it matters
This declaration supplies the direct bridge from the Hodge scaffold to the BSD placeholder inside the M-006 program, supporting the larger RS chain that routes algebraic-geometry inputs toward analytic rank statements via the phi-ladder. No downstream uses are recorded, so the precise integration into proofs of the full conjectures remains open.
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