hodge_from_ledger
Equates the Recognition Science Hodge-conjecture structural placeholder to the Birch-Swinnerton-Dyer ledger condition. Algebraic geometers exploring RS routes to algebraicity statements cite this alias when chaining to rank and L-function properties. The definition is a direct one-line alias to the upstream placeholder.
claimDefine the proposition $H$ for the Hodge conjecture scaffold by $H :=$ the Birch-Swinnerton-Dyer ledger proposition, where the latter asserts that the golden ratio is irrational.
background
Module M-006 formalizes a structural RS scaffold for Hodge-type algebraicity statements. It imports the Birch-Swinnerton-Dyer structure module to supply the necessary ledger input. The upstream bsd_from_ledger is a placeholder proposition defined as the irrationality of phi, described in its doc-comment as the RS route connecting rank and L-value vanishing order.
proof idea
Implemented as a one-line definition that directly aliases the upstream bsd_from_ledger placeholder.
why it matters in Recognition Science
This definition supplies the base proposition for the Hodge conjecture scaffold in Recognition Science. It feeds the theorems hodge_implies_bsd and hodge_structure, which use it to link Hodge-type algebraicity to BSD structural input. The placement advances the M-006 program by establishing a structural bridge between algebraic geometry and the phi-based ledger.
scope and limits
- Does not prove any instance of the classical Hodge conjecture for projective varieties.
- Does not identify specific algebraic cycles or their Hodge classes.
- Does not incorporate the forcing chain steps T0 through T8 or the Recognition Composition Law.
- Does not produce numerical predictions from the phi-ladder or mass formula.
Lean usage
theorem hodge_implies_bsd (h : hodge_from_ledger) : bsd_from_ledger := hformal statement (Lean)
17def hodge_from_ledger : Prop := bsd_from_ledger
proof body
Definition body.
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