HodgeConjStructuralCert
plain-language theorem explainer
The HodgeConjStructuralCert structure packages the three discrete facts that open the Recognition Science treatment of the Hodge conjecture: the set of Hodge types has cardinality five, the discrete dimension equals three, and the Q3 lattice therefore contains eight vertices. A researcher examining the discrete ledger version of the Millennium problem would cite this certificate when stating the structural prerequisites before attempting the bridge to algebraic cycles. The declaration is a plain structure definition with no computational steps.
Claim. A structural certificate for the discrete Hodge conjecture consists of the assertions that the set of Hodge types in degree 2 has cardinality five, that the discrete Hodge dimension equals three, and that two raised to the discrete Hodge dimension equals eight.
background
Recognition Science translates the Hodge conjecture as the biconditional that homology classes stable under coarse-graining equal algebraic cycles; on the discrete ledger this becomes the statement that J-cost stable classes equal algebraic cycles on the Q3 lattice. The module introduces five canonical Hodge types at degree 2 (primitive, non-primitive, effective, ample, nef) that match the configuration dimension at spatial dimension three. Upstream, discreteHodgeDimension is defined as the constant 3, reflecting the spatial dimension forced by the eight-tick octave.
proof idea
This declaration is a structure definition that directly encodes the three required equalities; the first field records the Fintype cardinality of the five Hodge types, the second field records the constant discrete dimension, and the third field records the resulting vertex count on Q3.
why it matters
This certificate supplies the structural opening for the Hodge conjecture in RS and is used to construct the concrete instance hodgeConjStructuralCert. It records the discrete prerequisites drawn from the five Hodge types and the D=3 dimension, addressing the translation in biggest-questions.md §XXIII where the bridge to algebraic geometry remains open. The structure ties into the Recognition Science forcing chain at T8, where three spatial dimensions emerge from the self-similar fixed point.
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