hodgeConjStructuralCert
plain-language theorem explainer
This definition assembles the structural certificate for the discrete Hodge conjecture by supplying the five Hodge types at dimension 3, the discrete dimension value, and the eight Q3 vertices. Mathematicians examining Recognition Science translations of the Hodge conjecture would cite it as the verified discrete ledger base. The construction proceeds by direct field assignment from pre-established cardinality and equality results.
Claim. The structural certificate for the discrete Hodge conjecture asserts that the number of Hodge types equals 5, the discrete Hodge dimension equals 3, and the Q3 lattice satisfies $2^3 = 8$.
background
Recognition Science recasts the Hodge conjecture as the claim that homology classes stable under coarse-graining coincide with algebraic cycles on the recognition lattice. In this module a coarse-grained recognition class is a sub-manifold of the lattice and algebraic cycles are identified with J-cost zeros; the discrete Hodge statement then equates J-cost stable classes with algebraic cycles on Q3. Five canonical Hodge types appear in degree 2, matching configDim D = 5, while the eight-tick octave forces D = 3 and therefore eight vertices on Q3.
proof idea
The definition constructs the HodgeConjStructuralCert structure directly. It populates the five_hodge_types field with the theorem establishing cardinality of HodgeType as 5, sets the discrete dimension field by reflexivity, and fills the Q3 vertices field with the theorem proving two to the power of the dimension equals eight.
why it matters
This definition supplies the verified discrete certificate that opens the Recognition Science treatment of the Hodge conjecture, as referenced in the module documentation to biggest-questions.md §XXIII. It encodes the discrete ledger version in which stable homology classes equal algebraic cycles and aligns with the eight-tick octave and D = 3 from the forcing chain. The bridge from this discrete ledger to continuous algebraic geometry remains open.
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