LinkingInvariantHypothesis
plain-language theorem explainer
The declaration supplies a hypothesis interface asserting that the D-sphere admits nontrivial linking of embedded circles, expressed through reduced cohomology in degree D-2. It is cited in proofs that only three spatial dimensions satisfy the combined Recognition Science constraints on linking, Kepler selection, and synchronization. The definition is a direct one-line delegation to the Alexander duality predicate for circle linking.
Claim. For a natural number $D$, the hypothesis asserts that the $D$-sphere admits pairs of disjoint embedded circles with nonzero linking number. Equivalently, the reduced cohomology group $H^{D-2}(S^1)$ is nontrivial.
background
This declaration sits in the DraftV1 module, which mirrors paper statements from Draft_v1.tex by supplying hypothesis interfaces for results that rely on external mathematics such as Alexander duality for complements of embeddings. The key upstream definition states that the D-sphere admits circle linking precisely when the reduced cohomology of the circle in degree D-2 is nontrivial, following Hatcher Theorem 3.44; this replaces earlier tautological definitions with a cohomology-based predicate. Additional upstream structures supply the nuclear density tiers, J-cost calibration from the ledger factorization, and the spectral emergence that forces SU(3) x SU(2) x U(1) gauge content together with exactly three generations.
proof idea
The definition is a one-line wrapper that directly invokes the Alexander duality characterization of circle linking on the D-sphere.
why it matters
This hypothesis interface is invoked by the main dimensional rigidity theorems, including dimensional_rigidity_main which concludes D equals 3 from the linking hypothesis together with ConstraintK, and the full forward theorem dimensional_rigidity_full_forward. It fills the paper proposition on the linking selection principle, supporting the framework landmark that exactly three spatial dimensions are forced. The open question it touches is the full certification of the linking invariant without external topology assumptions.
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