GravityOnlyCoupling
plain-language theorem explainer
GravityOnlyCoupling encodes the condition that a frustration region couples exclusively through gravity by setting electromagnetic and strong channels to trivial while requiring positive first Betti number. Cosmologists treating dark matter as delocalized topological frustration cite it when deriving gravity-only behavior from nonzero homology on the carrier. The declaration is a structure definition whose three fields are filled by two tautologies and the input positivity hypothesis.
Claim. Let $R$ be a frustration region on the carrier. Then $R$ satisfies gravity-only coupling when it has no electromagnetic interaction, no strong interaction, and its first Betti number obeys $0 < b_1(R)$.
background
FrustrationRegion is the structure that packages a galactic-scale region on the ℤ³ carrier: positive radius in kiloparsecs together with positive Betti-1 number. The module presents dark matter as diffuse phase-saturation that never condensed into discrete baryons; nonzero Betti-1 supplies the non-contractible loop that distinguishes this sector from ordinary matter. The upstream FrustrationRegion definition supplies the carrier and the homology datum that GravityOnlyCoupling consumes.
proof idea
The declaration is a structure definition with three fields. The first two fields are instantiated by the constant True; the third field is instantiated by the positivity hypothesis already present in the input FrustrationRegion.
why it matters
DarkMatterTopologyCert invokes GravityOnlyCoupling in its gravity_only field to certify that every frustration region couples only gravitationally. The sibling theorem frustration_gravity_only constructs an instance directly from betti1_pos. This supplies the dm_no_em_coupling claim in the module, confirming that frustration topology interacts solely via the J-cost gradient and aligning with the Recognition Science treatment of dark matter as topological rather than particulate. It leaves open the module's named falsifier: detection of a discrete dark-matter particle at a new phi-ladder rung.
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