IndisputableMonolith.Gravity.D2ScalarDirichletPartial
Partial bridge between scalar Dirichlet limits and quadrature convergence for D2 gravity probes. Supplies iff characterizations linking nonempty Dirichlet limits to filter tendsto, plus uniform-probe reductions that scale Dirichlet data into quadrature integrals. Gravity analysts cite it when converting continuum curvature-bearing limits into discrete quadrature statements. The argument is mostly equivalence packaging over the upstream quadrature-limit module.
claimFor scalar Dirichlet data on D2 probes, the Dirichlet limit set is nonempty if and only if the associated net tends to a limit; that limit coincides with the quadrature limit precisely when the quadrature integrals converge. For uniform probes, the quadrature integral equals a scaled Dirichlet integral, so the scalar Dirichlet limit exists iff the quadrature net tends.
background
Recognition Science gravity work treats curvature-bearing probes through scalar Dirichlet energies and their continuum limits. The upstream module D2ScalarDirichletQuadratureLimit packages the curvature-bearing conditional: claimed equivalences are theorems with no internal gaps, while the scalar Dirichlet limit for curvature-bearing probes stays a named open analytic input.
This partial module sits one layer closer to usable rewrite rules. It introduces the scalar Dirichlet limit object, the tendsto characterization of nonempty limit sets, and the identification of that limit with a quadrature limit when the discrete integrals converge. Uniform probes receive a scaling identity that turns Dirichlet integrals into quadrature integrals, so limit existence becomes a pure convergence statement on the quadrature side.
Notation follows the D2 gravity stack: Dirichlet energies on probe configurations, filter-theoretic tendsto for nets of approximations, and quadrature sums as the discrete stand-in for the continuum integral.
proof idea
The module is a thin equivalence layer over the quadrature-limit development. Nonemptiness of the scalar Dirichlet limit set is rewritten as tendsto of the approximating net; existence of a Dirichlet limit is then equated with tendsto of the quadrature integrals once the two limit notions are identified. For uniform probes, a direct scaling identity equates the quadrature integral to a constant multiple of the Dirichlet integral, so the Dirichlet-limit iff quadrature-tendsto statement is a one-step transport of the general equivalence. No deep analytic estimates live here; the work is packaging and specialization.
why it matters in Recognition Science
In the gravity domain this module turns the upstream curvature-bearing conditional into rewrite lemmas that downstream D2 arguments can apply without re-deriving filter and scaling bookkeeping. The uniform-probe identities are the practical entry point: once a probe family is uniform, continuum Dirichlet convergence collapses to a quadrature tendsto check. Used-by edges are not yet recorded on this page, so the module presently feeds the local gravity stack rather than a named parent theorem. It does not close the open analytic input (existence of the scalar Dirichlet limit for curvature-bearing probes); it only organizes the consequences of that limit when it is assumed or established elsewhere.
scope and limits
- Does not prove existence of the scalar Dirichlet limit for curvature-bearing probes.
- Does not supply new analytic estimates beyond equivalence and scaling identities.
- Does not treat non-uniform probes except through the general tendsto iff forms.
- Does not discharge the named open analytic input left by the upstream quadrature-limit module.
- Does not claim a downstream parent theorem; used-by is currently empty.