{
  "schema": "open-mathematics-proof-routes/v1",
  "course": "LG-MF",
  "lesson": "LG-MF-06",
  "license": "CC0-1.0",
  "source": "src/dimension-formulas-for-congruence-subgroups.md",
  "reader": "LG-MF-06.html",
  "results": [
    {
      "id": "LG-MF-06-smooth-completion",
      "statement": "The integral inner-product space of smooth sections of a smooth Hermitian bundle on a compact Riemann surface has a complete Cauchy-class completion, with bounded linear extensions.",
      "proof": "LG-MF-06.html#a-1-3-completion-by-cauchy-classes"
    },
    {
      "id": "LG-MF-06-fourier-completion",
      "statement": "For positive periods a,b and every integer s>=0, the smooth periodic Sobolev completion is isometrically the weighted coefficient space with weights (1+m^2+n^2)^s; Fourier polynomials are dense.",
      "proof": "LG-MF-06.html#a-1-7-the-fourier-sobolev-completions-and-derivative-maps"
    },
    {
      "id": "LG-MF-06-weak-localization",
      "statement": "Global smooth L2 and integer Sobolev completions inject into chart test distributions; bounded localization and the explicit dbar Fourier multiplier recognize completed weak derivatives and local smooth solutions.",
      "proof": "LG-MF-06.html#a-1-8-injection-of-l-2-completions-into-distributions"
    },
    {
      "id": "LG-MF-06-continuous-inclusion",
      "statement": "A continuous compactly supported chart coefficient admits uniform smooth compactly supported approximants and therefore represents the same test pairings in the smooth L2 completion.",
      "proof": "LG-MF-06.html#a-1-6-fourier-density-and-parseval-in-the-smooth-completion"
    }
  ],
  "earlier_proofs": [
    {
      "lesson": "LG-MF-01",
      "result": "Lemma 0.3",
      "reader": "LG-MF-01.html#analytic-facts-for-subsequent-lessons"
    }
  ],
  "retained_prerequisites": [
    "Real completeness and elementary smooth calculus",
    "Existence and finite additivity of continuous Riemann integrals",
    "Compactness and smooth chart cutoffs",
    "Coordinate-integral change of variables",
    "Smooth-boundary Stokes approximation in R1"
  ],
  "scope": "These proofs construct smooth completions and their specified weak-test maps. They do not supply general measurable L2 representation, algebraic cohomology/GAGA or the arithmetic inputs of Section6.1."
}