{
  "schema": "AN04-restored-compact-solvability-proof-map/v1",
  "proofs": [
    {
      "id": "KS:F1",
      "source": "closed-graphs-and-smooth-quotients.md",
      "source_sha256": "aff83e2fd15807aaad2bd8a8426bbf783222b7c69655aeffcde38a1cffd84329",
      "proof_locator": "### 14.4.",
      "dependencies": [
        "TG:B1",
        "TG:B2"
      ],
      "scope": "Full Frechet Baire closure neighborhood for each actual balanced convex source neighborhood"
    },
    {
      "id": "KS:F2",
      "source": "closed-graphs-and-smooth-quotients.md",
      "source_sha256": "aff83e2fd15807aaad2bd8a8426bbf783222b7c69655aeffcde38a1cffd84329",
      "proof_locator": "We now remove the closure",
      "dependencies": [
        "KS:F1",
        "TG:B2"
      ],
      "scope": "Every seminorm retained in the actual summable correction, open mapping and inverse without linear selection"
    },
    {
      "id": "KS:F3",
      "source": "closed-graphs-and-smooth-quotients.md",
      "source_sha256": "aff83e2fd15807aaad2bd8a8426bbf783222b7c69655aeffcde38a1cffd84329",
      "proof_locator": "### 14.5.",
      "dependencies": [
        "KS:F2",
        "TG:B2"
      ],
      "scope": "Full closed graph proof and Banach-domain estimate BF13"
    },
    {
      "id": "KS:F4",
      "source": "closed-graphs-and-smooth-quotients.md",
      "source_sha256": "aff83e2fd15807aaad2bd8a8426bbf783222b7c69655aeffcde38a1cffd84329",
      "proof_locator": "### 14.7.",
      "dependencies": [
        "CE:H1"
      ],
      "scope": "Original seminorm null quotient, exact real/complex extension and all zero-space cases"
    },
    {
      "id": "KS:F5",
      "source": "closed-graphs-and-smooth-quotients.md",
      "source_sha256": "aff83e2fd15807aaad2bd8a8426bbf783222b7c69655aeffcde38a1cffd84329",
      "proof_locator": "### 14.8.",
      "dependencies": [
        "TG:B2"
      ],
      "scope": "Actual closed-subspace quotient, both exact metrics, every seminorm and complete representative lift"
    },
    {
      "id": "KS:S1",
      "source": "fixed-support-sobolev-compactness.md",
      "source_sha256": "bcc6cc2dacdb0e266a89df56957d763d28f2a7407c97a515191e5480b9c5ef17",
      "proof_locator": "## S1.",
      "dependencies": [
        "P3:L1",
        "P3:L2",
        "P3:L3",
        "P3:M4",
        "P3:M5",
        "P3:M8",
        "U001:Q3",
        "U001:Q4",
        "U001:P14.2-powers",
        "RC:K1"
      ],
      "scope": "Complete all-real Fourier norms, anti-duality, exact Peetre weights, negative order for compact distributions and every smooth derivative"
    },
    {
      "id": "KS:S2",
      "source": "fixed-support-sobolev-compactness.md",
      "source_sha256": "bcc6cc2dacdb0e266a89df56957d763d28f2a7407c97a515191e5480b9c5ef17",
      "proof_locator": "## S2.",
      "dependencies": [
        "KS:S1",
        "RC:K2",
        "CT:T1",
        "U001:F0-COMP"
      ],
      "scope": "Actual smooth compact-kernel action at any two real orders via absolutely convergent weighted finite-rank tails"
    },
    {
      "id": "KS:S3",
      "source": "fixed-support-sobolev-compactness.md",
      "source_sha256": "bcc6cc2dacdb0e266a89df56957d763d28f2a7407c97a515191e5480b9c5ef17",
      "proof_locator": "## S3.",
      "dependencies": [
        "KS:S1",
        "KS:S2",
        "CE:G1",
        "IF:M2",
        "PS:PS5",
        "CT:T1",
        "U001:P14.3"
      ],
      "scope": "Equivalent finite-chart norms, actual fixed-support Hilbert completion and compact inclusion at every real order"
    },
    {
      "id": "KS:S4",
      "source": "fixed-support-sobolev-compactness.md",
      "source_sha256": "bcc6cc2dacdb0e266a89df56957d763d28f2a7407c97a515191e5480b9c5ef17",
      "proof_locator": "## S4.",
      "dependencies": [
        "KS:S1",
        "KS:S3",
        "CE:H1",
        "CT:T1",
        "U001:P2"
      ],
      "scope": "Finite-chart reciprocal representation on any subspace, complete orthogonal projection and finite dimension from compact unit ball"
    },
    {
      "id": "KS:I1",
      "source": "fixed-support-sobolev-compactness.md",
      "source_sha256": "bcc6cc2dacdb0e266a89df56957d763d28f2a7407c97a515191e5480b9c5ef17",
      "proof_locator": "## 1.",
      "dependencies": [
        "KS:S1",
        "IF:M2",
        "CE:G1",
        "PS:PS5",
        "P3:M7"
      ],
      "scope": "Full retained closed-manifold I1 with its actual compact-manifold scope and complete local Fourier justification"
    },
    {
      "id": "KS:I2",
      "source": "fixed-support-sobolev-compactness.md",
      "source_sha256": "bcc6cc2dacdb0e266a89df56957d763d28f2a7407c97a515191e5480b9c5ef17",
      "proof_locator": "For every \\(\\delta>0\\)",
      "dependencies": [
        "KS:I1",
        "KS:S2",
        "KS:S3",
        "CT:T1"
      ],
      "scope": "Full retained I2, actual all-order compact-kernel proof and explicit fixed-support version"
    },
    {
      "id": "KS:I3",
      "source": "fixed-support-sobolev-compactness.md",
      "source_sha256": "bcc6cc2dacdb0e266a89df56957d763d28f2a7407c97a515191e5480b9c5ef17",
      "proof_locator": "The anti-dual bundle",
      "dependencies": [
        "KS:I1",
        "KS:S4",
        "IF:M2"
      ],
      "scope": "Full retained reciprocal-weight anti-duality, chart gluing and actual annihilator statement"
    },
    {
      "id": "KS:G0",
      "source": "compact-set-solvability-from-characteristic-escape.md",
      "source_sha256": "d044e99e8a37a53a60c1633076a8175d93efac305ac656b108d6e2280aa0e047",
      "proof_locator": "To express this condition",
      "dependencies": [
        "PS:PS5",
        "G2:G0",
        "G2:G41",
        "RP:P1",
        "U001:P14.2-powers"
      ],
      "scope": "Actual positive norm, all-real homogeneous normalization and unchanged oriented maximal strips"
    },
    {
      "id": "KS:G1",
      "source": "compact-set-solvability-from-characteristic-escape.md",
      "source_sha256": "d044e99e8a37a53a60c1633076a8175d93efac305ac656b108d6e2280aa0e047",
      "proof_locator": "The cosphere bundle over",
      "dependencies": [
        "KS:G0",
        "PS:PS5",
        "G2:NF1",
        "G2:NF5",
        "G2:NF7",
        "U001:F0-COMP",
        "U001:P15.1",
        "U001:L5.4.1"
      ],
      "scope": "Compact projected continuation, all finite lift exponential bounds and characteristic subsets without regular-level assumption"
    },
    {
      "id": "KS:G2",
      "source": "compact-set-solvability-from-characteristic-escape.md",
      "source_sha256": "d044e99e8a37a53a60c1633076a8175d93efac305ac656b108d6e2280aa0e047",
      "proof_locator": "It also explains the radial case.",
      "dependencies": [
        "KS:G1",
        "G2:NF1"
      ],
      "scope": "Radial and stationary exclusion under escape and inability to reach a projected zero in finite time"
    },
    {
      "id": "KS:G3",
      "source": "compact-set-solvability-from-characteristic-escape.md",
      "source_sha256": "d044e99e8a37a53a60c1633076a8175d93efac305ac656b108d6e2280aa0e047",
      "proof_locator": "**Lemma 1.1.**",
      "dependencies": [
        "KS:G1",
        "KS:G2",
        "PS:PS5",
        "G2:NF5",
        "G2:NF7",
        "U001:F0-COMP"
      ],
      "scope": "Explicit nested compact neighborhoods from complement exhaustion, compact limiting curves and persistence of escape"
    },
    {
      "id": "KS:R0",
      "source": "compact-set-solvability-from-characteristic-escape.md",
      "source_sha256": "d044e99e8a37a53a60c1633076a8175d93efac305ac656b108d6e2280aa0e047",
      "proof_locator": "To prove this, the principal symbol",
      "dependencies": [
        "KS:G2",
        "SP:P1",
        "SP:P5",
        "SP:S3",
        "P2:OP3",
        "IF:M2",
        "KS:S3",
        "U001:F0-COMP"
      ],
      "scope": "All-real fixed-support regularity from finite escape, elliptic and characteristic directions, complete common-cutoff Fourier norm"
    },
    {
      "id": "KS:R1",
      "source": "compact-set-solvability-from-characteristic-escape.md",
      "source_sha256": "d044e99e8a37a53a60c1633076a8175d93efac305ac656b108d6e2280aa0e047",
      "proof_locator": "whenever the quantities on the right",
      "dependencies": [
        "KS:R0",
        "KS:S3",
        "KS:F3",
        "CE:G1",
        "P2:OP5"
      ],
      "scope": "Actual Banach graph space, correct H^(r-2) continuity and one-lower-order closed graph estimate"
    },
    {
      "id": "KS:N0",
      "source": "compact-set-solvability-from-characteristic-escape.md",
      "source_sha256": "d044e99e8a37a53a60c1633076a8175d93efac305ac656b108d6e2280aa0e047",
      "proof_locator": "Smooth propagation and escape imply",
      "dependencies": [
        "KS:R0",
        "KS:R1",
        "KS:S1",
        "KS:S3",
        "KS:S4",
        "RP:P0"
      ],
      "scope": "Smooth adjoint kernel, closed L2 subspace, compact unit ball and finite dimension"
    },
    {
      "id": "KS:N1",
      "source": "compact-set-solvability-from-characteristic-escape.md",
      "source_sha256": "d044e99e8a37a53a60c1633076a8175d93efac305ac656b108d6e2280aa0e047",
      "proof_locator": "For any real \\(r\\), let",
      "dependencies": [
        "KS:N0",
        "KS:R1",
        "KS:S3",
        "KS:S4"
      ],
      "scope": "Every real-order orthogonal complement coercivity with pairwise compactness and graph estimates"
    },
    {
      "id": "KS:N2",
      "source": "compact-set-solvability-from-characteristic-escape.md",
      "source_sha256": "d044e99e8a37a53a60c1633076a8175d93efac305ac656b108d6e2280aa0e047",
      "proof_locator": "The obstruction space annihilates",
      "dependencies": [
        "KS:N0",
        "RC:K8",
        "P2:OP3",
        "P2:OP5"
      ],
      "scope": "Necessary exact distribution/compact-smooth-test adjoint pairing"
    },
    {
      "id": "KS:N3",
      "source": "compact-set-solvability-from-characteristic-escape.md",
      "source_sha256": "d044e99e8a37a53a60c1633076a8175d93efac305ac656b108d6e2280aa0e047",
      "proof_locator": "## 4. Enlarge",
      "dependencies": [
        "KS:G3",
        "KS:N0",
        "U001:P2"
      ],
      "scope": "Descending finite-dimensional kernels stabilize on an actual neighborhood at precisely N(K)"
    },
    {
      "id": "KS:D0",
      "source": "compact-set-solvability-from-characteristic-escape.md",
      "source_sha256": "d044e99e8a37a53a60c1633076a8175d93efac305ac656b108d6e2280aa0e047",
      "proof_locator": "**Theorem 5.1.**",
      "dependencies": [
        "KS:D1",
        "KS:D2",
        "KS:J9",
        "KS:N0",
        "KS:N2",
        "KS:N3"
      ],
      "scope": "Full compact-neighborhood existence theorem, all real Sobolev orders, arbitrary lower terms and one smooth solution"
    },
    {
      "id": "KS:D1",
      "source": "compact-set-solvability-from-characteristic-escape.md",
      "source_sha256": "d044e99e8a37a53a60c1633076a8175d93efac305ac656b108d6e2280aa0e047",
      "proof_locator": "**Proof of the Sobolev assertion.**",
      "dependencies": [
        "KS:N1",
        "KS:N3",
        "KS:S1",
        "KS:S4",
        "CE:H1"
      ],
      "scope": "Correct reciprocal orders, actual annihilation-invariant estimate and well-defined conjugate-linear functional"
    },
    {
      "id": "KS:D2",
      "source": "compact-set-solvability-from-characteristic-escape.md",
      "source_sha256": "d044e99e8a37a53a60c1633076a8175d93efac305ac656b108d6e2280aa0e047",
      "proof_locator": "For completeness, realize",
      "dependencies": [
        "KS:D1",
        "KS:S4",
        "IF:M2",
        "CE:H1"
      ],
      "scope": "Full finite-chart extension and represented global distribution without chart boundary defects or linear selection"
    },
    {
      "id": "KS:J0",
      "source": "compact-set-solvability-from-characteristic-escape.md",
      "source_sha256": "d044e99e8a37a53a60c1633076a8175d93efac305ac656b108d6e2280aa0e047",
      "proof_locator": "We now prove the smooth assertion",
      "dependencies": [
        "KS:F5",
        "TG:B2",
        "TG:B3",
        "PS:PS5"
      ],
      "scope": "Actual closed flat-jet quotient with every boundary derivative and complete topology"
    },
    {
      "id": "KS:J1",
      "source": "compact-set-solvability-from-characteristic-escape.md",
      "source_sha256": "d044e99e8a37a53a60c1633076a8175d93efac305ac656b108d6e2280aa0e047",
      "proof_locator": "### 6.1.",
      "dependencies": [
        "KS:J0",
        "RC:K1",
        "RC:K8",
        "U001:P14.3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "P3:M4"
      ],
      "scope": "Exact compact-support dual and flat-function annihilation via full shrinking-cutoff/Taylor bounds without boundary regularity"
    },
    {
      "id": "KS:J2",
      "source": "compact-set-solvability-from-characteristic-escape.md",
      "source_sha256": "d044e99e8a37a53a60c1633076a8175d93efac305ac656b108d6e2280aa0e047",
      "proof_locator": "It is a closed Fréchet subspace.",
      "dependencies": [
        "KS:J0",
        "KS:J1",
        "KS:N2",
        "KS:S1",
        "CE:G1",
        "P2:OP5"
      ],
      "scope": "Actual finite-codimension annihilator, proper smooth operator continuity and range map"
    },
    {
      "id": "KS:J3",
      "source": "compact-set-solvability-from-characteristic-escape.md",
      "source_sha256": "d044e99e8a37a53a60c1633076a8175d93efac305ac656b108d6e2280aa0e047",
      "proof_locator": "Choose a compact exhaustion",
      "dependencies": [
        "KS:J0",
        "KS:J2",
        "PS:PS5",
        "KS:S1",
        "KS:S3",
        "TG:B2",
        "TG:B3",
        "IF:M2"
      ],
      "scope": "Increasing actual Hilbert chart seminorms, full smooth completeness and exact quotient topology"
    },
    {
      "id": "KS:J4",
      "source": "compact-set-solvability-from-characteristic-escape.md",
      "source_sha256": "d044e99e8a37a53a60c1633076a8175d93efac305ac656b108d6e2280aa0e047",
      "proof_locator": "If \\(\\ell\\in Y_A",
      "dependencies": [
        "KS:J1",
        "KS:J2",
        "KS:F4",
        "KS:N0",
        "U001:P2"
      ],
      "scope": "Every continuous quotient-dual extension and precise finite-dimensional ambiguity N(A)"
    },
    {
      "id": "KS:J5",
      "source": "compact-set-solvability-from-characteristic-escape.md",
      "source_sha256": "d044e99e8a37a53a60c1633076a8175d93efac305ac656b108d6e2280aa0e047",
      "proof_locator": "Suppose this is finite.",
      "dependencies": [
        "KS:J4",
        "KS:S4",
        "KS:S1",
        "IF:M2",
        "CE:G1"
      ],
      "scope": "Finite source-seminorm bound represents P-star v in the actual H-minus-j support norm"
    },
    {
      "id": "KS:J6",
      "source": "compact-set-solvability-from-characteristic-escape.md",
      "source_sha256": "d044e99e8a37a53a60c1633076a8175d93efac305ac656b108d6e2280aa0e047",
      "proof_locator": "Apply (CS3) and (CS8)",
      "dependencies": [
        "KS:J5",
        "KS:R0",
        "KS:N1",
        "KS:J3",
        "KS:S1",
        "KS:F5"
      ],
      "scope": "Modulo-kernel bound, later quotient seminorm and zero-dual-bound case"
    },
    {
      "id": "KS:J7",
      "source": "compact-set-solvability-from-characteristic-escape.md",
      "source_sha256": "d044e99e8a37a53a60c1633076a8175d93efac305ac656b108d6e2280aa0e047",
      "proof_locator": "### 6.3.",
      "dependencies": [
        "KS:J6",
        "KS:F4",
        "KS:F5",
        "TG:B2"
      ],
      "scope": "Full balanced-convex Hahn-Banach separation and actual closure neighborhood without normability"
    },
    {
      "id": "KS:J8",
      "source": "compact-set-solvability-from-characteristic-escape.md",
      "source_sha256": "d044e99e8a37a53a60c1633076a8175d93efac305ac656b108d6e2280aa0e047",
      "proof_locator": "We apply the closure-removal part",
      "dependencies": [
        "KS:J7",
        "KS:J3",
        "KS:F2",
        "TG:B2"
      ],
      "scope": "Explicit scaled closure neighborhoods, every original seminorm tail and actual target residual convergence"
    },
    {
      "id": "KS:J9",
      "source": "compact-set-solvability-from-characteristic-escape.md",
      "source_sha256": "d044e99e8a37a53a60c1633076a8175d93efac305ac656b108d6e2280aa0e047",
      "proof_locator": "If the original \\(f\\) is smooth",
      "dependencies": [
        "KS:J8",
        "KS:N3",
        "KS:J0"
      ],
      "scope": "One smooth solution, actual flat difference and equation on an open neighborhood of K"
    },
    {
      "id": "KS:M1",
      "source": "compact-set-solvability-from-characteristic-escape.md",
      "source_sha256": "d044e99e8a37a53a60c1633076a8175d93efac305ac656b108d6e2280aa0e047",
      "proof_locator": "Escape is sufficient;",
      "dependencies": [
        "KS:G0",
        "G2:G0",
        "U001:P16.1",
        "U001:P16.2",
        "U001:P15.1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Full trapped circular model and exact smooth periodic inverse including both endpoints and nonzero scalar phase"
    },
    {
      "id": "KS:M2",
      "source": "compact-set-solvability-from-characteristic-escape.md",
      "source_sha256": "d044e99e8a37a53a60c1633076a8175d93efac305ac656b108d6e2280aa0e047",
      "proof_locator": "The finite-dimensional obstruction can",
      "dependencies": [
        "KS:N2",
        "RP:C1",
        "P2:OP3",
        "P2:OP5",
        "KS:S4",
        "U001:P14.3"
      ],
      "scope": "Proper elliptic rank-one counterexample, exact adjoint and full compact kernel identification"
    },
    {
      "id": "KS:V1",
      "source": "compact-set-solvability-from-characteristic-escape.md",
      "source_sha256": "d044e99e8a37a53a60c1633076a8175d93efac305ac656b108d6e2280aa0e047",
      "proof_locator": "### Exact characteristic models",
      "dependencies": [
        "KS:G0",
        "KS:M1",
        "G2:G0",
        "U001:P16.1",
        "U001:P16.2"
      ],
      "scope": "Exact escaping and trapped Hamilton coordinates, true compact sets and explicitly scaled covector illustration"
    },
    {
      "id": "KS:X1",
      "source": "compact-set-solvability-from-characteristic-escape.md",
      "source_sha256": "d044e99e8a37a53a60c1633076a8175d93efac305ac656b108d6e2280aa0e047",
      "proof_locator": "**1. The order in the dual extension.**",
      "dependencies": [
        "KS:D1",
        "KS:D2"
      ],
      "scope": "Unchanged complete fractional-order exercise and every duality sign"
    },
    {
      "id": "KS:X2",
      "source": "compact-set-solvability-from-characteristic-escape.md",
      "source_sha256": "d044e99e8a37a53a60c1633076a8175d93efac305ac656b108d6e2280aa0e047",
      "proof_locator": "**2. A smooth obstruction",
      "dependencies": [
        "KS:M2",
        "KS:N2",
        "RP:C0",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Unchanged complete rank-one necessity and explicit global smooth primitive with allowed constant tail"
    },
    {
      "id": "KS:X3",
      "source": "compact-set-solvability-from-characteristic-escape.md",
      "source_sha256": "d044e99e8a37a53a60c1633076a8175d93efac305ac656b108d6e2280aa0e047",
      "proof_locator": "**3. Summability in a smooth topology.**",
      "dependencies": [
        "KS:J8",
        "KS:J9",
        "KS:F2"
      ],
      "scope": "Unchanged full all-seminorm correction exercise and precise absence of a linear right inverse claim"
    }
  ],
  "earlier_maps": [
    "../20261004-free-stationary-phase/proof-map.json",
    "../20261004-free-intrinsic-graph/proof-map.json",
    "../20261004-free-tangent-zoom/proof-map.json",
    "../20261004-free-canonical-composition/proof-map.json",
    "../20261005-restored-oscillatory/proof-map.json",
    "../20261005-restored-phase-space/proof-map.json",
    "../20261005-restored-phase-equivalence/proof-map.json",
    "../20261005-restored-intrinsic-regularity/proof-map.json",
    "../20261005-restored-tangent-zoom/proof-map.json",
    "../20261005-restored-gaussian-symbols/proof-map.json",
    "../20261005-restored-linear-reduction/proof-map.json",
    "../20261005-restored-analytic-composition/proof-map.json",
    "../20261005-restored-subprincipal-transport/proof-map.json",
    "../20261005-restored-graph-egorov/proof-map.json",
    "../20261005-restored-corank-continuity/proof-map.json",
    "../20261005-restored-submanifolds/proof-map.json",
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