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      "id": "AN04-GLOBAL-P04",
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    {
      "id": "AN04-GLOBAL-P05",
      "obligation": "Every compact adjoint input with smooth forcing is smooth, including a possibly nonzero smooth kernel.",
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    {
      "id": "AN04-GLOBAL-P06",
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    {
      "id": "AN04-GLOBAL-P07",
      "obligation": "Distinct minimal excursion interiors are disjoint; shared compact endpoints are harmless.",
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    {
      "id": "AN04-GLOBAL-P08",
      "obligation": "Full half-interval realization at actual unattained order minus j.",
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    {
      "id": "AN04-GLOBAL-P09",
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      "id": "AN04-GLOBAL-P10",
      "obligation": "Finite distribution order gives one negative Sobolev order near K with explicit integrable Fourier weight.",
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      "id": "AN04-GLOBAL-P11",
      "obligation": "Two different original endpoint regularities and full forcing-regular shortened interval give the contradiction.",
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      "id": "AN04-GLOBAL-P12",
      "obligation": "Interleaved compact bounds exhaust X; the empty initial bound uses GS3.",
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    {
      "id": "AN04-GLOBAL-P13",
      "obligation": "Actual complete C0/forcing graph space V_j and distributional limit compatibility.",
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    {
      "id": "AN04-GLOBAL-P14",
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      "id": "AN04-GLOBAL-P15",
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      "id": "AN04-GLOBAL-P16",
      "obligation": "Finite-stage estimate extension and correctly normalized failure sequence.",
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        "GL:E1"
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    {
      "id": "AN04-GLOBAL-P17",
      "obligation": "Countable bump averages detect every continuous outer section.",
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    {
      "id": "AN04-GLOBAL-P18",
      "obligation": "One support cutoff, full smooth proper continuity and the strict finite-stage contradiction.",
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    {
      "id": "AN04-GLOBAL-P19",
      "obligation": "Summable constant losses give one finite global estimate constant.",
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    {
      "id": "AN04-GLOBAL-P20",
      "obligation": "Seminorm increments vanish on increasing interior supports; the limit is finite-order continuous on every compact test space.",
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    {
      "id": "AN04-GLOBAL-P21",
      "obligation": "One conjugate-linear image functional, correct D plus l1 seminorm and exact arbitrary-subspace extension.",
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        "GL:H1"
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    {
      "id": "AN04-GLOBAL-P22",
      "obligation": "The represented input is a distribution; bounded l1 coefficients have the required conjugation.",
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    {
      "id": "AN04-GLOBAL-P23",
      "obligation": "Locally finite smooth residual proves A for one global solution.",
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    {
      "id": "AN04-GLOBAL-P24",
      "obligation": "All three implications for every real m, with no external weak closed-range substitute.",
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    {
      "id": "AN04-GLOBAL-P25",
      "obligation": "Full elliptic identity example and nonoptimal generic gain.",
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    {
      "id": "AN04-GLOBAL-P26",
      "obligation": "Complete slit-domain principal-type, compact-endpoint and positive-distance obstruction.",
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    {
      "id": "AN04-GLOBAL-P27",
      "obligation": "Test-space continuity differs from global smooth topology; full locally finite residual check.",
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}