Explain monotone upper-bound provenance

app.py

@@ -1445,6 +1445,18 @@ def detail_row(label: str, value: str) -> str:
 
 
 def visual_provenance(record: dict[str, Any], visual_record: dict[str, Any]) -> str:
+    source_case = str(visual_record.get("monotone_upper_bound_from_case") or "").strip()
+    if source_case:
+        visual_metric = metric_value(visual_record)
+        shown_metric = metric_value(record)
+        suffix = ""
+        if visual_metric != shown_metric:
+            suffix = f" Coordinate JSON evaluator metric: {esc(metric_symbol(visual_record))} = {esc(visual_metric)}; Friedman reported metric: {esc(metric_symbol(record))} = {esc(shown_metric)}."
+        return (
+            f"Verified monotone upper-bound layout derived by deleting items from {esc(source_case)}; "
+            "the Friedman-listed finder remains credited for this row."
+        ) + suffix
+
     if visual_record.get("frontend_seed"):
         visual_metric = metric_value(visual_record)
         shown_metric = metric_value(record)

data/solutions/generated/cirinhex_006_3e5660c8b6310c57.json

@@ -0,0 +1,98 @@
+{
+  "case": "cirinhex@6",
+  "container": {
+    "circumradius": 3.1547526785063353,
+    "orientation_radians": 0.0,
+    "side_length": 3.154752678506335,
+    "sides": 6,
+    "type": "regular_polygon"
+  },
+  "friedman_reference": {
+    "analytical_or_proved": true,
+    "case": "cirinhex@6",
+    "credit": "Trivial",
+    "date": "",
+    "family_updated": "",
+    "metric_expression": "2 + 2/\u221a3 = 3.154+",
+    "metric_symbol": "s",
+    "metric_value": 3.1547005383792515,
+    "reference_text": "s = 2 + 2/\u221a3 = 3.154+ | Trivial. |",
+    "source_image": "hc6.gif",
+    "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc6.gif",
+    "source_page": "https://erich-friedman.github.io/packing/cirinhex/",
+    "title": "Circles in Hexagons"
+  },
+  "item": {
+    "radius": 1.0,
+    "type": "circle"
+  },
+  "metadata": {
+    "friedman_reference": {
+      "analytical_or_proved": true,
+      "case": "cirinhex@6",
+      "credit": "Trivial",
+      "date": "",
+      "family_updated": "",
+      "metric_expression": "2 + 2/\u221a3 = 3.154+",
+      "metric_symbol": "s",
+      "metric_value": 3.1547005383792515,
+      "reference_text": "s = 2 + 2/\u221a3 = 3.154+ | Trivial. |",
+      "source_image": "hc6.gif",
+      "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc6.gif",
+      "source_page": "https://erich-friedman.github.io/packing/cirinhex/",
+      "title": "Circles in Hexagons"
+    },
+    "generator": "monotone_subset_upper_bound",
+    "metric_recovery_tolerance_rel": 0.001,
+    "monotone_upper_bound_from_case": "cirinhex@7",
+    "monotone_upper_bound_from_record": "7368090471750227",
+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from cirinhex@7; this prevents displaying a dominated coordinate reconstruction.",
+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 3.1547005383792515,
+    "source_url": "https://erich-friedman.github.io/packing/cirinhex/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
+    "submitted_by": "Trivial"
+  },
+  "placements": [
+    {
+      "id": 1,
+      "rotation_radians": -0.23322506594830267,
+      "x": -1.0000309000324779,
+      "y": -1.7320801817404443
+    },
+    {
+      "id": 2,
+      "rotation_radians": -0.045304852292609214,
+      "x": 1.0000097039055962,
+      "y": -1.7320929631920046
+    },
+    {
+      "id": 3,
+      "rotation_radians": -0.25463607947563405,
+      "x": 2.000042262428875,
+      "y": -1.2974602376825998e-05
+    },
+    {
+      "id": 4,
+      "rotation_radians": 0.20046369676565526,
+      "x": -2.0000412968186594,
+      "y": 1.176064020196015e-05
+    },
+    {
+      "id": 5,
+      "rotation_radians": 0.0417618583579787,
+      "x": -5.888372798050501e-07,
+      "y": -1.9188932050271395e-06
+    },
+    {
+      "id": 6,
+      "rotation_radians": 0.023035387174397022,
+      "x": 1.0000323094283496,
+      "y": 1.7320814401895983
+    }
+  ],
+  "problem": "6 equal items for Circles in Hexagons",
+  "setup": "cirinhex",
+  "source_image": "https://erich-friedman.github.io/packing/cirinhex/hc6.gif",
+  "source_page": "https://erich-friedman.github.io/packing/cirinhex/"
+}

data/solutions/generated/cirintri_002_befbb923c21388e1.json

@@ -0,0 +1,74 @@
+{
+  "case": "cirintri@2",
+  "container": {
+    "circumradius": 3.1547403089836683,
+    "orientation_radians": 0.0,
+    "side_length": 5.464170499845252,
+    "sides": 3,
+    "type": "regular_polygon"
+  },
+  "friedman_reference": {
+    "analytical_or_proved": true,
+    "case": "cirintri@2",
+    "credit": "Trivial",
+    "date": "",
+    "family_updated": "",
+    "metric_expression": "2 + 2\u221a3 = 5.464+",
+    "metric_symbol": "s",
+    "metric_value": 5.464101615137754,
+    "reference_text": "s = 2 + 2\u221a3 = 5.464+ | Trivial.",
+    "source_image": "ct2.gif",
+    "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct2.gif",
+    "source_page": "https://erich-friedman.github.io/packing/cirintri/",
+    "title": "Circles in Triangles"
+  },
+  "item": {
+    "radius": 1.0,
+    "type": "circle"
+  },
+  "metadata": {
+    "friedman_reference": {
+      "analytical_or_proved": true,
+      "case": "cirintri@2",
+      "credit": "Trivial",
+      "date": "",
+      "family_updated": "",
+      "metric_expression": "2 + 2\u221a3 = 5.464+",
+      "metric_symbol": "s",
+      "metric_value": 5.464101615137754,
+      "reference_text": "s = 2 + 2\u221a3 = 5.464+ | Trivial.",
+      "source_image": "ct2.gif",
+      "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct2.gif",
+      "source_page": "https://erich-friedman.github.io/packing/cirintri/",
+      "title": "Circles in Triangles"
+    },
+    "generator": "monotone_subset_upper_bound",
+    "metric_recovery_tolerance_rel": 0.001,
+    "monotone_upper_bound_from_case": "cirintri@3",
+    "monotone_upper_bound_from_record": "fcee9d6e8b7c5ce0",
+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from cirintri@3; this prevents displaying a dominated coordinate reconstruction.",
+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 5.464101615137754,
+    "source_url": "https://erich-friedman.github.io/packing/cirintri/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
+    "submitted_by": "Trivial"
+  },
+  "placements": [
+    {
+      "id": 1,
+      "rotation_radians": 1.9406897221311108,
+      "x": -0.5773672542051238,
+      "y": -1.0000326436766995
+    },
+    {
+      "id": 2,
+      "rotation_radians": 4.510088484258312,
+      "x": 1.1547340029331585,
+      "y": -2.0483411347167786e-06
+    }
+  ],
+  "problem": "2 equal items for Circles in Triangles",
+  "setup": "cirintri",
+  "source_image": "https://erich-friedman.github.io/packing/cirintri/ct2.gif",
+  "source_page": "https://erich-friedman.github.io/packing/cirintri/"
+}

data/solutions/generated/cirintri_005_64d422ca71110b0d.json

@@ -0,0 +1,92 @@
+{
+  "case": "cirintri@5",
+  "container": {
+    "circumradius": 4.30945637922671,
+    "orientation_radians": 0.0,
+    "side_length": 7.4641974018224735,
+    "sides": 3,
+    "type": "regular_polygon"
+  },
+  "friedman_reference": {
+    "analytical_or_proved": true,
+    "case": "cirintri@5",
+    "credit": "Milano",
+    "date": "1987",
+    "family_updated": "",
+    "metric_expression": "4 + 2\u221a3 = 7.464+",
+    "metric_symbol": "s",
+    "metric_value": 7.464101615137754,
+    "reference_text": "s = 4 + 2\u221a3 = 7.464+ | Proved by Milano in 1987.",
+    "source_image": "ct5.gif",
+    "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct5.gif",
+    "source_page": "https://erich-friedman.github.io/packing/cirintri/",
+    "title": "Circles in Triangles"
+  },
+  "item": {
+    "radius": 1.0,
+    "type": "circle"
+  },
+  "metadata": {
+    "friedman_reference": {
+      "analytical_or_proved": true,
+      "case": "cirintri@5",
+      "credit": "Milano",
+      "date": "1987",
+      "family_updated": "",
+      "metric_expression": "4 + 2\u221a3 = 7.464+",
+      "metric_symbol": "s",
+      "metric_value": 7.464101615137754,
+      "reference_text": "s = 4 + 2\u221a3 = 7.464+ | Proved by Milano in 1987.",
+      "source_image": "ct5.gif",
+      "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct5.gif",
+      "source_page": "https://erich-friedman.github.io/packing/cirintri/",
+      "title": "Circles in Triangles"
+    },
+    "generator": "monotone_subset_upper_bound",
+    "metric_recovery_tolerance_rel": 0.001,
+    "monotone_upper_bound_from_case": "cirintri@6",
+    "monotone_upper_bound_from_record": "09bddcb97a85d368",
+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from cirintri@6; this prevents displaying a dominated coordinate reconstruction.",
+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 7.464101615137754,
+    "source_url": "https://erich-friedman.github.io/packing/cirintri/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
+    "submitted_by": "Milano"
+  },
+  "placements": [
+    {
+      "id": 1,
+      "rotation_radians": 0.10032113679130471,
+      "x": -1.1547261902240415,
+      "y": -2.0000440827371597
+    },
+    {
+      "id": 2,
+      "rotation_radians": 1.836113755980503,
+      "x": 0.5773533672475022,
+      "y": -1.00002705878021
+    },
+    {
+      "id": 3,
+      "rotation_radians": 2.221704687086976,
+      "x": 2.309450337987728,
+      "y": 3.0425669316776003e-07
+    },
+    {
+      "id": 4,
+      "rotation_radians": 4.5510667197791035,
+      "x": -1.1547264683521061,
+      "y": -9.201608967254509e-06
+    },
+    {
+      "id": 5,
+      "rotation_radians": 2.8863577663880133,
+      "x": -1.1547265791003039,
+      "y": 2.0000444340075796
+    }
+  ],
+  "problem": "5 equal items for Circles in Triangles",
+  "setup": "cirintri",
+  "source_image": "https://erich-friedman.github.io/packing/cirintri/ct5.gif",
+  "source_page": "https://erich-friedman.github.io/packing/cirintri/"
+}

data/solutions/generated/domindom_007_a480ce1181b75833.json

@@ -0,0 +1,103 @@
+{
+  "case": "domindom@7",
+  "container": {
+    "height": 3.0000496969324035,
+    "type": "rectangle",
+    "width": 6.000099393864807
+  },
+  "friedman_reference": {
+    "analytical_or_proved": true,
+    "case": "domindom@7",
+    "credit": "Trivial",
+    "date": "",
+    "family_updated": "",
+    "metric_expression": "3",
+    "metric_symbol": "s",
+    "metric_value": 3.0,
+    "reference_text": "7.-9. | | s = 3 | Trivial.",
+    "source_image": "9.gif",
+    "source_image_url": "https://erich-friedman.github.io/packing/domindom/9.gif",
+    "source_page": "https://erich-friedman.github.io/packing/domindom/",
+    "title": "Dominoes in Dominoes"
+  },
+  "item": {
+    "height": 1.0,
+    "type": "rectangle",
+    "width": 2.0
+  },
+  "metadata": {
+    "friedman_reference": {
+      "analytical_or_proved": true,
+      "case": "domindom@7",
+      "credit": "Trivial",
+      "date": "",
+      "family_updated": "",
+      "metric_expression": "3",
+      "metric_symbol": "s",
+      "metric_value": 3.0,
+      "reference_text": "7.-9. | | s = 3 | Trivial.",
+      "source_image": "9.gif",
+      "source_image_url": "https://erich-friedman.github.io/packing/domindom/9.gif",
+      "source_page": "https://erich-friedman.github.io/packing/domindom/",
+      "title": "Dominoes in Dominoes"
+    },
+    "generator": "monotone_subset_upper_bound",
+    "metric_recovery_tolerance_rel": 0.001,
+    "monotone_upper_bound_from_case": "domindom@9",
+    "monotone_upper_bound_from_record": "43b9630822b40990",
+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from domindom@9; this prevents displaying a dominated coordinate reconstruction.",
+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 3.0,
+    "source_url": "https://erich-friedman.github.io/packing/domindom/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
+    "submitted_by": "Trivial"
+  },
+  "placements": [
+    {
+      "id": 1,
+      "rotation_radians": 9.263873246125958e-07,
+      "x": -2.000033839253429,
+      "y": -1.0000190423982707
+    },
+    {
+      "id": 2,
+      "rotation_radians": 3.141595415874885,
+      "x": -4.3116620003076745e-06,
+      "y": -1.0000174852818229
+    },
+    {
+      "id": 3,
+      "rotation_radians": 3.619286297433364e-06,
+      "x": 2.000032982352467,
+      "y": -1.00002035441768
+    },
+    {
+      "id": 4,
+      "rotation_radians": 6.283187837994955,
+      "x": -2.000035922701681,
+      "y": 3.819382934038143e-06
+    },
+    {
+      "id": 5,
+      "rotation_radians": 6.283188040756427,
+      "x": -2.8313438263452805e-06,
+      "y": 4.819324933834533e-06
+    },
+    {
+      "id": 6,
+      "rotation_radians": 3.7960562003022356e-06,
+      "x": 2.0000386719337584,
+      "y": -3.907581416887514e-06
+    },
+    {
+      "id": 7,
+      "rotation_radians": 6.28318764872637,
+      "x": -2.000025907176924,
+      "y": 1.0000218025466947
+    }
+  ],
+  "problem": "7 equal items for Dominoes in Dominoes",
+  "setup": "domindom",
+  "source_image": "https://erich-friedman.github.io/packing/domindom/9.gif",
+  "source_page": "https://erich-friedman.github.io/packing/domindom/"
+}

data/solutions/generated/domindom_008_a4f6452b97bb5cc0.json

@@ -0,0 +1,109 @@
+{
+  "case": "domindom@8",
+  "container": {
+    "height": 3.0000496969324035,
+    "type": "rectangle",
+    "width": 6.000099393864807
+  },
+  "friedman_reference": {
+    "analytical_or_proved": true,
+    "case": "domindom@8",
+    "credit": "Trivial",
+    "date": "",
+    "family_updated": "",
+    "metric_expression": "3",
+    "metric_symbol": "s",
+    "metric_value": 3.0,
+    "reference_text": "7.-9. | | s = 3 | Trivial.",
+    "source_image": "9.gif",
+    "source_image_url": "https://erich-friedman.github.io/packing/domindom/9.gif",
+    "source_page": "https://erich-friedman.github.io/packing/domindom/",
+    "title": "Dominoes in Dominoes"
+  },
+  "item": {
+    "height": 1.0,
+    "type": "rectangle",
+    "width": 2.0
+  },
+  "metadata": {
+    "friedman_reference": {
+      "analytical_or_proved": true,
+      "case": "domindom@8",
+      "credit": "Trivial",
+      "date": "",
+      "family_updated": "",
+      "metric_expression": "3",
+      "metric_symbol": "s",
+      "metric_value": 3.0,
+      "reference_text": "7.-9. | | s = 3 | Trivial.",
+      "source_image": "9.gif",
+      "source_image_url": "https://erich-friedman.github.io/packing/domindom/9.gif",
+      "source_page": "https://erich-friedman.github.io/packing/domindom/",
+      "title": "Dominoes in Dominoes"
+    },
+    "generator": "monotone_subset_upper_bound",
+    "metric_recovery_tolerance_rel": 0.001,
+    "monotone_upper_bound_from_case": "domindom@9",
+    "monotone_upper_bound_from_record": "43b9630822b40990",
+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from domindom@9; this prevents displaying a dominated coordinate reconstruction.",
+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 3.0,
+    "source_url": "https://erich-friedman.github.io/packing/domindom/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
+    "submitted_by": "Trivial"
+  },
+  "placements": [
+    {
+      "id": 1,
+      "rotation_radians": 9.263873246125958e-07,
+      "x": -2.000033839253429,
+      "y": -1.0000190423982707
+    },
+    {
+      "id": 2,
+      "rotation_radians": 3.141595415874885,
+      "x": -4.3116620003076745e-06,
+      "y": -1.0000174852818229
+    },
+    {
+      "id": 3,
+      "rotation_radians": 3.619286297433364e-06,
+      "x": 2.000032982352467,
+      "y": -1.00002035441768
+    },
+    {
+      "id": 4,
+      "rotation_radians": 6.283187837994955,
+      "x": -2.000035922701681,
+      "y": 3.819382934038143e-06
+    },
+    {
+      "id": 5,
+      "rotation_radians": 6.283188040756427,
+      "x": -2.8313438263452805e-06,
+      "y": 4.819324933834533e-06
+    },
+    {
+      "id": 6,
+      "rotation_radians": 3.7960562003022356e-06,
+      "x": 2.0000386719337584,
+      "y": -3.907581416887514e-06
+    },
+    {
+      "id": 7,
+      "rotation_radians": 6.28318764872637,
+      "x": -2.000025907176924,
+      "y": 1.0000218025466947
+    },
+    {
+      "id": 8,
+      "rotation_radians": 3.1415956249077306,
+      "x": 1.4882322040403512e-05,
+      "y": 1.0000215017270928
+    }
+  ],
+  "problem": "8 equal items for Dominoes in Dominoes",
+  "setup": "domindom",
+  "source_image": "https://erich-friedman.github.io/packing/domindom/9.gif",
+  "source_page": "https://erich-friedman.github.io/packing/domindom/"
+}

data/solutions/generated/domindom_014_6f795311544960ee.json

@@ -0,0 +1,145 @@
+{
+  "case": "domindom@14",
+  "container": {
+    "height": 4.000080890451122,
+    "type": "rectangle",
+    "width": 8.000161780902245
+  },
+  "friedman_reference": {
+    "analytical_or_proved": true,
+    "case": "domindom@14",
+    "credit": "Trivial",
+    "date": "",
+    "family_updated": "",
+    "metric_expression": "4",
+    "metric_symbol": "s",
+    "metric_value": 4.0,
+    "reference_text": "14.-16 | | s = 4 | Trivial.",
+    "source_image": "16.gif",
+    "source_image_url": "https://erich-friedman.github.io/packing/domindom/16.gif",
+    "source_page": "https://erich-friedman.github.io/packing/domindom/",
+    "title": "Dominoes in Dominoes"
+  },
+  "item": {
+    "height": 1.0,
+    "type": "rectangle",
+    "width": 2.0
+  },
+  "metadata": {
+    "friedman_reference": {
+      "analytical_or_proved": true,
+      "case": "domindom@14",
+      "credit": "Trivial",
+      "date": "",
+      "family_updated": "",
+      "metric_expression": "4",
+      "metric_symbol": "s",
+      "metric_value": 4.0,
+      "reference_text": "14.-16 | | s = 4 | Trivial.",
+      "source_image": "16.gif",
+      "source_image_url": "https://erich-friedman.github.io/packing/domindom/16.gif",
+      "source_page": "https://erich-friedman.github.io/packing/domindom/",
+      "title": "Dominoes in Dominoes"
+    },
+    "generator": "monotone_subset_upper_bound",
+    "metric_recovery_tolerance_rel": 0.001,
+    "monotone_upper_bound_from_case": "domindom@16",
+    "monotone_upper_bound_from_record": "5102d8436800a7ff",
+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from domindom@16; this prevents displaying a dominated coordinate reconstruction.",
+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 4.0,
+    "source_url": "https://erich-friedman.github.io/packing/domindom/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
+    "submitted_by": "Trivial"
+  },
+  "placements": [
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+    {
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+      "x": -0.9999956531073101,
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+    {
+      "id": 7,
+      "rotation_radians": -1.6237729528698304e-06,
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+      "y": -0.5000088554865022
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+      "rotation_radians": -2.223663485252913e-06,
+      "x": 3.0000715669140843,
+      "y": -0.500009200115029
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+    {
+      "id": 9,
+      "rotation_radians": -1.928971444131956e-06,
+      "x": -3.0000588262125683,
+      "y": 0.5000092891905134
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+    {
+      "id": 10,
+      "rotation_radians": -3.5734767730511228e-06,
+      "x": -1.0000154886954458,
+      "y": 0.5000114110302204
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+    {
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+      "rotation_radians": -2.0633687007540427e-06,
+      "x": 1.0000221646621588,
+      "y": 0.5000114365475254
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+    {
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+      "rotation_radians": -1.5254541101185505e-06,
+      "x": 3.000063012720547,
+      "y": 0.5000149417918678
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+    {
+      "id": 13,
+      "rotation_radians": -2.5255375621672033e-06,
+      "x": -3.0000662461592107,
+      "y": 1.5000297167676229
+    },
+    {
+      "id": 14,
+      "rotation_radians": -2.7791420435910494e-06,
+      "x": -1.0000308198438532,
+      "y": 1.5000325676233306
+    }
+  ],
+  "problem": "14 equal items for Dominoes in Dominoes",
+  "setup": "domindom",
+  "source_image": "https://erich-friedman.github.io/packing/domindom/16.gif",
+  "source_page": "https://erich-friedman.github.io/packing/domindom/"
+}

data/solutions/generated/domindom_015_dda9136316a32e15.json

@@ -0,0 +1,151 @@
+{
+  "case": "domindom@15",
+  "container": {
+    "height": 4.000080890451122,
+    "type": "rectangle",
+    "width": 8.000161780902245
+  },
+  "friedman_reference": {
+    "analytical_or_proved": true,
+    "case": "domindom@15",
+    "credit": "Trivial",
+    "date": "",
+    "family_updated": "",
+    "metric_expression": "4",
+    "metric_symbol": "s",
+    "metric_value": 4.0,
+    "reference_text": "14.-16 | | s = 4 | Trivial.",
+    "source_image": "16.gif",
+    "source_image_url": "https://erich-friedman.github.io/packing/domindom/16.gif",
+    "source_page": "https://erich-friedman.github.io/packing/domindom/",
+    "title": "Dominoes in Dominoes"
+  },
+  "item": {
+    "height": 1.0,
+    "type": "rectangle",
+    "width": 2.0
+  },
+  "metadata": {
+    "friedman_reference": {
+      "analytical_or_proved": true,
+      "case": "domindom@15",
+      "credit": "Trivial",
+      "date": "",
+      "family_updated": "",
+      "metric_expression": "4",
+      "metric_symbol": "s",
+      "metric_value": 4.0,
+      "reference_text": "14.-16 | | s = 4 | Trivial.",
+      "source_image": "16.gif",
+      "source_image_url": "https://erich-friedman.github.io/packing/domindom/16.gif",
+      "source_page": "https://erich-friedman.github.io/packing/domindom/",
+      "title": "Dominoes in Dominoes"
+    },
+    "generator": "monotone_subset_upper_bound",
+    "metric_recovery_tolerance_rel": 0.001,
+    "monotone_upper_bound_from_case": "domindom@16",
+    "monotone_upper_bound_from_record": "5102d8436800a7ff",
+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from domindom@16; this prevents displaying a dominated coordinate reconstruction.",
+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 4.0,
+    "source_url": "https://erich-friedman.github.io/packing/domindom/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
+    "submitted_by": "Trivial"
+  },
+  "placements": [
+    {
+      "id": 1,
+      "rotation_radians": -4.6972688014620373e-07,
+      "x": -3.000061323109711,
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+    {
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+      "x": -1.000021531822367,
+      "y": -1.5000330720011108
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+    {
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+      "x": 1.0000165752754013,
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+    {
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+      "y": -1.500031885010241
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+    {
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+      "x": -3.0000428974193705,
+      "y": -0.5000096770564719
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+      "x": -0.9999956531073101,
+      "y": -0.500008129526475
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+      "id": 7,
+      "rotation_radians": -1.6237729528698304e-06,
+      "x": 1.0000413402407151,
+      "y": -0.5000088554865022
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+    {
+      "id": 8,
+      "rotation_radians": -2.223663485252913e-06,
+      "x": 3.0000715669140843,
+      "y": -0.500009200115029
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+    {
+      "id": 9,
+      "rotation_radians": -1.928971444131956e-06,
+      "x": -3.0000588262125683,
+      "y": 0.5000092891905134
+    },
+    {
+      "id": 10,
+      "rotation_radians": -3.5734767730511228e-06,
+      "x": -1.0000154886954458,
+      "y": 0.5000114110302204
+    },
+    {
+      "id": 11,
+      "rotation_radians": -2.0633687007540427e-06,
+      "x": 1.0000221646621588,
+      "y": 0.5000114365475254
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+    {
+      "id": 12,
+      "rotation_radians": -1.5254541101185505e-06,
+      "x": 3.000063012720547,
+      "y": 0.5000149417918678
+    },
+    {
+      "id": 13,
+      "rotation_radians": -2.5255375621672033e-06,
+      "x": -3.0000662461592107,
+      "y": 1.5000297167676229
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+    {
+      "id": 14,
+      "rotation_radians": -2.7791420435910494e-06,
+      "x": -1.0000308198438532,
+      "y": 1.5000325676233306
+    },
+    {
+      "id": 15,
+      "rotation_radians": -3.2813575841223767e-06,
+      "x": 1.0000101503778438,
+      "y": 1.5000329269853367
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+  ],
+  "problem": "15 equal items for Dominoes in Dominoes",
+  "setup": "domindom",
+  "source_image": "https://erich-friedman.github.io/packing/domindom/16.gif",
+  "source_page": "https://erich-friedman.github.io/packing/domindom/"
+}

data/solutions/generated/hexincir_006_0d33963cb4bcb842.json

@@ -0,0 +1,97 @@
+{
+  "case": "hexincir@6",
+  "container": {
+    "radius": 2.6457937463486663,
+    "type": "circle"
+  },
+  "friedman_reference": {
+    "analytical_or_proved": true,
+    "case": "hexincir@6",
+    "credit": "Trivial",
+    "date": "",
+    "family_updated": "",
+    "metric_expression": "\u221a7 = 2.645+",
+    "metric_symbol": "r",
+    "metric_value": 2.6457513110645907,
+    "reference_text": "r = \u221a7 = 2.645+ | Trivial. |",
+    "source_image": "7.png",
+    "source_image_url": "https://erich-friedman.github.io/packing/hexincir/7.png",
+    "source_page": "https://erich-friedman.github.io/packing/hexincir/",
+    "title": "Hexagons in Circles"
+  },
+  "item": {
+    "circumradius": 1.0000000000000002,
+    "side_length": 1.0,
+    "sides": 6,
+    "type": "regular_polygon"
+  },
+  "metadata": {
+    "friedman_reference": {
+      "analytical_or_proved": true,
+      "case": "hexincir@6",
+      "credit": "Trivial",
+      "date": "",
+      "family_updated": "",
+      "metric_expression": "\u221a7 = 2.645+",
+      "metric_symbol": "r",
+      "metric_value": 2.6457513110645907,
+      "reference_text": "r = \u221a7 = 2.645+ | Trivial. |",
+      "source_image": "7.png",
+      "source_image_url": "https://erich-friedman.github.io/packing/hexincir/7.png",
+      "source_page": "https://erich-friedman.github.io/packing/hexincir/",
+      "title": "Hexagons in Circles"
+    },
+    "generator": "monotone_subset_upper_bound",
+    "metric_recovery_tolerance_rel": 0.001,
+    "monotone_upper_bound_from_case": "hexincir@7",
+    "monotone_upper_bound_from_record": "5cd27a740cc9b2a5",
+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from hexincir@7; this prevents displaying a dominated coordinate reconstruction.",
+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 2.6457513110645907,
+    "source_url": "https://erich-friedman.github.io/packing/hexincir/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
+    "submitted_by": "Trivial"
+  },
+  "placements": [
+    {
+      "id": 1,
+      "rotation_radians": 6.3330598593198735,
+      "x": -1.4549954137239105,
+      "y": -0.939747069682325
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+    {
+      "id": 2,
+      "rotation_radians": 3.1914712204700044,
+      "x": 0.08633526573246775,
+      "y": -1.7299351830412515
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+    {
+      "id": 3,
+      "rotation_radians": 4.238668164264897,
+      "x": 1.5413350295213695,
+      "y": -0.7901966101691185
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+    {
+      "id": 4,
+      "rotation_radians": 5.285871326554258,
+      "x": -1.5413388940335084,
+      "y": 0.7901856736152597
+    },
+    {
+      "id": 5,
+      "rotation_radians": 1.0970775663574264,
+      "x": -1.4678146157623213e-06,
+      "y": -2.29284965591799e-06
+    },
+    {
+      "id": 6,
+      "rotation_radians": 1.0970757782442058,
+      "x": 1.4549931140732282,
+      "y": 0.939739135674936
+    }
+  ],
+  "problem": "6 equal items for Hexagons in Circles",
+  "setup": "hexincir",
+  "source_image": "https://erich-friedman.github.io/packing/hexincir/7.png",
+  "source_page": "https://erich-friedman.github.io/packing/hexincir/"
+}

data/solutions/generated/hexincir_018_3bf05115c19cc6d5.json

@@ -0,0 +1,169 @@
+{
+  "case": "hexincir@18",
+  "container": {
+    "radius": 4.358986819791536,
+    "type": "circle"
+  },
+  "friedman_reference": {
+    "analytical_or_proved": true,
+    "case": "hexincir@18",
+    "credit": "Trivial",
+    "date": "",
+    "family_updated": "",
+    "metric_expression": "\u221a19 = 4.35889+",
+    "metric_symbol": "r",
+    "metric_value": 4.358898943540673,
+    "reference_text": "r = \u221a19 = 4.35889+ | Trivial.",
+    "source_image": "19.png",
+    "source_image_url": "https://erich-friedman.github.io/packing/hexincir/19.png",
+    "source_page": "https://erich-friedman.github.io/packing/hexincir/",
+    "title": "Hexagons in Circles"
+  },
+  "item": {
+    "circumradius": 1.0000000000000002,
+    "side_length": 1.0,
+    "sides": 6,
+    "type": "regular_polygon"
+  },
+  "metadata": {
+    "friedman_reference": {
+      "analytical_or_proved": true,
+      "case": "hexincir@18",
+      "credit": "Trivial",
+      "date": "",
+      "family_updated": "",
+      "metric_expression": "\u221a19 = 4.35889+",
+      "metric_symbol": "r",
+      "metric_value": 4.358898943540673,
+      "reference_text": "r = \u221a19 = 4.35889+ | Trivial.",
+      "source_image": "19.png",
+      "source_image_url": "https://erich-friedman.github.io/packing/hexincir/19.png",
+      "source_page": "https://erich-friedman.github.io/packing/hexincir/",
+      "title": "Hexagons in Circles"
+    },
+    "generator": "monotone_subset_upper_bound",
+    "metric_recovery_tolerance_rel": 0.001,
+    "monotone_upper_bound_from_case": "hexincir@19",
+    "monotone_upper_bound_from_record": "b4bf200670caf09c",
+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from hexincir@19; this prevents displaying a dominated coordinate reconstruction.",
+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 4.358898943540673,
+    "source_url": "https://erich-friedman.github.io/packing/hexincir/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
+    "submitted_by": "Trivial"
+  },
+  "placements": [
+    {
+      "id": 1,
+      "rotation_radians": 5.2399976638384365,
+      "x": -2.993096147482499,
+      "y": -1.744107671477441
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+    {
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+      "rotation_radians": 2.0984018494288867,
+      "x": -1.6677528334111735,
+      "y": -2.915577274740411
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+    {
+      "id": 3,
+      "rotation_radians": 1.0512116602031218,
+      "x": 0.013891971052751357,
+      "y": -3.464152351237459
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+    {
+      "id": 4,
+      "rotation_radians": 3.145600457722643,
+      "x": 1.691129402709777,
+      "y": -2.902180436005106
+    },
+    {
+      "id": 5,
+      "rotation_radians": 2.098400068373407,
+      "x": 3.0069888858544633,
+      "y": -1.720053117635764
+    },
+    {
+      "id": 6,
+      "rotation_radians": 3.14560045873877,
+      "x": -3.3589161081697547,
+      "y": -0.013470553824323309
+    },
+    {
+      "id": 7,
+      "rotation_radians": 4.192809995835601,
+      "x": -1.3226921165278105,
+      "y": -1.170418572303279
+    },
+    {
+      "id": 8,
+      "rotation_radians": 5.240003404360133,
+      "x": 0.3522663278341521,
+      "y": -1.7306949421585403
+    },
+    {
+      "id": 9,
+      "rotation_radians": 3.145611740979831,
+      "x": 1.674958299259687,
+      "y": -0.5602757350145259
+    },
+    {
+      "id": 10,
+      "rotation_radians": 1.0512016870378174,
+      "x": 3.3588324067012905,
+      "y": 0.01346187631256563
+    },
+    {
+      "id": 11,
+      "rotation_radians": 4.1927980473842705,
+      "x": -3.0069911264837548,
+      "y": 1.7200456374958961
+    },
+    {
+      "id": 12,
+      "rotation_radians": 0.004022977779067363,
+      "x": -1.6749557739777987,
+      "y": 0.5602757508940872
+    },
+    {
+      "id": 13,
+      "rotation_radians": 0.00402115019557869,
+      "x": -1.952759009739562e-06,
+      "y": -1.781864378349866e-07
+    },
+    {
+      "id": 14,
+      "rotation_radians": 4.1928019448498866,
+      "x": 1.3226935656547805,
+      "y": 1.1704218333625105
+    },
+    {
+      "id": 15,
+      "rotation_radians": 2.098397030345533,
+      "x": 2.9931018352468324,
+      "y": 1.7441011531233162
+    },
+    {
+      "id": 16,
+      "rotation_radians": 0.004010292918059301,
+      "x": -1.691076788634328,
+      "y": 2.902089381699464
+    },
+    {
+      "id": 17,
+      "rotation_radians": 5.240005872706958,
+      "x": -0.35226794414100643,
+      "y": 1.7306961587426004
+    },
+    {
+      "id": 18,
+      "rotation_radians": 3.1456036906204337,
+      "x": -0.013894218336600163,
+      "y": 3.464154275882162
+    }
+  ],
+  "problem": "18 equal items for Hexagons in Circles",
+  "setup": "hexincir",
+  "source_image": "https://erich-friedman.github.io/packing/hexincir/19.png",
+  "source_page": "https://erich-friedman.github.io/packing/hexincir/"
+}

data/solutions/generated/hexinhex_002_c8ccd859ecc810e4.json

@@ -0,0 +1,76 @@
+{
+  "case": "hexinhex@2",
+  "container": {
+    "circumradius": 2.0000156772159707,
+    "orientation_radians": 0.0,
+    "side_length": 2.0000156772159703,
+    "sides": 6,
+    "type": "regular_polygon"
+  },
+  "friedman_reference": {
+    "analytical_or_proved": true,
+    "case": "hexinhex@2",
+    "credit": "Trivial",
+    "date": "",
+    "family_updated": "",
+    "metric_expression": "2",
+    "metric_symbol": "s",
+    "metric_value": 2.0,
+    "reference_text": "2. | | s = 2 | Trivial.",
+    "source_image": "2.gif",
+    "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/2.gif",
+    "source_page": "https://erich-friedman.github.io/packing/hexinhex/",
+    "title": "Hexagons in Hexagons"
+  },
+  "item": {
+    "circumradius": 1.0000000000000002,
+    "side_length": 1.0,
+    "sides": 6,
+    "type": "regular_polygon"
+  },
+  "metadata": {
+    "friedman_reference": {
+      "analytical_or_proved": true,
+      "case": "hexinhex@2",
+      "credit": "Trivial",
+      "date": "",
+      "family_updated": "",
+      "metric_expression": "2",
+      "metric_symbol": "s",
+      "metric_value": 2.0,
+      "reference_text": "2. | | s = 2 | Trivial.",
+      "source_image": "2.gif",
+      "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/2.gif",
+      "source_page": "https://erich-friedman.github.io/packing/hexinhex/",
+      "title": "Hexagons in Hexagons"
+    },
+    "generator": "monotone_subset_upper_bound",
+    "metric_recovery_tolerance_rel": 0.001,
+    "monotone_upper_bound_from_case": "hexinhex@3",
+    "monotone_upper_bound_from_record": "2e22e12b6d11de4d",
+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from hexinhex@3; this prevents displaying a dominated coordinate reconstruction.",
+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 2.0,
+    "source_url": "https://erich-friedman.github.io/packing/hexinhex/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
+    "submitted_by": "Trivial"
+  },
+  "placements": [
+    {
+      "id": 1,
+      "rotation_radians": -8.73652531392515e-06,
+      "x": -1.0000075000240263,
+      "y": 5.11231640226178e-07
+    },
+    {
+      "id": 2,
+      "rotation_radians": -4.568606117872516e-06,
+      "x": 0.5000030248779901,
+      "y": 0.8660316868560286
+    }
+  ],
+  "problem": "2 equal items for Hexagons in Hexagons",
+  "setup": "hexinhex",
+  "source_image": "https://erich-friedman.github.io/packing/hexinhex/2.gif",
+  "source_page": "https://erich-friedman.github.io/packing/hexinhex/"
+}

data/solutions/generated/hexinhex_006_93e84dfdad8f0036.json

@@ -0,0 +1,100 @@
+{
+  "case": "hexinhex@6",
+  "container": {
+    "circumradius": 2.886797667992364,
+    "orientation_radians": 0.0,
+    "side_length": 2.8867976679923637,
+    "sides": 6,
+    "type": "regular_polygon"
+  },
+  "friedman_reference": {
+    "analytical_or_proved": true,
+    "case": "hexinhex@6",
+    "credit": "Maurizio Morandi",
+    "date": "April 2015",
+    "family_updated": "",
+    "metric_expression": "5/\u221a3 = 2.886+",
+    "metric_symbol": "s",
+    "metric_value": 2.8867513459481287,
+    "reference_text": "6-7. | | s = 5/\u221a3 = 2.886+ | Found by Maurizio Morandi | in April 2015.",
+    "source_image": "7.gif",
+    "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/7.gif",
+    "source_page": "https://erich-friedman.github.io/packing/hexinhex/",
+    "title": "Hexagons in Hexagons"
+  },
+  "item": {
+    "circumradius": 1.0000000000000002,
+    "side_length": 1.0,
+    "sides": 6,
+    "type": "regular_polygon"
+  },
+  "metadata": {
+    "friedman_reference": {
+      "analytical_or_proved": true,
+      "case": "hexinhex@6",
+      "credit": "Maurizio Morandi",
+      "date": "April 2015",
+      "family_updated": "",
+      "metric_expression": "5/\u221a3 = 2.886+",
+      "metric_symbol": "s",
+      "metric_value": 2.8867513459481287,
+      "reference_text": "6-7. | | s = 5/\u221a3 = 2.886+ | Found by Maurizio Morandi | in April 2015.",
+      "source_image": "7.gif",
+      "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/7.gif",
+      "source_page": "https://erich-friedman.github.io/packing/hexinhex/",
+      "title": "Hexagons in Hexagons"
+    },
+    "generator": "monotone_subset_upper_bound",
+    "metric_recovery_tolerance_rel": 0.001,
+    "monotone_upper_bound_from_case": "hexinhex@7",
+    "monotone_upper_bound_from_record": "d2a43d4ca2836467",
+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from hexinhex@7; this prevents displaying a dominated coordinate reconstruction.",
+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 2.8867513459481287,
+    "source_url": "https://erich-friedman.github.io/packing/hexinhex/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
+    "submitted_by": "Maurizio Morandi"
+  },
+  "placements": [
+    {
+      "id": 1,
+      "rotation_radians": 0.5226417137671361,
+      "x": -8.279927955112656e-07,
+      "y": -4.711760804674806e-07
+    },
+    {
+      "id": 2,
+      "rotation_radians": 0.5226452546369924,
+      "x": 0.8660426984788827,
+      "y": 1.500032424149046
+    },
+    {
+      "id": 3,
+      "rotation_radians": 0.5226480261281576,
+      "x": 1.732086250541385,
+      "y": 8.58826264170227e-08
+    },
+    {
+      "id": 4,
+      "rotation_radians": 0.5226486434017761,
+      "x": -0.8660447233947516,
+      "y": 1.5000348035197788
+    },
+    {
+      "id": 5,
+      "rotation_radians": 0.522645168867023,
+      "x": 0.8660436687488999,
+      "y": -1.5000283075524146
+    },
+    {
+      "id": 6,
+      "rotation_radians": 0.5226481118300205,
+      "x": -1.7320900713881753,
+      "y": 1.7675679263296385e-06
+    }
+  ],
+  "problem": "6 equal items for Hexagons in Hexagons",
+  "setup": "hexinhex",
+  "source_image": "https://erich-friedman.github.io/packing/hexinhex/7.gif",
+  "source_page": "https://erich-friedman.github.io/packing/hexinhex/"
+}

data/solutions/generated/hexinhex_018_2eb7c5be1ce30256.json

@@ -0,0 +1,172 @@
+{
+  "case": "hexinhex@18",
+  "container": {
+    "circumradius": 4.618904616698239,
+    "orientation_radians": 0.0,
+    "side_length": 4.618904616698238,
+    "sides": 6,
+    "type": "regular_polygon"
+  },
+  "friedman_reference": {
+    "analytical_or_proved": true,
+    "case": "hexinhex@18",
+    "credit": "Maurizio Morandi",
+    "date": "April 2015",
+    "family_updated": "",
+    "metric_expression": "8/\u221a3 = 4.618+",
+    "metric_symbol": "s",
+    "metric_value": 4.618802153517006,
+    "reference_text": "18-19. | | s = 8/\u221a3 = 4.618+ | Found by Maurizio Morandi | in April 2015. |",
+    "source_image": "19.gif",
+    "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/19.gif",
+    "source_page": "https://erich-friedman.github.io/packing/hexinhex/",
+    "title": "Hexagons in Hexagons"
+  },
+  "item": {
+    "circumradius": 1.0000000000000002,
+    "side_length": 1.0,
+    "sides": 6,
+    "type": "regular_polygon"
+  },
+  "metadata": {
+    "friedman_reference": {
+      "analytical_or_proved": true,
+      "case": "hexinhex@18",
+      "credit": "Maurizio Morandi",
+      "date": "April 2015",
+      "family_updated": "",
+      "metric_expression": "8/\u221a3 = 4.618+",
+      "metric_symbol": "s",
+      "metric_value": 4.618802153517006,
+      "reference_text": "18-19. | | s = 8/\u221a3 = 4.618+ | Found by Maurizio Morandi | in April 2015. |",
+      "source_image": "19.gif",
+      "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/19.gif",
+      "source_page": "https://erich-friedman.github.io/packing/hexinhex/",
+      "title": "Hexagons in Hexagons"
+    },
+    "generator": "monotone_subset_upper_bound",
+    "metric_recovery_tolerance_rel": 0.001,
+    "monotone_upper_bound_from_case": "hexinhex@19",
+    "monotone_upper_bound_from_record": "4b11a1a09f2a5b70",
+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from hexinhex@19; this prevents displaying a dominated coordinate reconstruction.",
+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 4.618802153517006,
+    "source_url": "https://erich-friedman.github.io/packing/hexinhex/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
+    "submitted_by": "Maurizio Morandi"
+  },
+  "placements": [
+    {
+      "id": 1,
+      "rotation_radians": 0.5234558836128137,
+      "x": -3.2842232400588377e-06,
+      "y": -2.550710319198415e-06
+    },
+    {
+      "id": 2,
+      "rotation_radians": 0.5234571589934958,
+      "x": 0.8660481685111352,
+      "y": 1.500029948325956
+    },
+    {
+      "id": 3,
+      "rotation_radians": 0.5234570967640373,
+      "x": 1.7320938297963193,
+      "y": -8.064443215692669e-06
+    },
+    {
+      "id": 4,
+      "rotation_radians": 0.5234554583723405,
+      "x": -0.8660438438856551,
+      "y": 1.5000387372562904
+    },
+    {
+      "id": 5,
+      "rotation_radians": 0.5234545041986823,
+      "x": 0.8660399818573424,
+      "y": -1.500042129809262
+    },
+    {
+      "id": 6,
+      "rotation_radians": 0.5234579503833225,
+      "x": -0.8660544847172335,
+      "y": -1.500035394945919
+    },
+    {
+      "id": 7,
+      "rotation_radians": 0.52346195389581,
+      "x": -1.7320951414134365,
+      "y": 5.430451211204787e-06
+    },
+    {
+      "id": 8,
+      "rotation_radians": 0.5234557148680753,
+      "x": 2.5981460007653134,
+      "y": 1.5000306261704976
+    },
+    {
+      "id": 9,
+      "rotation_radians": 0.5234592140135343,
+      "x": 2.5981379335626027,
+      "y": -1.5000421196868152
+    },
+    {
+      "id": 10,
+      "rotation_radians": 0.5234590773684019,
+      "x": 9.140162784597355e-06,
+      "y": 3.00007374315004
+    },
+    {
+      "id": 11,
+      "rotation_radians": 0.5234566299912738,
+      "x": 1.7320973103430997,
+      "y": 3.0000666873211483
+    },
+    {
+      "id": 12,
+      "rotation_radians": 0.5234534012478698,
+      "x": 3.46418783791881,
+      "y": -4.057655083085431e-06
+    },
+    {
+      "id": 13,
+      "rotation_radians": 0.5234663480425249,
+      "x": -2.5981382202967835,
+      "y": 1.5000450150446436
+    },
+    {
+      "id": 14,
+      "rotation_radians": 0.5234535534152779,
+      "x": -7.838950909804222e-06,
+      "y": -3.0000777144674013
+    },
+    {
+      "id": 15,
+      "rotation_radians": 0.5234618407713834,
+      "x": -1.7320865313307823,
+      "y": 3.0000746422542264
+    },
+    {
+      "id": 16,
+      "rotation_radians": 0.5234563132987613,
+      "x": 1.7320881414458684,
+      "y": -3.000074376021685
+    },
+    {
+      "id": 17,
+      "rotation_radians": 0.5234645330794415,
+      "x": -2.598149015132673,
+      "y": -1.5000313054053336
+    },
+    {
+      "id": 18,
+      "rotation_radians": 0.5234692642158965,
+      "x": -3.4641888581596616,
+      "y": 8.83368252839888e-06
+    }
+  ],
+  "problem": "18 equal items for Hexagons in Hexagons",
+  "setup": "hexinhex",
+  "source_image": "https://erich-friedman.github.io/packing/hexinhex/19.gif",
+  "source_page": "https://erich-friedman.github.io/packing/hexinhex/"
+}

data/solutions/generated/hexinhex_020_197dfdbe3f1f6905.json

@@ -0,0 +1,184 @@
+{
+  "case": "hexinhex@20",
+  "container": {
+    "circumradius": 5.000067508172298,
+    "orientation_radians": 0.0,
+    "side_length": 5.000067508172297,
+    "sides": 6,
+    "type": "regular_polygon"
+  },
+  "friedman_reference": {
+    "analytical_or_proved": true,
+    "case": "hexinhex@20",
+    "credit": "Trivial",
+    "date": "",
+    "family_updated": "",
+    "metric_expression": "5",
+    "metric_symbol": "s",
+    "metric_value": 5.0,
+    "reference_text": "20-21. | | s = 5 | Trivial. | |",
+    "source_image": "21.gif",
+    "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/21.gif",
+    "source_page": "https://erich-friedman.github.io/packing/hexinhex/",
+    "title": "Hexagons in Hexagons"
+  },
+  "item": {
+    "circumradius": 1.0000000000000002,
+    "side_length": 1.0,
+    "sides": 6,
+    "type": "regular_polygon"
+  },
+  "metadata": {
+    "friedman_reference": {
+      "analytical_or_proved": true,
+      "case": "hexinhex@20",
+      "credit": "Trivial",
+      "date": "",
+      "family_updated": "",
+      "metric_expression": "5",
+      "metric_symbol": "s",
+      "metric_value": 5.0,
+      "reference_text": "20-21. | | s = 5 | Trivial. | |",
+      "source_image": "21.gif",
+      "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/21.gif",
+      "source_page": "https://erich-friedman.github.io/packing/hexinhex/",
+      "title": "Hexagons in Hexagons"
+    },
+    "generator": "monotone_subset_upper_bound",
+    "metric_recovery_tolerance_rel": 0.001,
+    "monotone_upper_bound_from_case": "hexinhex@21",
+    "monotone_upper_bound_from_record": "4a431b677be571c9",
+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from hexinhex@21; this prevents displaying a dominated coordinate reconstruction.",
+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 5.0,
+    "source_url": "https://erich-friedman.github.io/packing/hexinhex/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
+    "submitted_by": "Trivial"
+  },
+  "placements": [
+    {
+      "id": 1,
+      "rotation_radians": 7.901581286981117e-07,
+      "x": -0.5000174463538043,
+      "y": 0.8660408791733488
+    },
+    {
+      "id": 2,
+      "rotation_radians": -8.371924427954672e-07,
+      "x": -0.49998926754271145,
+      "y": -0.8660342103698403
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+    {
+      "id": 3,
+      "rotation_radians": -2.660678494033981e-07,
+      "x": 1.0000192810757436,
+      "y": 1.7876577640023134e-05
+    },
+    {
+      "id": 4,
+      "rotation_radians": -7.438935821621196e-08,
+      "x": -2.000028448902631,
+      "y": -1.3886483582868472e-05
+    },
+    {
+      "id": 5,
+      "rotation_radians": -3.2787528884077454e-07,
+      "x": 1.0000286771626443,
+      "y": -1.7320685106263434
+    },
+    {
+      "id": 6,
+      "rotation_radians": -5.648917492132542e-07,
+      "x": 1.0000042693678135,
+      "y": 1.732087424964717
+    },
+    {
+      "id": 7,
+      "rotation_radians": 7.623293417932666e-07,
+      "x": -1.9999814394233528,
+      "y": -1.7320956192726793
+    },
+    {
+      "id": 8,
+      "rotation_radians": 5.726382936736113e-07,
+      "x": -2.0001063176145357,
+      "y": 1.7320682172424777
+    },
+    {
+      "id": 9,
+      "rotation_radians": -6.540268107444737e-07,
+      "x": -0.49996917052936474,
+      "y": -2.598161234269701
+    },
+    {
+      "id": 10,
+      "rotation_radians": 5.893385085121173e-07,
+      "x": -0.5000298003785586,
+      "y": 2.5981094128802877
+    },
+    {
+      "id": 11,
+      "rotation_radians": -1.9986158879680022e-07,
+      "x": 2.500045944099602,
+      "y": 0.8660744448134621
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+    {
+      "id": 12,
+      "rotation_radians": 5.977223461539629e-08,
+      "x": 2.500039054190333,
+      "y": -0.8660052621280638
+    },
+    {
+      "id": 13,
+      "rotation_radians": -1.3478963937982246e-08,
+      "x": -3.500092072845965,
+      "y": 0.8659632371807578
+    },
+    {
+      "id": 14,
+      "rotation_radians": 4.1132012924842904e-07,
+      "x": -3.50001719472001,
+      "y": -0.8661073175269502
+    },
+    {
+      "id": 15,
+      "rotation_radians": 5.546656121720664e-07,
+      "x": 0.9999791170267553,
+      "y": 3.464158149492293
+    },
+    {
+      "id": 16,
+      "rotation_radians": -4.0502975026486896e-07,
+      "x": 1.0000797359239124,
+      "y": -3.464151263809527
+    },
+    {
+      "id": 17,
+      "rotation_radians": -4.772253186330887e-08,
+      "x": -2.000035262364772,
+      "y": 3.464146559555753
+    },
+    {
+      "id": 18,
+      "rotation_radians": 9.1237704507367e-07,
+      "x": -2.000020084800537,
+      "y": -3.4641558808448685
+    },
+    {
+      "id": 19,
+      "rotation_radians": 2.3453726621205355e-07,
+      "x": 2.500069476967483,
+      "y": -2.5980694889362432
+    },
+    {
+      "id": 20,
+      "rotation_radians": 1.69584433962456e-07,
+      "x": 2.500015099600881,
+      "y": 2.598150980951056
+    }
+  ],
+  "problem": "20 equal items for Hexagons in Hexagons",
+  "setup": "hexinhex",
+  "source_image": "https://erich-friedman.github.io/packing/hexinhex/21.gif",
+  "source_page": "https://erich-friedman.github.io/packing/hexinhex/"
+}

data/solutions/generated/hexinhex_029_baea309b7bc15617.json

@@ -0,0 +1,238 @@
+{
+  "case": "hexinhex@29",
+  "container": {
+    "circumradius": 6.000078501746011,
+    "orientation_radians": 0.0,
+    "side_length": 6.00007850174601,
+    "sides": 6,
+    "type": "regular_polygon"
+  },
+  "friedman_reference": {
+    "analytical_or_proved": true,
+    "case": "hexinhex@29",
+    "credit": "Trivial",
+    "date": "",
+    "family_updated": "",
+    "metric_expression": "6",
+    "metric_symbol": "s",
+    "metric_value": 6.0,
+    "reference_text": "29-31. | | s = 6 | Trivial. |",
+    "source_image": "31.gif",
+    "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/31.gif",
+    "source_page": "https://erich-friedman.github.io/packing/hexinhex/",
+    "title": "Hexagons in Hexagons"
+  },
+  "item": {
+    "circumradius": 1.0000000000000002,
+    "side_length": 1.0,
+    "sides": 6,
+    "type": "regular_polygon"
+  },
+  "metadata": {
+    "friedman_reference": {
+      "analytical_or_proved": true,
+      "case": "hexinhex@29",
+      "credit": "Trivial",
+      "date": "",
+      "family_updated": "",
+      "metric_expression": "6",
+      "metric_symbol": "s",
+      "metric_value": 6.0,
+      "reference_text": "29-31. | | s = 6 | Trivial. |",
+      "source_image": "31.gif",
+      "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/31.gif",
+      "source_page": "https://erich-friedman.github.io/packing/hexinhex/",
+      "title": "Hexagons in Hexagons"
+    },
+    "generator": "monotone_subset_upper_bound",
+    "metric_recovery_tolerance_rel": 0.001,
+    "monotone_upper_bound_from_case": "hexinhex@30",
+    "monotone_upper_bound_from_record": "100db5f9f28894cc",
+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from hexinhex@30; this prevents displaying a dominated coordinate reconstruction.",
+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 6.0,
+    "source_url": "https://erich-friedman.github.io/packing/hexinhex/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
+    "submitted_by": "Trivial"
+  },
+  "placements": [
+    {
+      "id": 1,
+      "rotation_radians": -2.1568093060581115e-06,
+      "x": -0.5000046790830123,
+      "y": -0.8660375027573861
+    },
+    {
+      "id": 2,
+      "rotation_radians": -6.91313726895934e-07,
+      "x": 1.0000168855896254,
+      "y": 6.305863012131126e-06
+    },
+    {
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+      "x": 2.500037478603552,
+      "y": -4.330182414425288
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+    {
+      "id": 25,
+      "rotation_radians": 1.4954656685310197e-06,
+      "x": -3.5000399678531604,
+      "y": -2.5981301368837437
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+    {
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+      "x": -3.5000607330501903,
+      "y": 2.5981032880488897
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+    {
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+      "rotation_radians": 9.76757375278175e-07,
+      "x": -0.5000231083448415,
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+    {
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+      "rotation_radians": 9.842621830853817e-07,
+      "x": -5.000068695617427,
+      "y": -7.1268798689748576e-06
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+  ],
+  "problem": "29 equal items for Hexagons in Hexagons",
+  "setup": "hexinhex",
+  "source_image": "https://erich-friedman.github.io/packing/hexinhex/31.gif",
+  "source_page": "https://erich-friedman.github.io/packing/hexinhex/"
+}

data/solutions/generated/hexinsqu_007_6492fb111d24a3e4.json

@@ -0,0 +1,106 @@
+{
+  "case": "hexinsqu@7",
+  "container": {
+    "circumradius": 3.6742599744526214,
+    "orientation_radians": 0.0,
+    "side_length": 5.196188287555518,
+    "sides": 4,
+    "type": "regular_polygon"
+  },
+  "friedman_reference": {
+    "analytical_or_proved": true,
+    "case": "hexinsqu@7",
+    "credit": "Trivial",
+    "date": "",
+    "family_updated": "",
+    "metric_expression": "3\u221a3 = 5.196+",
+    "metric_symbol": "s",
+    "metric_value": 5.196152422706632,
+    "reference_text": "s = 3\u221a3 = 5.196+ | Trivial. |",
+    "source_image": "8.png",
+    "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/8.png",
+    "source_page": "https://erich-friedman.github.io/packing/hexinsqu/",
+    "title": "Hexagons in Squares"
+  },
+  "item": {
+    "circumradius": 1.0000000000000002,
+    "side_length": 1.0,
+    "sides": 6,
+    "type": "regular_polygon"
+  },
+  "metadata": {
+    "friedman_reference": {
+      "analytical_or_proved": true,
+      "case": "hexinsqu@7",
+      "credit": "Trivial",
+      "date": "",
+      "family_updated": "",
+      "metric_expression": "3\u221a3 = 5.196+",
+      "metric_symbol": "s",
+      "metric_value": 5.196152422706632,
+      "reference_text": "s = 3\u221a3 = 5.196+ | Trivial. |",
+      "source_image": "8.png",
+      "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/8.png",
+      "source_page": "https://erich-friedman.github.io/packing/hexinsqu/",
+      "title": "Hexagons in Squares"
+    },
+    "generator": "monotone_subset_upper_bound",
+    "metric_recovery_tolerance_rel": 0.001,
+    "monotone_upper_bound_from_case": "hexinsqu@8",
+    "monotone_upper_bound_from_record": "b1252a56565d2ec0",
+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from hexinsqu@8; this prevents displaying a dominated coordinate reconstruction.",
+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 5.196152422706632,
+    "source_url": "https://erich-friedman.github.io/packing/hexinsqu/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
+    "submitted_by": "Trivial"
+  },
+  "placements": [
+    {
+      "id": 1,
+      "rotation_radians": 0.7854016570627027,
+      "x": 1.0170145342200057,
+      "y": 1.0170134987742179
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+    {
+      "id": 2,
+      "rotation_radians": 0.8221574350601246,
+      "x": 0.7126124858342775,
+      "y": -0.7084831354689171
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+    {
+      "id": 3,
+      "rotation_radians": 0.7853995123906887,
+      "x": 2.3547411738843413,
+      "y": -0.09476567117498827
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+    {
+      "id": 4,
+      "rotation_radians": -0.26179766104249264,
+      "x": -0.09475265327530596,
+      "y": 2.354754819846214
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+    {
+      "id": 5,
+      "rotation_radians": 0.7854059125449453,
+      "x": 0.09478137113003707,
+      "y": -2.3547267913368533
+    },
+    {
+      "id": 6,
+      "rotation_radians": 0.7854052231788333,
+      "x": -2.3547120325827744,
+      "y": 0.09479659533686603
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+    {
+      "id": 7,
+      "rotation_radians": 1.8326012962028284,
+      "x": -1.1299361893550244,
+      "y": -1.1299358322158408
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+  ],
+  "problem": "7 equal items for Hexagons in Squares",
+  "setup": "hexinsqu",
+  "source_image": "https://erich-friedman.github.io/packing/hexinsqu/8.png",
+  "source_page": "https://erich-friedman.github.io/packing/hexinsqu/"
+}

data/solutions/generated/hexintri_013_3b4102a5f66851b2.json

@@ -0,0 +1,142 @@
+{
+  "case": "hexintri@13",
+  "container": {
+    "circumradius": 6.000104811629913,
+    "orientation_radians": 0.0,
+    "side_length": 10.392486384481497,
+    "sides": 3,
+    "type": "regular_polygon"
+  },
+  "friedman_reference": {
+    "analytical_or_proved": true,
+    "case": "hexintri@13",
+    "credit": "Trivial",
+    "date": "",
+    "family_updated": "",
+    "metric_expression": "6\u221a3 = 10.3923+",
+    "metric_symbol": "s",
+    "metric_value": 10.392304845413264,
+    "reference_text": "s = 6\u221a3 = 10.3923+ | Trivial. |",
+    "source_image": "15.gif",
+    "source_image_url": "https://erich-friedman.github.io/packing/hexintri/15.gif",
+    "source_page": "https://erich-friedman.github.io/packing/hexintri/",
+    "title": "Hexagons in Triangles"
+  },
+  "item": {
+    "circumradius": 1.0000000000000002,
+    "side_length": 1.0,
+    "sides": 6,
+    "type": "regular_polygon"
+  },
+  "metadata": {
+    "friedman_reference": {
+      "analytical_or_proved": true,
+      "case": "hexintri@13",
+      "credit": "Trivial",
+      "date": "",
+      "family_updated": "",
+      "metric_expression": "6\u221a3 = 10.3923+",
+      "metric_symbol": "s",
+      "metric_value": 10.392304845413264,
+      "reference_text": "s = 6\u221a3 = 10.3923+ | Trivial. |",
+      "source_image": "15.gif",
+      "source_image_url": "https://erich-friedman.github.io/packing/hexintri/15.gif",
+      "source_page": "https://erich-friedman.github.io/packing/hexintri/",
+      "title": "Hexagons in Triangles"
+    },
+    "generator": "monotone_subset_upper_bound",
+    "metric_recovery_tolerance_rel": 0.001,
+    "monotone_upper_bound_from_case": "hexintri@15",
+    "monotone_upper_bound_from_record": "83460d360e93ff66",
+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from hexintri@15; this prevents displaying a dominated coordinate reconstruction.",
+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 10.392304845413264,
+    "source_url": "https://erich-friedman.github.io/packing/hexintri/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
+    "submitted_by": "Trivial"
+  },
+  "placements": [
+    {
+      "id": 1,
+      "rotation_radians": -0.0008773216340713351,
+      "x": -2.0000433075228283,
+      "y": -3.464175834546559
+    },
+    {
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+      "rotation_radians": -0.0008786916702983763,
+      "x": -0.5000075332102101,
+      "y": -2.598128899162436
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+    {
+      "id": 3,
+      "rotation_radians": -0.0008769437212764101,
+      "x": 1.0000309298712058,
+      "y": -1.7320837124519806
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+    {
+      "id": 4,
+      "rotation_radians": -0.0008724118720011458,
+      "x": 2.5000772782478062,
+      "y": -0.8660374676439979
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+    {
+      "id": 5,
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+      "x": -2.000042996234925,
+      "y": -1.7320829376753015
+    },
+    {
+      "id": 6,
+      "rotation_radians": -0.0008756453918099686,
+      "x": -0.5000066927092914,
+      "y": -0.8660378227886485
+    },
+    {
+      "id": 7,
+      "rotation_radians": -0.0008770382555270698,
+      "x": 1.0000309646052692,
+      "y": 3.883886587223335e-06
+    },
+    {
+      "id": 8,
+      "rotation_radians": -0.552254332594487,
+      "x": 4.19396558651501,
+      "y": 0.01863022051388554
+    },
+    {
+      "id": 9,
+      "rotation_radians": -0.0008768327269119654,
+      "x": -2.0000441233364357,
+      "y": 1.2698259020990741e-05
+    },
+    {
+      "id": 10,
+      "rotation_radians": -0.0008778702033141064,
+      "x": -0.5000087757146733,
+      "y": 0.8660540879660805
+    },
+    {
+      "id": 11,
+      "rotation_radians": -0.0008764852889457795,
+      "x": 1.0000276836712718,
+      "y": 1.7320925544954868
+    },
+    {
+      "id": 12,
+      "rotation_radians": -1.0480709334341858,
+      "x": 2.5000732913403003,
+      "y": 0.8660432719307646
+    },
+    {
+      "id": 13,
+      "rotation_radians": 1.0463206307111212,
+      "x": -2.000046949475865,
+      "y": 1.7321102095159302
+    }
+  ],
+  "problem": "13 equal items for Hexagons in Triangles",
+  "setup": "hexintri",
+  "source_image": "https://erich-friedman.github.io/packing/hexintri/15.gif",
+  "source_page": "https://erich-friedman.github.io/packing/hexintri/"
+}

data/solutions/generated/hexintri_014_aa2a4511d2c644b1.json

@@ -0,0 +1,148 @@
+{
+  "case": "hexintri@14",
+  "container": {
+    "circumradius": 6.000104811629913,
+    "orientation_radians": 0.0,
+    "side_length": 10.392486384481497,
+    "sides": 3,
+    "type": "regular_polygon"
+  },
+  "friedman_reference": {
+    "analytical_or_proved": true,
+    "case": "hexintri@14",
+    "credit": "Trivial",
+    "date": "",
+    "family_updated": "",
+    "metric_expression": "6\u221a3 = 10.3923+",
+    "metric_symbol": "s",
+    "metric_value": 10.392304845413264,
+    "reference_text": "s = 6\u221a3 = 10.3923+ | Trivial. |",
+    "source_image": "15.gif",
+    "source_image_url": "https://erich-friedman.github.io/packing/hexintri/15.gif",
+    "source_page": "https://erich-friedman.github.io/packing/hexintri/",
+    "title": "Hexagons in Triangles"
+  },
+  "item": {
+    "circumradius": 1.0000000000000002,
+    "side_length": 1.0,
+    "sides": 6,
+    "type": "regular_polygon"
+  },
+  "metadata": {
+    "friedman_reference": {
+      "analytical_or_proved": true,
+      "case": "hexintri@14",
+      "credit": "Trivial",
+      "date": "",
+      "family_updated": "",
+      "metric_expression": "6\u221a3 = 10.3923+",
+      "metric_symbol": "s",
+      "metric_value": 10.392304845413264,
+      "reference_text": "s = 6\u221a3 = 10.3923+ | Trivial. |",
+      "source_image": "15.gif",
+      "source_image_url": "https://erich-friedman.github.io/packing/hexintri/15.gif",
+      "source_page": "https://erich-friedman.github.io/packing/hexintri/",
+      "title": "Hexagons in Triangles"
+    },
+    "generator": "monotone_subset_upper_bound",
+    "metric_recovery_tolerance_rel": 0.001,
+    "monotone_upper_bound_from_case": "hexintri@15",
+    "monotone_upper_bound_from_record": "83460d360e93ff66",
+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from hexintri@15; this prevents displaying a dominated coordinate reconstruction.",
+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 10.392304845413264,
+    "source_url": "https://erich-friedman.github.io/packing/hexintri/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
+    "submitted_by": "Trivial"
+  },
+  "placements": [
+    {
+      "id": 1,
+      "rotation_radians": -0.0008773216340713351,
+      "x": -2.0000433075228283,
+      "y": -3.464175834546559
+    },
+    {
+      "id": 2,
+      "rotation_radians": -0.0008786916702983763,
+      "x": -0.5000075332102101,
+      "y": -2.598128899162436
+    },
+    {
+      "id": 3,
+      "rotation_radians": -0.0008769437212764101,
+      "x": 1.0000309298712058,
+      "y": -1.7320837124519806
+    },
+    {
+      "id": 4,
+      "rotation_radians": -0.0008724118720011458,
+      "x": 2.5000772782478062,
+      "y": -0.8660374676439979
+    },
+    {
+      "id": 5,
+      "rotation_radians": -0.0008780149727463441,
+      "x": -2.000042996234925,
+      "y": -1.7320829376753015
+    },
+    {
+      "id": 6,
+      "rotation_radians": -0.0008756453918099686,
+      "x": -0.5000066927092914,
+      "y": -0.8660378227886485
+    },
+    {
+      "id": 7,
+      "rotation_radians": -0.0008770382555270698,
+      "x": 1.0000309646052692,
+      "y": 3.883886587223335e-06
+    },
+    {
+      "id": 8,
+      "rotation_radians": -0.552254332594487,
+      "x": 4.19396558651501,
+      "y": 0.01863022051388554
+    },
+    {
+      "id": 9,
+      "rotation_radians": -0.0008768327269119654,
+      "x": -2.0000441233364357,
+      "y": 1.2698259020990741e-05
+    },
+    {
+      "id": 10,
+      "rotation_radians": -0.0008778702033141064,
+      "x": -0.5000087757146733,
+      "y": 0.8660540879660805
+    },
+    {
+      "id": 11,
+      "rotation_radians": -0.0008764852889457795,
+      "x": 1.0000276836712718,
+      "y": 1.7320925544954868
+    },
+    {
+      "id": 12,
+      "rotation_radians": -1.0480709334341858,
+      "x": 2.5000732913403003,
+      "y": 0.8660432719307646
+    },
+    {
+      "id": 13,
+      "rotation_radians": 1.0463206307111212,
+      "x": -2.000046949475865,
+      "y": 1.7321102095159302
+    },
+    {
+      "id": 14,
+      "rotation_radians": 0.5561699208755143,
+      "x": -2.1094060182784053,
+      "y": 3.6453113976413682
+    }
+  ],
+  "problem": "14 equal items for Hexagons in Triangles",
+  "setup": "hexintri",
+  "source_image": "https://erich-friedman.github.io/packing/hexintri/15.gif",
+  "source_page": "https://erich-friedman.github.io/packing/hexintri/"
+}

data/solutions/generated/peninpen_005_5517dc706e76853c.json

@@ -0,0 +1,94 @@
+{
+  "case": "peninpen@5",
+  "container": {
+    "circumradius": 2.2270650746212577,
+    "orientation_radians": 0.0,
+    "side_length": 2.618072013516023,
+    "sides": 5,
+    "type": "regular_polygon"
+  },
+  "friedman_reference": {
+    "analytical_or_proved": true,
+    "case": "peninpen@5",
+    "credit": "Trivial",
+    "date": "",
+    "family_updated": "",
+    "metric_expression": "(3+\u221a5)/2 = 2.61803+",
+    "metric_symbol": "s",
+    "metric_value": 2.618033988749895,
+    "reference_text": "s = (3+\u221a5)/2 = 2.61803+ | Trivial.",
+    "source_image": "6.png",
+    "source_image_url": "https://erich-friedman.github.io/packing/peninpen/6.png",
+    "source_page": "https://erich-friedman.github.io/packing/peninpen/",
+    "title": "Pentagons in Pentagons"
+  },
+  "item": {
+    "circumradius": 0.8506508083520399,
+    "side_length": 1.0,
+    "sides": 5,
+    "type": "regular_polygon"
+  },
+  "metadata": {
+    "friedman_reference": {
+      "analytical_or_proved": true,
+      "case": "peninpen@5",
+      "credit": "Trivial",
+      "date": "",
+      "family_updated": "",
+      "metric_expression": "(3+\u221a5)/2 = 2.61803+",
+      "metric_symbol": "s",
+      "metric_value": 2.618033988749895,
+      "reference_text": "s = (3+\u221a5)/2 = 2.61803+ | Trivial.",
+      "source_image": "6.png",
+      "source_image_url": "https://erich-friedman.github.io/packing/peninpen/6.png",
+      "source_page": "https://erich-friedman.github.io/packing/peninpen/",
+      "title": "Pentagons in Pentagons"
+    },
+    "generator": "monotone_subset_upper_bound",
+    "metric_recovery_tolerance_rel": 0.001,
+    "monotone_upper_bound_from_case": "peninpen@6",
+    "monotone_upper_bound_from_record": "198bd12691483df1",
+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from peninpen@6; this prevents displaying a dominated coordinate reconstruction.",
+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 2.618033988749895,
+    "source_url": "https://erich-friedman.github.io/packing/peninpen/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
+    "submitted_by": "Trivial"
+  },
+  "placements": [
+    {
+      "id": 1,
+      "rotation_radians": 5.859687189350455e-06,
+      "x": -1.1135369273268207,
+      "y": -0.8090295495630315
+    },
+    {
+      "id": 2,
+      "rotation_radians": 5.689788196098776e-06,
+      "x": 0.42533021122830145,
+      "y": -1.3090405128356537
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+    {
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+      "x": 1.3764035897498028,
+      "y": -3.44180539106247e-06
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+    {
+      "id": 4,
+      "rotation_radians": 5.611088362108696e-06,
+      "x": -1.1135336974442216,
+      "y": 0.8090323048616854
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+    {
+      "id": 5,
+      "rotation_radians": 0.6283270502148597,
+      "x": -1.1594982472853596e-06,
+      "y": -2.7978315551800707e-06
+    }
+  ],
+  "problem": "5 equal items for Pentagons in Pentagons",
+  "setup": "peninpen",
+  "source_image": "https://erich-friedman.github.io/packing/peninpen/6.png",
+  "source_page": "https://erich-friedman.github.io/packing/peninpen/"
+}

data/solutions/generated/squindom_001_b71122905cd4c5c7.json

@@ -0,0 +1,68 @@
+{
+  "case": "squindom@1",
+  "container": {
+    "height": 1.0000058501509195,
+    "type": "rectangle",
+    "width": 2.000011700301839
+  },
+  "friedman_reference": {
+    "analytical_or_proved": true,
+    "case": "squindom@1",
+    "credit": "Trivial",
+    "date": "",
+    "family_updated": "",
+    "metric_expression": "1",
+    "metric_symbol": "s",
+    "metric_value": 1.0,
+    "reference_text": "1-2. | | s = 1 | Trivial. |",
+    "source_image": "2.gif",
+    "source_image_url": "https://erich-friedman.github.io/packing/squindom/2.gif",
+    "source_page": "https://erich-friedman.github.io/packing/squindom/",
+    "title": "Squares in Dominoes"
+  },
+  "item": {
+    "circumradius": 0.7071067811865476,
+    "side_length": 1.0,
+    "sides": 4,
+    "type": "regular_polygon"
+  },
+  "metadata": {
+    "friedman_reference": {
+      "analytical_or_proved": true,
+      "case": "squindom@1",
+      "credit": "Trivial",
+      "date": "",
+      "family_updated": "",
+      "metric_expression": "1",
+      "metric_symbol": "s",
+      "metric_value": 1.0,
+      "reference_text": "1-2. | | s = 1 | Trivial. |",
+      "source_image": "2.gif",
+      "source_image_url": "https://erich-friedman.github.io/packing/squindom/2.gif",
+      "source_page": "https://erich-friedman.github.io/packing/squindom/",
+      "title": "Squares in Dominoes"
+    },
+    "generator": "monotone_subset_upper_bound",
+    "metric_recovery_tolerance_rel": 0.001,
+    "monotone_upper_bound_from_case": "squindom@2",
+    "monotone_upper_bound_from_record": "7684110dce2e5290",
+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from squindom@2; this prevents displaying a dominated coordinate reconstruction.",
+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 1.0,
+    "source_url": "https://erich-friedman.github.io/packing/squindom/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
+    "submitted_by": "Trivial"
+  },
+  "placements": [
+    {
+      "id": 1,
+      "rotation_radians": 3.9269915138000475,
+      "x": -0.500005155407247,
+      "y": -3.7224941878791494e-07
+    }
+  ],
+  "problem": "1 equal items for Squares in Dominoes",
+  "setup": "squindom",
+  "source_image": "https://erich-friedman.github.io/packing/squindom/2.gif",
+  "source_page": "https://erich-friedman.github.io/packing/squindom/"
+}

data/solutions/generated/squindom_005_30146b12b3cd6b1f.json

@@ -0,0 +1,92 @@
+{
+  "case": "squindom@5",
+  "container": {
+    "height": 2.000021366856493,
+    "type": "rectangle",
+    "width": 4.000042733712986
+  },
+  "friedman_reference": {
+    "analytical_or_proved": true,
+    "case": "squindom@5",
+    "credit": "Trivial",
+    "date": "",
+    "family_updated": "",
+    "metric_expression": "2",
+    "metric_symbol": "s",
+    "metric_value": 2.0,
+    "reference_text": "5-8. | | s = 2 | Trivial. |",
+    "source_image": "8.gif",
+    "source_image_url": "https://erich-friedman.github.io/packing/squindom/8.gif",
+    "source_page": "https://erich-friedman.github.io/packing/squindom/",
+    "title": "Squares in Dominoes"
+  },
+  "item": {
+    "circumradius": 0.7071067811865476,
+    "side_length": 1.0,
+    "sides": 4,
+    "type": "regular_polygon"
+  },
+  "metadata": {
+    "friedman_reference": {
+      "analytical_or_proved": true,
+      "case": "squindom@5",
+      "credit": "Trivial",
+      "date": "",
+      "family_updated": "",
+      "metric_expression": "2",
+      "metric_symbol": "s",
+      "metric_value": 2.0,
+      "reference_text": "5-8. | | s = 2 | Trivial. |",
+      "source_image": "8.gif",
+      "source_image_url": "https://erich-friedman.github.io/packing/squindom/8.gif",
+      "source_page": "https://erich-friedman.github.io/packing/squindom/",
+      "title": "Squares in Dominoes"
+    },
+    "generator": "monotone_subset_upper_bound",
+    "metric_recovery_tolerance_rel": 0.001,
+    "monotone_upper_bound_from_case": "squindom@6",
+    "monotone_upper_bound_from_record": "095d951ace7a918a",
+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from squindom@6; this prevents displaying a dominated coordinate reconstruction.",
+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 2.0,
+    "source_url": "https://erich-friedman.github.io/packing/squindom/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
+    "submitted_by": "Trivial"
+  },
+  "placements": [
+    {
+      "id": 1,
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+      "y": -0.5000057922707816
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+    {
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+    {
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+      "x": -1.48061341767279,
+      "y": 0.5000046125708607
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+    {
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+      "x": -0.10290492456776802,
+      "y": 0.5000058081885114
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+  ],
+  "problem": "5 equal items for Squares in Dominoes",
+  "setup": "squindom",
+  "source_image": "https://erich-friedman.github.io/packing/squindom/8.gif",
+  "source_page": "https://erich-friedman.github.io/packing/squindom/"
+}

data/solutions/generated/squindom_007_24268bca52468f25.json

@@ -0,0 +1,104 @@
+{
+  "case": "squindom@7",
+  "container": {
+    "height": 2.0000277726402436,
+    "type": "rectangle",
+    "width": 4.000055545280487
+  },
+  "friedman_reference": {
+    "analytical_or_proved": true,
+    "case": "squindom@7",
+    "credit": "Trivial",
+    "date": "",
+    "family_updated": "",
+    "metric_expression": "2",
+    "metric_symbol": "s",
+    "metric_value": 2.0,
+    "reference_text": "5-8. | | s = 2 | Trivial. |",
+    "source_image": "8.gif",
+    "source_image_url": "https://erich-friedman.github.io/packing/squindom/8.gif",
+    "source_page": "https://erich-friedman.github.io/packing/squindom/",
+    "title": "Squares in Dominoes"
+  },
+  "item": {
+    "circumradius": 0.7071067811865476,
+    "side_length": 1.0,
+    "sides": 4,
+    "type": "regular_polygon"
+  },
+  "metadata": {
+    "friedman_reference": {
+      "analytical_or_proved": true,
+      "case": "squindom@7",
+      "credit": "Trivial",
+      "date": "",
+      "family_updated": "",
+      "metric_expression": "2",
+      "metric_symbol": "s",
+      "metric_value": 2.0,
+      "reference_text": "5-8. | | s = 2 | Trivial. |",
+      "source_image": "8.gif",
+      "source_image_url": "https://erich-friedman.github.io/packing/squindom/8.gif",
+      "source_page": "https://erich-friedman.github.io/packing/squindom/",
+      "title": "Squares in Dominoes"
+    },
+    "generator": "monotone_subset_upper_bound",
+    "metric_recovery_tolerance_rel": 0.001,
+    "monotone_upper_bound_from_case": "squindom@8",
+    "monotone_upper_bound_from_record": "0b83841ee591689c",
+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from squindom@8; this prevents displaying a dominated coordinate reconstruction.",
+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 2.0,
+    "source_url": "https://erich-friedman.github.io/packing/squindom/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
+    "submitted_by": "Trivial"
+  },
+  "placements": [
+    {
+      "id": 1,
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+      "x": 1.5000149119527248,
+      "y": -0.5000097739490859
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+      "x": -1.5000251801567914,
+      "y": -0.5000060956347121
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+    {
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+      "x": 0.5000184314486544,
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+    {
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+      "y": 0.5000096521148194
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+    {
+      "id": 6,
+      "rotation_radians": 2.356198903146586,
+      "x": -0.49999402074017957,
+      "y": 0.5000089601704955
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+    {
+      "id": 7,
+      "rotation_radians": 0.7854026230935394,
+      "x": 1.5000253378425243,
+      "y": 0.5000061105664929
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+  ],
+  "problem": "7 equal items for Squares in Dominoes",
+  "setup": "squindom",
+  "source_image": "https://erich-friedman.github.io/packing/squindom/8.gif",
+  "source_page": "https://erich-friedman.github.io/packing/squindom/"
+}

data/solutions/generated/squindom_014_821200c01d9d607e.json

@@ -0,0 +1,146 @@
+{
+  "case": "squindom@14",
+  "container": {
+    "height": 3.0000460387568455,
+    "type": "rectangle",
+    "width": 6.000092077513691
+  },
+  "friedman_reference": {
+    "analytical_or_proved": true,
+    "case": "squindom@14",
+    "credit": "Trivial",
+    "date": "",
+    "family_updated": "",
+    "metric_expression": "3",
+    "metric_symbol": "s",
+    "metric_value": 3.0,
+    "reference_text": "14-18. | | s = 3 | Trivial. |",
+    "source_image": "18.gif",
+    "source_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif",
+    "source_page": "https://erich-friedman.github.io/packing/squindom/",
+    "title": "Squares in Dominoes"
+  },
+  "item": {
+    "circumradius": 0.7071067811865476,
+    "side_length": 1.0,
+    "sides": 4,
+    "type": "regular_polygon"
+  },
+  "metadata": {
+    "friedman_reference": {
+      "analytical_or_proved": true,
+      "case": "squindom@14",
+      "credit": "Trivial",
+      "date": "",
+      "family_updated": "",
+      "metric_expression": "3",
+      "metric_symbol": "s",
+      "metric_value": 3.0,
+      "reference_text": "14-18. | | s = 3 | Trivial. |",
+      "source_image": "18.gif",
+      "source_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif",
+      "source_page": "https://erich-friedman.github.io/packing/squindom/",
+      "title": "Squares in Dominoes"
+    },
+    "generator": "monotone_subset_upper_bound",
+    "metric_recovery_tolerance_rel": 0.001,
+    "monotone_upper_bound_from_case": "squindom@15",
+    "monotone_upper_bound_from_record": "6d686963e9ef0ece",
+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from squindom@15; this prevents displaying a dominated coordinate reconstruction.",
+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 3.0,
+    "source_url": "https://erich-friedman.github.io/packing/squindom/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
+    "submitted_by": "Trivial"
+  },
+  "placements": [
+    {
+      "id": 1,
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+      "x": -1.500023186559018,
+      "y": -1.0000172895613988
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+    {
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+      "y": 4.82353857784986e-06
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+    {
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+      "rotation_radians": -0.7854018133207131,
+      "x": 0.500007365737834,
+      "y": -1.0000121873659318
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+    {
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+      "rotation_radians": -0.7854032603221883,
+      "x": 2.4834764761082972,
+      "y": 4.6483989210692585e-06
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+    {
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+      "rotation_radians": -0.7854000377880068,
+      "x": -1.8477993087555635,
+      "y": 1.320162991670803e-06
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+    {
+      "id": 10,
+      "rotation_radians": -0.7750279134518145,
+      "x": 0.42648274905728634,
+      "y": 0.5479367032730736
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+    {
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+      "rotation_radians": -0.7854029720343415,
+      "x": 1.4370258118849715,
+      "y": 5.131322940837433e-06
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+    {
+      "id": 12,
+      "rotation_radians": -0.7853999151118402,
+      "x": 2.4716303464317853,
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+    {
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+      "rotation_radians": -0.7853831426036192,
+      "x": -2.4332487703098646,
+      "y": 1.000014470810833
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+    {
+      "id": 14,
+      "rotation_radians": -0.785401236560056,
+      "x": -0.9382967443764308,
+      "y": 1.0000202440221884
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+  ],
+  "problem": "14 equal items for Squares in Dominoes",
+  "setup": "squindom",
+  "source_image": "https://erich-friedman.github.io/packing/squindom/18.gif",
+  "source_page": "https://erich-friedman.github.io/packing/squindom/"
+}

data/solutions/generated/squindom_016_e980aaa270c37d2c.json

@@ -0,0 +1,158 @@
+{
+  "case": "squindom@16",
+  "container": {
+    "height": 3.0000601159570133,
+    "type": "rectangle",
+    "width": 6.0001202319140265
+  },
+  "friedman_reference": {
+    "analytical_or_proved": true,
+    "case": "squindom@16",
+    "credit": "Trivial",
+    "date": "",
+    "family_updated": "",
+    "metric_expression": "3",
+    "metric_symbol": "s",
+    "metric_value": 3.0,
+    "reference_text": "14-18. | | s = 3 | Trivial. |",
+    "source_image": "18.gif",
+    "source_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif",
+    "source_page": "https://erich-friedman.github.io/packing/squindom/",
+    "title": "Squares in Dominoes"
+  },
+  "item": {
+    "circumradius": 0.7071067811865476,
+    "side_length": 1.0,
+    "sides": 4,
+    "type": "regular_polygon"
+  },
+  "metadata": {
+    "friedman_reference": {
+      "analytical_or_proved": true,
+      "case": "squindom@16",
+      "credit": "Trivial",
+      "date": "",
+      "family_updated": "",
+      "metric_expression": "3",
+      "metric_symbol": "s",
+      "metric_value": 3.0,
+      "reference_text": "14-18. | | s = 3 | Trivial. |",
+      "source_image": "18.gif",
+      "source_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif",
+      "source_page": "https://erich-friedman.github.io/packing/squindom/",
+      "title": "Squares in Dominoes"
+    },
+    "generator": "monotone_subset_upper_bound",
+    "metric_recovery_tolerance_rel": 0.001,
+    "monotone_upper_bound_from_case": "squindom@18",
+    "monotone_upper_bound_from_record": "a6900e1fd3f925c6",
+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from squindom@18; this prevents displaying a dominated coordinate reconstruction.",
+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 3.0,
+    "source_url": "https://erich-friedman.github.io/packing/squindom/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
+    "submitted_by": "Trivial"
+  },
+  "placements": [
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+  "problem": "16 equal items for Squares in Dominoes",
+  "setup": "squindom",
+  "source_image": "https://erich-friedman.github.io/packing/squindom/18.gif",
+  "source_page": "https://erich-friedman.github.io/packing/squindom/"
+}

data/solutions/generated/squindom_017_ccddf4810f297da5.json

@@ -0,0 +1,164 @@
+{
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+    "case": "squindom@17",
+    "credit": "Trivial",
+    "date": "",
+    "family_updated": "",
+    "metric_expression": "3",
+    "metric_symbol": "s",
+    "metric_value": 3.0,
+    "reference_text": "14-18. | | s = 3 | Trivial. |",
+    "source_image": "18.gif",
+    "source_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif",
+    "source_page": "https://erich-friedman.github.io/packing/squindom/",
+    "title": "Squares in Dominoes"
+  },
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+    "sides": 4,
+    "type": "regular_polygon"
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+      "analytical_or_proved": true,
+      "case": "squindom@17",
+      "credit": "Trivial",
+      "date": "",
+      "family_updated": "",
+      "metric_expression": "3",
+      "metric_symbol": "s",
+      "metric_value": 3.0,
+      "reference_text": "14-18. | | s = 3 | Trivial. |",
+      "source_image": "18.gif",
+      "source_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif",
+      "source_page": "https://erich-friedman.github.io/packing/squindom/",
+      "title": "Squares in Dominoes"
+    },
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+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 3.0,
+    "source_url": "https://erich-friedman.github.io/packing/squindom/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
+    "submitted_by": "Trivial"
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+  "setup": "squindom",
+  "source_image": "https://erich-friedman.github.io/packing/squindom/18.gif",
+  "source_page": "https://erich-friedman.github.io/packing/squindom/"
+}

data/solutions/generated/squindom_020_7351f2aa5e1553d9.json

@@ -0,0 +1,182 @@
+{
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+    "case": "squindom@20",
+    "credit": "Trivial",
+    "date": "",
+    "family_updated": "",
+    "metric_expression": "7/2 = 3.5",
+    "metric_symbol": "s",
+    "metric_value": 3.5,
+    "reference_text": "20-21. | | s = 7/2 = 3.5 | Trivial. |",
+    "source_image": "21.gif",
+    "source_image_url": "https://erich-friedman.github.io/packing/squindom/21.gif",
+    "source_page": "https://erich-friedman.github.io/packing/squindom/",
+    "title": "Squares in Dominoes"
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+  "metadata": {
+    "friedman_reference": {
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+      "case": "squindom@20",
+      "credit": "Trivial",
+      "date": "",
+      "family_updated": "",
+      "metric_expression": "7/2 = 3.5",
+      "metric_symbol": "s",
+      "metric_value": 3.5,
+      "reference_text": "20-21. | | s = 7/2 = 3.5 | Trivial. |",
+      "source_image": "21.gif",
+      "source_image_url": "https://erich-friedman.github.io/packing/squindom/21.gif",
+      "source_page": "https://erich-friedman.github.io/packing/squindom/",
+      "title": "Squares in Dominoes"
+    },
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+    "monotone_upper_bound_from_case": "squindom@21",
+    "monotone_upper_bound_from_record": "f91a28177536f6d3",
+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from squindom@21; this prevents displaying a dominated coordinate reconstruction.",
+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 3.5,
+    "source_url": "https://erich-friedman.github.io/packing/squindom/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
+    "submitted_by": "Trivial"
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+  "problem": "20 equal items for Squares in Dominoes",
+  "setup": "squindom",
+  "source_image": "https://erich-friedman.github.io/packing/squindom/21.gif",
+  "source_page": "https://erich-friedman.github.io/packing/squindom/"
+}

data/solutions/generated/squindom_027_bc414cdd10a510fa.json

@@ -0,0 +1,224 @@
+{
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+    "width": 8.000171889803864
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+  "friedman_reference": {
+    "analytical_or_proved": true,
+    "case": "squindom@27",
+    "credit": "Trivial",
+    "date": "",
+    "family_updated": "",
+    "metric_expression": "4",
+    "metric_symbol": "s",
+    "metric_value": 4.0,
+    "reference_text": "27-32. | | s = 4 | Trivial. | |",
+    "source_image": "32.gif",
+    "source_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif",
+    "source_page": "https://erich-friedman.github.io/packing/squindom/",
+    "title": "Squares in Dominoes"
+  },
+  "item": {
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+    "sides": 4,
+    "type": "regular_polygon"
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+  "metadata": {
+    "friedman_reference": {
+      "analytical_or_proved": true,
+      "case": "squindom@27",
+      "credit": "Trivial",
+      "date": "",
+      "family_updated": "",
+      "metric_expression": "4",
+      "metric_symbol": "s",
+      "metric_value": 4.0,
+      "reference_text": "27-32. | | s = 4 | Trivial. | |",
+      "source_image": "32.gif",
+      "source_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif",
+      "source_page": "https://erich-friedman.github.io/packing/squindom/",
+      "title": "Squares in Dominoes"
+    },
+    "generator": "monotone_subset_upper_bound",
+    "metric_recovery_tolerance_rel": 0.001,
+    "monotone_upper_bound_from_case": "squindom@30",
+    "monotone_upper_bound_from_record": "d67a491f7b4498c0",
+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from squindom@30; this prevents displaying a dominated coordinate reconstruction.",
+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 4.0,
+    "source_url": "https://erich-friedman.github.io/packing/squindom/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
+    "submitted_by": "Trivial"
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+}

data/solutions/generated/squindom_028_a26ece2a03ab477f.json

@@ -0,0 +1,230 @@
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+    "source_page": "https://erich-friedman.github.io/packing/squindom/",
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+}

data/solutions/generated/squindom_029_a006f9c77930af4d.json

@@ -0,0 +1,236 @@
+{
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+      "reference_text": "27-32. | | s = 4 | Trivial. | |",
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+}

data/solutions/generated/squindom_035_7d2266788c6a0b7a.json

@@ -0,0 +1,272 @@
+{
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+    "family_updated": "",
+    "metric_expression": "9/2 = 4.5",
+    "metric_symbol": "s",
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+    "reference_text": "35-36. | | s = 9/2 = 4.5 | Trivial. |",
+    "source_image": "36.gif",
+    "source_image_url": "https://erich-friedman.github.io/packing/squindom/36.gif",
+    "source_page": "https://erich-friedman.github.io/packing/squindom/",
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+      "metric_expression": "9/2 = 4.5",
+      "metric_symbol": "s",
+      "metric_value": 4.5,
+      "reference_text": "35-36. | | s = 9/2 = 4.5 | Trivial. |",
+      "source_image": "36.gif",
+      "source_image_url": "https://erich-friedman.github.io/packing/squindom/36.gif",
+      "source_page": "https://erich-friedman.github.io/packing/squindom/",
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+}

data/solutions/generated/squindom_043_83b582ff79edcebf.json

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data/solutions/generated/squindom_045_efa26e844da7d857.json

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data/solutions/generated/squindom_046_7fb3175fb0fba7a7.json

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data/solutions/generated/squindom_047_a146ed2b5a4b6cf9.json

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+  "problem": "47 equal items for Squares in Dominoes",
+  "setup": "squindom",
+  "source_image": "https://erich-friedman.github.io/packing/squindom/50.gif",
+  "source_page": "https://erich-friedman.github.io/packing/squindom/"
+}

data/solutions/generated/squindom_049_598e94e082ebd7d9.json

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+{
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+    "case": "squindom@49",
+    "credit": "Trivial",
+    "date": "",
+    "family_updated": "",
+    "metric_expression": "5",
+    "metric_symbol": "s",
+    "metric_value": 5.0,
+    "reference_text": "43-50. | | s = 5 | Trivial. |",
+    "source_image": "50.gif",
+    "source_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif",
+    "source_page": "https://erich-friedman.github.io/packing/squindom/",
+    "title": "Squares in Dominoes"
+  },
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+  "metadata": {
+    "friedman_reference": {
+      "analytical_or_proved": true,
+      "case": "squindom@49",
+      "credit": "Trivial",
+      "date": "",
+      "family_updated": "",
+      "metric_expression": "5",
+      "metric_symbol": "s",
+      "metric_value": 5.0,
+      "reference_text": "43-50. | | s = 5 | Trivial. |",
+      "source_image": "50.gif",
+      "source_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif",
+      "source_page": "https://erich-friedman.github.io/packing/squindom/",
+      "title": "Squares in Dominoes"
+    },
+    "generator": "monotone_subset_upper_bound",
+    "metric_recovery_tolerance_rel": 0.001,
+    "monotone_upper_bound_from_case": "squindom@50",
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+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from squindom@50; this prevents displaying a dominated coordinate reconstruction.",
+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 5.0,
+    "source_url": "https://erich-friedman.github.io/packing/squindom/",
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+  "problem": "49 equal items for Squares in Dominoes",
+  "setup": "squindom",
+  "source_image": "https://erich-friedman.github.io/packing/squindom/50.gif",
+  "source_page": "https://erich-friedman.github.io/packing/squindom/"
+}

data/solutions/generated/squinoct_016_4405320aab52a39c.json

@@ -0,0 +1,160 @@
+{
+  "case": "squinoct@16",
+  "container": {
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+    "orientation_radians": 0.0,
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+    "sides": 8,
+    "type": "regular_polygon"
+  },
+  "friedman_reference": {
+    "analytical_or_proved": true,
+    "case": "squinoct@16",
+    "credit": "Trivial",
+    "date": "",
+    "family_updated": "",
+    "metric_expression": "5\u221a2 \u2013 5 = 2.07106+",
+    "metric_symbol": "s",
+    "metric_value": 2.071067811865475,
+    "reference_text": "s = 5\u221a2 \u2013 5 = 2.07106+ | Trivial. |",
+    "source_image": "16.gif",
+    "source_image_url": "https://erich-friedman.github.io/packing/squinoct/16.gif",
+    "source_page": "https://erich-friedman.github.io/packing/squinoct/",
+    "title": "Squares in Octagons"
+  },
+  "item": {
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+    "sides": 4,
+    "type": "regular_polygon"
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+  "metadata": {
+    "friedman_reference": {
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+      "case": "squinoct@16",
+      "credit": "Trivial",
+      "date": "",
+      "family_updated": "",
+      "metric_expression": "5\u221a2 \u2013 5 = 2.07106+",
+      "metric_symbol": "s",
+      "metric_value": 2.071067811865475,
+      "reference_text": "s = 5\u221a2 \u2013 5 = 2.07106+ | Trivial. |",
+      "source_image": "16.gif",
+      "source_image_url": "https://erich-friedman.github.io/packing/squinoct/16.gif",
+      "source_page": "https://erich-friedman.github.io/packing/squinoct/",
+      "title": "Squares in Octagons"
+    },
+    "generator": "monotone_subset_upper_bound",
+    "metric_recovery_tolerance_rel": 0.001,
+    "monotone_upper_bound_from_case": "squinoct@17",
+    "monotone_upper_bound_from_record": "ab660f0510a3dd1c",
+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from squinoct@17; this prevents displaying a dominated coordinate reconstruction.",
+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 2.071067811865475,
+    "source_url": "https://erich-friedman.github.io/packing/squinoct/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
+    "submitted_by": "Trivial"
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+      "y": 0.712928221707734
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+      "rotation_radians": -0.3927002465656748,
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+      "y": -1.6654196423250718
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+      "x": 0.27509344456537743,
+      "y": -2.0508709494533184
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+  ],
+  "problem": "16 equal items for Squares in Octagons",
+  "setup": "squinoct",
+  "source_image": "https://erich-friedman.github.io/packing/squinoct/16.gif",
+  "source_page": "https://erich-friedman.github.io/packing/squinoct/"
+}

data/solutions/generated/squintri_005_21b5dd023d9a44d0.json

@@ -0,0 +1,94 @@
+{
+  "case": "squintri@5",
+  "container": {
+    "circumradius": 2.4881047104532623,
+    "orientation_radians": 0.0,
+    "side_length": 4.3095237730565,
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+    "type": "regular_polygon"
+  },
+  "friedman_reference": {
+    "analytical_or_proved": true,
+    "case": "squintri@5",
+    "credit": "Erich Friedman",
+    "date": "1997",
+    "family_updated": "",
+    "metric_expression": "2 + 4 / \u221a3 = 4.309+",
+    "metric_symbol": "s",
+    "metric_value": 4.309401076758503,
+    "reference_text": "s = 2 + 4 / \u221a3 = 4.309+ | Found by Erich Friedman | in 1997.",
+    "source_image": "ts6.gif",
+    "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts6.gif",
+    "source_page": "https://erich-friedman.github.io/packing/squintri/",
+    "title": "Squares in Triangles"
+  },
+  "item": {
+    "circumradius": 0.7071067811865476,
+    "side_length": 1.0,
+    "sides": 4,
+    "type": "regular_polygon"
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+  "metadata": {
+    "friedman_reference": {
+      "analytical_or_proved": true,
+      "case": "squintri@5",
+      "credit": "Erich Friedman",
+      "date": "1997",
+      "family_updated": "",
+      "metric_expression": "2 + 4 / \u221a3 = 4.309+",
+      "metric_symbol": "s",
+      "metric_value": 4.309401076758503,
+      "reference_text": "s = 2 + 4 / \u221a3 = 4.309+ | Found by Erich Friedman | in 1997.",
+      "source_image": "ts6.gif",
+      "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts6.gif",
+      "source_page": "https://erich-friedman.github.io/packing/squintri/",
+      "title": "Squares in Triangles"
+    },
+    "generator": "monotone_subset_upper_bound",
+    "metric_recovery_tolerance_rel": 0.001,
+    "monotone_upper_bound_from_case": "squintri@6",
+    "monotone_upper_bound_from_record": "cafd65072dcc4c6f",
+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from squintri@6; this prevents displaying a dominated coordinate reconstruction.",
+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 4.309401076758503,
+    "source_url": "https://erich-friedman.github.io/packing/squintri/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
+    "submitted_by": "Erich Friedman"
+  },
+  "placements": [
+    {
+      "id": 1,
+      "rotation_radians": -0.7854019217904592,
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+      "y": -0.07734415958360119
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+      "x": 0.3050407863102119,
+      "y": 0.6830291876013032
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+  ],
+  "problem": "5 equal items for Squares in Triangles",
+  "setup": "squintri",
+  "source_image": "https://erich-friedman.github.io/packing/squintri/ts6.gif",
+  "source_page": "https://erich-friedman.github.io/packing/squintri/"
+}

data/solutions/generated/squintri_020_33507e9f74d78419.json

@@ -0,0 +1,184 @@
+{
+  "case": "squintri@20",
+  "container": {
+    "circumradius": 4.398847304000316,
+    "orientation_radians": 0.0,
+    "side_length": 7.619027025265926,
+    "sides": 3,
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+  "friedman_reference": {
+    "analytical_or_proved": true,
+    "case": "squintri@20",
+    "credit": "Erich Friedman",
+    "date": "1997",
+    "family_updated": "",
+    "metric_expression": "3 + 8 / \u221a3 = 7.618+",
+    "metric_symbol": "s",
+    "metric_value": 7.618802153517006,
+    "reference_text": "s = 3 + 8 / \u221a3 = 7.618+ | Found by Erich Friedman | in 1997.",
+    "source_image": "ts21.gif",
+    "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts21.gif",
+    "source_page": "https://erich-friedman.github.io/packing/squintri/",
+    "title": "Squares in Triangles"
+  },
+  "item": {
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+    "side_length": 1.0,
+    "sides": 4,
+    "type": "regular_polygon"
+  },
+  "metadata": {
+    "friedman_reference": {
+      "analytical_or_proved": true,
+      "case": "squintri@20",
+      "credit": "Erich Friedman",
+      "date": "1997",
+      "family_updated": "",
+      "metric_expression": "3 + 8 / \u221a3 = 7.618+",
+      "metric_symbol": "s",
+      "metric_value": 7.618802153517006,
+      "reference_text": "s = 3 + 8 / \u221a3 = 7.618+ | Found by Erich Friedman | in 1997.",
+      "source_image": "ts21.gif",
+      "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts21.gif",
+      "source_page": "https://erich-friedman.github.io/packing/squintri/",
+      "title": "Squares in Triangles"
+    },
+    "generator": "monotone_subset_upper_bound",
+    "metric_recovery_tolerance_rel": 0.001,
+    "monotone_upper_bound_from_case": "squintri@21",
+    "monotone_upper_bound_from_record": "a2e24601c2557198",
+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from squintri@21; this prevents displaying a dominated coordinate reconstruction.",
+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 7.618802153517006,
+    "source_url": "https://erich-friedman.github.io/packing/squintri/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
+    "submitted_by": "Erich Friedman"
+  },
+  "placements": [
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+  ],
+  "problem": "20 equal items for Squares in Triangles",
+  "setup": "squintri",
+  "source_image": "https://erich-friedman.github.io/packing/squintri/ts21.gif",
+  "source_page": "https://erich-friedman.github.io/packing/squintri/"
+}

data/solutions/generated/squintri_030_58fcbf20aa7765b3.json

@@ -0,0 +1,244 @@
+{
+  "case": "squintri@30",
+  "container": {
+    "circumradius": 5.375037381839187,
+    "orientation_radians": 0.0,
+    "side_length": 9.309837837927466,
+    "sides": 3,
+    "type": "regular_polygon"
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+  "friedman_reference": {
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+    "case": "squintri@30",
+    "credit": "David W. Cantrell",
+    "date": "July 2002",
+    "family_updated": "",
+    "metric_expression": "9.301+",
+    "metric_symbol": "s",
+    "metric_value": 9.301,
+    "reference_text": "s = 9.301+ | Found by David W. Cantrell | in July 2002.",
+    "source_image": "ts30.gif",
+    "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts30.gif",
+    "source_page": "https://erich-friedman.github.io/packing/squintri/",
+    "title": "Squares in Triangles"
+  },
+  "item": {
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+    "side_length": 1.0,
+    "sides": 4,
+    "type": "regular_polygon"
+  },
+  "metadata": {
+    "friedman_reference": {
+      "analytical_or_proved": false,
+      "case": "squintri@30",
+      "credit": "David W. Cantrell",
+      "date": "July 2002",
+      "family_updated": "",
+      "metric_expression": "9.301+",
+      "metric_symbol": "s",
+      "metric_value": 9.301,
+      "reference_text": "s = 9.301+ | Found by David W. Cantrell | in July 2002.",
+      "source_image": "ts30.gif",
+      "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts30.gif",
+      "source_page": "https://erich-friedman.github.io/packing/squintri/",
+      "title": "Squares in Triangles"
+    },
+    "generator": "monotone_subset_upper_bound",
+    "metric_recovery_tolerance_rel": 0.001,
+    "monotone_upper_bound_from_case": "squintri@31",
+    "monotone_upper_bound_from_record": "22b5f5f5707858bb",
+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from squintri@31; this prevents displaying a dominated coordinate reconstruction.",
+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 9.301,
+    "source_url": "https://erich-friedman.github.io/packing/squintri/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
+    "submitted_by": "David W. Cantrell"
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+  "setup": "squintri",
+  "source_image": "https://erich-friedman.github.io/packing/squintri/ts30.gif",
+  "source_page": "https://erich-friedman.github.io/packing/squintri/"
+}

data/solutions/generated/squintri_033_8d116d694ad5570a.json

@@ -0,0 +1,262 @@
+{
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+    "credit": "Erich Friedman",
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+    "metric_symbol": "s",
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+    "reference_text": "s = 5 + 8 / \u221a3 = 9.618+ | Found by Erich Friedman | in 1997.",
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+    "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts34.gif",
+    "source_page": "https://erich-friedman.github.io/packing/squintri/",
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+      "case": "squintri@33",
+      "credit": "Erich Friedman",
+      "date": "1997",
+      "family_updated": "",
+      "metric_expression": "5 + 8 / \u221a3 = 9.618+",
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+      "reference_text": "s = 5 + 8 / \u221a3 = 9.618+ | Found by Erich Friedman | in 1997.",
+      "source_image": "ts34.gif",
+      "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts34.gif",
+      "source_page": "https://erich-friedman.github.io/packing/squintri/",
+      "title": "Squares in Triangles"
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+  "problem": "33 equal items for Squares in Triangles",
+  "setup": "squintri",
+  "source_image": "https://erich-friedman.github.io/packing/squintri/ts34.gif",
+  "source_page": "https://erich-friedman.github.io/packing/squintri/"
+}

data/solutions/generated/squintri_035_454defabaa6b59ca.json

@@ -0,0 +1,274 @@
+{
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+    "case": "squintri@35",
+    "credit": "Erich Friedman",
+    "date": "1997",
+    "family_updated": "",
+    "metric_expression": "4 + 10 / \u221a3 = 9.773+",
+    "metric_symbol": "s",
+    "metric_value": 9.773502691896258,
+    "reference_text": "s = 4 + 10 / \u221a3 = 9.773+ | Found by Erich Friedman | in 1997.",
+    "source_image": "ts36.gif",
+    "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts36.gif",
+    "source_page": "https://erich-friedman.github.io/packing/squintri/",
+    "title": "Squares in Triangles"
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+      "case": "squintri@35",
+      "credit": "Erich Friedman",
+      "date": "1997",
+      "family_updated": "",
+      "metric_expression": "4 + 10 / \u221a3 = 9.773+",
+      "metric_symbol": "s",
+      "metric_value": 9.773502691896258,
+      "reference_text": "s = 4 + 10 / \u221a3 = 9.773+ | Found by Erich Friedman | in 1997.",
+      "source_image": "ts36.gif",
+      "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts36.gif",
+      "source_page": "https://erich-friedman.github.io/packing/squintri/",
+      "title": "Squares in Triangles"
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+    "generator": "monotone_subset_upper_bound",
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+    "monotone_upper_bound_from_case": "squintri@36",
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+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from squintri@36; this prevents displaying a dominated coordinate reconstruction.",
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+    "reference_side": 9.773502691896258,
+    "source_url": "https://erich-friedman.github.io/packing/squintri/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
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+  "problem": "35 equal items for Squares in Triangles",
+  "setup": "squintri",
+  "source_image": "https://erich-friedman.github.io/packing/squintri/ts36.gif",
+  "source_page": "https://erich-friedman.github.io/packing/squintri/"
+}

data/solutions/generated/triincir_005_a3d09cb09f1298d4.json

@@ -0,0 +1,91 @@
+{
+  "case": "triincir@5",
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+  "friedman_reference": {
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+    "case": "triincir@5",
+    "credit": "Trivial",
+    "date": "",
+    "family_updated": "",
+    "metric_expression": "1",
+    "metric_symbol": "r",
+    "metric_value": 1.0,
+    "reference_text": "r = 1 | Trivial.",
+    "source_image": "tc5.gif",
+    "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc5.gif",
+    "source_page": "https://erich-friedman.github.io/packing/triincir/",
+    "title": "Triangles in Circles"
+  },
+  "item": {
+    "circumradius": 0.5773502691896258,
+    "side_length": 1.0,
+    "sides": 3,
+    "type": "regular_polygon"
+  },
+  "metadata": {
+    "friedman_reference": {
+      "analytical_or_proved": true,
+      "case": "triincir@5",
+      "credit": "Trivial",
+      "date": "",
+      "family_updated": "",
+      "metric_expression": "1",
+      "metric_symbol": "r",
+      "metric_value": 1.0,
+      "reference_text": "r = 1 | Trivial.",
+      "source_image": "tc5.gif",
+      "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc5.gif",
+      "source_page": "https://erich-friedman.github.io/packing/triincir/",
+      "title": "Triangles in Circles"
+    },
+    "generator": "monotone_subset_upper_bound",
+    "metric_recovery_tolerance_rel": 0.001,
+    "monotone_upper_bound_from_case": "triincir@6",
+    "monotone_upper_bound_from_record": "fe93ba87f23c6c01",
+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from triincir@6; this prevents displaying a dominated coordinate reconstruction.",
+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 1.0,
+    "source_url": "https://erich-friedman.github.io/packing/triincir/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
+    "submitted_by": "Trivial"
+  },
+  "placements": [
+    {
+      "id": 1,
+      "rotation_radians": 6.854987231039776,
+      "x": -0.4855164253270816,
+      "y": -0.31244473325270194
+    },
+    {
+      "id": 2,
+      "rotation_radians": 5.807789720115666,
+      "x": 0.02782796705922122,
+      "y": -0.5766902001807842
+    },
+    {
+      "id": 3,
+      "rotation_radians": 2.6661974658004457,
+      "x": 0.5133440868386314,
+      "y": -0.26424177756156353
+    },
+    {
+      "id": 4,
+      "rotation_radians": 1.6189996671280515,
+      "x": -0.513345565252066,
+      "y": 0.264246554365814
+    },
+    {
+      "id": 5,
+      "rotation_radians": 2.6661974041529044,
+      "x": -0.02783165797009881,
+      "y": 0.5766931444053651
+    }
+  ],
+  "problem": "5 equal items for Triangles in Circles",
+  "setup": "triincir",
+  "source_image": "https://erich-friedman.github.io/packing/triincir/tc5.gif",
+  "source_page": "https://erich-friedman.github.io/packing/triincir/"
+}

data/solutions/generated/triinhex_004_d2b5a3b06543cb98.json

@@ -0,0 +1,88 @@
+{
+  "case": "triinhex@4",
+  "container": {
+    "circumradius": 1.0000186366904509,
+    "orientation_radians": 0.0,
+    "side_length": 1.0000186366904507,
+    "sides": 6,
+    "type": "regular_polygon"
+  },
+  "friedman_reference": {
+    "analytical_or_proved": true,
+    "case": "triinhex@4",
+    "credit": "Trivial",
+    "date": "",
+    "family_updated": "",
+    "metric_expression": "1",
+    "metric_symbol": "s",
+    "metric_value": 1.0,
+    "reference_text": "s = 1 | Trivial.",
+    "source_image": "4.gif",
+    "source_image_url": "https://erich-friedman.github.io/packing/triinhex/4.gif",
+    "source_page": "https://erich-friedman.github.io/packing/triinhex/",
+    "title": "Triangles in Hexagons"
+  },
+  "item": {
+    "circumradius": 0.5773502691896258,
+    "side_length": 1.0,
+    "sides": 3,
+    "type": "regular_polygon"
+  },
+  "metadata": {
+    "friedman_reference": {
+      "analytical_or_proved": true,
+      "case": "triinhex@4",
+      "credit": "Trivial",
+      "date": "",
+      "family_updated": "",
+      "metric_expression": "1",
+      "metric_symbol": "s",
+      "metric_value": 1.0,
+      "reference_text": "s = 1 | Trivial.",
+      "source_image": "4.gif",
+      "source_image_url": "https://erich-friedman.github.io/packing/triinhex/4.gif",
+      "source_page": "https://erich-friedman.github.io/packing/triinhex/",
+      "title": "Triangles in Hexagons"
+    },
+    "generator": "monotone_subset_upper_bound",
+    "metric_recovery_tolerance_rel": 0.001,
+    "monotone_upper_bound_from_case": "triinhex@5",
+    "monotone_upper_bound_from_record": "0d61e48787b7d5d1",
+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from triinhex@5; this prevents displaying a dominated coordinate reconstruction.",
+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 1.0,
+    "source_url": "https://erich-friedman.github.io/packing/triinhex/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
+    "submitted_by": "Trivial"
+  },
+  "placements": [
+    {
+      "id": 1,
+      "rotation_radians": -2.61800105286312,
+      "x": 1.5395755069990018e-06,
+      "y": -0.5773615760130512
+    },
+    {
+      "id": 2,
+      "rotation_radians": 0.523590389909893,
+      "x": 0.5000091035718225,
+      "y": -0.28867948487928635
+    },
+    {
+      "id": 3,
+      "rotation_radians": -2.6180032106769118,
+      "x": -0.5000133572734049,
+      "y": 0.2886710400622856
+    },
+    {
+      "id": 4,
+      "rotation_radians": -2.6180044057315026,
+      "x": 0.5000059440711798,
+      "y": 0.28868444903714346
+    }
+  ],
+  "problem": "4 equal items for Triangles in Hexagons",
+  "setup": "triinhex",
+  "source_image": "https://erich-friedman.github.io/packing/triinhex/4.gif",
+  "source_page": "https://erich-friedman.github.io/packing/triinhex/"
+}

data/solutions/generated/triinhex_033_a1101d95622ba18c.json

@@ -0,0 +1,262 @@
+{
+  "case": "triinhex@33",
+  "container": {
+    "circumradius": 2.5000778510013064,
+    "orientation_radians": 0.0,
+    "side_length": 2.500077851001306,
+    "sides": 6,
+    "type": "regular_polygon"
+  },
+  "friedman_reference": {
+    "analytical_or_proved": false,
+    "case": "triinhex@33",
+    "credit": "Emerson Connelly",
+    "date": "April 2026",
+    "family_updated": "",
+    "metric_expression": "2.498+",
+    "metric_symbol": "s",
+    "metric_value": 2.498,
+    "reference_text": "s = 2.498+ | Found by Emerson Connelly | in April 2026.",
+    "source_image": "33.gif",
+    "source_image_url": "https://erich-friedman.github.io/packing/triinhex/33.gif",
+    "source_page": "https://erich-friedman.github.io/packing/triinhex/",
+    "title": "Triangles in Hexagons"
+  },
+  "item": {
+    "circumradius": 0.5773502691896258,
+    "side_length": 1.0,
+    "sides": 3,
+    "type": "regular_polygon"
+  },
+  "metadata": {
+    "friedman_reference": {
+      "analytical_or_proved": false,
+      "case": "triinhex@33",
+      "credit": "Emerson Connelly",
+      "date": "April 2026",
+      "family_updated": "",
+      "metric_expression": "2.498+",
+      "metric_symbol": "s",
+      "metric_value": 2.498,
+      "reference_text": "s = 2.498+ | Found by Emerson Connelly | in April 2026.",
+      "source_image": "33.gif",
+      "source_image_url": "https://erich-friedman.github.io/packing/triinhex/33.gif",
+      "source_page": "https://erich-friedman.github.io/packing/triinhex/",
+      "title": "Triangles in Hexagons"
+    },
+    "generator": "monotone_subset_upper_bound",
+    "metric_recovery_tolerance_rel": 0.001,
+    "monotone_upper_bound_from_case": "triinhex@34",
+    "monotone_upper_bound_from_record": "a712744ff3b5bcc9",
+    "notes": "Monotone upper-bound coordinate JSON derived by deleting items from triinhex@34; this prevents displaying a dominated coordinate reconstruction.",
+    "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
+    "reference_side": 2.498,
+    "source_url": "https://erich-friedman.github.io/packing/triinhex/",
+    "submitted_at": "2026-05-01T18:14:49+00:00",
+    "submitted_by": "Emerson Connelly"
+  },
+  "placements": [
+    {
+      "id": 1,
+      "rotation_radians": 1.5707820368974008,
+      "x": 0.24999585579224345,
+      "y": -0.1443415158584702
+    },
+    {
+      "id": 2,
+      "rotation_radians": -1.5708105035071722,
+      "x": -0.25001926033074506,
+      "y": 0.1443384386382384
+    },
+    {
+      "id": 3,
+      "rotation_radians": -1.570810440573233,
+      "x": 0.2500118498006379,
+      "y": -0.7217092866218728
+    },
+    {
+      "id": 4,
+      "rotation_radians": -1.570810673621609,
+      "x": 0.5000131878169409,
+      "y": 0.5773634763856955
+    },
+    {
+      "id": 5,
+      "rotation_radians": -1.5708104601760577,
+      "x": -0.7500166120610576,
+      "y": -0.7217144067360562
+    },
+    {
+      "id": 6,
+      "rotation_radians": -1.5708038572440912,
+      "x": -0.2500086725774649,
+      "y": 1.0103977328238378
+    },
+    {
+      "id": 7,
+      "rotation_radians": 1.5707888249485136,
+      "x": -0.7500204833436255,
+      "y": 0.7217106424470385
+    },
+    {
+      "id": 8,
+      "rotation_radians": 1.5707823235581515,
+      "x": -0.7500303027524092,
+      "y": -0.14434836707006837
+    },
+    {
+      "id": 9,
+      "rotation_radians": 1.5707825003198397,
+      "x": 0.5000173258159155,
+      "y": 1.154730951448823
+    },
+    {
+      "id": 10,
+      "rotation_radians": 1.5707818079722116,
+      "x": -0.25000136851044696,
+      "y": -1.0103934692939822
+    },
+    {
+      "id": 11,
+      "rotation_radians": -1.5708101527721678,
+      "x": 1.0000357771086577,
+      "y": -0.28867436604089936
+    },
+    {
+      "id": 12,
+      "rotation_radians": 1.5707824591898962,
+      "x": 0.7500276388404962,
+      "y": -1.0103930416335438
+    },
+    {
+      "id": 13,
+      "rotation_radians": 1.5707821785643816,
+      "x": 1.000029961794482,
+      "y": 0.28868829936898804
+    },
+    {
+      "id": 14,
+      "rotation_radians": 1.5707912415806538,
+      "x": -1.750055863810582,
+      "y": 0.7217079710527934
+    },
+    {
+      "id": 15,
+      "rotation_radians": -1.5708106231893202,
+      "x": -1.2500456503332784,
+      "y": 0.14433304312200526
+    },
+    {
+      "id": 16,
+      "rotation_radians": -1.5708032125206675,
+      "x": -1.250038107006699,
+      "y": 1.0103907296201537
+    },
+    {
+      "id": 17,
+      "rotation_radians": 1.5707963267545721,
+      "x": 1.898155482926013,
+      "y": 0.4651915085043393
+    },
+    {
+      "id": 18,
+      "rotation_radians": 1.5707825118299594,
+      "x": 0.2500450505070686,
+      "y": -1.8764440396097806
+    },
+    {
+      "id": 19,
+      "rotation_radians": -1.5708103003283267,
+      "x": -0.24997218620071285,
+      "y": -1.5877610603504544
+    },
+    {
+      "id": 20,
+      "rotation_radians": -1.570810851687786,
+      "x": 1.3532976299170438,
+      "y": 0.8315508528018319
+    },
+    {
+      "id": 21,
+      "rotation_radians": -1.5708099712428505,
+      "x": 1.9645372210100804,
+      "y": -0.22713386525182022
+    },
+    {
+      "id": 22,
+      "rotation_radians": -1.570810755797554,
+      "x": 0.7500605986169268,
+      "y": -1.5877627775654515
+    },
+    {
+      "id": 23,
+      "rotation_radians": -1.5708103067750077,
+      "x": -1.7500453171576065,
+      "y": -0.7217181904016983
+    },
+    {
+      "id": 24,
+      "rotation_radians": -1.5708036376638406,
+      "x": -0.7500249278741333,
+      "y": 1.8764458992387094
+    },
+    {
+      "id": 25,
+      "rotation_radians": -1.5708109741631284,
+      "x": -1.2500201688118435,
+      "y": -1.5877663765691419
+    },
+    {
+      "id": 26,
+      "rotation_radians": -1.5708003535265032,
+      "x": 0.25001568551138365,
+      "y": 1.876447032881754
+    },
+    {
+      "id": 27,
+      "rotation_radians": 1.5707825147511743,
+      "x": -1.2500295658711273,
+      "y": -1.0104005088230064
+    },
+    {
+      "id": 28,
+      "rotation_radians": 1.5707889989441273,
+      "x": -0.2500069224693436,
+      "y": 1.5877648773066506
+    },
+    {
+      "id": 29,
+      "rotation_radians": 1.5707889707679246,
+      "x": -1.2500404241936998,
+      "y": 1.5877604220987174
+    },
+    {
+      "id": 30,
+      "rotation_radians": 1.5707818230280484,
+      "x": -1.7500586464932677,
+      "y": -0.14434841133404017
+    },
+    {
+      "id": 31,
+      "rotation_radians": 1.5707843544985376,
+      "x": -0.7499937353611469,
+      "y": -1.8764426346950283
+    },
+    {
+      "id": 32,
+      "rotation_radians": 1.5707963268853689,
+      "x": 1.2500568910921024,
+      "y": 1.5877324038717813
+    },
+    {
+      "id": 33,
+      "rotation_radians": -1.5708091209744035,
+      "x": 1.5000650938916413,
+      "y": -1.1547072894263115
+    }
+  ],
+  "problem": "33 equal items for Triangles in Hexagons",
+  "setup": "triinhex",
+  "source_image": "https://erich-friedman.github.io/packing/triinhex/33.gif",
+  "source_page": "https://erich-friedman.github.io/packing/triinhex/"
+}

data/solutions/generated/triintri_001_b09a5ce4a2c03ad3.json

@@ -51,7 +51,7 @@
     "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
     "reference_side": 1.0,
     "source_url": "https://erich-friedman.github.io/packing/triintri/",
-    "submitted_at": "2026-05-01T18:08:37+00:00",
+    "submitted_at": "2026-05-01T18:15:00+00:00",
     "submitted_by": "Trivial"
   },
   "placements": [

data/solutions/generated/triintri_002_2d6f654db11412d8.json

@@ -51,7 +51,7 @@
     "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
     "reference_side": 2.0,
     "source_url": "https://erich-friedman.github.io/packing/triintri/",
-    "submitted_at": "2026-05-01T18:08:37+00:00",
+    "submitted_at": "2026-05-01T18:15:00+00:00",
     "submitted_by": "Trivial"
   },
   "placements": [

data/solutions/generated/triintri_003_76381d42f8f17f41.json

@@ -51,7 +51,7 @@
     "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
     "reference_side": 2.0,
     "source_url": "https://erich-friedman.github.io/packing/triintri/",
-    "submitted_at": "2026-05-01T18:08:37+00:00",
+    "submitted_at": "2026-05-01T18:15:00+00:00",
     "submitted_by": "Trivial"
   },
   "placements": [

data/solutions/generated/triintri_004_be1b0b4b1802d8fa.json

@@ -51,7 +51,7 @@
     "public_credit_policy": "@1 cases are credited as Trivial; Nathan Roll is credited only for non-image-seeded coordinate improvements above 0.15% relative to the Friedman reference metric.",
     "reference_side": 2.0,
     "source_url": "https://erich-friedman.github.io/packing/triintri/",
-    "submitted_at": "2026-05-01T18:08:37+00:00",
+    "submitted_at": "2026-05-01T18:15:00+00:00",
     "submitted_by": "Trivial"
   },
   "placements": [