diff --git a/app.py b/app.py
index a5708276eb3f4c3824fa9cde82c2876a38a743e7..8faeeb85c7b85c78a82b08b68988e84dffd2c443 100644
--- a/app.py
+++ b/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)
diff --git a/data/evaluation_results.jsonl b/data/evaluation_results.jsonl
index d5878fb4e6790a04422098e725b062ba687cf054..520c049158e37f5df406b5b2622e10d7bd4a74a8 100644
--- a/data/evaluation_results.jsonl
+++ b/data/evaluation_results.jsonl
@@ -23,6 +23,7 @@
{"case": "cirinhex@3", "container": {"circumradius": 2.3094165045351462, "orientation_radians": 0.0, "side_length": 2.309416504535146, "sides": 6, "type": "regular_polygon"}, "density": 0.6801656739683151, "errors": [], "item": {"radius": 1.0, "type": "circle"}, "max_boundary_excess": -1.3364075199806535e-06, "max_pair_overlap_depth": -1.818559246702378e-05, "n": 3, "ok": true, "record_id": "fa4a26f729d80904", "setup": "cirinhex", "side": 2.309416504535146, "solution_hash": "fa4a26f729d80904", "solution_path": "data/solutions/generated/cirinhex_003_fa4a26f729d80904.json", "tolerance": 1e-08, "warnings": []}
{"case": "cirinhex@4", "container": {"circumradius": 2.666604151094754, "orientation_radians": 0.0, "side_length": 2.6666041510947536, "sides": 6, "type": "regular_polygon"}, "density": 0.6802066538449848, "errors": [], "item": {"radius": 1.0, "type": "circle"}, "max_boundary_excess": -3.3940250920849735e-07, "max_pair_overlap_depth": -4.6158989302380604e-05, "n": 4, "ok": true, "record_id": "4ea68db180a8d4c4", "setup": "cirinhex", "side": 2.6666041510947536, "solution_hash": "4ea68db180a8d4c4", "solution_path": "data/solutions/generated/cirinhex_004_4ea68db180a8d4c4.json", "tolerance": 1e-08, "warnings": []}
{"case": "cirinhex@5", "container": {"circumradius": 2.999411672594878, "orientation_radians": 0.0, "side_length": 2.9994116725948774, "sides": 6, "type": "regular_polygon"}, "density": 0.672041103262025, "errors": [], "item": {"radius": 1.0, "type": "circle"}, "max_boundary_excess": -2.6369105032841844e-07, "max_pair_overlap_depth": -3.862655436481788e-05, "n": 5, "ok": true, "record_id": "eb88c54ef20a6355", "setup": "cirinhex", "side": 2.9994116725948774, "solution_hash": "eb88c54ef20a6355", "solution_path": "data/solutions/generated/cirinhex_005_eb88c54ef20a6355.json", "tolerance": 1e-08, "warnings": []}
+{"case": "cirinhex@6", "container": {"circumradius": 3.1547526785063353, "orientation_radians": 0.0, "side_length": 3.154752678506335, "sides": 6, "type": "regular_polygon"}, "density": 0.7289850151429867, "errors": [], "item": {"radius": 1.0, "type": "circle"}, "max_boundary_excess": -1.857578518338343e-06, "max_pair_overlap_depth": -3.893260533160969e-05, "n": 6, "ok": true, "record_id": "3e5660c8b6310c57", "setup": "cirinhex", "side": 3.154752678506335, "solution_hash": "3e5660c8b6310c57", "solution_path": "data/solutions/generated/cirinhex_006_3e5660c8b6310c57.json", "tolerance": 1e-08, "warnings": []}
{"case": "cirinhex@6", "container": {"circumradius": 3.154774248194912, "orientation_radians": 0.0, "side_length": 3.1547742481949115, "sides": 6, "type": "regular_polygon"}, "density": 0.7289750468062609, "errors": [], "item": {"radius": 1.0, "type": "circle"}, "max_boundary_excess": -4.7088474341983044e-06, "max_pair_overlap_depth": -4.5158069021944414e-05, "n": 6, "ok": true, "record_id": "9b7724fab73c49c1", "setup": "cirinhex", "side": 3.1547742481949115, "solution_hash": "9b7724fab73c49c1", "solution_path": "data/solutions/generated/cirinhex_006_9b7724fab73c49c1.json", "tolerance": 1e-08, "warnings": []}
{"case": "cirinhex@7", "container": {"circumradius": 3.1547526785063353, "orientation_radians": 0.0, "side_length": 3.154752678506335, "sides": 6, "type": "regular_polygon"}, "density": 0.8504825176668179, "errors": [], "item": {"radius": 1.0, "type": "circle"}, "max_boundary_excess": -1.857578518338343e-06, "max_pair_overlap_depth": -3.893260533160969e-05, "n": 7, "ok": true, "record_id": "7368090471750227", "setup": "cirinhex", "side": 3.154752678506335, "solution_hash": "7368090471750227", "solution_path": "data/solutions/generated/cirinhex_007_7368090471750227.json", "tolerance": 1e-08, "warnings": []}
{"case": "cirinhex@8", "container": {"circumradius": 3.709450333181874, "orientation_radians": 0.0, "side_length": 3.7094503331818736, "sides": 6, "type": "regular_polygon"}, "density": 0.7030218969047355, "errors": [], "item": {"radius": 1.0, "type": "circle"}, "max_boundary_excess": -2.1028444985304873e-08, "max_pair_overlap_depth": -2.2199913737530608e-05, "n": 8, "ok": true, "record_id": "afebd04e4720460b", "setup": "cirinhex", "side": 3.7094503331818736, "solution_hash": "afebd04e4720460b", "solution_path": "data/solutions/generated/cirinhex_008_afebd04e4720460b.json", "tolerance": 1e-08, "warnings": []}
@@ -91,9 +92,11 @@
{"case": "cirinsqu@23", "container": {"circumradius": 6.881779845968298, "orientation_radians": 0.0, "side_length": 9.732306391434195, "sides": 4, "type": "regular_polygon"}, "density": 0.7628623175184639, "errors": [], "item": {"radius": 1.0, "type": "circle"}, "max_boundary_excess": -1.8298221013957061e-06, "max_pair_overlap_depth": -2.6081243190478887e-05, "n": 23, "ok": true, "record_id": "cfa99060ee2203a1", "setup": "cirinsqu", "side": 9.732306391434195, "solution_hash": "cfa99060ee2203a1", "solution_path": "data/solutions/generated/cirinsqu_023_cfa99060ee2203a1.json", "tolerance": 1e-08, "warnings": []}
{"case": "cirinsqu@24", "container": {"circumradius": 6.9748433868661355, "orientation_radians": 0.0, "side_length": 9.86391811313438, "sides": 4, "type": "regular_polygon"}, "density": 0.7749295071503051, "errors": [], "item": {"radius": 1.0, "type": "circle"}, "max_boundary_excess": -8.506144878683841e-06, "max_pair_overlap_depth": -4.0021533562306644e-05, "n": 24, "ok": true, "record_id": "e2eca298edbdb88c", "setup": "cirinsqu", "side": 9.86391811313438, "solution_hash": "e2eca298edbdb88c", "solution_path": "data/solutions/generated/cirinsqu_024_e2eca298edbdb88c.json", "tolerance": 1e-08, "warnings": []}
{"case": "cirintri@1", "container": {"circumradius": 2.000005772855109, "orientation_radians": 0.0, "side_length": 3.464111614016108, "sides": 3, "type": "regular_polygon"}, "density": 0.6045962978262086, "errors": [], "item": {"radius": 1.0, "type": "circle"}, "max_boundary_excess": -1.7091087705090047e-06, "max_pair_overlap_depth": 0.0, "n": 1, "ok": true, "record_id": "10a7a380c7f5376a", "setup": "cirintri", "side": 3.464111614016108, "solution_hash": "10a7a380c7f5376a", "solution_path": "data/solutions/generated/cirintri_001_10a7a380c7f5376a.json", "tolerance": 1e-08, "warnings": []}
+{"case": "cirintri@2", "container": {"circumradius": 3.1547403089836683, "orientation_radians": 0.0, "side_length": 5.464170499845252, "sides": 3, "type": "regular_polygon"}, "density": 0.4859938211704954, "errors": [], "item": {"radius": 1.0, "type": "circle"}, "max_boundary_excess": -1.0755666179917966e-07, "max_pair_overlap_depth": -5.898827690131725e-05, "n": 2, "ok": true, "record_id": "befbb923c21388e1", "setup": "cirintri", "side": 5.464170499845252, "solution_hash": "befbb923c21388e1", "solution_path": "data/solutions/generated/cirintri_002_befbb923c21388e1.json", "tolerance": 1e-08, "warnings": []}
{"case": "cirintri@2", "container": {"circumradius": 3.1547697246602167, "orientation_radians": 0.0, "side_length": 5.464221449291572, "sides": 3, "type": "regular_polygon"}, "density": 0.48598475821399, "errors": [], "item": {"radius": 1.0, "type": "circle"}, "max_boundary_excess": -3.7587277048611867e-06, "max_pair_overlap_depth": -7.068795172848752e-05, "n": 2, "ok": true, "record_id": "f5cde1b8295da3ad", "setup": "cirintri", "side": 5.464221449291572, "solution_hash": "f5cde1b8295da3ad", "solution_path": "data/solutions/generated/cirintri_002_f5cde1b8295da3ad.json", "tolerance": 1e-08, "warnings": []}
{"case": "cirintri@3", "container": {"circumradius": 3.1547403089836683, "orientation_radians": 0.0, "side_length": 5.464170499845252, "sides": 3, "type": "regular_polygon"}, "density": 0.7289907317557431, "errors": [], "item": {"radius": 1.0, "type": "circle"}, "max_boundary_excess": -1.0755666179917966e-07, "max_pair_overlap_depth": -5.0474041760395494e-05, "n": 3, "ok": true, "record_id": "fcee9d6e8b7c5ce0", "setup": "cirintri", "side": 5.464170499845252, "solution_hash": "fcee9d6e8b7c5ce0", "solution_path": "data/solutions/generated/cirintri_003_fcee9d6e8b7c5ce0.json", "tolerance": 1e-08, "warnings": []}
{"case": "cirintri@4", "container": {"circumradius": 4.000062000129495, "orientation_radians": 0.0, "side_length": 6.9283106176498706, "sides": 3, "type": "regular_polygon"}, "density": 0.6045810458812538, "errors": [], "item": {"radius": 1.0, "type": "circle"}, "max_boundary_excess": -3.7330840119143005e-06, "max_pair_overlap_depth": -4.49553106092182e-05, "n": 4, "ok": true, "record_id": "9e0c39beb3cc8aa7", "setup": "cirintri", "side": 6.9283106176498706, "solution_hash": "9e0c39beb3cc8aa7", "solution_path": "data/solutions/generated/cirintri_004_9e0c39beb3cc8aa7.json", "tolerance": 1e-08, "warnings": []}
+{"case": "cirintri@5", "container": {"circumradius": 4.30945637922671, "orientation_radians": 0.0, "side_length": 7.4641974018224735, "sides": 3, "type": "regular_polygon"}, "density": 0.6511079649804111, "errors": [], "item": {"radius": 1.0, "type": "circle"}, "max_boundary_excess": -1.6105130511423482e-06, "max_pair_overlap_depth": -3.3410124574029254e-05, "n": 5, "ok": true, "record_id": "64d422ca71110b0d", "setup": "cirintri", "side": 7.4641974018224735, "solution_hash": "64d422ca71110b0d", "solution_path": "data/solutions/generated/cirintri_005_64d422ca71110b0d.json", "tolerance": 1e-08, "warnings": []}
{"case": "cirintri@5", "container": {"circumradius": 4.309487478333852, "orientation_radians": 0.0, "side_length": 7.464251267056113, "sides": 3, "type": "regular_polygon"}, "density": 0.6510985676665063, "errors": [], "item": {"radius": 1.0, "type": "circle"}, "max_boundary_excess": -2.12146637612598e-06, "max_pair_overlap_depth": -3.650753517803196e-05, "n": 5, "ok": true, "record_id": "ad32c3b8e4c714b5", "setup": "cirintri", "side": 7.464251267056113, "solution_hash": "ad32c3b8e4c714b5", "solution_path": "data/solutions/generated/cirintri_005_ad32c3b8e4c714b5.json", "tolerance": 1e-08, "warnings": []}
{"case": "cirintri@6", "container": {"circumradius": 4.30945637922671, "orientation_radians": 0.0, "side_length": 7.4641974018224735, "sides": 3, "type": "regular_polygon"}, "density": 0.7813295579764934, "errors": [], "item": {"radius": 1.0, "type": "circle"}, "max_boundary_excess": -1.6105130511423482e-06, "max_pair_overlap_depth": -3.3410124574029254e-05, "n": 6, "ok": true, "record_id": "09bddcb97a85d368", "setup": "cirintri", "side": 7.4641974018224735, "solution_hash": "09bddcb97a85d368", "solution_path": "data/solutions/generated/cirintri_006_09bddcb97a85d368.json", "tolerance": 1e-08, "warnings": []}
{"case": "cirintri@7", "container": {"circumradius": 5.154804028982868, "orientation_radians": 0.0, "side_length": 8.928382481259078, "sides": 3, "type": "regular_polygon"}, "density": 0.6370913423934993, "errors": [], "item": {"radius": 1.0, "type": "circle"}, "max_boundary_excess": -6.6242968528307244e-06, "max_pair_overlap_depth": -4.768068922000879e-05, "n": 7, "ok": true, "record_id": "1d479488f6f25560", "setup": "cirintri", "side": 8.928382481259078, "solution_hash": "1d479488f6f25560", "solution_path": "data/solutions/generated/cirintri_007_1d479488f6f25560.json", "tolerance": 1e-08, "warnings": []}
@@ -112,13 +115,17 @@
{"case": "domindom@5", "container": {"height": 2.500118873959463, "type": "rectangle", "width": 5.000237747918926}, "density": 0.7999239260919105, "errors": [], "item": {"height": 1.0, "type": "rectangle", "width": 2.0}, "max_boundary_excess": -4.6075584103633105e-07, "max_pair_overlap_depth": -2.297221660940707e-05, "n": 5, "ok": true, "record_id": "af0995e19f0dc8ce", "setup": "domindom", "side": 5.000237747918926, "solution_hash": "af0995e19f0dc8ce", "solution_path": "data/solutions/generated/domindom_005_af0995e19f0dc8ce.json", "tolerance": 1e-08, "warnings": []}
{"case": "domindom@6", "container": {"height": 2.9170183771068423, "type": "rectangle", "width": 5.834036754213685}, "density": 0.7051360525077088, "errors": [], "item": {"height": 1.0, "type": "rectangle", "width": 2.0}, "max_boundary_excess": -5.01302900812739e-06, "max_pair_overlap_depth": -7.57681649177222e-06, "n": 6, "ok": true, "record_id": "0f3d5defa1e042a8", "setup": "domindom", "side": 5.834036754213685, "solution_hash": "0f3d5defa1e042a8", "solution_path": "data/solutions/generated/domindom_006_0f3d5defa1e042a8.json", "tolerance": 1e-08, "warnings": []}
{"case": "domindom@7", "container": {"height": 3.000099429737261, "type": "rectangle", "width": 6.000198859474522}, "density": 0.777726224180711, "errors": [], "item": {"height": 1.0, "type": "rectangle", "width": 2.0}, "max_boundary_excess": -3.3331344280185253e-06, "max_pair_overlap_depth": -2.1961883454491016e-05, "n": 7, "ok": true, "record_id": "7b5335d8b8c4a5db", "setup": "domindom", "side": 6.000198859474522, "solution_hash": "7b5335d8b8c4a5db", "solution_path": "data/solutions/generated/domindom_007_7b5335d8b8c4a5db.json", "tolerance": 1e-08, "warnings": []}
+{"case": "domindom@7", "container": {"height": 3.0000496969324035, "type": "rectangle", "width": 6.000099393864807}, "density": 0.7777520096383135, "errors": [], "item": {"height": 1.0, "type": "rectangle", "width": 2.0}, "max_boundary_excess": -7.043740939405296e-07, "max_pair_overlap_depth": -7.019236833216702e-06, "n": 7, "ok": true, "record_id": "a480ce1181b75833", "setup": "domindom", "side": 6.000099393864807, "solution_hash": "a480ce1181b75833", "solution_path": "data/solutions/generated/domindom_007_a480ce1181b75833.json", "tolerance": 1e-08, "warnings": []}
{"case": "domindom@8", "container": {"height": 3.0000753074216044, "type": "rectangle", "width": 6.000150614843209}, "density": 0.8888442639489805, "errors": [], "item": {"height": 1.0, "type": "rectangle", "width": 2.0}, "max_boundary_excess": -4.80012014980602e-06, "max_pair_overlap_depth": -1.8965164302886528e-05, "n": 8, "ok": true, "record_id": "953d10b3e926402a", "setup": "domindom", "side": 6.000150614843209, "solution_hash": "953d10b3e926402a", "solution_path": "data/solutions/generated/domindom_008_953d10b3e926402a.json", "tolerance": 1e-08, "warnings": []}
+{"case": "domindom@8", "container": {"height": 3.0000496969324035, "type": "rectangle", "width": 6.000099393864807}, "density": 0.8888594395866439, "errors": [], "item": {"height": 1.0, "type": "rectangle", "width": 2.0}, "max_boundary_excess": -3.7542337882889854e-07, "max_pair_overlap_depth": -7.019236833216702e-06, "n": 8, "ok": true, "record_id": "a4f6452b97bb5cc0", "setup": "domindom", "side": 6.000099393864807, "solution_hash": "a4f6452b97bb5cc0", "solution_path": "data/solutions/generated/domindom_008_a4f6452b97bb5cc0.json", "tolerance": 1e-08, "warnings": []}
{"case": "domindom@9", "container": {"height": 3.0000496969324035, "type": "rectangle", "width": 6.000099393864807}, "density": 0.9999668695349745, "errors": [], "item": {"height": 1.0, "type": "rectangle", "width": 2.0}, "max_boundary_excess": -3.7542337882889854e-07, "max_pair_overlap_depth": -3.0527855267026283e-06, "n": 9, "ok": true, "record_id": "43b9630822b40990", "setup": "domindom", "side": 6.000099393864807, "solution_hash": "43b9630822b40990", "solution_path": "data/solutions/generated/domindom_009_43b9630822b40990.json", "tolerance": 1e-08, "warnings": []}
{"case": "domindom@10", "container": {"height": 3.500249869013882, "type": "rectangle", "width": 7.000499738027764}, "density": 0.8162099861183555, "errors": [], "item": {"height": 1.0, "type": "rectangle", "width": 2.0}, "max_boundary_excess": -1.5739397918324016e-07, "max_pair_overlap_depth": -9.291197802940854e-05, "n": 10, "ok": true, "record_id": "d7508c45c3f18a7c", "setup": "domindom", "side": 7.000499738027764, "solution_hash": "d7508c45c3f18a7c", "solution_path": "data/solutions/generated/domindom_010_d7508c45c3f18a7c.json", "tolerance": 1e-08, "warnings": []}
{"case": "domindom@11", "container": {"height": 3.8015321723096425, "type": "rectangle", "width": 7.603064344619285}, "density": 0.7611589259525644, "errors": [], "item": {"height": 1.0, "type": "rectangle", "width": 2.0}, "max_boundary_excess": -3.0757791549262947e-06, "max_pair_overlap_depth": -0.0001019023981195799, "n": 11, "ok": true, "record_id": "0fa9d3de612fbf03", "setup": "domindom", "side": 7.603064344619285, "solution_hash": "0fa9d3de612fbf03", "solution_path": "data/solutions/generated/domindom_011_0fa9d3de612fbf03.json", "tolerance": 1e-08, "warnings": []}
{"case": "domindom@12", "container": {"height": 3.916663137487305, "type": "rectangle", "width": 7.83332627497461}, "density": 0.7822558234911755, "errors": [], "item": {"height": 1.0, "type": "rectangle", "width": 2.0}, "max_boundary_excess": -4.0673284962267076e-07, "max_pair_overlap_depth": -1.4217486933421064e-05, "n": 12, "ok": true, "record_id": "3a75d33bd56228b0", "setup": "domindom", "side": 7.83332627497461, "solution_hash": "3a75d33bd56228b0", "solution_path": "data/solutions/generated/domindom_012_3a75d33bd56228b0.json", "tolerance": 1e-08, "warnings": []}
+{"case": "domindom@14", "container": {"height": 4.000080890451122, "type": "rectangle", "width": 8.000161780902245}, "density": 0.8749646115011096, "errors": [], "item": {"height": 1.0, "type": "rectangle", "width": 2.0}, "max_boundary_excess": -5.0905786346255866e-06, "max_pair_overlap_depth": -4.621969549978289e-06, "n": 14, "ok": true, "record_id": "6f795311544960ee", "setup": "domindom", "side": 8.000161780902245, "solution_hash": "6f795311544960ee", "solution_path": "data/solutions/generated/domindom_014_6f795311544960ee.json", "tolerance": 1e-08, "warnings": []}
{"case": "domindom@14", "container": {"height": 4.000117921926035, "type": "rectangle", "width": 8.00023584385207}, "density": 0.8749484114386541, "errors": [], "item": {"height": 1.0, "type": "rectangle", "width": 2.0}, "max_boundary_excess": -4.966859551203129e-06, "max_pair_overlap_depth": -1.688780940467538e-05, "n": 14, "ok": true, "record_id": "8be6cbfdbea40909", "setup": "domindom", "side": 8.00023584385207, "solution_hash": "8be6cbfdbea40909", "solution_path": "data/solutions/generated/domindom_014_8be6cbfdbea40909.json", "tolerance": 1e-08, "warnings": []}
{"case": "domindom@15", "container": {"height": 4.000112654115248, "type": "rectangle", "width": 8.000225308230496}, "density": 0.9374471956142247, "errors": [], "item": {"height": 1.0, "type": "rectangle", "width": 2.0}, "max_boundary_excess": -2.415030228686277e-06, "max_pair_overlap_depth": -1.0456826571081379e-05, "n": 15, "ok": true, "record_id": "d2b8e4757dde349c", "setup": "domindom", "side": 8.000225308230496, "solution_hash": "d2b8e4757dde349c", "solution_path": "data/solutions/generated/domindom_015_d2b8e4757dde349c.json", "tolerance": 1e-08, "warnings": []}
+{"case": "domindom@15", "container": {"height": 4.000080890451122, "type": "rectangle", "width": 8.000161780902245}, "density": 0.9374620837511888, "errors": [], "item": {"height": 1.0, "type": "rectangle", "width": 2.0}, "max_boundary_excess": -4.23688533190969e-06, "max_pair_overlap_depth": -4.621969549978289e-06, "n": 15, "ok": true, "record_id": "dda9136316a32e15", "setup": "domindom", "side": 8.000161780902245, "solution_hash": "dda9136316a32e15", "solution_path": "data/solutions/generated/domindom_015_dda9136316a32e15.json", "tolerance": 1e-08, "warnings": []}
{"case": "domindom@16", "container": {"height": 4.000080890451122, "type": "rectangle", "width": 8.000161780902245}, "density": 0.9999595560012681, "errors": [], "item": {"height": 1.0, "type": "rectangle", "width": 2.0}, "max_boundary_excess": -4.23688533190969e-06, "max_pair_overlap_depth": -4.621969549978289e-06, "n": 16, "ok": true, "record_id": "5102d8436800a7ff", "setup": "domindom", "side": 8.000161780902245, "solution_hash": "5102d8436800a7ff", "solution_path": "data/solutions/generated/domindom_016_5102d8436800a7ff.json", "tolerance": 1e-08, "warnings": []}
{"case": "domindom@17", "container": {"height": 4.500205860758932, "type": "rectangle", "width": 9.000411721517864}, "density": 0.8394293686085971, "errors": [], "item": {"height": 1.0, "type": "rectangle", "width": 2.0}, "max_boundary_excess": -4.953267209728551e-07, "max_pair_overlap_depth": -5.000451830872166e-05, "n": 17, "ok": true, "record_id": "3b388ddd068a843f", "setup": "domindom", "side": 9.000411721517864, "solution_hash": "3b388ddd068a843f", "solution_path": "data/solutions/generated/domindom_017_3b388ddd068a843f.json", "tolerance": 1e-08, "warnings": []}
{"case": "domindom@18", "container": {"height": 4.504321147560441, "type": "rectangle", "width": 9.008642295120882}, "density": 0.887184224626392, "errors": [], "item": {"height": 1.0, "type": "rectangle", "width": 2.0}, "max_boundary_excess": -2.4781434060372476e-06, "max_pair_overlap_depth": -4.894490537976637e-05, "n": 18, "ok": true, "record_id": "6a5fcb6ec4e55187", "setup": "domindom", "side": 9.008642295120882, "solution_hash": "6a5fcb6ec4e55187", "solution_path": "data/solutions/generated/domindom_018_6a5fcb6ec4e55187.json", "tolerance": 1e-08, "warnings": []}
@@ -151,6 +158,7 @@
{"case": "hexincir@3", "container": {"radius": 2.000056249552998, "type": "circle"}, "density": 0.6202101203168955, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -9.970405754344824e-06, "max_pair_overlap_depth": -7.230141190991447e-05, "n": 3, "ok": true, "record_id": "194e619a74e29452", "setup": "hexincir", "side": 4.000112499105996, "solution_hash": "194e619a74e29452", "solution_path": "data/solutions/generated/hexincir_003_194e619a74e29452.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexincir@4", "container": {"radius": 2.2396783874729134, "type": "circle"}, "density": 0.6594633839026793, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -4.544625404889757e-07, "max_pair_overlap_depth": -5.057462804924917e-05, "n": 4, "ok": true, "record_id": "355b11dfc363f5ba", "setup": "hexincir", "side": 4.479356774945827, "solution_hash": "355b11dfc363f5ba", "solution_path": "data/solutions/generated/hexincir_004_355b11dfc363f5ba.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexincir@5", "container": {"radius": 2.4719739785189008, "type": "circle"}, "density": 0.6766813833384198, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -7.673907177263573e-06, "max_pair_overlap_depth": -5.983134966366932e-05, "n": 5, "ok": true, "record_id": "9427b00c077aafc3", "setup": "hexincir", "side": 4.9439479570378015, "solution_hash": "9427b00c077aafc3", "solution_path": "data/solutions/generated/hexincir_005_9427b00c077aafc3.json", "tolerance": 1e-08, "warnings": []}
+{"case": "hexincir@6", "container": {"radius": 2.6457937463486663, "type": "circle"}, "density": 0.7088286989393032, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -1.370930279076532e-06, "max_pair_overlap_depth": -2.6143388138111234e-05, "n": 6, "ok": true, "record_id": "0d33963cb4bcb842", "setup": "hexincir", "side": 5.2915874926973325, "solution_hash": "0d33963cb4bcb842", "solution_path": "data/solutions/generated/hexincir_006_0d33963cb4bcb842.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexincir@6", "container": {"radius": 2.6458488830398026, "type": "circle"}, "density": 0.7087991567669789, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -2.0221671226394733e-07, "max_pair_overlap_depth": -3.6674166167594e-05, "n": 6, "ok": true, "record_id": "d7d900dd0bba32a3", "setup": "hexincir", "side": 5.291697766079605, "solution_hash": "d7d900dd0bba32a3", "solution_path": "data/solutions/generated/hexincir_006_d7d900dd0bba32a3.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexincir@7", "container": {"radius": 2.6457937463486663, "type": "circle"}, "density": 0.8269668154291873, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -1.370930279076532e-06, "max_pair_overlap_depth": -2.6143388138111234e-05, "n": 7, "ok": true, "record_id": "5cd27a740cc9b2a5", "setup": "hexincir", "side": 5.2915874926973325, "solution_hash": "5cd27a740cc9b2a5", "solution_path": "data/solutions/generated/hexincir_007_5cd27a740cc9b2a5.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexincir@8", "container": {"radius": 2.984772928023753, "type": "circle"}, "density": 0.7426247287224678, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -3.6889173538057207e-06, "max_pair_overlap_depth": -1.732806256671271e-05, "n": 8, "ok": true, "record_id": "7d27528d78a05a43", "setup": "hexincir", "side": 5.969545856047506, "solution_hash": "7d27528d78a05a43", "solution_path": "data/solutions/generated/hexincir_008_7d27528d78a05a43.json", "tolerance": 1e-08, "warnings": []}
@@ -164,6 +172,7 @@
{"case": "hexincir@16", "container": {"radius": 4.141580910601899, "type": "circle"}, "density": 0.7714178714001686, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -6.8109720468001456e-06, "max_pair_overlap_depth": -4.18854370716204e-05, "n": 16, "ok": true, "record_id": "928ee7a851bce6e5", "setup": "hexincir", "side": 8.283161821203798, "solution_hash": "928ee7a851bce6e5", "solution_path": "data/solutions/generated/hexincir_016_928ee7a851bce6e5.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexincir@17", "container": {"radius": 4.303381676380651, "type": "circle"}, "density": 0.7591563034967428, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -5.855951803468429e-06, "max_pair_overlap_depth": -4.543663468803416e-05, "n": 17, "ok": true, "record_id": "e582b760dc061cd3", "setup": "hexincir", "side": 8.606763352761302, "solution_hash": "e582b760dc061cd3", "solution_path": "data/solutions/generated/hexincir_017_e582b760dc061cd3.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexincir@18", "container": {"radius": 4.3590845423028055, "type": "circle"}, "density": 0.7834006630179079, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -4.696366953105269e-06, "max_pair_overlap_depth": -1.0490734119961154e-05, "n": 18, "ok": true, "record_id": "12eaac899664a007", "setup": "hexincir", "side": 8.718169084605611, "solution_hash": "12eaac899664a007", "solution_path": "data/solutions/generated/hexincir_018_12eaac899664a007.json", "tolerance": 1e-08, "warnings": []}
+{"case": "hexincir@18", "container": {"radius": 4.358986819791536, "type": "circle"}, "density": 0.7834357889503433, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -5.636539265552187e-06, "max_pair_overlap_depth": -6.973398227638761e-06, "n": 18, "ok": true, "record_id": "3bf05115c19cc6d5", "setup": "hexincir", "side": 8.717973639583072, "solution_hash": "3bf05115c19cc6d5", "solution_path": "data/solutions/generated/hexincir_018_3bf05115c19cc6d5.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexincir@19", "container": {"radius": 4.358986819791536, "type": "circle"}, "density": 0.8269599994475847, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -5.636539265552187e-06, "max_pair_overlap_depth": -6.973398227638761e-06, "n": 19, "ok": true, "record_id": "b4bf200670caf09c", "setup": "hexincir", "side": 8.717973639583072, "solution_hash": "b4bf200670caf09c", "solution_path": "data/solutions/generated/hexincir_019_b4bf200670caf09c.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexincir@20", "container": {"radius": 4.627526064628178, "type": "circle"}, "density": 0.7723857667872704, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -1.8288872105642895e-06, "max_pair_overlap_depth": -4.792351042670617e-06, "n": 20, "ok": true, "record_id": "30108dc5d1ea9a14", "setup": "hexincir", "side": 9.255052129256356, "solution_hash": "30108dc5d1ea9a14", "solution_path": "data/solutions/generated/hexincir_020_30108dc5d1ea9a14.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexincir@21", "container": {"radius": 4.68340545778963, "type": "circle"}, "density": 0.7917677222663352, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -4.498160996568856e-06, "max_pair_overlap_depth": -3.8704072840012316e-06, "n": 21, "ok": true, "record_id": "49a047c29bb184c3", "setup": "hexincir", "side": 9.36681091557926, "solution_hash": "49a047c29bb184c3", "solution_path": "data/solutions/generated/hexincir_021_49a047c29bb184c3.json", "tolerance": 1e-08, "warnings": []}
@@ -189,10 +198,12 @@
{"case": "hexindom@20", "container": {"height": 5.801664108762803, "type": "rectangle", "width": 11.603328217525606}, "density": 0.7718745532343245, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -6.575652109486185e-06, "max_pair_overlap_depth": -1.0063478294242145e-05, "n": 20, "ok": true, "record_id": "49ce661db444ec25", "setup": "hexindom", "side": 11.603328217525606, "solution_hash": "49ce661db444ec25", "solution_path": "data/solutions/generated/hexindom_020_49ce661db444ec25.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexinhex@1", "container": {"circumradius": 1.0000067750358794, "orientation_radians": 0.0, "side_length": 1.0000067750358792, "sides": 6, "type": "regular_polygon"}, "density": 0.9999864500659439, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -3.374323059598708e-06, "max_pair_overlap_depth": 0.0, "n": 1, "ok": true, "record_id": "fd6c6df7b6a9fd4f", "setup": "hexinhex", "side": 1.0000067750358792, "solution_hash": "fd6c6df7b6a9fd4f", "solution_path": "data/solutions/generated/hexinhex_001_fd6c6df7b6a9fd4f.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexinhex@2", "container": {"circumradius": 2.000032241791776, "orientation_radians": 0.0, "side_length": 2.0000322417917755, "sides": 6, "type": "regular_polygon"}, "density": 0.49998387949392886, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -4.177681809212628e-06, "max_pair_overlap_depth": -3.6384174144801684e-05, "n": 2, "ok": true, "record_id": "4c0cc36e970c84b1", "setup": "hexinhex", "side": 2.0000322417917755, "solution_hash": "4c0cc36e970c84b1", "solution_path": "data/solutions/generated/hexinhex_002_4c0cc36e970c84b1.json", "tolerance": 1e-08, "warnings": []}
+{"case": "hexinhex@2", "container": {"circumradius": 2.0000156772159707, "orientation_radians": 0.0, "side_length": 2.0000156772159703, "sides": 6, "type": "regular_polygon"}, "density": 0.4999921614841797, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -2.457810528986454e-06, "max_pair_overlap_depth": -9.916736621090457e-06, "n": 2, "ok": true, "record_id": "c8ccd859ecc810e4", "setup": "hexinhex", "side": 2.0000156772159703, "solution_hash": "c8ccd859ecc810e4", "solution_path": "data/solutions/generated/hexinhex_002_c8ccd859ecc810e4.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexinhex@3", "container": {"circumradius": 2.0000156772159707, "orientation_radians": 0.0, "side_length": 2.0000156772159703, "sides": 6, "type": "regular_polygon"}, "density": 0.7499882422262695, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -2.457810528986454e-06, "max_pair_overlap_depth": -9.916736621090457e-06, "n": 3, "ok": true, "record_id": "2e22e12b6d11de4d", "setup": "hexinhex", "side": 2.0000156772159703, "solution_hash": "2e22e12b6d11de4d", "solution_path": "data/solutions/generated/hexinhex_003_2e22e12b6d11de4d.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexinhex@4", "container": {"circumradius": 2.3333540432955684, "orientation_radians": 0.0, "side_length": 2.333354043295568, "sides": 6, "type": "regular_polygon"}, "density": 0.7346808358825446, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -1.969618748468479e-06, "max_pair_overlap_depth": -1.5832978702556844e-05, "n": 4, "ok": true, "record_id": "73ddab3f27a95ffe", "setup": "hexinhex", "side": 2.333354043295568, "solution_hash": "73ddab3f27a95ffe", "solution_path": "data/solutions/generated/hexinhex_004_73ddab3f27a95ffe.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexinhex@5", "container": {"circumradius": 2.6666930086059697, "orientation_radians": 0.0, "side_length": 2.6666930086059693, "sides": 6, "type": "regular_polygon"}, "density": 0.7031111089487749, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -1.5309295562371306e-06, "max_pair_overlap_depth": -6.88869177611906e-06, "n": 5, "ok": true, "record_id": "f45300c86d046ee7", "setup": "hexinhex", "side": 2.6666930086059693, "solution_hash": "f45300c86d046ee7", "solution_path": "data/solutions/generated/hexinhex_005_f45300c86d046ee7.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexinhex@6", "container": {"circumradius": 2.8868841804633925, "orientation_radians": 0.0, "side_length": 2.886884180463392, "sides": 6, "type": "regular_polygon"}, "density": 0.7199337426480286, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -8.09670685382364e-06, "max_pair_overlap_depth": -5.069396536394599e-05, "n": 6, "ok": true, "record_id": "06b9c79146f8c119", "setup": "hexinhex", "side": 2.886884180463392, "solution_hash": "06b9c79146f8c119", "solution_path": "data/solutions/generated/hexinhex_006_06b9c79146f8c119.json", "tolerance": 1e-08, "warnings": []}
+{"case": "hexinhex@6", "container": {"circumradius": 2.886797667992364, "orientation_radians": 0.0, "side_length": 2.8867976679923637, "sides": 6, "type": "regular_polygon"}, "density": 0.7199768937015341, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -5.6806989039887556e-06, "max_pair_overlap_depth": -3.096762286192156e-05, "n": 6, "ok": true, "record_id": "93e84dfdad8f0036", "setup": "hexinhex", "side": 2.8867976679923637, "solution_hash": "93e84dfdad8f0036", "solution_path": "data/solutions/generated/hexinhex_006_93e84dfdad8f0036.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexinhex@7", "container": {"circumradius": 2.886797667992364, "orientation_radians": 0.0, "side_length": 2.8867976679923637, "sides": 6, "type": "regular_polygon"}, "density": 0.8399730426517898, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -5.6806989039887556e-06, "max_pair_overlap_depth": -2.9189085207215193e-05, "n": 7, "ok": true, "record_id": "d2a43d4ca2836467", "setup": "hexinhex", "side": 2.8867976679923637, "solution_hash": "d2a43d4ca2836467", "solution_path": "data/solutions/generated/hexinhex_007_d2a43d4ca2836467.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexinhex@8", "container": {"circumradius": 3.3145783682856225, "orientation_radians": 0.0, "side_length": 3.314578368285622, "sides": 6, "type": "regular_polygon"}, "density": 0.728171041468435, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -6.71875974277703e-06, "max_pair_overlap_depth": -2.9528072121864923e-05, "n": 8, "ok": true, "record_id": "8d9e02005a596470", "setup": "hexinhex", "side": 3.314578368285622, "solution_hash": "8d9e02005a596470", "solution_path": "data/solutions/generated/hexinhex_008_8d9e02005a596470.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexinhex@9", "container": {"circumradius": 3.6291431939654926, "orientation_radians": 0.0, "side_length": 3.629143193965492, "sides": 6, "type": "regular_polygon"}, "density": 0.683335999337671, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -9.454488069704802e-06, "max_pair_overlap_depth": -2.836888994439679e-05, "n": 9, "ok": true, "record_id": "21c530fbe612f27c", "setup": "hexinhex", "side": 3.629143193965492, "solution_hash": "21c530fbe612f27c", "solution_path": "data/solutions/generated/hexinhex_009_21c530fbe612f27c.json", "tolerance": 1e-08, "warnings": []}
@@ -204,8 +215,10 @@
{"case": "hexinhex@15", "container": {"circumradius": 4.448689692280499, "orientation_radians": 0.0, "side_length": 4.448689692280498, "sides": 6, "type": "regular_polygon"}, "density": 0.7579263950656321, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -4.564398033090082e-06, "max_pair_overlap_depth": -2.5468766324809877e-06, "n": 15, "ok": true, "record_id": "536c412ab623afbc", "setup": "hexinhex", "side": 4.448689692280498, "solution_hash": "536c412ab623afbc", "solution_path": "data/solutions/generated/hexinhex_015_536c412ab623afbc.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexinhex@16", "container": {"circumradius": 4.528188920895578, "orientation_radians": 0.0, "side_length": 4.528188920895577, "sides": 6, "type": "regular_polygon"}, "density": 0.7803167098453238, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -2.541715273786282e-06, "max_pair_overlap_depth": -2.3206185233659582e-05, "n": 16, "ok": true, "record_id": "17cb432379a18c18", "setup": "hexinhex", "side": 4.528188920895577, "solution_hash": "17cb432379a18c18", "solution_path": "data/solutions/generated/hexinhex_016_17cb432379a18c18.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexinhex@17", "container": {"circumradius": 4.614071706266547, "orientation_radians": 0.0, "side_length": 4.614071706266546, "sides": 6, "type": "regular_polygon"}, "density": 0.7985097849440846, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -9.290243996089003e-06, "max_pair_overlap_depth": -4.642493336670661e-05, "n": 17, "ok": true, "record_id": "df809b176b000a6d", "setup": "hexinhex", "side": 4.614071706266546, "solution_hash": "df809b176b000a6d", "solution_path": "data/solutions/generated/hexinhex_017_df809b176b000a6d.json", "tolerance": 1e-08, "warnings": []}
+{"case": "hexinhex@18", "container": {"circumradius": 4.618904616698239, "orientation_radians": 0.0, "side_length": 4.618904616698238, "sides": 6, "type": "regular_polygon"}, "density": 0.8437125658646701, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -8.772589960788204e-06, "max_pair_overlap_depth": -3.612300978728111e-05, "n": 18, "ok": true, "record_id": "2eb7c5be1ce30256", "setup": "hexinhex", "side": 4.618904616698238, "solution_hash": "2eb7c5be1ce30256", "solution_path": "data/solutions/generated/hexinhex_018_2eb7c5be1ce30256.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexinhex@18", "container": {"circumradius": 4.618907031487258, "orientation_radians": 0.0, "side_length": 4.618907031487257, "sides": 6, "type": "regular_polygon"}, "density": 0.8437116836701195, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": 1.0443685916789865e-09, "max_pair_overlap_depth": -3.080790913889686e-05, "n": 18, "ok": true, "record_id": "e320d39029a817c1", "setup": "hexinhex", "side": 4.618907031487257, "solution_hash": "e320d39029a817c1", "solution_path": "data/solutions/generated/hexinhex_018_e320d39029a817c1.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexinhex@19", "container": {"circumradius": 4.618904616698239, "orientation_radians": 0.0, "side_length": 4.618904616698238, "sides": 6, "type": "regular_polygon"}, "density": 0.8905854861904853, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -8.772589960788204e-06, "max_pair_overlap_depth": -3.612300978728111e-05, "n": 19, "ok": true, "record_id": "4b11a1a09f2a5b70", "setup": "hexinhex", "side": 4.618904616698238, "solution_hash": "4b11a1a09f2a5b70", "solution_path": "data/solutions/generated/hexinhex_019_4b11a1a09f2a5b70.json", "tolerance": 1e-08, "warnings": []}
+{"case": "hexinhex@20", "container": {"circumradius": 5.000067508172298, "orientation_radians": 0.0, "side_length": 5.000067508172297, "sides": 6, "type": "regular_polygon"}, "density": 0.7999783978223634, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -1.5389132812752791e-06, "max_pair_overlap_depth": -3.519050301936666e-06, "n": 20, "ok": true, "record_id": "197dfdbe3f1f6905", "setup": "hexinhex", "side": 5.000067508172297, "solution_hash": "197dfdbe3f1f6905", "solution_path": "data/solutions/generated/hexinhex_020_197dfdbe3f1f6905.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexinhex@20", "container": {"circumradius": 5.000157249896444, "orientation_radians": 0.0, "side_length": 5.000157249896443, "sides": 6, "type": "regular_polygon"}, "density": 0.7999496824068819, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -5.445035556179186e-06, "max_pair_overlap_depth": -1.6551811964360397e-06, "n": 20, "ok": true, "record_id": "2830e2dec38749ef", "setup": "hexinhex", "side": 5.000157249896443, "solution_hash": "2830e2dec38749ef", "solution_path": "data/solutions/generated/hexinhex_020_2830e2dec38749ef.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexinhex@21", "container": {"circumradius": 5.000067508172298, "orientation_radians": 0.0, "side_length": 5.000067508172297, "sides": 6, "type": "regular_polygon"}, "density": 0.8399773177134815, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -1.5389132812752791e-06, "max_pair_overlap_depth": -3.519050301936666e-06, "n": 21, "ok": true, "record_id": "4a431b677be571c9", "setup": "hexinhex", "side": 5.000067508172297, "solution_hash": "4a431b677be571c9", "solution_path": "data/solutions/generated/hexinhex_021_4a431b677be571c9.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexinhex@22", "container": {"circumradius": 5.285900869804883, "orientation_radians": 0.0, "side_length": 5.285900869804882, "sides": 6, "type": "regular_polygon"}, "density": 0.7873804951730353, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -4.575542655386671e-06, "max_pair_overlap_depth": -1.0001604064946434e-05, "n": 22, "ok": true, "record_id": "003a713177cb171c", "setup": "hexinhex", "side": 5.285900869804882, "solution_hash": "003a713177cb171c", "solution_path": "data/solutions/generated/hexinhex_022_003a713177cb171c.json", "tolerance": 1e-08, "warnings": []}
@@ -215,6 +228,7 @@
{"case": "hexinhex@26", "container": {"circumradius": 5.71328227842848, "orientation_radians": 0.0, "side_length": 5.713282278428479, "sides": 6, "type": "regular_polygon"}, "density": 0.7965297187067493, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -2.9624619447323227e-06, "max_pair_overlap_depth": -1.2105377510118753e-05, "n": 26, "ok": true, "record_id": "8ed8c9fb94f18119", "setup": "hexinhex", "side": 5.713282278428479, "solution_hash": "8ed8c9fb94f18119", "solution_path": "data/solutions/generated/hexinhex_026_8ed8c9fb94f18119.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexinhex@27", "container": {"circumradius": 5.716567746994859, "orientation_radians": 0.0, "side_length": 5.716567746994858, "sides": 6, "type": "regular_polygon"}, "density": 0.82621496088318, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -4.56387472302006e-06, "max_pair_overlap_depth": -5.6693937127594296e-05, "n": 27, "ok": true, "record_id": "4e9628a9f49def85", "setup": "hexinhex", "side": 5.716567746994858, "solution_hash": "4e9628a9f49def85", "solution_path": "data/solutions/generated/hexinhex_027_4e9628a9f49def85.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexinhex@28", "container": {"circumradius": 5.9118497830949535, "orientation_radians": 0.0, "side_length": 5.911849783094953, "sides": 6, "type": "regular_polygon"}, "density": 0.8011452289129106, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -7.424277796084301e-06, "max_pair_overlap_depth": -3.209757705180394e-05, "n": 28, "ok": true, "record_id": "ba522f97744aed1a", "setup": "hexinhex", "side": 5.911849783094953, "solution_hash": "ba522f97744aed1a", "solution_path": "data/solutions/generated/hexinhex_028_ba522f97744aed1a.json", "tolerance": 1e-08, "warnings": []}
+{"case": "hexinhex@29", "container": {"circumradius": 6.000078501746011, "orientation_radians": 0.0, "side_length": 6.00007850174601, "sides": 6, "type": "regular_polygon"}, "density": 0.8055344767966968, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -3.7272811687230685e-07, "max_pair_overlap_depth": -4.232627726974414e-06, "n": 29, "ok": true, "record_id": "baea309b7bc15617", "setup": "hexinhex", "side": 6.00007850174601, "solution_hash": "baea309b7bc15617", "solution_path": "data/solutions/generated/hexinhex_029_baea309b7bc15617.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexinhex@29", "container": {"circumradius": 6.000134351852798, "orientation_radians": 0.0, "side_length": 6.000134351852797, "sides": 6, "type": "regular_polygon"}, "density": 0.8055194808067645, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -5.231638156342025e-06, "max_pair_overlap_depth": -6.307567709251671e-06, "n": 29, "ok": true, "record_id": "dba25a591b8992bc", "setup": "hexinhex", "side": 6.000134351852797, "solution_hash": "dba25a591b8992bc", "solution_path": "data/solutions/generated/hexinhex_029_dba25a591b8992bc.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexinhex@30", "container": {"circumradius": 6.000078501746011, "orientation_radians": 0.0, "side_length": 6.00007850174601, "sides": 6, "type": "regular_polygon"}, "density": 0.8333115277207207, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -3.7272811687230685e-07, "max_pair_overlap_depth": -4.232627726974414e-06, "n": 30, "ok": true, "record_id": "100db5f9f28894cc", "setup": "hexinhex", "side": 6.00007850174601, "solution_hash": "100db5f9f28894cc", "solution_path": "data/solutions/generated/hexinhex_030_100db5f9f28894cc.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexinhex@31", "container": {"circumradius": 6.000103172861179, "orientation_radians": 0.0, "side_length": 6.000103172861178, "sides": 6, "type": "regular_polygon"}, "density": 0.8610814974425702, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -1.0312532034243658e-06, "max_pair_overlap_depth": -8.80446118189937e-06, "n": 31, "ok": true, "record_id": "d503e2b08764a35e", "setup": "hexinhex", "side": 6.000103172861178, "solution_hash": "d503e2b08764a35e", "solution_path": "data/solutions/generated/hexinhex_031_d503e2b08764a35e.json", "tolerance": 1e-08, "warnings": []}
@@ -224,6 +238,7 @@
{"case": "hexinsqu@4", "container": {"circumradius": 2.63404475327656, "orientation_radians": 0.0, "side_length": 3.725101813981404, "sides": 4, "type": "regular_polygon"}, "density": 0.748920266367969, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -7.577436704193743e-06, "max_pair_overlap_depth": -5.3695636117151446e-05, "n": 4, "ok": true, "record_id": "33ddf042dd02d191", "setup": "hexinsqu", "side": 3.725101813981404, "solution_hash": "33ddf042dd02d191", "solution_path": "data/solutions/generated/hexinsqu_004_33ddf042dd02d191.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexinsqu@5", "container": {"circumradius": 3.1380721042625472, "orientation_radians": 0.0, "side_length": 4.437904129552771, "sides": 4, "type": "regular_polygon"}, "density": 0.6595778437263831, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -6.5429089870357805e-06, "max_pair_overlap_depth": -1.5874388830661168e-05, "n": 5, "ok": true, "record_id": "38f2dc66ba941978", "setup": "hexinsqu", "side": 4.437904129552771, "solution_hash": "38f2dc66ba941978", "solution_path": "data/solutions/generated/hexinsqu_005_38f2dc66ba941978.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexinsqu@6", "container": {"circumradius": 3.4040378375484592, "orientation_radians": 0.0, "side_length": 4.8140364766922135, "sides": 4, "type": "regular_polygon"}, "density": 0.6726426228424899, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -2.6841191118288626e-06, "max_pair_overlap_depth": -6.696342223433405e-05, "n": 6, "ok": true, "record_id": "31291850c6887128", "setup": "hexinsqu", "side": 4.8140364766922135, "solution_hash": "31291850c6887128", "solution_path": "data/solutions/generated/hexinsqu_006_31291850c6887128.json", "tolerance": 1e-08, "warnings": []}
+{"case": "hexinsqu@7", "container": {"circumradius": 3.6742599744526214, "orientation_radians": 0.0, "side_length": 5.196188287555518, "sides": 4, "type": "regular_polygon"}, "density": 0.673566015856675, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -1.033193436423119e-06, "max_pair_overlap_depth": -1.0846799477937097e-05, "n": 7, "ok": true, "record_id": "6492fb111d24a3e4", "setup": "hexinsqu", "side": 5.196188287555518, "solution_hash": "6492fb111d24a3e4", "solution_path": "data/solutions/generated/hexinsqu_007_6492fb111d24a3e4.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexinsqu@7", "container": {"circumradius": 3.6742982337516255, "orientation_radians": 0.0, "side_length": 5.196242394375057, "sides": 4, "type": "regular_polygon"}, "density": 0.6735519886722757, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -5.667089398456013e-06, "max_pair_overlap_depth": -2.5433886154724306e-05, "n": 7, "ok": true, "record_id": "a4c696c014daab3c", "setup": "hexinsqu", "side": 5.196242394375057, "solution_hash": "a4c696c014daab3c", "solution_path": "data/solutions/generated/hexinsqu_007_a4c696c014daab3c.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexinsqu@8", "container": {"circumradius": 3.6742599744526214, "orientation_radians": 0.0, "side_length": 5.196188287555518, "sides": 4, "type": "regular_polygon"}, "density": 0.7697897324076285, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -1.033193436423119e-06, "max_pair_overlap_depth": -1.0846799477937097e-05, "n": 8, "ok": true, "record_id": "b1252a56565d2ec0", "setup": "hexinsqu", "side": 5.196188287555518, "solution_hash": "b1252a56565d2ec0", "solution_path": "data/solutions/generated/hexinsqu_008_b1252a56565d2ec0.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexinsqu@9", "container": {"circumradius": 3.9020218720568334, "orientation_radians": 0.0, "side_length": 5.5182922521392275, "sides": 4, "type": "regular_polygon"}, "density": 0.7678652278206223, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -6.874395903155008e-06, "max_pair_overlap_depth": -4.408050364324012e-05, "n": 9, "ok": true, "record_id": "5b4ef4c36dcfba05", "setup": "hexinsqu", "side": 5.5182922521392275, "solution_hash": "5b4ef4c36dcfba05", "solution_path": "data/solutions/generated/hexinsqu_009_5b4ef4c36dcfba05.json", "tolerance": 1e-08, "warnings": []}
@@ -255,8 +270,10 @@
{"case": "hexintri@10", "container": {"circumradius": 5.000144303841564, "orientation_radians": 0.0, "side_length": 8.660503979429702, "sides": 3, "type": "regular_polygon"}, "density": 0.7999538247696885, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -1.2645686941820143e-06, "max_pair_overlap_depth": -4.384611814706174e-05, "n": 10, "ok": true, "record_id": "5df6cec7a90f52fe", "setup": "hexintri", "side": 8.660503979429702, "solution_hash": "5df6cec7a90f52fe", "solution_path": "data/solutions/generated/hexintri_010_5df6cec7a90f52fe.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexintri@11", "container": {"circumradius": 5.5133415455578865, "orientation_radians": 0.0, "side_length": 9.549387676386578, "sides": 3, "type": "regular_polygon"}, "density": 0.7237571814877352, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -3.6199514834868076e-06, "max_pair_overlap_depth": -2.6419225191420814e-05, "n": 11, "ok": true, "record_id": "d582cb032b9c575f", "setup": "hexintri", "side": 9.549387676386578, "solution_hash": "d582cb032b9c575f", "solution_path": "data/solutions/generated/hexintri_011_d582cb032b9c575f.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexintri@12", "container": {"circumradius": 5.773835799112101, "orientation_radians": 0.0, "side_length": 10.000576958622208, "sides": 3, "type": "regular_polygon"}, "density": 0.7199169251480841, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -6.006918638920166e-07, "max_pair_overlap_depth": -2.082412255655086e-05, "n": 12, "ok": true, "record_id": "b6e20361c3af5615", "setup": "hexintri", "side": 10.000576958622208, "solution_hash": "b6e20361c3af5615", "solution_path": "data/solutions/generated/hexintri_012_b6e20361c3af5615.json", "tolerance": 1e-08, "warnings": []}
+{"case": "hexintri@13", "container": {"circumradius": 6.000104811629913, "orientation_radians": 0.0, "side_length": 10.392486384481497, "sides": 3, "type": "regular_polygon"}, "density": 0.7221969904539467, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -6.672744116747984e-07, "max_pair_overlap_depth": -2.8784199847526563e-05, "n": 13, "ok": true, "record_id": "3b4102a5f66851b2", "setup": "hexintri", "side": 10.392486384481497, "solution_hash": "3b4102a5f66851b2", "solution_path": "data/solutions/generated/hexintri_013_3b4102a5f66851b2.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexintri@13", "container": {"circumradius": 6.000163746844505, "orientation_radians": 0.0, "side_length": 10.392588463267526, "sides": 3, "type": "regular_polygon"}, "density": 0.7221828032992709, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -1.7733151165742505e-06, "max_pair_overlap_depth": -4.4289278687159594e-05, "n": 13, "ok": true, "record_id": "da55bd8ffef5f50e", "setup": "hexintri", "side": 10.392588463267526, "solution_hash": "da55bd8ffef5f50e", "solution_path": "data/solutions/generated/hexintri_013_da55bd8ffef5f50e.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexintri@14", "container": {"circumradius": 6.000208334029342, "orientation_radians": 0.0, "side_length": 10.39266569053703, "sides": 3, "type": "regular_polygon"}, "density": 0.7777237680646841, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -8.442529134834328e-06, "max_pair_overlap_depth": -4.275788154761706e-05, "n": 14, "ok": true, "record_id": "272118e9284c579f", "setup": "hexintri", "side": 10.39266569053703, "solution_hash": "272118e9284c579f", "solution_path": "data/solutions/generated/hexintri_014_272118e9284c579f.json", "tolerance": 1e-08, "warnings": []}
+{"case": "hexintri@14", "container": {"circumradius": 6.000104811629913, "orientation_radians": 0.0, "side_length": 10.392486384481497, "sides": 3, "type": "regular_polygon"}, "density": 0.7777506051042504, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -6.672744116747984e-07, "max_pair_overlap_depth": -2.8784199847526563e-05, "n": 14, "ok": true, "record_id": "aa2a4511d2c644b1", "setup": "hexintri", "side": 10.392486384481497, "solution_hash": "aa2a4511d2c644b1", "solution_path": "data/solutions/generated/hexintri_014_aa2a4511d2c644b1.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexintri@15", "container": {"circumradius": 6.000104811629913, "orientation_radians": 0.0, "side_length": 10.392486384481497, "sides": 3, "type": "regular_polygon"}, "density": 0.833304219754554, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -6.672744116747984e-07, "max_pair_overlap_depth": -2.8784199847526563e-05, "n": 15, "ok": true, "record_id": "83460d360e93ff66", "setup": "hexintri", "side": 10.392486384481497, "solution_hash": "83460d360e93ff66", "solution_path": "data/solutions/generated/hexintri_015_83460d360e93ff66.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexintri@16", "container": {"circumradius": 6.562948539050667, "orientation_radians": 0.0, "side_length": 11.36736031709569, "sides": 3, "type": "regular_polygon"}, "density": 0.7429369875675369, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -6.657075146936789e-07, "max_pair_overlap_depth": -2.5616124350591463e-06, "n": 16, "ok": true, "record_id": "89d5bef670451c83", "setup": "hexintri", "side": 11.36736031709569, "solution_hash": "89d5bef670451c83", "solution_path": "data/solutions/generated/hexintri_16_89d5bef670451c83.json", "tolerance": 1e-08, "warnings": []}
{"case": "hexintri@17", "container": {"circumradius": 6.721674200378152, "orientation_radians": 0.0, "side_length": 11.642281226979867, "sides": 3, "type": "regular_polygon"}, "density": 0.7525303158572544, "errors": [], "item": {"circumradius": 1.0000000000000002, "side_length": 1.0, "sides": 6, "type": "regular_polygon"}, "max_boundary_excess": -1.447216764354318e-05, "max_pair_overlap_depth": -1.3408176938511929e-05, "n": 17, "ok": true, "record_id": "8137d039fd339f48", "setup": "hexintri", "side": 11.642281226979867, "solution_hash": "8137d039fd339f48", "solution_path": "data/solutions/generated/hexintri_017_8137d039fd339f48.json", "tolerance": 1e-08, "warnings": []}
@@ -321,6 +338,7 @@
{"case": "peninpen@3", "container": {"circumradius": 1.9021465733927407, "orientation_radians": 0.0, "side_length": 2.2361074070778306, "sides": 5, "type": "regular_polygon"}, "density": 0.599978840427638, "errors": [], "item": {"circumradius": 0.8506508083520399, "side_length": 1.0, "sides": 5, "type": "regular_polygon"}, "max_boundary_excess": -2.7662857614262655e-06, "max_pair_overlap_depth": -2.7755970794667603e-05, "n": 3, "ok": true, "record_id": "bf261a4dce178a90", "setup": "peninpen", "side": 2.2361074070778306, "solution_hash": "bf261a4dce178a90", "solution_path": "data/solutions/generated/peninpen_003_bf261a4dce178a90.json", "tolerance": 1e-08, "warnings": []}
{"case": "peninpen@4", "container": {"circumradius": 2.081779715517815, "orientation_radians": 0.0, "side_length": 2.4472788306059834, "sides": 5, "type": "regular_polygon"}, "density": 0.6678717661629443, "errors": [], "item": {"circumradius": 0.8506508083520399, "side_length": 1.0, "sides": 5, "type": "regular_polygon"}, "max_boundary_excess": -1.0634336001709255e-06, "max_pair_overlap_depth": -3.575601486860425e-05, "n": 4, "ok": true, "record_id": "85435733af14070a", "setup": "peninpen", "side": 2.4472788306059834, "solution_hash": "85435733af14070a", "solution_path": "data/solutions/generated/peninpen_004_85435733af14070a.json", "tolerance": 1e-08, "warnings": []}
{"case": "peninpen@5", "container": {"circumradius": 2.2270761068691023, "orientation_radians": 0.0, "side_length": 2.6180849827011885, "sides": 5, "type": "regular_polygon"}, "density": 0.7294617516267693, "errors": [], "item": {"circumradius": 0.8506508083520399, "side_length": 1.0, "sides": 5, "type": "regular_polygon"}, "max_boundary_excess": -2.513169742800457e-06, "max_pair_overlap_depth": -2.4501936757292953e-06, "n": 5, "ok": true, "record_id": "28fa9f787595145b", "setup": "peninpen", "side": 2.6180849827011885, "solution_hash": "28fa9f787595145b", "solution_path": "data/solutions/generated/peninpen_005_28fa9f787595145b.json", "tolerance": 1e-08, "warnings": []}
+{"case": "peninpen@5", "container": {"circumradius": 2.2270650746212577, "orientation_radians": 0.0, "side_length": 2.618072013516023, "sides": 5, "type": "regular_polygon"}, "density": 0.7294689787373387, "errors": [], "item": {"circumradius": 0.8506508083520399, "side_length": 1.0, "sides": 5, "type": "regular_polygon"}, "max_boundary_excess": -2.4414928963119693e-06, "max_pair_overlap_depth": -1.898562579671914e-05, "n": 5, "ok": true, "record_id": "5517dc706e76853c", "setup": "peninpen", "side": 2.618072013516023, "solution_hash": "5517dc706e76853c", "solution_path": "data/solutions/generated/peninpen_005_5517dc706e76853c.json", "tolerance": 1e-08, "warnings": []}
{"case": "peninpen@6", "container": {"circumradius": 2.2270650746212577, "orientation_radians": 0.0, "side_length": 2.618072013516023, "sides": 5, "type": "regular_polygon"}, "density": 0.8753627744848066, "errors": [], "item": {"circumradius": 0.8506508083520399, "side_length": 1.0, "sides": 5, "type": "regular_polygon"}, "max_boundary_excess": -2.4414928963119693e-06, "max_pair_overlap_depth": -1.898562579671914e-05, "n": 6, "ok": true, "record_id": "198bd12691483df1", "setup": "peninpen", "side": 2.618072013516023, "solution_hash": "198bd12691483df1", "solution_path": "data/solutions/generated/peninpen_006_198bd12691483df1.json", "tolerance": 1e-08, "warnings": []}
{"case": "peninpen@7", "container": {"circumradius": 2.6189969470021492, "orientation_radians": 0.0, "side_length": 3.0788155624937503, "sides": 5, "type": "regular_polygon"}, "density": 0.7384663216832239, "errors": [], "item": {"circumradius": 0.8506508083520399, "side_length": 1.0, "sides": 5, "type": "regular_polygon"}, "max_boundary_excess": -6.964189962454e-06, "max_pair_overlap_depth": -1.5079265303863387e-05, "n": 7, "ok": true, "record_id": "5b6283ab3dc4c3b5", "setup": "peninpen", "side": 3.0788155624937503, "solution_hash": "5b6283ab3dc4c3b5", "solution_path": "data/solutions/generated/peninpen_007_5b6283ab3dc4c3b5.json", "tolerance": 1e-08, "warnings": []}
{"case": "peninpen@8", "container": {"circumradius": 2.837192223521861, "orientation_radians": 0.0, "side_length": 3.3353194938100796, "sides": 5, "type": "regular_polygon"}, "density": 0.7191427449408644, "errors": [], "item": {"circumradius": 0.8506508083520399, "side_length": 1.0, "sides": 5, "type": "regular_polygon"}, "max_boundary_excess": -6.630985964589326e-06, "max_pair_overlap_depth": -3.1607294870367175e-05, "n": 8, "ok": true, "record_id": "de5f4ff90a59b8bf", "setup": "peninpen", "side": 3.3353194938100796, "solution_hash": "de5f4ff90a59b8bf", "solution_path": "data/solutions/generated/peninpen_008_de5f4ff90a59b8bf.json", "tolerance": 1e-08, "warnings": []}
@@ -403,11 +421,14 @@
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{"case": "squincir@35", "container": {"radius": 3.640654908255323, "type": "circle"}, "density": 0.8405414638481816, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -4.121617296348745e-07, "max_pair_overlap_depth": -2.1344301219083928e-05, "n": 35, "ok": true, "record_id": "f2d91d98b4c2ee39", "setup": "squincir", "side": 7.281309816510646, "solution_hash": "f2d91d98b4c2ee39", "solution_path": "data/solutions/generated/squincir_035_f2d91d98b4c2ee39.json", "tolerance": 1e-08, "warnings": []}
{"case": "squindom@1", "container": {"height": 1.0000093204700702, "type": "rectangle", "width": 2.0000186409401404}, "density": 0.499990679660235, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -3.7525252651882113e-06, "max_pair_overlap_depth": 0.0, "n": 1, "ok": true, "record_id": "557f8de873a17a8e", "setup": "squindom", "side": 2.0000186409401404, "solution_hash": "557f8de873a17a8e", "solution_path": "data/solutions/generated/squindom_001_557f8de873a17a8e.json", "tolerance": 1e-08, "warnings": []}
+{"case": "squindom@1", "container": {"height": 1.0000058501509195, "type": "rectangle", "width": 2.000011700301839}, "density": 0.4999941499004167, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -3.463373909884382e-07, "max_pair_overlap_depth": 0.0, "n": 1, "ok": true, "record_id": "b71122905cd4c5c7", "setup": "squindom", "side": 2.000011700301839, "solution_hash": "b71122905cd4c5c7", "solution_path": "data/solutions/generated/squindom_001_b71122905cd4c5c7.json", "tolerance": 1e-08, "warnings": []}
{"case": "squindom@2", "container": {"height": 1.0000058501509195, "type": "rectangle", "width": 2.000011700301839}, "density": 0.9999882998008334, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -3.463373909884382e-07, "max_pair_overlap_depth": -9.41881114351023e-06, "n": 2, "ok": true, "record_id": "7684110dce2e5290", "setup": "squindom", "side": 2.000011700301839, "solution_hash": "7684110dce2e5290", "solution_path": "data/solutions/generated/squindom_002_7684110dce2e5290.json", "tolerance": 1e-08, "warnings": []}
{"case": "squindom@3", "container": {"height": 1.5000280665315933, "type": "rectangle", "width": 3.0000561330631865}, "density": 0.6666417193387711, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -2.0617527398680835e-06, "max_pair_overlap_depth": -1.9554412996725112e-05, "n": 3, "ok": true, "record_id": "3047ca63de76a853", "setup": "squindom", "side": 3.0000561330631865, "solution_hash": "3047ca63de76a853", "solution_path": "data/solutions/generated/squindom_003_3047ca63de76a853.json", "tolerance": 1e-08, "warnings": []}
{"case": "squindom@4", "container": {"height": 1.9303615097026952, "type": "rectangle", "width": 3.8607230194053903}, "density": 0.5367260778952853, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -1.561542731343124e-06, "max_pair_overlap_depth": -3.757775137364083e-05, "n": 4, "ok": true, "record_id": "8d067d60c7138964", "setup": "squindom", "side": 3.8607230194053903, "solution_hash": "8d067d60c7138964", "solution_path": "data/solutions/generated/squindom_004_8d067d60c7138964.json", "tolerance": 1e-08, "warnings": []}
{"case": "squindom@5", "container": {"height": 2.000023397984937, "type": "rectangle", "width": 4.000046795969874}, "density": 0.6249853765160349, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -3.3736510935966635e-06, "max_pair_overlap_depth": -1.2206465427119814e-05, "n": 5, "ok": true, "record_id": "05f6000ecc64a077", "setup": "squindom", "side": 4.000046795969874, "solution_hash": "05f6000ecc64a077", "solution_path": "data/solutions/generated/squindom_005_05f6000ecc64a077.json", "tolerance": 1e-08, "warnings": []}
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@@ -416,26 +437,34 @@
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{"case": "squindom@34", "container": {"height": 4.472382543731088, "type": "rectangle", "width": 8.944765087462176}, "density": 0.8499062715939519, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -2.603876101403557e-07, "max_pair_overlap_depth": -3.616786977911346e-06, "n": 34, "ok": true, "record_id": "b631e5a58c1517e1", "setup": "squindom", "side": 8.944765087462176, "solution_hash": "b631e5a58c1517e1", "solution_path": "data/solutions/generated/squindom_034_b631e5a58c1517e1.json", "tolerance": 1e-08, "warnings": []}
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{"case": "squindom@35", "container": {"height": 4.504210097373901, "type": "rectangle", "width": 9.008420194747803}, "density": 0.8625827503428114, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -1.9606355010637344e-06, "max_pair_overlap_depth": -1.1537592481902781e-05, "n": 35, "ok": true, "record_id": "dcd4cfae3ff101a2", "setup": "squindom", "side": 9.008420194747803, "solution_hash": "dcd4cfae3ff101a2", "solution_path": "data/solutions/generated/squindom_035_dcd4cfae3ff101a2.json", "tolerance": 1e-08, "warnings": []}
{"case": "squindom@36", "container": {"height": 4.503911109790387, "type": "rectangle", "width": 9.007822219580774}, "density": 0.8873457711540725, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -9.310647293148122e-07, "max_pair_overlap_depth": -3.943598063660403e-06, "n": 36, "ok": true, "record_id": "e0fc61bd01777ea3", "setup": "squindom", "side": 9.007822219580774, "solution_hash": "e0fc61bd01777ea3", "solution_path": "data/solutions/generated/squindom_036_e0fc61bd01777ea3.json", "tolerance": 1e-08, "warnings": []}
{"case": "squindom@37", "container": {"height": 4.749914140956527, "type": "rectangle", "width": 9.499828281913054}, "density": 0.8199742411033116, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -4.702594714522945e-06, "max_pair_overlap_depth": -1.6080659807293785e-05, "n": 37, "ok": true, "record_id": "09a846134d0a3602", "setup": "squindom", "side": 9.499828281913054, "solution_hash": "09a846134d0a3602", "solution_path": "data/solutions/generated/squindom_037_09a846134d0a3602.json", "tolerance": 1e-08, "warnings": []}
@@ -447,11 +476,16 @@
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{"case": "squindom@42", "container": {"height": 4.978534605225135, "type": "rectangle", "width": 9.95706921045027}, "density": 0.8472590848618945, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -3.54667148005916e-06, "max_pair_overlap_depth": -1.5216861767419765e-05, "n": 42, "ok": true, "record_id": "7faea8ae450b67a1", "setup": "squindom", "side": 9.95706921045027, "solution_hash": "7faea8ae450b67a1", "solution_path": "data/solutions/generated/squindom_042_7faea8ae450b67a1.json", "tolerance": 1e-08, "warnings": []}
{"case": "squindom@43", "container": {"height": 5.000325272095715, "type": "rectangle", "width": 10.00065054419143}, "density": 0.859888117316887, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -5.1461540806485566e-08, "max_pair_overlap_depth": -2.005039219099558e-06, "n": 43, "ok": true, "record_id": "7f9527ccddc9d648", "setup": "squindom", "side": 10.00065054419143, "solution_hash": "7f9527ccddc9d648", "solution_path": "data/solutions/generated/squindom_043_7f9527ccddc9d648.json", "tolerance": 1e-08, "warnings": []}
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{"case": "squindom@44", "container": {"height": 5.000104558237924, "type": "rectangle", "width": 10.000209116475848}, "density": 0.8799631966546829, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -4.245467354557775e-06, "max_pair_overlap_depth": -3.962242465904708e-06, "n": 44, "ok": true, "record_id": "1ffd758f1f058de5", "setup": "squindom", "side": 10.000209116475848, "solution_hash": "1ffd758f1f058de5", "solution_path": "data/solutions/generated/squindom_044_1ffd758f1f058de5.json", "tolerance": 1e-08, "warnings": []}
{"case": "squindom@45", "container": {"height": 5.000138556559188, "type": "rectangle", "width": 10.000277113118376}, "density": 0.8999501217119913, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -1.0485935568738114e-06, "max_pair_overlap_depth": -1.361329106774889e-06, "n": 45, "ok": true, "record_id": "d085ed9682ca2968", "setup": "squindom", "side": 10.000277113118376, "solution_hash": "d085ed9682ca2968", "solution_path": "data/solutions/generated/squindom_045_d085ed9682ca2968.json", "tolerance": 1e-08, "warnings": []}
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{"case": "squindom@46", "container": {"height": 5.000284354260023, "type": "rectangle", "width": 10.000568708520046}, "density": 0.9198953665582859, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -1.1757865436834436e-06, "max_pair_overlap_depth": -7.423155479191479e-07, "n": 46, "ok": true, "record_id": "3b408e84cd206068", "setup": "squindom", "side": 10.000568708520046, "solution_hash": "3b408e84cd206068", "solution_path": "data/solutions/generated/squindom_046_3b408e84cd206068.json", "tolerance": 1e-08, "warnings": []}
+{"case": "squindom@46", "container": {"height": 5.000125376863833, "type": "rectangle", "width": 10.000250753727666}, "density": 0.9199538630494686, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -1.0555756748686917e-06, "max_pair_overlap_depth": -4.3576353303542703e-07, "n": 46, "ok": true, "record_id": "7fb3175fb0fba7a7", "setup": "squindom", "side": 10.000250753727666, "solution_hash": "7fb3175fb0fba7a7", "solution_path": "data/solutions/generated/squindom_046_7fb3175fb0fba7a7.json", "tolerance": 1e-08, "warnings": []}
{"case": "squindom@47", "container": {"height": 5.000415845954438, "type": "rectangle", "width": 10.000831691908877}, "density": 0.9398436614252309, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -2.3105818880431173e-06, "max_pair_overlap_depth": -3.7244704524930228e-06, "n": 47, "ok": true, "record_id": "2525e331a392c6b9", "setup": "squindom", "side": 10.000831691908877, "solution_hash": "2525e331a392c6b9", "solution_path": "data/solutions/generated/squindom_047_2525e331a392c6b9.json", "tolerance": 1e-08, "warnings": []}
+{"case": "squindom@47", "container": {"height": 5.000125376863833, "type": "rectangle", "width": 10.000250753727666}, "density": 0.9399528600722832, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -1.0555756748686917e-06, "max_pair_overlap_depth": -4.3576353303542703e-07, "n": 47, "ok": true, "record_id": "a146ed2b5a4b6cf9", "setup": "squindom", "side": 10.000250753727666, "solution_hash": "a146ed2b5a4b6cf9", "solution_path": "data/solutions/generated/squindom_047_a146ed2b5a4b6cf9.json", "tolerance": 1e-08, "warnings": []}
{"case": "squindom@48", "container": {"height": 5.000125376863833, "type": "rectangle", "width": 10.000250753727666}, "density": 0.9599518570950978, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -1.0555756748686917e-06, "max_pair_overlap_depth": -4.3576353303542703e-07, "n": 48, "ok": true, "record_id": "0433c6294c4f3fe7", "setup": "squindom", "side": 10.000250753727666, "solution_hash": "0433c6294c4f3fe7", "solution_path": "data/solutions/generated/squindom_048_0433c6294c4f3fe7.json", "tolerance": 1e-08, "warnings": []}
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{"case": "squindom@49", "container": {"height": 5.000357344416605, "type": "rectangle", "width": 10.00071468883321}, "density": 0.9798599360041962, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -4.316744310450815e-06, "max_pair_overlap_depth": -1.1644813818767474e-06, "n": 49, "ok": true, "record_id": "c84a987ae840ebe3", "setup": "squindom", "side": 10.00071468883321, "solution_hash": "c84a987ae840ebe3", "solution_path": "data/solutions/generated/squindom_049_c84a987ae840ebe3.json", "tolerance": 1e-08, "warnings": []}
{"case": "squindom@50", "container": {"height": 5.000172880899938, "type": "rectangle", "width": 10.000345761799876}, "density": 0.9999308512263965, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -9.912590304139712e-09, "max_pair_overlap_depth": -7.234911549747203e-06, "n": 50, "ok": true, "record_id": "df0259815cc5320a", "setup": "squindom", "side": 10.000345761799876, "solution_hash": "df0259815cc5320a", "solution_path": "data/solutions/generated/squindom_050_df0259815cc5320a.json", "tolerance": 1e-08, "warnings": []}
{"case": "squinhex@1", "container": {"circumradius": 0.7886816291028929, "orientation_radians": 0.0, "side_length": 0.7886816291028927, "sides": 6, "type": "regular_polygon"}, "density": 0.6187919623349352, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -5.500249687614274e-06, "max_pair_overlap_depth": 0.0, "n": 1, "ok": true, "record_id": "7a4ae27d520be7d9", "setup": "squinhex", "side": 0.7886816291028927, "solution_hash": "7a4ae27d520be7d9", "solution_path": "data/solutions/generated/squinhex_001_7a4ae27d520be7d9.json", "tolerance": 1e-08, "warnings": []}
@@ -514,6 +548,7 @@
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{"case": "squinoct@15", "container": {"circumradius": 2.6563780118357685, "orientation_radians": 0.0, "side_length": 2.03310371045693, "sides": 8, "type": "regular_polygon"}, "density": 0.7515649396279209, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -1.7983161874182644e-07, "max_pair_overlap_depth": -1.9704810269915995e-05, "n": 15, "ok": true, "record_id": "c898d8b3cd2b55db", "setup": "squinoct", "side": 2.03310371045693, "solution_hash": "c898d8b3cd2b55db", "solution_path": "data/solutions/generated/squinoct_015_c898d8b3cd2b55db.json", "tolerance": 1e-08, "warnings": []}
{"case": "squinoct@16", "container": {"circumradius": 2.7060344677951713, "orientation_radians": 0.0, "side_length": 2.0711091164681905, "sides": 8, "type": "regular_polygon"}, "density": 0.7725175260502497, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -1.5220970331887e-06, "max_pair_overlap_depth": -8.060079695892952e-07, "n": 16, "ok": true, "record_id": "3a2e6d99239355fd", "setup": "squinoct", "side": 2.0711091164681905, "solution_hash": "3a2e6d99239355fd", "solution_path": "data/solutions/generated/squinoct_016_3a2e6d99239355fd.json", "tolerance": 1e-08, "warnings": []}
+{"case": "squinoct@16", "container": {"circumradius": 2.7060211339575333, "orientation_radians": 0.0, "side_length": 2.0710989111906826, "sides": 8, "type": "regular_polygon"}, "density": 0.7725251391826925, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": 1.368661750365744e-09, "max_pair_overlap_depth": -1.0480566122961221e-05, "n": 16, "ok": true, "record_id": "4405320aab52a39c", "setup": "squinoct", "side": 2.0710989111906826, "solution_hash": "4405320aab52a39c", "solution_path": "data/solutions/generated/squinoct_016_4405320aab52a39c.json", "tolerance": 1e-08, "warnings": []}
{"case": "squinoct@17", "container": {"circumradius": 2.7060211339575333, "orientation_radians": 0.0, "side_length": 2.0710989111906826, "sides": 8, "type": "regular_polygon"}, "density": 0.8208079603816107, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": 1.368661750365744e-09, "max_pair_overlap_depth": -1.0480566122961221e-05, "n": 17, "ok": true, "record_id": "ab660f0510a3dd1c", "setup": "squinoct", "side": 2.0710989111906826, "solution_hash": "ab660f0510a3dd1c", "solution_path": "data/solutions/generated/squinoct_017_ab660f0510a3dd1c.json", "tolerance": 1e-08, "warnings": []}
{"case": "squinoct@18", "container": {"circumradius": 2.832106182259099, "orientation_radians": 0.0, "side_length": 2.167600229298605, "sides": 8, "type": "regular_polygon"}, "density": 0.7934296897658486, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -2.9711774532196686e-06, "max_pair_overlap_depth": -1.4522556675777842e-05, "n": 18, "ok": true, "record_id": "a7275e2ab7024187", "setup": "squinoct", "side": 2.167600229298605, "solution_hash": "a7275e2ab7024187", "solution_path": "data/solutions/generated/squinoct_018_a7275e2ab7024187.json", "tolerance": 1e-08, "warnings": []}
{"case": "squinoct@19", "container": {"circumradius": 2.909638760965531, "orientation_radians": 0.0, "side_length": 2.226941095977593, "sides": 8, "type": "regular_polygon"}, "density": 0.793469906795261, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -1.0543459428412958e-06, "max_pair_overlap_depth": -1.4850274468458657e-05, "n": 19, "ok": true, "record_id": "330906f1db929cdb", "setup": "squinoct", "side": 2.226941095977593, "solution_hash": "330906f1db929cdb", "solution_path": "data/solutions/generated/squinoct_019_330906f1db929cdb.json", "tolerance": 1e-08, "warnings": []}
@@ -579,6 +614,7 @@
{"case": "squintri@2", "container": {"circumradius": 1.8214143375875054, "orientation_radians": 0.0, "side_length": 3.15478217433597, "sides": 3, "type": "regular_polygon"}, "density": 0.4640775964340414, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -6.2952441585539525e-06, "max_pair_overlap_depth": -3.4919698730116744e-05, "n": 2, "ok": true, "record_id": "9fcb7fc91bdb6f8d", "setup": "squintri", "side": 3.15478217433597, "solution_hash": "9fcb7fc91bdb6f8d", "solution_path": "data/solutions/generated/squintri_002_9fcb7fc91bdb6f8d.json", "tolerance": 1e-08, "warnings": []}
{"case": "squintri@3", "container": {"circumradius": 1.8660645071778983, "orientation_radians": 0.0, "side_length": 3.232118536633098, "sides": 3, "type": "regular_polygon"}, "density": 0.6632023535454387, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -2.0862631378148764e-06, "max_pair_overlap_depth": -4.0896047962125925e-05, "n": 3, "ok": true, "record_id": "6a0fd09e2fb037d5", "setup": "squintri", "side": 3.232118536633098, "solution_hash": "6a0fd09e2fb037d5", "solution_path": "data/solutions/generated/squintri_003_6a0fd09e2fb037d5.json", "tolerance": 1e-08, "warnings": []}
{"case": "squintri@4", "container": {"circumradius": 2.3987748817273813, "orientation_radians": 0.0, "side_length": 4.154799971071848, "sides": 3, "type": "regular_polygon"}, "density": 0.5351297741951115, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -3.3541096500101422e-06, "max_pair_overlap_depth": -2.3979848717270436e-05, "n": 4, "ok": true, "record_id": "a14dd3a45e182f53", "setup": "squintri", "side": 4.154799971071848, "solution_hash": "a14dd3a45e182f53", "solution_path": "data/solutions/generated/squintri_004_a14dd3a45e182f53.json", "tolerance": 1e-08, "warnings": []}
+{"case": "squintri@5", "container": {"circumradius": 2.4881047104532623, "orientation_radians": 0.0, "side_length": 4.3095237730565, "sides": 3, "type": "regular_polygon"}, "density": 0.6217428601610049, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -2.6885485560193324e-06, "max_pair_overlap_depth": -3.4490596049874256e-05, "n": 5, "ok": true, "record_id": "21b5dd023d9a44d0", "setup": "squintri", "side": 4.3095237730565, "solution_hash": "21b5dd023d9a44d0", "solution_path": "data/solutions/generated/squintri_005_21b5dd023d9a44d0.json", "tolerance": 1e-08, "warnings": []}
{"case": "squintri@5", "container": {"circumradius": 2.4881098691354784, "orientation_radians": 0.0, "side_length": 4.3095327081561985, "sides": 3, "type": "regular_polygon"}, "density": 0.6217402820027407, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -2.818148398440643e-06, "max_pair_overlap_depth": -2.4242157156018074e-05, "n": 5, "ok": true, "record_id": "b7ec125fb5317da3", "setup": "squintri", "side": 4.3095327081561985, "solution_hash": "b7ec125fb5317da3", "solution_path": "data/solutions/generated/squintri_005_b7ec125fb5317da3.json", "tolerance": 1e-08, "warnings": []}
{"case": "squintri@6", "container": {"circumradius": 2.4881047104532623, "orientation_radians": 0.0, "side_length": 4.3095237730565, "sides": 3, "type": "regular_polygon"}, "density": 0.7460914321932061, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -2.6885485560193324e-06, "max_pair_overlap_depth": -3.2625800104213454e-05, "n": 6, "ok": true, "record_id": "cafd65072dcc4c6f", "setup": "squintri", "side": 4.3095237730565, "solution_hash": "cafd65072dcc4c6f", "solution_path": "data/solutions/generated/squintri_006_cafd65072dcc4c6f.json", "tolerance": 1e-08, "warnings": []}
{"case": "squintri@7", "container": {"circumradius": 2.9761478728006363, "orientation_radians": 0.0, "side_length": 5.154839326528738, "sides": 3, "type": "regular_polygon"}, "density": 0.6083690864087284, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -6.7967605582714725e-06, "max_pair_overlap_depth": -3.132565940933141e-05, "n": 7, "ok": true, "record_id": "b90420d34e1bd6e3", "setup": "squintri", "side": 5.154839326528738, "solution_hash": "b90420d34e1bd6e3", "solution_path": "data/solutions/generated/squintri_007_b90420d34e1bd6e3.json", "tolerance": 1e-08, "warnings": []}
@@ -594,6 +630,7 @@
{"case": "squintri@17", "container": {"circumradius": 4.217401538358075, "orientation_radians": 0.0, "side_length": 7.304753740355329, "sides": 3, "type": "regular_polygon"}, "density": 0.7357616143895036, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": 2.4850519352755694e-10, "max_pair_overlap_depth": -2.8059876787889237e-05, "n": 17, "ok": true, "record_id": "3a2c987432da5f1d", "setup": "squintri", "side": 7.304753740355329, "solution_hash": "3a2c987432da5f1d", "solution_path": "data/solutions/generated/squintri_017_3a2c987432da5f1d.json", "tolerance": 1e-08, "warnings": []}
{"case": "squintri@18", "container": {"circumradius": 4.220148513555036, "orientation_radians": 0.0, "side_length": 7.3095116409635965, "sides": 3, "type": "regular_polygon"}, "density": 0.7780278532013658, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -1.1021248752761939e-06, "max_pair_overlap_depth": -8.579145527987109e-06, "n": 18, "ok": true, "record_id": "e729e13f2fcc5856", "setup": "squintri", "side": 7.3095116409635965, "solution_hash": "e729e13f2fcc5856", "solution_path": "data/solutions/generated/squintri_018_e729e13f2fcc5856.json", "tolerance": 1e-08, "warnings": []}
{"case": "squintri@19", "container": {"circumradius": 4.309532681383897, "orientation_radians": 0.0, "side_length": 7.464329561035447, "sides": 3, "type": "regular_polygon"}, "density": 0.787537700321716, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -3.0609380559736765e-06, "max_pair_overlap_depth": -1.836020690809681e-05, "n": 19, "ok": true, "record_id": "805d1e50c1db2842", "setup": "squintri", "side": 7.464329561035447, "solution_hash": "805d1e50c1db2842", "solution_path": "data/solutions/generated/squintri_019_805d1e50c1db2842.json", "tolerance": 1e-08, "warnings": []}
+{"case": "squintri@20", "container": {"circumradius": 4.398847304000316, "orientation_radians": 0.0, "side_length": 7.619027025265926, "sides": 3, "type": "regular_polygon"}, "density": 0.7956651409816211, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -2.8035391721895233e-06, "max_pair_overlap_depth": -9.7266150509423e-06, "n": 20, "ok": true, "record_id": "33507e9f74d78419", "setup": "squintri", "side": 7.619027025265926, "solution_hash": "33507e9f74d78419", "solution_path": "data/solutions/generated/squintri_020_33507e9f74d78419.json", "tolerance": 1e-08, "warnings": []}
{"case": "squintri@20", "container": {"circumradius": 4.39886945518194, "orientation_radians": 0.0, "side_length": 7.619065392237945, "sides": 3, "type": "regular_polygon"}, "density": 0.795657127614163, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -1.382631857088383e-07, "max_pair_overlap_depth": -1.1470843414018717e-05, "n": 20, "ok": true, "record_id": "f4d21a73470ce742", "setup": "squintri", "side": 7.619065392237945, "solution_hash": "f4d21a73470ce742", "solution_path": "data/solutions/generated/squintri_020_f4d21a73470ce742.json", "tolerance": 1e-08, "warnings": []}
{"case": "squintri@21", "container": {"circumradius": 4.398847304000316, "orientation_radians": 0.0, "side_length": 7.619027025265926, "sides": 3, "type": "regular_polygon"}, "density": 0.8354483980307021, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -2.8035391721895233e-06, "max_pair_overlap_depth": -9.7266150509423e-06, "n": 21, "ok": true, "record_id": "a2e24601c2557198", "setup": "squintri", "side": 7.619027025265926, "solution_hash": "a2e24601c2557198", "solution_path": "data/solutions/generated/squintri_021_a2e24601c2557198.json", "tolerance": 1e-08, "warnings": []}
{"case": "squintri@22", "container": {"circumradius": 4.708276091307807, "orientation_radians": 0.0, "side_length": 8.154973406206924, "sides": 3, "type": "regular_polygon"}, "density": 0.7639711289147475, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -2.4334644744109685e-07, "max_pair_overlap_depth": -2.16343036094635e-05, "n": 22, "ok": true, "record_id": "bbb028380d997350", "setup": "squintri", "side": 8.154973406206924, "solution_hash": "bbb028380d997350", "solution_path": "data/solutions/generated/squintri_022_bbb028380d997350.json", "tolerance": 1e-08, "warnings": []}
@@ -604,12 +641,15 @@
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{"case": "squintri@28", "container": {"circumradius": 5.065536966965852, "orientation_radians": 0.0, "side_length": 8.773767394403205, "sides": 3, "type": "regular_polygon"}, "density": 0.8400113660191094, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -3.0145694937175628e-06, "max_pair_overlap_depth": -1.3941973455278145e-05, "n": 28, "ok": true, "record_id": "0d20b359435ed501", "setup": "squintri", "side": 8.773767394403205, "solution_hash": "0d20b359435ed501", "solution_path": "data/solutions/generated/squintri_028_0d20b359435ed501.json", "tolerance": 1e-08, "warnings": []}
{"case": "squintri@29", "container": {"circumradius": 5.2855679241308415, "orientation_radians": 0.0, "side_length": 9.154872191450979, "sides": 3, "type": "regular_polygon"}, "density": 0.7990846545133872, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -2.2628787428294703e-06, "max_pair_overlap_depth": -1.875431036468811e-05, "n": 29, "ok": true, "record_id": "3947aab39c0a1a92", "setup": "squintri", "side": 9.154872191450979, "solution_hash": "3947aab39c0a1a92", "solution_path": "data/solutions/generated/squintri_029_3947aab39c0a1a92.json", "tolerance": 1e-08, "warnings": []}
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{"case": "squintri@30", "container": {"circumradius": 5.375274403108803, "orientation_radians": 0.0, "side_length": 9.310248370808917, "sides": 3, "type": "regular_polygon"}, "density": 0.7992784153759376, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -2.5690858985090003e-06, "max_pair_overlap_depth": -2.654303096472077e-05, "n": 30, "ok": true, "record_id": "e0b7505d8b6b1d46", "setup": "squintri", "side": 9.310248370808917, "solution_hash": "e0b7505d8b6b1d46", "solution_path": "data/solutions/generated/squintri_030_e0b7505d8b6b1d46.json", "tolerance": 1e-08, "warnings": []}
{"case": "squintri@31", "container": {"circumradius": 5.375037381839187, "orientation_radians": 0.0, "side_length": 9.309837837927466, "sides": 3, "type": "regular_polygon"}, "density": 0.825993871568094, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -2.6472234336534766e-06, "max_pair_overlap_depth": -3.090204228461957e-05, "n": 31, "ok": true, "record_id": "22b5f5f5707858bb", "setup": "squintri", "side": 9.309837837927466, "solution_hash": "22b5f5f5707858bb", "solution_path": "data/solutions/generated/squintri_031_22b5f5f5707858bb.json", "tolerance": 1e-08, "warnings": []}
{"case": "squintri@32", "container": {"circumradius": 5.464425438813088, "orientation_radians": 0.0, "side_length": 9.464662494196126, "sides": 3, "type": "regular_polygon"}, "density": 0.8249717531277396, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -3.1832282131171085e-07, "max_pair_overlap_depth": -1.699280632205813e-06, "n": 32, "ok": true, "record_id": "a44fbd99145b1ab3", "setup": "squintri", "side": 9.464662494196126, "solution_hash": "a44fbd99145b1ab3", "solution_path": "data/solutions/generated/squintri_032_a44fbd99145b1ab3.json", "tolerance": 1e-08, "warnings": []}
{"case": "squintri@33", "container": {"circumradius": 5.556459277935629, "orientation_radians": 0.0, "side_length": 9.624069779571988, "sides": 3, "type": "regular_polygon"}, "density": 0.8228028312642216, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -2.226435414550563e-06, "max_pair_overlap_depth": -3.0944328952564604e-06, "n": 33, "ok": true, "record_id": "67de1e08be2d150c", "setup": "squintri", "side": 9.624069779571988, "solution_hash": "67de1e08be2d150c", "solution_path": "data/solutions/generated/squintri_033_67de1e08be2d150c.json", "tolerance": 1e-08, "warnings": []}
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{"case": "squintri@34", "container": {"circumradius": 5.553663262151533, "orientation_radians": 0.0, "side_length": 9.619226938175167, "sides": 3, "type": "regular_polygon"}, "density": 0.8485900582279403, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -4.159334309452278e-06, "max_pair_overlap_depth": -1.741303470170763e-05, "n": 34, "ok": true, "record_id": "3661302cdd9d0eb0", "setup": "squintri", "side": 9.619226938175167, "solution_hash": "3661302cdd9d0eb0", "solution_path": "data/solutions/generated/squintri_034_3661302cdd9d0eb0.json", "tolerance": 1e-08, "warnings": []}
{"case": "squintri@35", "container": {"circumradius": 5.646061505097748, "orientation_radians": 0.0, "side_length": 9.779265389488105, "sides": 3, "type": "regular_polygon"}, "density": 0.8451911560111607, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -1.195518233524095e-06, "max_pair_overlap_depth": -2.612964804571405e-05, "n": 35, "ok": true, "record_id": "39d52a823b632afe", "setup": "squintri", "side": 9.779265389488105, "solution_hash": "39d52a823b632afe", "solution_path": "data/solutions/generated/squintri_035_39d52a823b632afe.json", "tolerance": 1e-08, "warnings": []}
+{"case": "squintri@35", "container": {"circumradius": 5.644515969725634, "orientation_radians": 0.0, "side_length": 9.776588443698708, "sides": 3, "type": "regular_polygon"}, "density": 0.8456540660893009, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -1.9270562690497914e-06, "max_pair_overlap_depth": -7.45737725771356e-06, "n": 35, "ok": true, "record_id": "454defabaa6b59ca", "setup": "squintri", "side": 9.776588443698708, "solution_hash": "454defabaa6b59ca", "solution_path": "data/solutions/generated/squintri_035_454defabaa6b59ca.json", "tolerance": 1e-08, "warnings": []}
{"case": "squintri@36", "container": {"circumradius": 5.644515969725634, "orientation_radians": 0.0, "side_length": 9.776588443698708, "sides": 3, "type": "regular_polygon"}, "density": 0.8698156108347095, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -1.9270562690497914e-06, "max_pair_overlap_depth": -7.45737725771356e-06, "n": 36, "ok": true, "record_id": "f0473aaa767ebe98", "setup": "squintri", "side": 9.776588443698708, "solution_hash": "f0473aaa767ebe98", "solution_path": "data/solutions/generated/squintri_036_f0473aaa767ebe98.json", "tolerance": 1e-08, "warnings": []}
{"case": "squintri@37", "container": {"circumradius": 5.863073513456338, "orientation_radians": 0.0, "side_length": 10.155141213817744, "sides": 3, "type": "regular_polygon"}, "density": 0.8285699034870297, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -3.3566887422690367e-06, "max_pair_overlap_depth": -1.2239987045248313e-05, "n": 37, "ok": true, "record_id": "8530be000ac2e9ed", "setup": "squintri", "side": 10.155141213817744, "solution_hash": "8530be000ac2e9ed", "solution_path": "data/solutions/generated/squintri_037_8530be000ac2e9ed.json", "tolerance": 1e-08, "warnings": []}
{"case": "squintri@38", "container": {"circumradius": 5.952347755555162, "orientation_radians": 0.0, "side_length": 10.309768736940113, "sides": 3, "type": "regular_polygon"}, "density": 0.8256293316995508, "errors": [], "item": {"circumradius": 0.7071067811865476, "side_length": 1.0, "sides": 4, "type": "regular_polygon"}, "max_boundary_excess": -6.231911901047403e-06, "max_pair_overlap_depth": -1.6045122417374102e-05, "n": 38, "ok": true, "record_id": "896b636f9d899bf5", "setup": "squintri", "side": 10.309768736940113, "solution_hash": "896b636f9d899bf5", "solution_path": "data/solutions/generated/squintri_038_896b636f9d899bf5.json", "tolerance": 1e-08, "warnings": []}
@@ -619,6 +659,7 @@
{"case": "triincir@2", "container": {"radius": 0.866038587120379, "type": "circle"}, "density": 0.3675414068402789, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": -3.072458078845486e-06, "max_pair_overlap_depth": -1.0890566688626091e-05, "n": 2, "ok": true, "record_id": "92f5511b689fe041", "setup": "triincir", "side": 1.732077174240758, "solution_hash": "92f5511b689fe041", "solution_path": "data/solutions/generated/triincir_002_92f5511b689fe041.json", "tolerance": 1e-08, "warnings": []}
{"case": "triincir@3", "container": {"radius": 0.9535486885675039, "type": "circle"}, "density": 0.45476420302551357, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": -2.4231654198736408e-06, "max_pair_overlap_depth": -2.0408752807413166e-05, "n": 3, "ok": true, "record_id": "6ede8225b5939550", "setup": "triincir", "side": 1.9070973771350077, "solution_hash": "6ede8225b5939550", "solution_path": "data/solutions/generated/triincir_003_6ede8225b5939550.json", "tolerance": 1e-08, "warnings": []}
{"case": "triincir@4", "container": {"radius": 0.9806773112916812, "type": "circle"}, "density": 0.5732690550709103, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": -4.642177417024662e-06, "max_pair_overlap_depth": -2.677155749965807e-05, "n": 4, "ok": true, "record_id": "7f9481f33de5ecda", "setup": "triincir", "side": 1.9613546225833625, "solution_hash": "7f9481f33de5ecda", "solution_path": "data/solutions/generated/triincir_004_7f9481f33de5ecda.json", "tolerance": 1e-08, "warnings": []}
+{"case": "triincir@5", "container": {"radius": 1.0000179617410019, "type": "circle"}, "density": 0.6891363628771783, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": -4.861140867618019e-08, "max_pair_overlap_depth": -1.1418495247272897e-05, "n": 5, "ok": true, "record_id": "a3d09cb09f1298d4", "setup": "triincir", "side": 2.0000359234820038, "solution_hash": "a3d09cb09f1298d4", "solution_path": "data/solutions/generated/triincir_005_a3d09cb09f1298d4.json", "tolerance": 1e-08, "warnings": []}
{"case": "triincir@5", "container": {"radius": 1.0000245730902964, "type": "circle"}, "density": 0.6891272508887964, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": -2.800898255106432e-06, "max_pair_overlap_depth": -9.00554050570603e-06, "n": 5, "ok": true, "record_id": "cf17d061ebfe392b", "setup": "triincir", "side": 2.0000491461805927, "solution_hash": "cf17d061ebfe392b", "solution_path": "data/solutions/generated/triincir_005_cf17d061ebfe392b.json", "tolerance": 1e-08, "warnings": []}
{"case": "triincir@6", "container": {"radius": 1.0000179617410019, "type": "circle"}, "density": 0.826963635452614, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": -4.861140867618019e-08, "max_pair_overlap_depth": -9.037179045745058e-06, "n": 6, "ok": true, "record_id": "fe93ba87f23c6c01", "setup": "triincir", "side": 2.0000359234820038, "solution_hash": "fe93ba87f23c6c01", "solution_path": "data/solutions/generated/triincir_006_fe93ba87f23c6c01.json", "tolerance": 1e-08, "warnings": []}
{"case": "triincir@7", "container": {"radius": 1.1524019079826915, "type": "circle"}, "density": 0.7265087780749782, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": -7.469968968942808e-06, "max_pair_overlap_depth": -4.662019983989185e-06, "n": 7, "ok": true, "record_id": "4d32cb6092e127b9", "setup": "triincir", "side": 2.304803815965383, "solution_hash": "4d32cb6092e127b9", "solution_path": "data/solutions/generated/triincir_007_4d32cb6092e127b9.json", "tolerance": 1e-08, "warnings": []}
@@ -670,6 +711,7 @@
{"case": "triinhex@2", "container": {"circumradius": 0.8660427548004997, "orientation_radians": 0.0, "side_length": 0.8660427548004996, "sides": 6, "type": "regular_polygon"}, "density": 0.4444266358884555, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": -8.75428637420658e-06, "max_pair_overlap_depth": -4.36783093215443e-06, "n": 2, "ok": true, "record_id": "132c3ed1fad59f87", "setup": "triinhex", "side": 0.8660427548004996, "solution_hash": "132c3ed1fad59f87", "solution_path": "data/solutions/generated/triinhex_002_132c3ed1fad59f87.json", "tolerance": 1e-08, "warnings": []}
{"case": "triinhex@3", "container": {"circumradius": 0.9893199470974533, "orientation_radians": 0.0, "side_length": 0.9893199470974532, "sides": 6, "type": "regular_polygon"}, "density": 0.5108536175549566, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": -6.001855317125196e-07, "max_pair_overlap_depth": -5.956334362088034e-06, "n": 3, "ok": true, "record_id": "3f19c46d9d07a54d", "setup": "triinhex", "side": 0.9893199470974532, "solution_hash": "3f19c46d9d07a54d", "solution_path": "data/solutions/generated/triinhex_003_3f19c46d9d07a54d.json", "tolerance": 1e-08, "warnings": []}
{"case": "triinhex@4", "container": {"circumradius": 1.0000239660140624, "orientation_radians": 0.0, "side_length": 1.0000239660140622, "sides": 6, "type": "regular_polygon"}, "density": 0.6666347131299533, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": -3.5130077190296347e-08, "max_pair_overlap_depth": -1.1324383645797624e-05, "n": 4, "ok": true, "record_id": "9a8f72cbad0da23e", "setup": "triinhex", "side": 1.0000239660140622, "solution_hash": "9a8f72cbad0da23e", "solution_path": "data/solutions/generated/triinhex_004_9a8f72cbad0da23e.json", "tolerance": 1e-08, "warnings": []}
+{"case": "triinhex@4", "container": {"circumradius": 1.0000186366904509, "orientation_radians": 0.0, "side_length": 1.0000186366904507, "sides": 6, "type": "regular_polygon"}, "density": 0.666641818440701, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": -1.0710555572355552e-06, "max_pair_overlap_depth": -2.545819687190834e-06, "n": 4, "ok": true, "record_id": "d2b5a3b06543cb98", "setup": "triinhex", "side": 1.0000186366904507, "solution_hash": "d2b5a3b06543cb98", "solution_path": "data/solutions/generated/triinhex_004_d2b5a3b06543cb98.json", "tolerance": 1e-08, "warnings": []}
{"case": "triinhex@5", "container": {"circumradius": 1.0000186366904509, "orientation_radians": 0.0, "side_length": 1.0000186366904507, "sides": 6, "type": "regular_polygon"}, "density": 0.8333022730508762, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": -1.0710555572355552e-06, "max_pair_overlap_depth": -2.545819687190834e-06, "n": 5, "ok": true, "record_id": "0d61e48787b7d5d1", "setup": "triinhex", "side": 1.0000186366904507, "solution_hash": "0d61e48787b7d5d1", "solution_path": "data/solutions/generated/triinhex_005_0d61e48787b7d5d1.json", "tolerance": 1e-08, "warnings": []}
{"case": "triinhex@6", "container": {"circumradius": 1.0000220591230227, "orientation_radians": 0.0, "side_length": 1.0000220591230224, "sides": 6, "type": "regular_polygon"}, "density": 0.9999558832137269, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": -3.751347439371281e-07, "max_pair_overlap_depth": -4.053173147537176e-06, "n": 6, "ok": true, "record_id": "163de15143701be8", "setup": "triinhex", "side": 1.0000220591230224, "solution_hash": "163de15143701be8", "solution_path": "data/solutions/generated/triinhex_006_163de15143701be8.json", "tolerance": 1e-08, "warnings": []}
{"case": "triinhex@7", "container": {"circumradius": 1.2773837447071252, "orientation_radians": 0.0, "side_length": 1.277383744707125, "sides": 6, "type": "regular_polygon"}, "density": 0.7149966699512036, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": -1.8980834118002716e-06, "max_pair_overlap_depth": -1.0312174508100846e-05, "n": 7, "ok": true, "record_id": "af96c8dfd69688ff", "setup": "triinhex", "side": 1.277383744707125, "solution_hash": "af96c8dfd69688ff", "solution_path": "data/solutions/generated/triinhex_007_af96c8dfd69688ff.json", "tolerance": 1e-08, "warnings": []}
@@ -700,6 +742,7 @@
{"case": "triinhex@31", "container": {"circumradius": 2.401165894947143, "orientation_radians": 0.0, "side_length": 2.4011658949471424, "sides": 6, "type": "regular_polygon"}, "density": 0.896119877899338, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": -1.9409739826535097e-07, "max_pair_overlap_depth": -2.8541100203227643e-06, "n": 31, "ok": true, "record_id": "cae7087451040120", "setup": "triinhex", "side": 2.4011658949471424, "solution_hash": "cae7087451040120", "solution_path": "data/solutions/generated/triinhex_31_cae7087451040120.json", "tolerance": 1e-08, "warnings": []}
{"case": "triinhex@32", "container": {"circumradius": 2.466336368233779, "orientation_radians": 0.0, "side_length": 2.4663363682337787, "sides": 6, "type": "regular_polygon"}, "density": 0.8767870241121234, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": -6.925122041572899e-07, "max_pair_overlap_depth": -1.2807310418028184e-06, "n": 32, "ok": true, "record_id": "8674581436e5e74d", "setup": "triinhex", "side": 2.4663363682337787, "solution_hash": "8674581436e5e74d", "solution_path": "data/solutions/generated/triinhex_032_8674581436e5e74d.json", "tolerance": 1e-08, "warnings": []}
{"case": "triinhex@33", "container": {"circumradius": 2.5000789881797005, "orientation_radians": 0.0, "side_length": 2.5000789881797, "sides": 6, "type": "regular_polygon"}, "density": 0.8799443949567897, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": -2.8634496562140136e-06, "max_pair_overlap_depth": -5.829272115620565e-06, "n": 33, "ok": true, "record_id": "5b836a27d2bb958c", "setup": "triinhex", "side": 2.5000789881797, "solution_hash": "5b836a27d2bb958c", "solution_path": "data/solutions/generated/triinhex_033_5b836a27d2bb958c.json", "tolerance": 1e-08, "warnings": []}
+{"case": "triinhex@33", "container": {"circumradius": 2.5000778510013064, "orientation_radians": 0.0, "side_length": 2.500077851001306, "sides": 6, "type": "regular_polygon"}, "density": 0.879945195455047, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": -1.2753913921730298e-06, "max_pair_overlap_depth": -7.960301433351802e-06, "n": 33, "ok": true, "record_id": "a1101d95622ba18c", "setup": "triinhex", "side": 2.500077851001306, "solution_hash": "a1101d95622ba18c", "solution_path": "data/solutions/generated/triinhex_033_a1101d95622ba18c.json", "tolerance": 1e-08, "warnings": []}
{"case": "triinhex@34", "container": {"circumradius": 2.5000778510013064, "orientation_radians": 0.0, "side_length": 2.500077851001306, "sides": 6, "type": "regular_polygon"}, "density": 0.9066102013779274, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": -1.2753913921730298e-06, "max_pair_overlap_depth": -7.960301433351802e-06, "n": 34, "ok": true, "record_id": "a712744ff3b5bcc9", "setup": "triinhex", "side": 2.500077851001306, "solution_hash": "a712744ff3b5bcc9", "solution_path": "data/solutions/generated/triinhex_034_a712744ff3b5bcc9.json", "tolerance": 1e-08, "warnings": []}
{"case": "triinhex@35", "container": {"circumradius": 2.500225007500126, "orientation_radians": 0.0, "side_length": 2.5002250075001258, "sides": 6, "type": "regular_polygon"}, "density": 0.93316535041203, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": -2.6207066397040256e-06, "max_pair_overlap_depth": -1.5837731757728513e-06, "n": 35, "ok": true, "record_id": "ec49778cf450c37b", "setup": "triinhex", "side": 2.5002250075001258, "solution_hash": "ec49778cf450c37b", "solution_path": "data/solutions/generated/triinhex_35_ec49778cf450c37b.json", "tolerance": 1e-08, "warnings": []}
{"case": "triinhex@36", "container": {"circumradius": 2.5902460640380607, "orientation_radians": 0.0, "side_length": 2.5902460640380602, "sides": 6, "type": "regular_polygon"}, "density": 0.8942711196026824, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": -5.744185746811731e-07, "max_pair_overlap_depth": -1.519675368655271e-06, "n": 36, "ok": true, "record_id": "d30ec5c19b8369f3", "setup": "triinhex", "side": 2.5902460640380602, "solution_hash": "d30ec5c19b8369f3", "solution_path": "data/solutions/record_hunt/triinhex_36_d30ec5c19b8369f3.json", "tolerance": 1e-08, "warnings": []}
@@ -845,11 +888,13 @@
{"case": "triintri@19", "container": {"circumradius": 2.694497112256744, "orientation_radians": 0.0, "side_length": 4.667005899276301, "sides": 3, "type": "regular_polygon"}, "density": 0.872322152073677, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": -5.331642222161759e-07, "max_pair_overlap_depth": -9.690072424628493e-06, "n": 19, "ok": true, "record_id": "bf4e89c9b4dd8ae2", "setup": "triintri", "side": 4.667005899276301, "solution_hash": "bf4e89c9b4dd8ae2", "solution_path": "data/solutions/generated/triintri_019_bf4e89c9b4dd8ae2.json", "tolerance": 1e-08, "warnings": []}
{"case": "triintri@20", "container": {"circumradius": 2.8083269652440315, "orientation_radians": 0.0, "side_length": 4.864164988068379, "sides": 3, "type": "regular_polygon"}, "density": 0.845304928058783, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": -2.619234998604407e-06, "max_pair_overlap_depth": -7.278337082583519e-06, "n": 20, "ok": true, "record_id": "4dff2c76e1273753", "setup": "triintri", "side": 4.864164988068379, "solution_hash": "4dff2c76e1273753", "solution_path": "data/solutions/generated/triintri_020_4dff2c76e1273753.json", "tolerance": 1e-08, "warnings": []}
{"case": "triintri@21", "container": {"circumradius": 2.8447132856898696, "orientation_radians": 0.0, "side_length": 4.927187943781052, "sides": 3, "type": "regular_polygon"}, "density": 0.8650098201747314, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": -2.4013824985741632e-06, "max_pair_overlap_depth": -4.2136074186321926e-06, "n": 21, "ok": true, "record_id": "4dc62e9e9421c0aa", "setup": "triintri", "side": 4.927187943781052, "solution_hash": "4dc62e9e9421c0aa", "solution_path": "data/solutions/generated/triintri_021_4dc62e9e9421c0aa.json", "tolerance": 1e-08, "warnings": []}
+{"case": "triintri@22", "container": {"circumradius": 2.886751345948129, "orientation_radians": 0.0, "side_length": 5.0, "sides": 3, "type": "regular_polygon"}, "density": 0.8800000000000002, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": 7.105427357601002e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "n": 22, "ok": true, "record_id": "16c9dbca1fb3bede", "setup": "triintri", "side": 5.0, "solution_hash": "16c9dbca1fb3bede", "solution_path": "data/solutions/generated/triintri_022_16c9dbca1fb3bede.json", "tolerance": 1e-08, "warnings": []}
{"case": "triintri@22", "container": {"circumradius": 2.8868751328982873, "orientation_radians": 0.0, "side_length": 5.000214405286989, "sides": 3, "type": "regular_polygon"}, "density": 0.8799245341930954, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": -5.380108740330726e-08, "max_pair_overlap_depth": -8.335028905959163e-07, "n": 22, "ok": true, "record_id": "2122d833c6fcab70", "setup": "triintri", "side": 5.000214405286989, "solution_hash": "2122d833c6fcab70", "solution_path": "data/solutions/generated/triintri_022_2122d833c6fcab70.json", "tolerance": 1e-08, "warnings": []}
{"case": "triintri@23", "container": {"circumradius": 2.886751345948129, "orientation_radians": 0.0, "side_length": 5.0, "sides": 3, "type": "regular_polygon"}, "density": 0.9200000000000002, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": 7.105427357601002e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "n": 23, "ok": true, "record_id": "4f1bdd5d6c07f589", "setup": "triintri", "side": 5.0, "solution_hash": "4f1bdd5d6c07f589", "solution_path": "data/solutions/generated/triintri_023_4f1bdd5d6c07f589.json", "tolerance": 1e-08, "warnings": []}
{"case": "triintri@24", "container": {"circumradius": 2.886751345948129, "orientation_radians": 0.0, "side_length": 5.0, "sides": 3, "type": "regular_polygon"}, "density": 0.9600000000000002, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": 7.105427357601002e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "n": 24, "ok": true, "record_id": "64d77fbf13ae0be7", "setup": "triintri", "side": 5.0, "solution_hash": "64d77fbf13ae0be7", "solution_path": "data/solutions/generated/triintri_024_64d77fbf13ae0be7.json", "tolerance": 1e-08, "warnings": []}
{"case": "triintri@25", "container": {"circumradius": 2.886751345948129, "orientation_radians": 0.0, "side_length": 5.0, "sides": 3, "type": "regular_polygon"}, "density": 1.0000000000000002, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": 7.105427357601002e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "n": 25, "ok": true, "record_id": "9c4806b8b37c5d37", "setup": "triintri", "side": 5.0, "solution_hash": "9c4806b8b37c5d37", "solution_path": "data/solutions/generated/triintri_025_9c4806b8b37c5d37.json", "tolerance": 1e-08, "warnings": []}
{"case": "triintri@26", "container": {"circumradius": 3.123348661570509, "orientation_radians": 0.0, "side_length": 5.409798571592372, "sides": 3, "type": "regular_polygon"}, "density": 0.8884053356293824, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": -2.296466577480461e-06, "max_pair_overlap_depth": -2.0400569136769242e-05, "n": 26, "ok": true, "record_id": "0af5a3765ecc4828", "setup": "triintri", "side": 5.409798571592372, "solution_hash": "0af5a3765ecc4828", "solution_path": "data/solutions/generated/triintri_026_0af5a3765ecc4828.json", "tolerance": 1e-08, "warnings": []}
+{"case": "triintri@27", "container": {"circumradius": 3.175523634890182, "orientation_radians": 0.0, "side_length": 5.500168276265596, "sides": 3, "type": "regular_polygon"}, "density": 0.8925073688875825, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": -2.296352348617567e-06, "max_pair_overlap_depth": -7.147352432124165e-08, "n": 27, "ok": true, "record_id": "70128a68bb429fb5", "setup": "triintri", "side": 5.500168276265596, "solution_hash": "70128a68bb429fb5", "solution_path": "data/solutions/generated/triintri_027_70128a68bb429fb5.json", "tolerance": 1e-08, "warnings": []}
{"case": "triintri@27", "container": {"circumradius": 3.175537421568583, "orientation_radians": 0.0, "side_length": 5.500192155493054, "sides": 3, "type": "regular_polygon"}, "density": 0.8924996192164427, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": -1.8365186496628e-06, "max_pair_overlap_depth": -1.8064988390742442e-06, "n": 27, "ok": true, "record_id": "f1c2be61e829f1e2", "setup": "triintri", "side": 5.500192155493054, "solution_hash": "f1c2be61e829f1e2", "solution_path": "data/solutions/generated/triintri_027_f1c2be61e829f1e2.json", "tolerance": 1e-08, "warnings": []}
{"case": "triintri@28", "container": {"circumradius": 3.175523634890182, "orientation_radians": 0.0, "side_length": 5.500168276265596, "sides": 3, "type": "regular_polygon"}, "density": 0.9255631973649003, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": -2.296352348617567e-06, "max_pair_overlap_depth": -7.147352432124165e-08, "n": 28, "ok": true, "record_id": "9562d7d6e075a347", "setup": "triintri", "side": 5.500168276265596, "solution_hash": "9562d7d6e075a347", "solution_path": "data/solutions/generated/triintri_028_9562d7d6e075a347.json", "tolerance": 1e-08, "warnings": []}
{"case": "triintri@29", "container": {"circumradius": 3.273034274723498, "orientation_radians": 0.0, "side_length": 5.669061658735449, "sides": 3, "type": "regular_polygon"}, "density": 0.9023512759650716, "errors": [], "item": {"circumradius": 0.5773502691896258, "side_length": 1.0, "sides": 3, "type": "regular_polygon"}, "max_boundary_excess": -1.9275914294015247e-06, "max_pair_overlap_depth": -5.088073942505211e-06, "n": 29, "ok": true, "record_id": "38c7a009f9059275", "setup": "triintri", "side": 5.669061658735449, "solution_hash": "38c7a009f9059275", "solution_path": "data/solutions/generated/triintri_29_38c7a009f9059275.json", "tolerance": 1e-08, "warnings": []}
diff --git a/data/records.jsonl b/data/records.jsonl
index 42b4161df83396de65a3a039fbb41aa3ecbf3ffe..e44226018ef0fdb8915aef6a5515d2a1db447690 100644
--- a/data/records.jsonl
+++ b/data/records.jsonl
@@ -23,6 +23,7 @@
{"case": "cirinhex@3", "density": 0.6801656739683151, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 2.309, "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc3.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "4/\u221a3 = 2.309+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.309401076758503, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "cirinhex@3", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4/\u221a3 = 2.309+", "metric_symbol": "s", "metric_value": 2.309, "reference_text": "s = 4/\u221a3 = 2.309+ | Trivial.", "source_image": "hc3.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc3.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 4/\u221a3 = 2.309+ | Trivial.", "friedman_source_image": "hc3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "fa4a26f729d80904", "improvement": -1.5427776642873425e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.3364075199806535e-06, "max_pair_overlap_depth": -1.818559246702378e-05, "metric_expression": "2.30942", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.309416504535146, "n": 3, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.309401076758503, "setup": "cirinhex", "side": 2.309416504535146, "solution_path": "data/solutions/generated/cirinhex_003_fa4a26f729d80904.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/fa4a26f729d80904.svg", "tolerance": 1e-08, "verified": true}
{"case": "cirinhex@4", "density": 0.6802066538449848, "friedman_analytical_or_proved": true, "friedman_credit": "Shahriar Manzoor", "friedman_date": "April 2002", "friedman_display_metric_value": 2.666, "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc4.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "2/\u221a3 + 4/\u221a7 = 2.666+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.6665584304161603, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "cirinhex@4", "credit": "Shahriar Manzoor", "date": "April 2002", "family_updated": "", "metric_expression": "2/\u221a3 + 4/\u221a7 = 2.666+", "metric_symbol": "s", "metric_value": 2.666, "reference_text": "s = 2/\u221a3 + 4/\u221a7 = 2.666+ | Found by Shahriar Manzoor | in April 2002.", "source_image": "hc4.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc4.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 2/\u221a3 + 4/\u221a7 = 2.666+ | Found by Shahriar Manzoor | in April 2002.", "friedman_source_image": "hc4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "4ea68db180a8d4c4", "improvement": -4.572067859331952e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.3940250920849735e-07, "max_pair_overlap_depth": -4.6158989302380604e-05, "metric_expression": "2.66660", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.6666041510947536, "n": 4, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.6665584304161603, "setup": "cirinhex", "side": 2.6666041510947536, "solution_path": "data/solutions/generated/cirinhex_004_4ea68db180a8d4c4.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Shahriar Manzoor", "svg_path": "data/svg/4ea68db180a8d4c4.svg", "tolerance": 1e-08, "verified": true}
{"case": "cirinhex@5", "density": 0.672041103262025, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "1999", "friedman_display_metric_value": 2.999, "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc5.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "2.999+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.999, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "cirinhex@5", "credit": "Erich Friedman", "date": "1999", "family_updated": "", "metric_expression": "2.999+", "metric_symbol": "s", "metric_value": 2.999, "reference_text": "s = 2.999+ | Found by Erich Friedman | in 1999.", "source_image": "hc5.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc5.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 2.999+ | Found by Erich Friedman | in 1999.", "friedman_source_image": "hc5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "eb88c54ef20a6355", "improvement": -0.00041167259487728813, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6369105032841844e-07, "max_pair_overlap_depth": -3.862655436481788e-05, "metric_expression": "2.99941", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.9994116725948774, "n": 5, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.999, "setup": "cirinhex", "side": 2.9994116725948774, "solution_path": "data/solutions/generated/cirinhex_005_eb88c54ef20a6355.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/eb88c54ef20a6355.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinhex@6", "density": 0.7289850151429867, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 3.154, "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc6.gif", "friedman_metric_expression": "2 + 2/\u221a3 = 3.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.1547005383792515, "friedman_metric_value_source": "exact_expression", "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"}, "friedman_reference_text": "s = 2 + 2/\u221a3 = 3.154+ | Trivial. |", "friedman_source_image": "hc6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "3e5660c8b6310c57", "improvement": -5.2140127083433185e-05, "max_boundary_excess": -1.857578518338343e-06, "max_pair_overlap_depth": -3.893260533160969e-05, "metric_expression": "3.15475", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.154752678506335, "monotone_upper_bound_from_case": "cirinhex@7", "monotone_upper_bound_from_record": "7368090471750227", "n": 6, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.1547005383792515, "setup": "cirinhex", "side": 3.154752678506335, "solution_path": "data/solutions/generated/cirinhex_006_3e5660c8b6310c57.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/3e5660c8b6310c57.svg", "tolerance": 1e-08, "verified": true}
{"case": "cirinhex@6", "density": 0.7289750468062609, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 3.154, "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc6.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "2 + 2/\u221a3 = 3.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.1547005383792515, "friedman_metric_value_source": "exact_expression", "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": 3.154, "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"}, "friedman_reference_text": "s = 2 + 2/\u221a3 = 3.154+ | Trivial. |", "friedman_source_image": "hc6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "9b7724fab73c49c1", "improvement": -7.370981566001689e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.7088474341983044e-06, "max_pair_overlap_depth": -4.5158069021944414e-05, "metric_expression": "3.15477", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.1547742481949115, "n": 6, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.1547005383792515, "setup": "cirinhex", "side": 3.1547742481949115, "solution_path": "data/solutions/generated/cirinhex_006_9b7724fab73c49c1.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/9b7724fab73c49c1.svg", "tolerance": 1e-08, "verified": true}
{"case": "cirinhex@7", "density": 0.8504825176668179, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 3.154, "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc7.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "2 + 2/\u221a3 = 3.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.1547005383792515, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "cirinhex@7", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2 + 2/\u221a3 = 3.154+", "metric_symbol": "s", "metric_value": 3.154, "reference_text": "s = 2 + 2/\u221a3 = 3.154+ | Trivial. |", "source_image": "hc7.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc7.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 2 + 2/\u221a3 = 3.154+ | Trivial. |", "friedman_source_image": "hc7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "7368090471750227", "improvement": -5.2140127083433185e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.857578518338343e-06, "max_pair_overlap_depth": -3.893260533160969e-05, "metric_expression": "3.15475", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.154752678506335, "n": 7, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.1547005383792515, "setup": "cirinhex", "side": 3.154752678506335, "solution_path": "data/solutions/generated/cirinhex_007_7368090471750227.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/7368090471750227.svg", "tolerance": 1e-08, "verified": true}
{"case": "cirinhex@8", "density": 0.7030218969047355, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_display_metric_value": 3.709, "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc8.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "3.709+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.709, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "cirinhex@8", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "3.709+", "metric_symbol": "s", "metric_value": 3.709, "reference_text": "s = 3.709+ | Found by David W. Cantrell | in July 2002.", "source_image": "hc8.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc8.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 3.709+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "hc8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "afebd04e4720460b", "improvement": -0.0004503331818734857, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.1028444985304873e-08, "max_pair_overlap_depth": -2.2199913737530608e-05, "metric_expression": "3.70945", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.7094503331818736, "n": 8, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.709, "setup": "cirinhex", "side": 3.7094503331818736, "solution_path": "data/solutions/generated/cirinhex_008_afebd04e4720460b.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/afebd04e4720460b.svg", "tolerance": 1e-08, "verified": true}
@@ -91,9 +92,11 @@
{"case": "cirinsqu@23", "density": 0.7628623175184639, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman Packing Center", "friedman_date": "", "friedman_display_metric_value": 9.727, "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs23.gif", "friedman_metric_expression": "2 + 2\u221a2 + 2\u221a6 = 9.727+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.727406610312546, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@23", "credit": "Erich Friedman Packing Center", "date": "", "family_updated": "", "metric_expression": "2 + 2\u221a2 + 2\u221a6 = 9.727+", "metric_symbol": "s", "metric_value": 9.727, "reference_text": "s = 2 + 2\u221a2 + 2\u221a6 = 9.727+", "source_image": "cs23.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs23.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 2 + 2\u221a2 + 2\u221a6 = 9.727+", "friedman_source_image": "cs23.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "cfa99060ee2203a1", "improvement": -0.004899781121649127, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.8298221013957061e-06, "max_pair_overlap_depth": -2.6081243190478887e-05, "metric_expression": "9.73231", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 9.732306391434195, "n": 23, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 9.727406610312546, "setup": "cirinsqu", "side": 9.732306391434195, "solution_path": "data/solutions/generated/cirinsqu_023_cfa99060ee2203a1.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman Packing Center", "svg_path": "data/svg/cfa99060ee2203a1.svg", "tolerance": 1e-08, "verified": true}
{"case": "cirinsqu@24", "density": 0.7749295071503051, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman Packing Center", "friedman_date": "", "friedman_display_metric_value": 9.863, "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs24.gif", "friedman_metric_expression": "6 + \u221a2 + \u221a6 = 9.863+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.863703305156273, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@24", "credit": "Erich Friedman Packing Center", "date": "", "family_updated": "", "metric_expression": "6 + \u221a2 + \u221a6 = 9.863+", "metric_symbol": "s", "metric_value": 9.863, "reference_text": "s = 6 + \u221a2 + \u221a6 = 9.863+", "source_image": "cs24.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs24.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 6 + \u221a2 + \u221a6 = 9.863+", "friedman_source_image": "cs24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "e2eca298edbdb88c", "improvement": -0.0002148079781072454, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.506144878683841e-06, "max_pair_overlap_depth": -4.0021533562306644e-05, "metric_expression": "9.86392", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 9.86391811313438, "n": 24, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 9.863703305156273, "setup": "cirinsqu", "side": 9.86391811313438, "solution_path": "data/solutions/generated/cirinsqu_024_e2eca298edbdb88c.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman Packing Center", "svg_path": "data/svg/e2eca298edbdb88c.svg", "tolerance": 1e-08, "verified": true}
{"case": "cirintri@1", "density": 0.6045962978262086, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 3.464, "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct1.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "2\u221a3 = 3.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.4641016151377544, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2\u221a3 = 3.464+", "metric_symbol": "s", "metric_value": 3.464, "reference_text": "s = 2\u221a3 = 3.464+ | Trivial.", "source_image": "ct1.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct1.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 2\u221a3 = 3.464+ | Trivial.", "friedman_source_image": "ct1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "10a7a380c7f5376a", "improvement": -9.998878353734142e-06, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.7091087705090047e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "3.46411", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.464111614016108, "n": 1, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.4641016151377544, "setup": "cirintri", "side": 3.464111614016108, "solution_path": "data/solutions/generated/cirintri_001_10a7a380c7f5376a.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/10a7a380c7f5376a.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirintri@2", "density": 0.4859938211704954, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 5.464, "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct2.gif", "friedman_metric_expression": "2 + 2\u221a3 = 5.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.464101615137754, "friedman_metric_value_source": "exact_expression", "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"}, "friedman_reference_text": "s = 2 + 2\u221a3 = 5.464+ | Trivial.", "friedman_source_image": "ct2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "befbb923c21388e1", "improvement": -6.888470749721876e-05, "max_boundary_excess": -1.0755666179917966e-07, "max_pair_overlap_depth": -5.898827690131725e-05, "metric_expression": "5.46417", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.464170499845252, "monotone_upper_bound_from_case": "cirintri@3", "monotone_upper_bound_from_record": "fcee9d6e8b7c5ce0", "n": 2, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.464101615137754, "setup": "cirintri", "side": 5.464170499845252, "solution_path": "data/solutions/generated/cirintri_002_befbb923c21388e1.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/befbb923c21388e1.svg", "tolerance": 1e-08, "verified": true}
{"case": "cirintri@2", "density": 0.48598475821399, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 5.464, "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct2.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "2 + 2\u221a3 = 5.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.464101615137754, "friedman_metric_value_source": "exact_expression", "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": 5.464, "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"}, "friedman_reference_text": "s = 2 + 2\u221a3 = 5.464+ | Trivial.", "friedman_source_image": "ct2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "f5cde1b8295da3ad", "improvement": -0.00011983415381777718, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.7587277048611867e-06, "max_pair_overlap_depth": -7.068795172848752e-05, "metric_expression": "5.46422", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.464221449291572, "n": 2, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.464101615137754, "setup": "cirintri", "side": 5.464221449291572, "solution_path": "data/solutions/generated/cirintri_002_f5cde1b8295da3ad.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/f5cde1b8295da3ad.svg", "tolerance": 1e-08, "verified": true}
{"case": "cirintri@3", "density": 0.7289907317557431, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 5.464, "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct3.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "2 + 2\u221a3 = 5.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.464101615137754, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@3", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2 + 2\u221a3 = 5.464+", "metric_symbol": "s", "metric_value": 5.464, "reference_text": "s = 2 + 2\u221a3 = 5.464+ | Trivial.", "source_image": "ct3.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct3.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 2 + 2\u221a3 = 5.464+ | Trivial.", "friedman_source_image": "ct3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "fcee9d6e8b7c5ce0", "improvement": -6.888470749721876e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0755666179917966e-07, "max_pair_overlap_depth": -5.0474041760395494e-05, "metric_expression": "5.46417", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.464170499845252, "n": 3, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.464101615137754, "setup": "cirintri", "side": 5.464170499845252, "solution_path": "data/solutions/generated/cirintri_003_fcee9d6e8b7c5ce0.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/fcee9d6e8b7c5ce0.svg", "tolerance": 1e-08, "verified": true}
{"case": "cirintri@4", "density": 0.6045810458812538, "friedman_analytical_or_proved": true, "friedman_credit": "Milano", "friedman_date": "1987", "friedman_display_metric_value": 6.928, "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct4.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "4\u221a3 = 6.928+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.928203230275509, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@4", "credit": "Milano", "date": "1987", "family_updated": "", "metric_expression": "4\u221a3 = 6.928+", "metric_symbol": "s", "metric_value": 6.928, "reference_text": "s = 4\u221a3 = 6.928+ | Proved by Milano in 1987.", "source_image": "ct4.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct4.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 4\u221a3 = 6.928+ | Proved by Milano in 1987.", "friedman_source_image": "ct4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "9e0c39beb3cc8aa7", "improvement": -0.00010738737436177814, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.7330840119143005e-06, "max_pair_overlap_depth": -4.49553106092182e-05, "metric_expression": "6.92831", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 6.9283106176498706, "n": 4, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 6.928203230275509, "setup": "cirintri", "side": 6.9283106176498706, "solution_path": "data/solutions/generated/cirintri_004_9e0c39beb3cc8aa7.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Milano", "svg_path": "data/svg/9e0c39beb3cc8aa7.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirintri@5", "density": 0.6511079649804111, "friedman_analytical_or_proved": true, "friedman_credit": "Milano", "friedman_date": "1987", "friedman_display_metric_value": 7.464, "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct5.gif", "friedman_metric_expression": "4 + 2\u221a3 = 7.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.464101615137754, "friedman_metric_value_source": "exact_expression", "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"}, "friedman_reference_text": "s = 4 + 2\u221a3 = 7.464+ | Proved by Milano in 1987.", "friedman_source_image": "ct5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "64d422ca71110b0d", "improvement": -9.578668471910134e-05, "max_boundary_excess": -1.6105130511423482e-06, "max_pair_overlap_depth": -3.3410124574029254e-05, "metric_expression": "7.46420", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 7.4641974018224735, "monotone_upper_bound_from_case": "cirintri@6", "monotone_upper_bound_from_record": "09bddcb97a85d368", "n": 5, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 7.464101615137754, "setup": "cirintri", "side": 7.4641974018224735, "solution_path": "data/solutions/generated/cirintri_005_64d422ca71110b0d.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Milano", "svg_path": "data/svg/64d422ca71110b0d.svg", "tolerance": 1e-08, "verified": true}
{"case": "cirintri@5", "density": 0.6510985676665063, "friedman_analytical_or_proved": true, "friedman_credit": "Milano", "friedman_date": "1987", "friedman_display_metric_value": 7.464, "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct5.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "4 + 2\u221a3 = 7.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.464101615137754, "friedman_metric_value_source": "exact_expression", "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": 7.464, "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"}, "friedman_reference_text": "s = 4 + 2\u221a3 = 7.464+ | Proved by Milano in 1987.", "friedman_source_image": "ct5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "ad32c3b8e4c714b5", "improvement": -0.0001496519183588063, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.12146637612598e-06, "max_pair_overlap_depth": -3.650753517803196e-05, "metric_expression": "7.46425", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 7.464251267056113, "n": 5, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 7.464101615137754, "setup": "cirintri", "side": 7.464251267056113, "solution_path": "data/solutions/generated/cirintri_005_ad32c3b8e4c714b5.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Milano", "svg_path": "data/svg/ad32c3b8e4c714b5.svg", "tolerance": 1e-08, "verified": true}
{"case": "cirintri@6", "density": 0.7813295579764934, "friedman_analytical_or_proved": true, "friedman_credit": "Oler/Groemer", "friedman_date": "1961", "friedman_display_metric_value": 7.464, "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct6.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "4 + 2\u221a3 = 7.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.464101615137754, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@6", "credit": "Oler/Groemer", "date": "1961", "family_updated": "", "metric_expression": "4 + 2\u221a3 = 7.464+", "metric_symbol": "s", "metric_value": 7.464, "reference_text": "s = 4 + 2\u221a3 = 7.464+ | Proved by Oler/Groemer in 1961.", "source_image": "ct6.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct6.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 4 + 2\u221a3 = 7.464+ | Proved by Oler/Groemer in 1961.", "friedman_source_image": "ct6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "09bddcb97a85d368", "improvement": -9.578668471910134e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.6105130511423482e-06, "max_pair_overlap_depth": -3.3410124574029254e-05, "metric_expression": "7.46420", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 7.4641974018224735, "n": 6, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 7.464101615137754, "setup": "cirintri", "side": 7.4641974018224735, "solution_path": "data/solutions/generated/cirintri_006_09bddcb97a85d368.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Oler/Groemer", "svg_path": "data/svg/09bddcb97a85d368.svg", "tolerance": 1e-08, "verified": true}
{"case": "cirintri@7", "density": 0.6370913423934993, "friedman_analytical_or_proved": true, "friedman_credit": "Melissen", "friedman_date": "1993", "friedman_display_metric_value": 8.928, "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct7.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "2 + 4\u221a3 = 8.928+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.928203230275509, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@7", "credit": "Melissen", "date": "1993", "family_updated": "", "metric_expression": "2 + 4\u221a3 = 8.928+", "metric_symbol": "s", "metric_value": 8.928, "reference_text": "s = 2 + 4\u221a3 = 8.928+ | Proved by Melissen in 1993.", "source_image": "ct7.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct7.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 2 + 4\u221a3 = 8.928+ | Proved by Melissen in 1993.", "friedman_source_image": "ct7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "1d479488f6f25560", "improvement": -0.00017925098356919023, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.6242968528307244e-06, "max_pair_overlap_depth": -4.768068922000879e-05, "metric_expression": "8.92838", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 8.928382481259078, "n": 7, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 8.928203230275509, "setup": "cirintri", "side": 8.928382481259078, "solution_path": "data/solutions/generated/cirintri_007_1d479488f6f25560.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Melissen", "svg_path": "data/svg/1d479488f6f25560.svg", "tolerance": 1e-08, "verified": true}
@@ -112,13 +115,17 @@
{"case": "domindom@5", "density": 0.7999239260919105, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 2.5, "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/5.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "5/2 = 2.5", "friedman_metric_symbol": "s", "friedman_metric_value": 2.5, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "domindom@5", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5/2 = 2.5", "metric_symbol": "s", "metric_value": 2.5, "reference_text": "5. | | s = 5/2 = 2.5 | Trivial.", "source_image": "5.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/5.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "5. | | s = 5/2 = 2.5 | Trivial.", "friedman_source_image": "5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "af0995e19f0dc8ce", "improvement": -0.00011887395946308388, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.6075584103633105e-07, "max_pair_overlap_depth": -2.297221660940707e-05, "metric_expression": "2.50012", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.500118873959463, "n": 5, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.5, "setup": "domindom", "side": 5.000237747918926, "solution_path": "data/solutions/generated/domindom_005_af0995e19f0dc8ce.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/af0995e19f0dc8ce.svg", "tolerance": 1e-08, "verified": true}
{"case": "domindom@6", "density": 0.7051360525077088, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "April 2026", "friedman_display_metric_value": 2.916, "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/6.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "2.916+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.916, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "domindom@6", "credit": "Erich Friedman", "date": "April 2026", "family_updated": "", "metric_expression": "2.916+", "metric_symbol": "s", "metric_value": 2.916, "reference_text": "6. | | s = 2.916+ | Found by Erich Friedman in April 2026.", "source_image": "6.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/6.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "6. | | s = 2.916+ | Found by Erich Friedman in April 2026.", "friedman_source_image": "6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "0f3d5defa1e042a8", "improvement": -0.001018377106842383, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.01302900812739e-06, "max_pair_overlap_depth": -7.57681649177222e-06, "metric_expression": "2.91702", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.9170183771068423, "n": 6, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.916, "setup": "domindom", "side": 5.834036754213685, "solution_path": "data/solutions/generated/domindom_006_0f3d5defa1e042a8.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/0f3d5defa1e042a8.svg", "tolerance": 1e-08, "verified": true}
{"case": "domindom@7", "density": 0.777726224180711, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 3.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/9.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "7.-9. | | s = 3 | Trivial.", "friedman_source_image": "9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "7b5335d8b8c4a5db", "improvement": -9.942973726095872e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.3331344280185253e-06, "max_pair_overlap_depth": -2.1961883454491016e-05, "metric_expression": "3.00010", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.000099429737261, "n": 7, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.0, "setup": "domindom", "side": 6.000198859474522, "solution_path": "data/solutions/generated/domindom_007_7b5335d8b8c4a5db.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/7b5335d8b8c4a5db.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domindom@7", "density": 0.7777520096383135, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 3.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/9.gif", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "7.-9. | | s = 3 | Trivial.", "friedman_source_image": "9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "a480ce1181b75833", "improvement": -4.9696932403531235e-05, "max_boundary_excess": -7.043740939405296e-07, "max_pair_overlap_depth": -7.019236833216702e-06, "metric_expression": "3.00005", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.0000496969324035, "monotone_upper_bound_from_case": "domindom@9", "monotone_upper_bound_from_record": "43b9630822b40990", "n": 7, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.0, "setup": "domindom", "side": 6.000099393864807, "solution_path": "data/solutions/generated/domindom_007_a480ce1181b75833.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/a480ce1181b75833.svg", "tolerance": 1e-08, "verified": true}
{"case": "domindom@8", "density": 0.8888442639489805, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 3.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/9.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "7.-9. | | s = 3 | Trivial.", "friedman_source_image": "9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "953d10b3e926402a", "improvement": -7.530742160444248e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.80012014980602e-06, "max_pair_overlap_depth": -1.8965164302886528e-05, "metric_expression": "3.00008", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.0000753074216044, "n": 8, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.0, "setup": "domindom", "side": 6.000150614843209, "solution_path": "data/solutions/generated/domindom_008_953d10b3e926402a.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/953d10b3e926402a.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domindom@8", "density": 0.8888594395866439, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 3.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/9.gif", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "7.-9. | | s = 3 | Trivial.", "friedman_source_image": "9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "a4f6452b97bb5cc0", "improvement": -4.9696932403531235e-05, "max_boundary_excess": -3.7542337882889854e-07, "max_pair_overlap_depth": -7.019236833216702e-06, "metric_expression": "3.00005", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.0000496969324035, "monotone_upper_bound_from_case": "domindom@9", "monotone_upper_bound_from_record": "43b9630822b40990", "n": 8, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.0, "setup": "domindom", "side": 6.000099393864807, "solution_path": "data/solutions/generated/domindom_008_a4f6452b97bb5cc0.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/a4f6452b97bb5cc0.svg", "tolerance": 1e-08, "verified": true}
{"case": "domindom@9", "density": 0.9999668695349745, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 3.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/9.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "domindom@9", "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"}, "friedman_reference_text": "7.-9. | | s = 3 | Trivial.", "friedman_source_image": "9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "43b9630822b40990", "improvement": -4.9696932403531235e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.7542337882889854e-07, "max_pair_overlap_depth": -3.0527855267026283e-06, "metric_expression": "3.00005", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.0000496969324035, "n": 9, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.0, "setup": "domindom", "side": 6.000099393864807, "solution_path": "data/solutions/generated/domindom_009_43b9630822b40990.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/43b9630822b40990.svg", "tolerance": 1e-08, "verified": true}
{"case": "domindom@10", "density": 0.8162099861183555, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 3.5, "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/10.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "7/2 = 3.5", "friedman_metric_symbol": "s", "friedman_metric_value": 3.5, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "domindom@10", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "7/2 = 3.5", "metric_symbol": "s", "metric_value": 3.5, "reference_text": "10. | | s = 7/2 = 3.5 | Trivial.", "source_image": "10.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/10.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "10. | | s = 7/2 = 3.5 | Trivial.", "friedman_source_image": "10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "d7508c45c3f18a7c", "improvement": -0.0002498690138819093, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.5739397918324016e-07, "max_pair_overlap_depth": -9.291197802940854e-05, "metric_expression": "3.50025", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.500249869013882, "n": 10, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.5, "setup": "domindom", "side": 7.000499738027764, "solution_path": "data/solutions/generated/domindom_010_d7508c45c3f18a7c.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/d7508c45c3f18a7c.svg", "tolerance": 1e-08, "verified": true}
{"case": "domindom@11", "density": 0.7611589259525644, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2026", "friedman_display_metric_value": 3.801, "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/11.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "(5+\u221a41)/3 = 3.801+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.801041412477616, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "domindom@11", "credit": "Maurizio Morandi", "date": "April 2026", "family_updated": "", "metric_expression": "(5+\u221a41)/3 = 3.801+", "metric_symbol": "s", "metric_value": 3.801, "reference_text": "11. | | s = (5+\u221a41)/3 = 3.801+ | Found by Maurizio Morandi in April 2026.", "source_image": "11.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/11.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "11. | | s = (5+\u221a41)/3 = 3.801+ | Found by Maurizio Morandi in April 2026.", "friedman_source_image": "11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "0fa9d3de612fbf03", "improvement": -0.0004907598320262885, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.0757791549262947e-06, "max_pair_overlap_depth": -0.0001019023981195799, "metric_expression": "3.80153", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.8015321723096425, "n": 11, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.801041412477616, "setup": "domindom", "side": 7.603064344619285, "solution_path": "data/solutions/generated/domindom_011_0fa9d3de612fbf03.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/0fa9d3de612fbf03.svg", "tolerance": 1e-08, "verified": true}
{"case": "domindom@12", "density": 0.7822558234911755, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2026", "friedman_display_metric_value": 3.914, "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/12.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "5/2+\u221a2 = 3.914+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.914213562373095, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "domindom@12", "credit": "Maurizio Morandi", "date": "April 2026", "family_updated": "", "metric_expression": "5/2+\u221a2 = 3.914+", "metric_symbol": "s", "metric_value": 3.914, "reference_text": "12. | | s = 5/2+\u221a2 = 3.914+ | Found by Maurizio Morandi in April 2026.", "source_image": "12.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/12.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "12. | | s = 5/2+\u221a2 = 3.914+ | Found by Maurizio Morandi in April 2026.", "friedman_source_image": "12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "3a75d33bd56228b0", "improvement": -0.0024495751142099564, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.0673284962267076e-07, "max_pair_overlap_depth": -1.4217486933421064e-05, "metric_expression": "3.91666", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.916663137487305, "n": 12, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.914213562373095, "setup": "domindom", "side": 7.83332627497461, "solution_path": "data/solutions/generated/domindom_012_3a75d33bd56228b0.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T03:07:58+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/3a75d33bd56228b0.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domindom@14", "density": 0.8749646115011096, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 4.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/16.gif", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "14.-16 | | s = 4 | Trivial.", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "6f795311544960ee", "improvement": -8.08904511222508e-05, "max_boundary_excess": -5.0905786346255866e-06, "max_pair_overlap_depth": -4.621969549978289e-06, "metric_expression": "4.00008", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.000080890451122, "monotone_upper_bound_from_case": "domindom@16", "monotone_upper_bound_from_record": "5102d8436800a7ff", "n": 14, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.0, "setup": "domindom", "side": 8.000161780902245, "solution_path": "data/solutions/generated/domindom_014_6f795311544960ee.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/6f795311544960ee.svg", "tolerance": 1e-08, "verified": true}
{"case": "domindom@14", "density": 0.8749484114386541, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 4.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/16.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "14.-16 | | s = 4 | Trivial.", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "8be6cbfdbea40909", "improvement": -0.00011792192603543583, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.966859551203129e-06, "max_pair_overlap_depth": -1.688780940467538e-05, "metric_expression": "4.00012", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.000117921926035, "n": 14, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.0, "setup": "domindom", "side": 8.00023584385207, "solution_path": "data/solutions/generated/domindom_014_8be6cbfdbea40909.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/8be6cbfdbea40909.svg", "tolerance": 1e-08, "verified": true}
{"case": "domindom@15", "density": 0.9374471956142247, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 4.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/16.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "14.-16 | | s = 4 | Trivial.", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "d2b8e4757dde349c", "improvement": -0.00011265411524785662, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.415030228686277e-06, "max_pair_overlap_depth": -1.0456826571081379e-05, "metric_expression": "4.00011", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.000112654115248, "n": 15, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.0, "setup": "domindom", "side": 8.000225308230496, "solution_path": "data/solutions/generated/domindom_015_d2b8e4757dde349c.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/d2b8e4757dde349c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domindom@15", "density": 0.9374620837511888, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 4.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/16.gif", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "14.-16 | | s = 4 | Trivial.", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "dda9136316a32e15", "improvement": -8.08904511222508e-05, "max_boundary_excess": -4.23688533190969e-06, "max_pair_overlap_depth": -4.621969549978289e-06, "metric_expression": "4.00008", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.000080890451122, "monotone_upper_bound_from_case": "domindom@16", "monotone_upper_bound_from_record": "5102d8436800a7ff", "n": 15, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.0, "setup": "domindom", "side": 8.000161780902245, "solution_path": "data/solutions/generated/domindom_015_dda9136316a32e15.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/dda9136316a32e15.svg", "tolerance": 1e-08, "verified": true}
{"case": "domindom@16", "density": 0.9999595560012681, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 4.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/16.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "domindom@16", "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"}, "friedman_reference_text": "14.-16 | | s = 4 | Trivial.", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "5102d8436800a7ff", "improvement": -8.08904511222508e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.23688533190969e-06, "max_pair_overlap_depth": -4.621969549978289e-06, "metric_expression": "4.00008", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.000080890451122, "n": 16, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.0, "setup": "domindom", "side": 8.000161780902245, "solution_path": "data/solutions/generated/domindom_016_5102d8436800a7ff.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/5102d8436800a7ff.svg", "tolerance": 1e-08, "verified": true}
{"case": "domindom@17", "density": 0.8394293686085971, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 4.5, "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/18.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "9/2 = 4.5", "friedman_metric_symbol": "s", "friedman_metric_value": 4.5, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "domindom@17", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "9/2 = 4.5", "metric_symbol": "s", "metric_value": 4.5, "reference_text": "17.-18 | | s = 9/2 = 4.5 | Trivial.", "source_image": "18.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/18.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "17.-18 | | s = 9/2 = 4.5 | Trivial.", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "3b388ddd068a843f", "improvement": -0.00020586075893191946, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.953267209728551e-07, "max_pair_overlap_depth": -5.000451830872166e-05, "metric_expression": "4.50021", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.500205860758932, "n": 17, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.5, "setup": "domindom", "side": 9.000411721517864, "solution_path": "data/solutions/generated/domindom_017_3b388ddd068a843f.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/3b388ddd068a843f.svg", "tolerance": 1e-08, "verified": true}
{"case": "domindom@18", "density": 0.887184224626392, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 4.5, "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/18.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "9/2 = 4.5", "friedman_metric_symbol": "s", "friedman_metric_value": 4.5, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "domindom@18", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "9/2 = 4.5", "metric_symbol": "s", "metric_value": 4.5, "reference_text": "17.-18 | | s = 9/2 = 4.5 | Trivial.", "source_image": "18.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/18.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "17.-18 | | s = 9/2 = 4.5 | Trivial.", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "6a5fcb6ec4e55187", "improvement": -0.004321147560441219, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4781434060372476e-06, "max_pair_overlap_depth": -4.894490537976637e-05, "metric_expression": "4.50432", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.504321147560441, "n": 18, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.5, "setup": "domindom", "side": 9.008642295120882, "solution_path": "data/solutions/generated/domindom_018_6a5fcb6ec4e55187.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/6a5fcb6ec4e55187.svg", "tolerance": 1e-08, "verified": true}
@@ -151,6 +158,7 @@
{"case": "hexincir@3", "density": 0.6202101203168955, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 2.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/3.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "r", "friedman_metric_value": 2.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "hexincir@3", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "r", "metric_value": 2.0, "reference_text": "r = 2 | Trivial.", "source_image": "3.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/3.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = 2 | Trivial.", "friedman_source_image": "3.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "194e619a74e29452", "improvement": -5.624955299809997e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.970405754344824e-06, "max_pair_overlap_depth": -7.230141190991447e-05, "metric_expression": "2.00006", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "r", "metric_value": 2.000056249552998, "n": 3, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.0, "setup": "hexincir", "side": 4.000112499105996, "solution_path": "data/solutions/generated/hexincir_003_194e619a74e29452.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/194e619a74e29452.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexincir@4", "density": 0.6594633839026793, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_display_metric_value": 2.239, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/4.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "(21\u2013\u221a145)/4 = 2.239+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.239601355301926, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "hexincir@4", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "(21\u2013\u221a145)/4 = 2.239+", "metric_symbol": "r", "metric_value": 2.239, "reference_text": "r = (21\u2013\u221a145)/4 = 2.239+ | Found by Maurizio Morandi | in March 2026.", "source_image": "4.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/4.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = (21\u2013\u221a145)/4 = 2.239+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "4.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "355b11dfc363f5ba", "improvement": -7.70321709873123e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.544625404889757e-07, "max_pair_overlap_depth": -5.057462804924917e-05, "metric_expression": "2.23968", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "r", "metric_value": 2.2396783874729134, "n": 4, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.239601355301926, "setup": "hexincir", "side": 4.479356774945827, "solution_path": "data/solutions/generated/hexincir_004_355b11dfc363f5ba.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/355b11dfc363f5ba.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexincir@5", "density": 0.6766813833384198, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_display_metric_value": 2.471, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/5.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "2.471+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.471, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "hexincir@5", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "2.471+", "metric_symbol": "r", "metric_value": 2.471, "reference_text": "r = 2.471+ | Found by Maurizio Morandi | in March 2026.", "source_image": "5.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/5.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = 2.471+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "5.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "9427b00c077aafc3", "improvement": -0.0009739785189006867, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.673907177263573e-06, "max_pair_overlap_depth": -5.983134966366932e-05, "metric_expression": "2.47197", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "r", "metric_value": 2.4719739785189008, "n": 5, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.471, "setup": "hexincir", "side": 4.9439479570378015, "solution_path": "data/solutions/generated/hexincir_005_9427b00c077aafc3.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/9427b00c077aafc3.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexincir@6", "density": 0.7088286989393032, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 2.645, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/7.png", "friedman_metric_expression": "\u221a7 = 2.645+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.6457513110645907, "friedman_metric_value_source": "exact_expression", "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"}, "friedman_reference_text": "r = \u221a7 = 2.645+ | Trivial. |", "friedman_source_image": "7.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "0d33963cb4bcb842", "improvement": -4.2435284075548196e-05, "max_boundary_excess": -1.370930279076532e-06, "max_pair_overlap_depth": -2.6143388138111234e-05, "metric_expression": "2.64579", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "r", "metric_value": 2.6457937463486663, "monotone_upper_bound_from_case": "hexincir@7", "monotone_upper_bound_from_record": "5cd27a740cc9b2a5", "n": 6, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.6457513110645907, "setup": "hexincir", "side": 5.2915874926973325, "solution_path": "data/solutions/generated/hexincir_006_0d33963cb4bcb842.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/0d33963cb4bcb842.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexincir@6", "density": 0.7087991567669789, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 2.645, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/7.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "\u221a7 = 2.645+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.6457513110645907, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "hexincir@6", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a7 = 2.645+", "metric_symbol": "r", "metric_value": 2.645, "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"}, "friedman_reference_text": "r = \u221a7 = 2.645+ | Trivial. |", "friedman_source_image": "7.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "d7d900dd0bba32a3", "improvement": -9.757197521187422e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.0221671226394733e-07, "max_pair_overlap_depth": -3.6674166167594e-05, "metric_expression": "2.64585", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "r", "metric_value": 2.6458488830398026, "n": 6, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.6457513110645907, "setup": "hexincir", "side": 5.291697766079605, "solution_path": "data/solutions/generated/hexincir_006_d7d900dd0bba32a3.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/d7d900dd0bba32a3.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexincir@7", "density": 0.8269668154291873, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 2.645, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/7.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "\u221a7 = 2.645+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.6457513110645907, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "hexincir@7", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a7 = 2.645+", "metric_symbol": "r", "metric_value": 2.645, "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"}, "friedman_reference_text": "r = \u221a7 = 2.645+ | Trivial. |", "friedman_source_image": "7.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "5cd27a740cc9b2a5", "improvement": -4.2435284075548196e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.370930279076532e-06, "max_pair_overlap_depth": -2.6143388138111234e-05, "metric_expression": "2.64579", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "r", "metric_value": 2.6457937463486663, "n": 7, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.6457513110645907, "setup": "hexincir", "side": 5.2915874926973325, "solution_path": "data/solutions/generated/hexincir_007_5cd27a740cc9b2a5.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/5cd27a740cc9b2a5.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexincir@8", "density": 0.7426247287224678, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_display_metric_value": 2.984, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/8.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "2.984+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.984, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "hexincir@8", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "2.984+", "metric_symbol": "r", "metric_value": 2.984, "reference_text": "r = 2.984+ | Found by Maurizio Morandi | in March 2026.", "source_image": "8.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/8.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = 2.984+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "8.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "7d27528d78a05a43", "improvement": -0.0007729280237529501, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.6889173538057207e-06, "max_pair_overlap_depth": -1.732806256671271e-05, "metric_expression": "2.98477", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "r", "metric_value": 2.984772928023753, "n": 8, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.984, "setup": "hexincir", "side": 5.969545856047506, "solution_path": "data/solutions/generated/hexincir_008_7d27528d78a05a43.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/7d27528d78a05a43.svg", "tolerance": 1e-08, "verified": true}
@@ -164,6 +172,7 @@
{"case": "hexincir@16", "density": 0.7714178714001686, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_display_metric_value": 4.13966, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/16.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4.13966+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.13966, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "hexincir@16", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4.13966+", "metric_symbol": "r", "metric_value": 4.13966, "reference_text": "r = 4.13966+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "16.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/16.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = 4.13966+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "16.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "928ee7a851bce6e5", "improvement": -0.0019209106018989175, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.8109720468001456e-06, "max_pair_overlap_depth": -4.18854370716204e-05, "metric_expression": "4.14158", "metric_recovery_tolerance_abs": 1e-05, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "r", "metric_value": 4.141580910601899, "n": 16, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.13966, "setup": "hexincir", "side": 8.283161821203798, "solution_path": "data/solutions/generated/hexincir_016_928ee7a851bce6e5.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/928ee7a851bce6e5.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexincir@17", "density": 0.7591563034967428, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_display_metric_value": 4.3017, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/17.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4.30170+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.3017, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "hexincir@17", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4.30170+", "metric_symbol": "r", "metric_value": 4.3017, "reference_text": "r = 4.30170+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "17.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/17.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = 4.30170+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "17.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "e582b760dc061cd3", "improvement": -0.001681676380650643, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.855951803468429e-06, "max_pair_overlap_depth": -4.543663468803416e-05, "metric_expression": "4.30338", "metric_recovery_tolerance_abs": 1e-05, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "r", "metric_value": 4.303381676380651, "n": 17, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.3017, "setup": "hexincir", "side": 8.606763352761302, "solution_path": "data/solutions/generated/hexincir_017_e582b760dc061cd3.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T03:07:58+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/e582b760dc061cd3.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexincir@18", "density": 0.7834006630179079, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 4.35889, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/19.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "\u221a19 = 4.35889+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.358898943540673, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "hexincir@18", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a19 = 4.35889+", "metric_symbol": "r", "metric_value": 4.35889, "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"}, "friedman_reference_text": "r = \u221a19 = 4.35889+ | Trivial.", "friedman_source_image": "19.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "12eaac899664a007", "improvement": -0.00018559876213242177, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.696366953105269e-06, "max_pair_overlap_depth": -1.0490734119961154e-05, "metric_expression": "4.35908", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "r", "metric_value": 4.3590845423028055, "n": 18, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.358898943540673, "setup": "hexincir", "side": 8.718169084605611, "solution_path": "data/solutions/generated/hexincir_018_12eaac899664a007.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/12eaac899664a007.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexincir@18", "density": 0.7834357889503433, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 4.35889, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/19.png", "friedman_metric_expression": "\u221a19 = 4.35889+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.358898943540673, "friedman_metric_value_source": "exact_expression", "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"}, "friedman_reference_text": "r = \u221a19 = 4.35889+ | Trivial.", "friedman_source_image": "19.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "3bf05115c19cc6d5", "improvement": -8.787625086270623e-05, "max_boundary_excess": -5.636539265552187e-06, "max_pair_overlap_depth": -6.973398227638761e-06, "metric_expression": "4.35899", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "r", "metric_value": 4.358986819791536, "monotone_upper_bound_from_case": "hexincir@19", "monotone_upper_bound_from_record": "b4bf200670caf09c", "n": 18, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.358898943540673, "setup": "hexincir", "side": 8.717973639583072, "solution_path": "data/solutions/generated/hexincir_018_3bf05115c19cc6d5.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/3bf05115c19cc6d5.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexincir@19", "density": 0.8269599994475847, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 4.35889, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/19.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "\u221a19 = 4.35889+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.358898943540673, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "hexincir@19", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a19 = 4.35889+", "metric_symbol": "r", "metric_value": 4.35889, "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"}, "friedman_reference_text": "r = \u221a19 = 4.35889+ | Trivial.", "friedman_source_image": "19.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "b4bf200670caf09c", "improvement": -8.787625086270623e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.636539265552187e-06, "max_pair_overlap_depth": -6.973398227638761e-06, "metric_expression": "4.35899", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "r", "metric_value": 4.358986819791536, "n": 19, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.358898943540673, "setup": "hexincir", "side": 8.717973639583072, "solution_path": "data/solutions/generated/hexincir_019_b4bf200670caf09c.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/b4bf200670caf09c.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexincir@20", "density": 0.7723857667872704, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_display_metric_value": 4.62598, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/20.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4.62598+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.62598, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "hexincir@20", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4.62598+", "metric_symbol": "r", "metric_value": 4.62598, "reference_text": "r = 4.62598+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "20.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/20.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = 4.62598+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "20.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "30108dc5d1ea9a14", "improvement": -0.0015460646281777457, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.8288872105642895e-06, "max_pair_overlap_depth": -4.792351042670617e-06, "metric_expression": "4.62753", "metric_recovery_tolerance_abs": 1e-05, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "r", "metric_value": 4.627526064628178, "n": 20, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.62598, "setup": "hexincir", "side": 9.255052129256356, "solution_path": "data/solutions/generated/hexincir_020_30108dc5d1ea9a14.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/30108dc5d1ea9a14.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexincir@21", "density": 0.7917677222663352, "friedman_analytical_or_proved": false, "friedman_credit": "Enea Victor", "friedman_date": "April 2026", "friedman_display_metric_value": 4.68339, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/21.png", "friedman_metric_expression": "4.68339+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.68339, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "hexincir@21", "credit": "Enea Victor", "date": "April 2026", "family_updated": "", "metric_expression": "4.68339+", "metric_symbol": "r", "metric_value": 4.68339, "reference_text": "r = 4.68339+ | Found by Enea Victor | in April 2026.", "source_image": "21.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/21.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = 4.68339+ | Found by Enea Victor | in April 2026.", "friedman_source_image": "21.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "49a047c29bb184c3", "improvement": -1.5457789629635954e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.498160996568856e-06, "max_pair_overlap_depth": -3.8704072840012316e-06, "metric_expression": "4.68341", "metric_recovery_tolerance_abs": 1e-05, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "r", "metric_value": 4.68340545778963, "n": 21, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.68339, "setup": "hexincir", "side": 9.36681091557926, "solution_path": "data/solutions/generated/hexincir_021_49a047c29bb184c3.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Enea Victor", "svg_path": "data/svg/49a047c29bb184c3.svg", "tolerance": 1e-08, "verified": true}
@@ -189,10 +198,12 @@
{"case": "hexindom@20", "density": 0.7718745532343245, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_display_metric_value": 5.8012, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/20.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "5.8012+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.8012, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "hexindom@20", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "5.8012+", "metric_symbol": "s", "metric_value": 5.8012, "reference_text": "20. | | s = 5.8012+ | Found by Jonathan Viquerat in April 2026.", "source_image": "20.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/20.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "20. | | s = 5.8012+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "20.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "49ce661db444ec25", "improvement": -0.0004641087628032281, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.575652109486185e-06, "max_pair_overlap_depth": -1.0063478294242145e-05, "metric_expression": "5.80166", "metric_recovery_tolerance_abs": 0.0001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.801664108762803, "n": 20, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.8012, "setup": "hexindom", "side": 11.603328217525606, "solution_path": "data/solutions/generated/hexindom_020_49ce661db444ec25.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/49ce661db444ec25.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexinhex@1", "density": 0.9999864500659439, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 1.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/1.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "1", "friedman_metric_symbol": "s", "friedman_metric_value": 1.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "s", "metric_value": 1.0, "reference_text": "1. | | s = 1 | Trivial.", "source_image": "1.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/1.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "1. | | s = 1 | Trivial.", "friedman_source_image": "1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "fd6c6df7b6a9fd4f", "improvement": -6.7750358792029175e-06, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.374323059598708e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "1.00001", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 1.0000067750358792, "n": 1, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 1.0, "setup": "hexinhex", "side": 1.0000067750358792, "solution_path": "data/solutions/generated/hexinhex_001_fd6c6df7b6a9fd4f.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/fd6c6df7b6a9fd4f.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexinhex@2", "density": 0.49998387949392886, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 2.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/2.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "2. | | s = 2 | Trivial.", "friedman_source_image": "2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "4c0cc36e970c84b1", "improvement": -3.2241791775522444e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.177681809212628e-06, "max_pair_overlap_depth": -3.6384174144801684e-05, "metric_expression": "2.00003", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.0000322417917755, "n": 2, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.0, "setup": "hexinhex", "side": 2.0000322417917755, "solution_path": "data/solutions/generated/hexinhex_002_4c0cc36e970c84b1.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/4c0cc36e970c84b1.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@2", "density": 0.4999921614841797, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 2.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/2.gif", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "2. | | s = 2 | Trivial.", "friedman_source_image": "2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "c8ccd859ecc810e4", "improvement": -1.5677215970288927e-05, "max_boundary_excess": -2.457810528986454e-06, "max_pair_overlap_depth": -9.916736621090457e-06, "metric_expression": "2.00002", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.0000156772159703, "monotone_upper_bound_from_case": "hexinhex@3", "monotone_upper_bound_from_record": "2e22e12b6d11de4d", "n": 2, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.0, "setup": "hexinhex", "side": 2.0000156772159703, "solution_path": "data/solutions/generated/hexinhex_002_c8ccd859ecc810e4.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/c8ccd859ecc810e4.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexinhex@3", "density": 0.7499882422262695, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 2.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/3.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@3", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "3. | | s = 2 | Trivial.", "source_image": "3.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/3.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "3. | | s = 2 | Trivial.", "friedman_source_image": "3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "2e22e12b6d11de4d", "improvement": -1.5677215970288927e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.457810528986454e-06, "max_pair_overlap_depth": -9.916736621090457e-06, "metric_expression": "2.00002", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.0000156772159703, "n": 3, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.0, "setup": "hexinhex", "side": 2.0000156772159703, "solution_path": "data/solutions/generated/hexinhex_003_2e22e12b6d11de4d.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/2e22e12b6d11de4d.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexinhex@4", "density": 0.7346808358825446, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 2.333, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/4.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "7/3 = 2.333+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.3333333333333335, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@4", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "7/3 = 2.333+", "metric_symbol": "s", "metric_value": 2.333, "reference_text": "4. | | s = 7/3 = 2.333+ | Trivial.", "source_image": "4.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/4.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "4. | | s = 7/3 = 2.333+ | Trivial.", "friedman_source_image": "4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "73ddab3f27a95ffe", "improvement": -2.0709962234466417e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.969618748468479e-06, "max_pair_overlap_depth": -1.5832978702556844e-05, "metric_expression": "2.33335", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.333354043295568, "n": 4, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.3333333333333335, "setup": "hexinhex", "side": 2.333354043295568, "solution_path": "data/solutions/generated/hexinhex_004_73ddab3f27a95ffe.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/73ddab3f27a95ffe.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexinhex@5", "density": 0.7031111089487749, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "March 2015", "friedman_display_metric_value": 2.666, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/5.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "8/3 = 2.666+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.6666666666666665, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@5", "credit": "Erich Friedman", "date": "March 2015", "family_updated": "", "metric_expression": "8/3 = 2.666+", "metric_symbol": "s", "metric_value": 2.666, "reference_text": "5. | | s = 8/3 = 2.666+ | Found by Erich Friedman | in March 2015.", "source_image": "5.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/5.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "5. | | s = 8/3 = 2.666+ | Found by Erich Friedman | in March 2015.", "friedman_source_image": "5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "f45300c86d046ee7", "improvement": -2.6341939302731987e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.5309295562371306e-06, "max_pair_overlap_depth": -6.88869177611906e-06, "metric_expression": "2.66669", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.6666930086059693, "n": 5, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.6666666666666665, "setup": "hexinhex", "side": 2.6666930086059693, "solution_path": "data/solutions/generated/hexinhex_005_f45300c86d046ee7.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/f45300c86d046ee7.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexinhex@6", "density": 0.7199337426480286, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2015", "friedman_display_metric_value": 2.886, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/7.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "5/\u221a3 = 2.886+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.8867513459481287, "friedman_metric_value_source": "exact_expression", "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": 2.886, "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"}, "friedman_reference_text": "6-7. | | s = 5/\u221a3 = 2.886+ | Found by Maurizio Morandi | in April 2015.", "friedman_source_image": "7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "06b9c79146f8c119", "improvement": -0.0001328345152633581, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.09670685382364e-06, "max_pair_overlap_depth": -5.069396536394599e-05, "metric_expression": "2.88688", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.886884180463392, "n": 6, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.8867513459481287, "setup": "hexinhex", "side": 2.886884180463392, "solution_path": "data/solutions/generated/hexinhex_006_06b9c79146f8c119.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/06b9c79146f8c119.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@6", "density": 0.7199768937015341, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2015", "friedman_display_metric_value": 2.886, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/7.gif", "friedman_metric_expression": "5/\u221a3 = 2.886+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.8867513459481287, "friedman_metric_value_source": "exact_expression", "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"}, "friedman_reference_text": "6-7. | | s = 5/\u221a3 = 2.886+ | Found by Maurizio Morandi | in April 2015.", "friedman_source_image": "7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "93e84dfdad8f0036", "improvement": -4.63220442350476e-05, "max_boundary_excess": -5.6806989039887556e-06, "max_pair_overlap_depth": -3.096762286192156e-05, "metric_expression": "2.88680", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.8867976679923637, "monotone_upper_bound_from_case": "hexinhex@7", "monotone_upper_bound_from_record": "d2a43d4ca2836467", "n": 6, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.8867513459481287, "setup": "hexinhex", "side": 2.8867976679923637, "solution_path": "data/solutions/generated/hexinhex_006_93e84dfdad8f0036.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/93e84dfdad8f0036.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexinhex@7", "density": 0.8399730426517898, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2015", "friedman_display_metric_value": 2.886, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/7.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "5/\u221a3 = 2.886+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.8867513459481287, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@7", "credit": "Maurizio Morandi", "date": "April 2015", "family_updated": "", "metric_expression": "5/\u221a3 = 2.886+", "metric_symbol": "s", "metric_value": 2.886, "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"}, "friedman_reference_text": "6-7. | | s = 5/\u221a3 = 2.886+ | Found by Maurizio Morandi | in April 2015.", "friedman_source_image": "7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "d2a43d4ca2836467", "improvement": -4.63220442350476e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.6806989039887556e-06, "max_pair_overlap_depth": -2.9189085207215193e-05, "metric_expression": "2.88680", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.8867976679923637, "n": 7, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.8867513459481287, "setup": "hexinhex", "side": 2.8867976679923637, "solution_path": "data/solutions/generated/hexinhex_007_d2a43d4ca2836467.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/d2a43d4ca2836467.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexinhex@8", "density": 0.728171041468435, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2015", "friedman_display_metric_value": 3.314, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/8.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "(51+5\u221a3)/18 = 3.314+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.3144585576580217, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@8", "credit": "Maurizio Morandi", "date": "April 2015", "family_updated": "", "metric_expression": "(51+5\u221a3)/18 = 3.314+", "metric_symbol": "s", "metric_value": 3.314, "reference_text": "8. | | s = (51+5\u221a3)/18 = 3.314+ | Found by Maurizio Morandi | in April 2015.", "source_image": "8.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/8.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "8. | | s = (51+5\u221a3)/18 = 3.314+ | Found by Maurizio Morandi | in April 2015.", "friedman_source_image": "8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "8d9e02005a596470", "improvement": -0.00011981062760035854, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.71875974277703e-06, "max_pair_overlap_depth": -2.9528072121864923e-05, "metric_expression": "3.31458", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.314578368285622, "n": 8, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.3144585576580217, "setup": "hexinhex", "side": 3.314578368285622, "solution_path": "data/solutions/generated/hexinhex_008_8d9e02005a596470.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/8d9e02005a596470.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexinhex@9", "density": 0.683335999337671, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2015", "friedman_display_metric_value": 3.628, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/9.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "(24+5\u221a3)/9 = 3.628+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.628917115316043, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@9", "credit": "Maurizio Morandi", "date": "April 2015", "family_updated": "", "metric_expression": "(24+5\u221a3)/9 = 3.628+", "metric_symbol": "s", "metric_value": 3.628, "reference_text": "9. | | s = (24+5\u221a3)/9 = 3.628+ | Found by Maurizio Morandi | in April 2015.", "source_image": "9.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/9.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "9. | | s = (24+5\u221a3)/9 = 3.628+ | Found by Maurizio Morandi | in April 2015.", "friedman_source_image": "9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "21c530fbe612f27c", "improvement": -0.00022607864944923506, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.454488069704802e-06, "max_pair_overlap_depth": -2.836888994439679e-05, "metric_expression": "3.62914", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.629143193965492, "n": 9, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.628917115316043, "setup": "hexinhex", "side": 3.629143193965492, "solution_path": "data/solutions/generated/hexinhex_009_21c530fbe612f27c.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/21c530fbe612f27c.svg", "tolerance": 1e-08, "verified": true}
@@ -204,8 +215,10 @@
{"case": "hexinhex@15", "density": 0.7579263950656321, "friedman_analytical_or_proved": true, "friedman_credit": "Timo Berthold et al", "friedman_date": "January 2026", "friedman_display_metric_value": 4.44727, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/15.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "4.44727+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.44727, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@15", "credit": "Timo Berthold et al", "date": "January 2026", "family_updated": "", "metric_expression": "4.44727+", "metric_symbol": "s", "metric_value": 4.44727, "reference_text": "15. | | s = 4.44727+ | Found by Timo Berthold et al | in January 2026. | Slightly improved by Alex Kravatsky et al | in February 2026.", "source_image": "15.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/15.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "15. | | s = 4.44727+ | Found by Timo Berthold et al | in January 2026. | Slightly improved by Alex Kravatsky et al | in February 2026.", "friedman_source_image": "15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "536c412ab623afbc", "improvement": -0.0014196922804980971, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.564398033090082e-06, "max_pair_overlap_depth": -2.5468766324809877e-06, "metric_expression": "4.44869", "metric_recovery_tolerance_abs": 1e-05, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.448689692280498, "n": 15, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.44727, "setup": "hexinhex", "side": 4.448689692280498, "solution_path": "data/solutions/generated/hexinhex_015_536c412ab623afbc.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Timo Berthold et al", "svg_path": "data/svg/536c412ab623afbc.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexinhex@16", "density": 0.7803167098453238, "friedman_analytical_or_proved": true, "friedman_credit": "Timo Berthold et al", "friedman_date": "January 2026", "friedman_display_metric_value": 4.52746, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/16.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "4.52746+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.52746, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@16", "credit": "Timo Berthold et al", "date": "January 2026", "family_updated": "", "metric_expression": "4.52746+", "metric_symbol": "s", "metric_value": 4.52746, "reference_text": "16. | | s = 4.52746+ | Found by Timo Berthold et al | in January 2026. | Slightly improved by Alex Kravatsky et al | in February 2026.", "source_image": "16.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/16.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "16. | | s = 4.52746+ | Found by Timo Berthold et al | in January 2026. | Slightly improved by Alex Kravatsky et al | in February 2026.", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "17cb432379a18c18", "improvement": -0.0007289208955771898, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.541715273786282e-06, "max_pair_overlap_depth": -2.3206185233659582e-05, "metric_expression": "4.52819", "metric_recovery_tolerance_abs": 1e-05, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.528188920895577, "n": 16, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.52746, "setup": "hexinhex", "side": 4.528188920895577, "solution_path": "data/solutions/generated/hexinhex_016_17cb432379a18c18.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Timo Berthold et al", "svg_path": "data/svg/17cb432379a18c18.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexinhex@17", "density": 0.7985097849440846, "friedman_analytical_or_proved": false, "friedman_credit": "Timo Berthold et al", "friedman_date": "February 2026", "friedman_display_metric_value": 4.6136, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/17.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "4.6136+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.6136, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "hexinhex@17", "credit": "Timo Berthold et al", "date": "February 2026", "family_updated": "", "metric_expression": "4.6136+", "metric_symbol": "s", "metric_value": 4.6136, "reference_text": "17. | | s = 4.6136+ | Found by Timo Berthold et al | and Alex Kravatsky et al | in February 2026.", "source_image": "17.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/17.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "17. | | s = 4.6136+ | Found by Timo Berthold et al | and Alex Kravatsky et al | in February 2026.", "friedman_source_image": "17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "df809b176b000a6d", "improvement": -0.0004717062665458016, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.290243996089003e-06, "max_pair_overlap_depth": -4.642493336670661e-05, "metric_expression": "4.61407", "metric_recovery_tolerance_abs": 0.0001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.614071706266546, "n": 17, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.6136, "setup": "hexinhex", "side": 4.614071706266546, "solution_path": "data/solutions/generated/hexinhex_017_df809b176b000a6d.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Timo Berthold et al", "svg_path": "data/svg/df809b176b000a6d.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@18", "density": 0.8437125658646701, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2015", "friedman_display_metric_value": 4.618, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/19.gif", "friedman_metric_expression": "8/\u221a3 = 4.618+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.618802153517006, "friedman_metric_value_source": "exact_expression", "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"}, "friedman_reference_text": "18-19. | | s = 8/\u221a3 = 4.618+ | Found by Maurizio Morandi | in April 2015. |", "friedman_source_image": "19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "2eb7c5be1ce30256", "improvement": -0.00010246318123208198, "max_boundary_excess": -8.772589960788204e-06, "max_pair_overlap_depth": -3.612300978728111e-05, "metric_expression": "4.61890", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.618904616698238, "monotone_upper_bound_from_case": "hexinhex@19", "monotone_upper_bound_from_record": "4b11a1a09f2a5b70", "n": 18, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.618802153517006, "setup": "hexinhex", "side": 4.618904616698238, "solution_path": "data/solutions/generated/hexinhex_018_2eb7c5be1ce30256.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/2eb7c5be1ce30256.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexinhex@18", "density": 0.8437116836701195, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2015", "friedman_display_metric_value": 4.618, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/19.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "8/\u221a3 = 4.618+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.618802153517006, "friedman_metric_value_source": "exact_expression", "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": 4.618, "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"}, "friedman_reference_text": "18-19. | | s = 8/\u221a3 = 4.618+ | Found by Maurizio Morandi | in April 2015. |", "friedman_source_image": "19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "e320d39029a817c1", "improvement": -0.00010487797025149348, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 1.0443685916789865e-09, "max_pair_overlap_depth": -3.080790913889686e-05, "metric_expression": "4.61891", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.618907031487257, "n": 18, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.618802153517006, "setup": "hexinhex", "side": 4.618907031487257, "solution_path": "data/solutions/generated/hexinhex_018_e320d39029a817c1.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/e320d39029a817c1.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexinhex@19", "density": 0.8905854861904853, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2015", "friedman_display_metric_value": 4.618, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/19.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "8/\u221a3 = 4.618+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.618802153517006, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@19", "credit": "Maurizio Morandi", "date": "April 2015", "family_updated": "", "metric_expression": "8/\u221a3 = 4.618+", "metric_symbol": "s", "metric_value": 4.618, "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"}, "friedman_reference_text": "18-19. | | s = 8/\u221a3 = 4.618+ | Found by Maurizio Morandi | in April 2015. |", "friedman_source_image": "19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "4b11a1a09f2a5b70", "improvement": -0.00010246318123208198, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.772589960788204e-06, "max_pair_overlap_depth": -3.612300978728111e-05, "metric_expression": "4.61890", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.618904616698238, "n": 19, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.618802153517006, "setup": "hexinhex", "side": 4.618904616698238, "solution_path": "data/solutions/generated/hexinhex_019_4b11a1a09f2a5b70.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/4b11a1a09f2a5b70.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@20", "density": 0.7999783978223634, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 5.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/21.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "20-21. | | s = 5 | Trivial. | |", "friedman_source_image": "21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "197dfdbe3f1f6905", "improvement": -6.750817229672634e-05, "max_boundary_excess": -1.5389132812752791e-06, "max_pair_overlap_depth": -3.519050301936666e-06, "metric_expression": "5.00007", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.000067508172297, "monotone_upper_bound_from_case": "hexinhex@21", "monotone_upper_bound_from_record": "4a431b677be571c9", "n": 20, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.0, "setup": "hexinhex", "side": 5.000067508172297, "solution_path": "data/solutions/generated/hexinhex_020_197dfdbe3f1f6905.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/197dfdbe3f1f6905.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexinhex@20", "density": 0.7999496824068819, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 5.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/21.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "20-21. | | s = 5 | Trivial. | |", "friedman_source_image": "21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "2830e2dec38749ef", "improvement": -0.00015724989644283482, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.445035556179186e-06, "max_pair_overlap_depth": -1.6551811964360397e-06, "metric_expression": "5.00016", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.000157249896443, "n": 20, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.0, "setup": "hexinhex", "side": 5.000157249896443, "solution_path": "data/solutions/generated/hexinhex_020_2830e2dec38749ef.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/2830e2dec38749ef.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexinhex@21", "density": 0.8399773177134815, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 5.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/21.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@21", "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"}, "friedman_reference_text": "20-21. | | s = 5 | Trivial. | |", "friedman_source_image": "21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "4a431b677be571c9", "improvement": -6.750817229672634e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.5389132812752791e-06, "max_pair_overlap_depth": -3.519050301936666e-06, "metric_expression": "5.00007", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.000067508172297, "n": 21, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.0, "setup": "hexinhex", "side": 5.000067508172297, "solution_path": "data/solutions/generated/hexinhex_021_4a431b677be571c9.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/4a431b677be571c9.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexinhex@22", "density": 0.7873804951730353, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2015", "friedman_display_metric_value": 5.285, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/22.gif", "friedman_metric_expression": "(39+8\u221a3)/10 = 5.285+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.285640646055102, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@22", "credit": "Maurizio Morandi", "date": "April 2015", "family_updated": "", "metric_expression": "(39+8\u221a3)/10 = 5.285+", "metric_symbol": "s", "metric_value": 5.285, "reference_text": "22. | | s = (39+8\u221a3)/10 = 5.285+ | Found by Maurizio Morandi | in April 2015.", "source_image": "22.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/22.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "22. | | s = (39+8\u221a3)/10 = 5.285+ | Found by Maurizio Morandi | in April 2015.", "friedman_source_image": "22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "003a713177cb171c", "improvement": -0.0002602237497804083, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.575542655386671e-06, "max_pair_overlap_depth": -1.0001604064946434e-05, "metric_expression": "5.28590", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.285900869804882, "n": 22, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.285640646055102, "setup": "hexinhex", "side": 5.285900869804882, "solution_path": "data/solutions/generated/hexinhex_022_003a713177cb171c.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/003a713177cb171c.svg", "tolerance": 1e-08, "verified": true}
@@ -215,6 +228,7 @@
{"case": "hexinhex@26", "density": 0.7965297187067493, "friedman_analytical_or_proved": false, "friedman_credit": "Alex Kravatsky et al", "friedman_date": "February 2026", "friedman_display_metric_value": 5.70972, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/26.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "5.70972+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.70972, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "hexinhex@26", "credit": "Alex Kravatsky et al", "date": "February 2026", "family_updated": "", "metric_expression": "5.70972+", "metric_symbol": "s", "metric_value": 5.70972, "reference_text": "26. | | s = 5.70972+ | Found by Alex Kravatsky et al | in February 2026.", "source_image": "26.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/26.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "26. | | s = 5.70972+ | Found by Alex Kravatsky et al | in February 2026.", "friedman_source_image": "26.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "8ed8c9fb94f18119", "improvement": -0.0035622784284790043, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.9624619447323227e-06, "max_pair_overlap_depth": -1.2105377510118753e-05, "metric_expression": "5.71328", "metric_recovery_tolerance_abs": 1e-05, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.713282278428479, "n": 26, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.70972, "setup": "hexinhex", "side": 5.713282278428479, "solution_path": "data/solutions/generated/hexinhex_026_8ed8c9fb94f18119.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Alex Kravatsky et al", "svg_path": "data/svg/8ed8c9fb94f18119.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexinhex@27", "density": 0.82621496088318, "friedman_analytical_or_proved": false, "friedman_credit": "Alex Kravatsky et al", "friedman_date": "February 2026", "friedman_display_metric_value": 5.71415, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/27.gif", "friedman_metric_expression": "5.71415+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.71415, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "hexinhex@27", "credit": "Alex Kravatsky et al", "date": "February 2026", "family_updated": "", "metric_expression": "5.71415+", "metric_symbol": "s", "metric_value": 5.71415, "reference_text": "27. | | s = 5.71415+ | Found by Alex Kravatsky et al | in February 2026.", "source_image": "27.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/27.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "27. | | s = 5.71415+ | Found by Alex Kravatsky et al | in February 2026.", "friedman_source_image": "27.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "4e9628a9f49def85", "improvement": -0.0024177469948583052, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.56387472302006e-06, "max_pair_overlap_depth": -5.6693937127594296e-05, "metric_expression": "5.71657", "metric_recovery_tolerance_abs": 1e-05, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.716567746994858, "n": 27, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.71415, "setup": "hexinhex", "side": 5.716567746994858, "solution_path": "data/solutions/generated/hexinhex_027_4e9628a9f49def85.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Alex Kravatsky et al", "svg_path": "data/svg/4e9628a9f49def85.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexinhex@28", "density": 0.8011452289129106, "friedman_analytical_or_proved": false, "friedman_credit": "Alex Kravatsky et al", "friedman_date": "February 2026", "friedman_display_metric_value": 5.90893, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/28.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "5.90893+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.90893, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "hexinhex@28", "credit": "Alex Kravatsky et al", "date": "February 2026", "family_updated": "", "metric_expression": "5.90893+", "metric_symbol": "s", "metric_value": 5.90893, "reference_text": "28. | | s = 5.90893+ | Found by Alex Kravatsky et al | in February 2026.", "source_image": "28.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/28.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "28. | | s = 5.90893+ | Found by Alex Kravatsky et al | in February 2026.", "friedman_source_image": "28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "ba522f97744aed1a", "improvement": -0.002919783094952777, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.424277796084301e-06, "max_pair_overlap_depth": -3.209757705180394e-05, "metric_expression": "5.91185", "metric_recovery_tolerance_abs": 1e-05, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.911849783094953, "n": 28, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.90893, "setup": "hexinhex", "side": 5.911849783094953, "solution_path": "data/solutions/generated/hexinhex_028_ba522f97744aed1a.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T02:51:36+00:00", "submitted_by": "Alex Kravatsky et al", "svg_path": "data/svg/ba522f97744aed1a.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@29", "density": 0.8055344767966968, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 6.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/31.gif", "friedman_metric_expression": "6", "friedman_metric_symbol": "s", "friedman_metric_value": 6.0, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "29-31. | | s = 6 | Trivial. |", "friedman_source_image": "31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "baea309b7bc15617", "improvement": -7.850174601031767e-05, "max_boundary_excess": -3.7272811687230685e-07, "max_pair_overlap_depth": -4.232627726974414e-06, "metric_expression": "6.00008", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 6.00007850174601, "monotone_upper_bound_from_case": "hexinhex@30", "monotone_upper_bound_from_record": "100db5f9f28894cc", "n": 29, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 6.0, "setup": "hexinhex", "side": 6.00007850174601, "solution_path": "data/solutions/generated/hexinhex_029_baea309b7bc15617.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/baea309b7bc15617.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexinhex@29", "density": 0.8055194808067645, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 6.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/31.gif", "friedman_metric_expression": "6", "friedman_metric_symbol": "s", "friedman_metric_value": 6.0, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "29-31. | | s = 6 | Trivial. |", "friedman_source_image": "31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "dba25a591b8992bc", "improvement": -0.00013435185279675466, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.231638156342025e-06, "max_pair_overlap_depth": -6.307567709251671e-06, "metric_expression": "6.00013", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 6.000134351852797, "n": 29, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 6.0, "setup": "hexinhex", "side": 6.000134351852797, "solution_path": "data/solutions/generated/hexinhex_029_dba25a591b8992bc.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/dba25a591b8992bc.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexinhex@30", "density": 0.8333115277207207, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 6.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/31.gif", "friedman_metric_expression": "6", "friedman_metric_symbol": "s", "friedman_metric_value": 6.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@30", "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"}, "friedman_reference_text": "29-31. | | s = 6 | Trivial. |", "friedman_source_image": "31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "100db5f9f28894cc", "improvement": -7.850174601031767e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.7272811687230685e-07, "max_pair_overlap_depth": -4.232627726974414e-06, "metric_expression": "6.00008", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 6.00007850174601, "n": 30, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 6.0, "setup": "hexinhex", "side": 6.00007850174601, "solution_path": "data/solutions/generated/hexinhex_030_100db5f9f28894cc.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/100db5f9f28894cc.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexinhex@31", "density": 0.8610814974425702, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 6.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/31.gif", "friedman_metric_expression": "6", "friedman_metric_symbol": "s", "friedman_metric_value": 6.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@31", "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"}, "friedman_reference_text": "29-31. | | s = 6 | Trivial. |", "friedman_source_image": "31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "d503e2b08764a35e", "improvement": -0.0001031728611780025, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0312532034243658e-06, "max_pair_overlap_depth": -8.80446118189937e-06, "metric_expression": "6.00010", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 6.000103172861178, "n": 31, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 6.0, "setup": "hexinhex", "side": 6.000103172861178, "solution_path": "data/solutions/generated/hexinhex_031_d503e2b08764a35e.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/d503e2b08764a35e.svg", "tolerance": 1e-08, "verified": true}
@@ -224,6 +238,7 @@
{"case": "hexinsqu@4", "density": 0.748920266367969, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_display_metric_value": 3.725, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/4.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "(7\u2013\u221a3)/\u221a2 = 3.725+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.7250025969142437, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@4", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "(7\u2013\u221a3)/\u221a2 = 3.725+", "metric_symbol": "s", "metric_value": 3.725, "reference_text": "s = (7\u2013\u221a3)/\u221a2 = 3.725+ | Found by Maurizio Morandi | in March 2026.", "source_image": "4.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/4.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = (7\u2013\u221a3)/\u221a2 = 3.725+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "4.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "33ddf042dd02d191", "improvement": -9.921706716031764e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.577436704193743e-06, "max_pair_overlap_depth": -5.3695636117151446e-05, "metric_expression": "3.72510", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.725101813981404, "n": 4, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.7250025969142437, "setup": "hexinsqu", "side": 3.725101813981404, "solution_path": "data/solutions/generated/hexinsqu_004_33ddf042dd02d191.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/33ddf042dd02d191.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexinsqu@5", "density": 0.6595778437263831, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_display_metric_value": 4.437, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/5.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "7(3\u2013\u221a3)/2 = 4.437+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.43782217350893, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@5", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "7(3\u2013\u221a3)/2 = 4.437+", "metric_symbol": "s", "metric_value": 4.437, "reference_text": "s = 7(3\u2013\u221a3)/2 = 4.437+ | Found by Maurizio Morandi | in March 2026.", "source_image": "5.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/5.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 7(3\u2013\u221a3)/2 = 4.437+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "5.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "38f2dc66ba941978", "improvement": -8.195604384120259e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.5429089870357805e-06, "max_pair_overlap_depth": -1.5874388830661168e-05, "metric_expression": "4.43790", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.437904129552771, "n": 5, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.43782217350893, "setup": "hexinsqu", "side": 4.437904129552771, "solution_path": "data/solutions/generated/hexinsqu_005_38f2dc66ba941978.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/38f2dc66ba941978.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexinsqu@6", "density": 0.6726426228424899, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_display_metric_value": 4.813, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/6.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "20(9\u20132\u221a3)/23 = 4.813+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.813824682488909, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@6", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "20(9\u20132\u221a3)/23 = 4.813+", "metric_symbol": "s", "metric_value": 4.813, "reference_text": "s = 20(9\u20132\u221a3)/23 = 4.813+ | Found by Maurizio Morandi | in March 2026.", "source_image": "6.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/6.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 20(9\u20132\u221a3)/23 = 4.813+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "6.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "31291850c6887128", "improvement": -0.00021179420330419418, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6841191118288626e-06, "max_pair_overlap_depth": -6.696342223433405e-05, "metric_expression": "4.81404", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.8140364766922135, "n": 6, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.813824682488909, "setup": "hexinsqu", "side": 4.8140364766922135, "solution_path": "data/solutions/generated/hexinsqu_006_31291850c6887128.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/31291850c6887128.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinsqu@7", "density": 0.673566015856675, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 5.196, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/8.png", "friedman_metric_expression": "3\u221a3 = 5.196+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.196152422706632, "friedman_metric_value_source": "exact_expression", "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"}, "friedman_reference_text": "s = 3\u221a3 = 5.196+ | Trivial. |", "friedman_source_image": "8.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "6492fb111d24a3e4", "improvement": -3.5864848886291156e-05, "max_boundary_excess": -1.033193436423119e-06, "max_pair_overlap_depth": -1.0846799477937097e-05, "metric_expression": "5.19619", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.196188287555518, "monotone_upper_bound_from_case": "hexinsqu@8", "monotone_upper_bound_from_record": "b1252a56565d2ec0", "n": 7, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.196152422706632, "setup": "hexinsqu", "side": 5.196188287555518, "solution_path": "data/solutions/generated/hexinsqu_007_6492fb111d24a3e4.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/6492fb111d24a3e4.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexinsqu@7", "density": 0.6735519886722757, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 5.196, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/8.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "3\u221a3 = 5.196+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.196152422706632, "friedman_metric_value_source": "exact_expression", "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": 5.196, "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"}, "friedman_reference_text": "s = 3\u221a3 = 5.196+ | Trivial. |", "friedman_source_image": "8.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "a4c696c014daab3c", "improvement": -8.99716684248375e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.667089398456013e-06, "max_pair_overlap_depth": -2.5433886154724306e-05, "metric_expression": "5.19624", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.196242394375057, "n": 7, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.196152422706632, "setup": "hexinsqu", "side": 5.196242394375057, "solution_path": "data/solutions/generated/hexinsqu_007_a4c696c014daab3c.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/a4c696c014daab3c.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexinsqu@8", "density": 0.7697897324076285, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 5.196, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/8.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "3\u221a3 = 5.196+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.196152422706632, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@8", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3\u221a3 = 5.196+", "metric_symbol": "s", "metric_value": 5.196, "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"}, "friedman_reference_text": "s = 3\u221a3 = 5.196+ | Trivial. |", "friedman_source_image": "8.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "b1252a56565d2ec0", "improvement": -3.5864848886291156e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.033193436423119e-06, "max_pair_overlap_depth": -1.0846799477937097e-05, "metric_expression": "5.19619", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.196188287555518, "n": 8, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.196152422706632, "setup": "hexinsqu", "side": 5.196188287555518, "solution_path": "data/solutions/generated/hexinsqu_008_b1252a56565d2ec0.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/b1252a56565d2ec0.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexinsqu@9", "density": 0.7678652278206223, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_display_metric_value": 5.518, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/9.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "(13\u20133\u221a3)/\u221a2 = 5.518+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.518153541250351, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@9", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "(13\u20133\u221a3)/\u221a2 = 5.518+", "metric_symbol": "s", "metric_value": 5.518, "reference_text": "s = (13\u20133\u221a3)/\u221a2 = 5.518+ | Found by Maurizio Morandi | in March 2026.", "source_image": "9.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/9.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = (13\u20133\u221a3)/\u221a2 = 5.518+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "9.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "5b4ef4c36dcfba05", "improvement": -0.00013871088887640326, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.874395903155008e-06, "max_pair_overlap_depth": -4.408050364324012e-05, "metric_expression": "5.51829", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.5182922521392275, "n": 9, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.518153541250351, "setup": "hexinsqu", "side": 5.5182922521392275, "solution_path": "data/solutions/generated/hexinsqu_009_5b4ef4c36dcfba05.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/5b4ef4c36dcfba05.svg", "tolerance": 1e-08, "verified": true}
@@ -255,8 +270,10 @@
{"case": "hexintri@10", "density": 0.7999538247696885, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 8.6602, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/10.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "5\u221a3 = 8.6602+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.660254037844387, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@10", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5\u221a3 = 8.6602+", "metric_symbol": "s", "metric_value": 8.6602, "reference_text": "s = 5\u221a3 = 8.6602+ | Trivial. |", "source_image": "10.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/10.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 5\u221a3 = 8.6602+ | Trivial. |", "friedman_source_image": "10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "5df6cec7a90f52fe", "improvement": -0.0002499415853147724, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.2645686941820143e-06, "max_pair_overlap_depth": -4.384611814706174e-05, "metric_expression": "8.66050", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 8.660503979429702, "n": 10, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 8.660254037844387, "setup": "hexintri", "side": 8.660503979429702, "solution_path": "data/solutions/generated/hexintri_010_5df6cec7a90f52fe.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/5df6cec7a90f52fe.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexintri@11", "density": 0.7237571814877352, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_display_metric_value": 9.54823, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/11.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "9.54823+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.54823, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "hexintri@11", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "9.54823+", "metric_symbol": "s", "metric_value": 9.54823, "reference_text": "s = 9.54823+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "11.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/11.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 9.54823+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "d582cb032b9c575f", "improvement": -0.0011576763865779327, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.6199514834868076e-06, "max_pair_overlap_depth": -2.6419225191420814e-05, "metric_expression": "9.54939", "metric_recovery_tolerance_abs": 1e-05, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 9.549387676386578, "n": 11, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 9.54823, "setup": "hexintri", "side": 9.549387676386578, "solution_path": "data/solutions/generated/hexintri_011_d582cb032b9c575f.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/d582cb032b9c575f.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexintri@12", "density": 0.7199169251480841, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 10.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/12.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "10", "friedman_metric_symbol": "s", "friedman_metric_value": 10.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@12", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "10", "metric_symbol": "s", "metric_value": 10.0, "reference_text": "s = 10 | Trivial. |", "source_image": "12.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/12.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 10 | Trivial. |", "friedman_source_image": "12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "b6e20361c3af5615", "improvement": -0.0005769586222079681, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.006918638920166e-07, "max_pair_overlap_depth": -2.082412255655086e-05, "metric_expression": "10.00058", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 10.000576958622208, "n": 12, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 10.0, "setup": "hexintri", "side": 10.000576958622208, "solution_path": "data/solutions/generated/hexintri_012_b6e20361c3af5615.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/b6e20361c3af5615.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexintri@13", "density": 0.7221969904539467, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 10.3923, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/15.gif", "friedman_metric_expression": "6\u221a3 = 10.3923+", "friedman_metric_symbol": "s", "friedman_metric_value": 10.392304845413264, "friedman_metric_value_source": "exact_expression", "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"}, "friedman_reference_text": "s = 6\u221a3 = 10.3923+ | Trivial. |", "friedman_source_image": "15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "3b4102a5f66851b2", "improvement": -0.0001815390682331497, "max_boundary_excess": -6.672744116747984e-07, "max_pair_overlap_depth": -2.8784199847526563e-05, "metric_expression": "10.39249", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 10.392486384481497, "monotone_upper_bound_from_case": "hexintri@15", "monotone_upper_bound_from_record": "83460d360e93ff66", "n": 13, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 10.392304845413264, "setup": "hexintri", "side": 10.392486384481497, "solution_path": "data/solutions/generated/hexintri_013_3b4102a5f66851b2.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/3b4102a5f66851b2.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexintri@13", "density": 0.7221828032992709, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 10.3923, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/15.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "6\u221a3 = 10.3923+", "friedman_metric_symbol": "s", "friedman_metric_value": 10.392304845413264, "friedman_metric_value_source": "exact_expression", "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": 10.3923, "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"}, "friedman_reference_text": "s = 6\u221a3 = 10.3923+ | Trivial. |", "friedman_source_image": "15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "da55bd8ffef5f50e", "improvement": -0.00028361785426156416, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.7733151165742505e-06, "max_pair_overlap_depth": -4.4289278687159594e-05, "metric_expression": "10.39259", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 10.392588463267526, "n": 13, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 10.392304845413264, "setup": "hexintri", "side": 10.392588463267526, "solution_path": "data/solutions/generated/hexintri_013_da55bd8ffef5f50e.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/da55bd8ffef5f50e.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexintri@14", "density": 0.7777237680646841, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 10.3923, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/15.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "6\u221a3 = 10.3923+", "friedman_metric_symbol": "s", "friedman_metric_value": 10.392304845413264, "friedman_metric_value_source": "exact_expression", "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": 10.3923, "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"}, "friedman_reference_text": "s = 6\u221a3 = 10.3923+ | Trivial. |", "friedman_source_image": "15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "272118e9284c579f", "improvement": -0.00036084512376532984, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.442529134834328e-06, "max_pair_overlap_depth": -4.275788154761706e-05, "metric_expression": "10.39267", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 10.39266569053703, "n": 14, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 10.392304845413264, "setup": "hexintri", "side": 10.39266569053703, "solution_path": "data/solutions/generated/hexintri_014_272118e9284c579f.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/272118e9284c579f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexintri@14", "density": 0.7777506051042504, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 10.3923, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/15.gif", "friedman_metric_expression": "6\u221a3 = 10.3923+", "friedman_metric_symbol": "s", "friedman_metric_value": 10.392304845413264, "friedman_metric_value_source": "exact_expression", "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"}, "friedman_reference_text": "s = 6\u221a3 = 10.3923+ | Trivial. |", "friedman_source_image": "15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "aa2a4511d2c644b1", "improvement": -0.0001815390682331497, "max_boundary_excess": -6.672744116747984e-07, "max_pair_overlap_depth": -2.8784199847526563e-05, "metric_expression": "10.39249", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 10.392486384481497, "monotone_upper_bound_from_case": "hexintri@15", "monotone_upper_bound_from_record": "83460d360e93ff66", "n": 14, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 10.392304845413264, "setup": "hexintri", "side": 10.392486384481497, "solution_path": "data/solutions/generated/hexintri_014_aa2a4511d2c644b1.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/aa2a4511d2c644b1.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexintri@15", "density": 0.833304219754554, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 10.3923, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/15.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "6\u221a3 = 10.3923+", "friedman_metric_symbol": "s", "friedman_metric_value": 10.392304845413264, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@15", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "6\u221a3 = 10.3923+", "metric_symbol": "s", "metric_value": 10.3923, "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"}, "friedman_reference_text": "s = 6\u221a3 = 10.3923+ | Trivial. |", "friedman_source_image": "15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "83460d360e93ff66", "improvement": -0.0001815390682331497, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.672744116747984e-07, "max_pair_overlap_depth": -2.8784199847526563e-05, "metric_expression": "10.39249", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 10.392486384481497, "n": 15, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 10.392304845413264, "setup": "hexintri", "side": 10.392486384481497, "solution_path": "data/solutions/generated/hexintri_015_83460d360e93ff66.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/83460d360e93ff66.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexintri@16", "density": 0.7429369875675369, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_display_metric_value": 11.36599, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/16.gif", "friedman_metric_expression": "11.36599+", "friedman_metric_symbol": "s", "friedman_metric_value": 11.36599, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "hexintri@16", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "11.36599+", "metric_symbol": "s", "metric_value": 11.36599, "reference_text": "s = 11.36599+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "16.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/16.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 11.36599+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "89d5bef670451c83", "improvement": -0.0013703170956897992, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.657075146936789e-07, "max_pair_overlap_depth": -2.5616124350591463e-06, "metric_expression": "11.36736", "metric_recovery_tolerance_abs": 1e-05, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 11.36736031709569, "n": 16, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 11.36599, "setup": "hexintri", "side": 11.36736031709569, "solution_path": "data/solutions/generated/hexintri_16_89d5bef670451c83.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T07:13:53+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/89d5bef670451c83.svg", "tolerance": 1e-08, "verified": true}
{"case": "hexintri@17", "density": 0.7525303158572544, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_display_metric_value": 11.63457, "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/17.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "11.63457+", "friedman_metric_symbol": "s", "friedman_metric_value": 11.63457, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "hexintri@17", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "11.63457+", "metric_symbol": "s", "metric_value": 11.63457, "reference_text": "s = 11.63457+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "17.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/17.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 11.63457+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "8137d039fd339f48", "improvement": -0.007711226979866481, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.447216764354318e-05, "max_pair_overlap_depth": -1.3408176938511929e-05, "metric_expression": "11.64228", "metric_recovery_tolerance_abs": 1e-05, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 11.642281226979867, "n": 17, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 11.63457, "setup": "hexintri", "side": 11.642281226979867, "solution_path": "data/solutions/generated/hexintri_017_8137d039fd339f48.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T03:07:58+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/8137d039fd339f48.svg", "tolerance": 1e-08, "verified": true}
@@ -321,6 +338,7 @@
{"case": "peninpen@3", "density": 0.599978840427638, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_display_metric_value": 2.23606, "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/3.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "\u221a5 = 2.23606+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.23606797749979, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "peninpen@3", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "\u221a5 = 2.23606+", "metric_symbol": "s", "metric_value": 2.23606, "reference_text": "s = \u221a5 = 2.23606+ | Found by Jonathan Viquerat in April 2026.", "source_image": "3.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/3.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = \u221a5 = 2.23606+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "3.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "bf261a4dce178a90", "improvement": -3.942957804081715e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.7662857614262655e-06, "max_pair_overlap_depth": -2.7755970794667603e-05, "metric_expression": "2.23611", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.2361074070778306, "n": 3, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.23606797749979, "setup": "peninpen", "side": 2.2361074070778306, "solution_path": "data/solutions/generated/peninpen_003_bf261a4dce178a90.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/bf261a4dce178a90.svg", "tolerance": 1e-08, "verified": true}
{"case": "peninpen@4", "density": 0.6678717661629443, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_display_metric_value": 2.44719, "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/4.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.44719+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.44719, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@4", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2.44719+", "metric_symbol": "s", "metric_value": 2.44719, "reference_text": "s = 2.44719+ | Found by Jonathan Viquerat in April 2026.", "source_image": "4.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/4.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 2.44719+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "4.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "85435733af14070a", "improvement": -8.88306059834143e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0634336001709255e-06, "max_pair_overlap_depth": -3.575601486860425e-05, "metric_expression": "2.44728", "metric_recovery_tolerance_abs": 1e-05, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.4472788306059834, "n": 4, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.44719, "setup": "peninpen", "side": 2.4472788306059834, "solution_path": "data/solutions/generated/peninpen_004_85435733af14070a.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/85435733af14070a.svg", "tolerance": 1e-08, "verified": true}
{"case": "peninpen@5", "density": 0.7294617516267693, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 2.61803, "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/6.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "(3+\u221a5)/2 = 2.61803+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.618033988749895, "friedman_metric_value_source": "exact_expression", "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": 2.61803, "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"}, "friedman_reference_text": "s = (3+\u221a5)/2 = 2.61803+ | Trivial.", "friedman_source_image": "6.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "28fa9f787595145b", "improvement": -5.099395129359863e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.513169742800457e-06, "max_pair_overlap_depth": -2.4501936757292953e-06, "metric_expression": "2.61808", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.6180849827011885, "n": 5, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.618033988749895, "setup": "peninpen", "side": 2.6180849827011885, "solution_path": "data/solutions/generated/peninpen_005_28fa9f787595145b.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/28fa9f787595145b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninpen@5", "density": 0.7294689787373387, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 2.61803, "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/6.png", "friedman_metric_expression": "(3+\u221a5)/2 = 2.61803+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.618033988749895, "friedman_metric_value_source": "exact_expression", "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"}, "friedman_reference_text": "s = (3+\u221a5)/2 = 2.61803+ | Trivial.", "friedman_source_image": "6.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "5517dc706e76853c", "improvement": -3.8024766128241794e-05, "max_boundary_excess": -2.4414928963119693e-06, "max_pair_overlap_depth": -1.898562579671914e-05, "metric_expression": "2.61807", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.618072013516023, "monotone_upper_bound_from_case": "peninpen@6", "monotone_upper_bound_from_record": "198bd12691483df1", "n": 5, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.618033988749895, "setup": "peninpen", "side": 2.618072013516023, "solution_path": "data/solutions/generated/peninpen_005_5517dc706e76853c.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/5517dc706e76853c.svg", "tolerance": 1e-08, "verified": true}
{"case": "peninpen@6", "density": 0.8753627744848066, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 2.61803, "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/6.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "(3+\u221a5)/2 = 2.61803+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.618033988749895, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "peninpen@6", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "(3+\u221a5)/2 = 2.61803+", "metric_symbol": "s", "metric_value": 2.61803, "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"}, "friedman_reference_text": "s = (3+\u221a5)/2 = 2.61803+ | Trivial.", "friedman_source_image": "6.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "198bd12691483df1", "improvement": -3.8024766128241794e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4414928963119693e-06, "max_pair_overlap_depth": -1.898562579671914e-05, "metric_expression": "2.61807", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.618072013516023, "n": 6, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.618033988749895, "setup": "peninpen", "side": 2.618072013516023, "solution_path": "data/solutions/generated/peninpen_006_198bd12691483df1.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/198bd12691483df1.svg", "tolerance": 1e-08, "verified": true}
{"case": "peninpen@7", "density": 0.7384663216832239, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_display_metric_value": 3.07866, "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/7.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.07866+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.07866, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@7", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.07866+", "metric_symbol": "s", "metric_value": 3.07866, "reference_text": "s = 3.07866+ | Found by Jonathan Viquerat in April 2026.", "source_image": "7.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/7.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 3.07866+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "7.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "5b6283ab3dc4c3b5", "improvement": -0.00015556249375014986, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.964189962454e-06, "max_pair_overlap_depth": -1.5079265303863387e-05, "metric_expression": "3.07882", "metric_recovery_tolerance_abs": 1e-05, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.0788155624937503, "n": 7, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.07866, "setup": "peninpen", "side": 3.0788155624937503, "solution_path": "data/solutions/generated/peninpen_007_5b6283ab3dc4c3b5.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/5b6283ab3dc4c3b5.svg", "tolerance": 1e-08, "verified": true}
{"case": "peninpen@8", "density": 0.7191427449408644, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_display_metric_value": 3.33514, "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/8.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.33514+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.33514, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@8", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.33514+", "metric_symbol": "s", "metric_value": 3.33514, "reference_text": "s = 3.33514+ | Found by Jonathan Viquerat in April 2026.", "source_image": "8.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/8.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 3.33514+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "8.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "de5f4ff90a59b8bf", "improvement": -0.00017949381007964504, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.630985964589326e-06, "max_pair_overlap_depth": -3.1607294870367175e-05, "metric_expression": "3.33532", "metric_recovery_tolerance_abs": 1e-05, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.3353194938100796, "n": 8, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.33514, "setup": "peninpen", "side": 3.3353194938100796, "solution_path": "data/solutions/generated/peninpen_008_de5f4ff90a59b8bf.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/de5f4ff90a59b8bf.svg", "tolerance": 1e-08, "verified": true}
@@ -403,11 +421,14 @@
{"case": "squincir@34", "density": 0.8206376457817698, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "December 2010", "friedman_display_metric_value": 3.629, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/34.gif", "friedman_metric_expression": "3.629+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.629, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "squincir@34", "credit": "Maurizio Morandi", "date": "December 2010", "family_updated": "", "metric_expression": "3.629+", "metric_symbol": "r", "metric_value": 3.629, "reference_text": "r = 3.629+ | Found by Maurizio Morandi | in December 2010.", "source_image": "34.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/34.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 3.629+ | Found by Maurizio Morandi | in December 2010.", "friedman_source_image": "34.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "aa2ddbda942a0ffd", "improvement": -0.0025230432891554244, "max_boundary_excess": -1.1187926634015355e-06, "max_pair_overlap_depth": -1.3727466190216475e-05, "metric_expression": "3.63152", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "r", "metric_value": 3.6315230432891554, "n": 34, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "original_submitted_by": "polygon-packer-c image seed", "record_type": "verified", "reference_side": 3.629, "setup": "squincir", "side": 7.263046086578311, "solution_path": "data/solutions/generated/squincir_34_aa2ddbda942a0ffd.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T17:13:54+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/aa2ddbda942a0ffd.svg", "tolerance": 1e-08, "verified": true}
{"case": "squincir@35", "density": 0.8405414638481816, "friedman_analytical_or_proved": true, "friedman_credit": "David Eppstein", "friedman_date": "March 2002", "friedman_display_metric_value": 3.64, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/35.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a53 / 2 = 3.640+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.640054944640259, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "squincir@35", "credit": "David Eppstein", "date": "March 2002", "family_updated": "", "metric_expression": "\u221a53 / 2 = 3.640+", "metric_symbol": "r", "metric_value": 3.64, "reference_text": "r = \u221a53 / 2 = 3.640+ | Found by David Eppstein | in March 2002.", "source_image": "35.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/35.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = \u221a53 / 2 = 3.640+ | Found by David Eppstein | in March 2002.", "friedman_source_image": "35.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "f2d91d98b4c2ee39", "improvement": -0.0005999636150639809, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.121617296348745e-07, "max_pair_overlap_depth": -2.1344301219083928e-05, "metric_expression": "3.64065", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "r", "metric_value": 3.640654908255323, "n": 35, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.640054944640259, "setup": "squincir", "side": 7.281309816510646, "solution_path": "data/solutions/generated/squincir_035_f2d91d98b4c2ee39.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "David Eppstein", "svg_path": "data/svg/f2d91d98b4c2ee39.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@1", "density": 0.499990679660235, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 1.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/2.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "1", "friedman_metric_symbol": "s", "friedman_metric_value": 1.0, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "1-2. | | s = 1 | Trivial. |", "friedman_source_image": "2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "557f8de873a17a8e", "improvement": -9.320470070184328e-06, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.7525252651882113e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "1.00001", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 1.0000093204700702, "n": 1, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 1.0, "setup": "squindom", "side": 2.0000186409401404, "solution_path": "data/solutions/generated/squindom_001_557f8de873a17a8e.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/557f8de873a17a8e.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@1", "density": 0.4999941499004167, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 1.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/2.gif", "friedman_metric_expression": "1", "friedman_metric_symbol": "s", "friedman_metric_value": 1.0, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "1-2. | | s = 1 | Trivial. |", "friedman_source_image": "2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "b71122905cd4c5c7", "improvement": -5.850150919473052e-06, "max_boundary_excess": -3.463373909884382e-07, "max_pair_overlap_depth": 0.0, "metric_expression": "1.00001", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 1.0000058501509195, "monotone_upper_bound_from_case": "squindom@2", "monotone_upper_bound_from_record": "7684110dce2e5290", "n": 1, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 1.0, "setup": "squindom", "side": 2.000011700301839, "solution_path": "data/solutions/generated/squindom_001_b71122905cd4c5c7.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/b71122905cd4c5c7.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@2", "density": 0.9999882998008334, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 1.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/2.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "1", "friedman_metric_symbol": "s", "friedman_metric_value": 1.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@2", "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"}, "friedman_reference_text": "1-2. | | s = 1 | Trivial. |", "friedman_source_image": "2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "7684110dce2e5290", "improvement": -5.850150919473052e-06, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.463373909884382e-07, "max_pair_overlap_depth": -9.41881114351023e-06, "metric_expression": "1.00001", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 1.0000058501509195, "n": 2, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 1.0, "setup": "squindom", "side": 2.000011700301839, "solution_path": "data/solutions/generated/squindom_002_7684110dce2e5290.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/7684110dce2e5290.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@3", "density": 0.6666417193387711, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 1.5, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/3.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3/2 = 1.5", "friedman_metric_symbol": "s", "friedman_metric_value": 1.5, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@3", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3/2 = 1.5", "metric_symbol": "s", "metric_value": 1.5, "reference_text": "3. | | s = 3/2 = 1.5 | Trivial. |", "source_image": "3.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/3.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "3. | | s = 3/2 = 1.5 | Trivial. |", "friedman_source_image": "3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "3047ca63de76a853", "improvement": -2.8066531593262667e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.0617527398680835e-06, "max_pair_overlap_depth": -1.9554412996725112e-05, "metric_expression": "1.50003", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 1.5000280665315933, "n": 3, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 1.5, "setup": "squindom", "side": 3.0000561330631865, "solution_path": "data/solutions/generated/squindom_003_3047ca63de76a853.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/3047ca63de76a853.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@4", "density": 0.5367260778952853, "friedman_analytical_or_proved": false, "friedman_credit": "Evert Stenlund", "friedman_date": "June 2010", "friedman_display_metric_value": 1.93, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/4.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "1.930+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.93, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "squindom@4", "credit": "Evert Stenlund", "date": "June 2010", "family_updated": "", "metric_expression": "1.930+", "metric_symbol": "s", "metric_value": 1.93, "reference_text": "4. | | s = 1.930+ | Found by Evert Stenlund | in June 2010.", "source_image": "4.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/4.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "4. | | s = 1.930+ | Found by Evert Stenlund | in June 2010.", "friedman_source_image": "4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "8d067d60c7138964", "improvement": -0.00036150970269521743, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.561542731343124e-06, "max_pair_overlap_depth": -3.757775137364083e-05, "metric_expression": "1.93036", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 1.9303615097026952, "n": 4, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 1.93, "setup": "squindom", "side": 3.8607230194053903, "solution_path": "data/solutions/generated/squindom_004_8d067d60c7138964.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Evert Stenlund", "svg_path": "data/svg/8d067d60c7138964.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@5", "density": 0.6249853765160349, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 2.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/8.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "5-8. | | s = 2 | Trivial. |", "friedman_source_image": "8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "05f6000ecc64a077", "improvement": -2.3397984937112426e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.3736510935966635e-06, "max_pair_overlap_depth": -1.2206465427119814e-05, "metric_expression": "2.00002", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.000023397984937, "n": 5, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.0, "setup": "squindom", "side": 4.000046795969874, "solution_path": "data/solutions/generated/squindom_005_05f6000ecc64a077.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/05f6000ecc64a077.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@5", "density": 0.6249866459286932, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 2.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/8.gif", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "5-8. | | s = 2 | Trivial. |", "friedman_source_image": "8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "30146b12b3cd6b1f", "improvement": -2.136685649301029e-05, "max_boundary_excess": -1.8852035799099554e-07, "max_pair_overlap_depth": -1.170836956105551e-05, "metric_expression": "2.00002", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.000021366856493, "monotone_upper_bound_from_case": "squindom@6", "monotone_upper_bound_from_record": "095d951ace7a918a", "n": 5, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.0, "setup": "squindom", "side": 4.000042733712986, "solution_path": "data/solutions/generated/squindom_005_30146b12b3cd6b1f.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/30146b12b3cd6b1f.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@6", "density": 0.749983975114432, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 2.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/8.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@6", "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"}, "friedman_reference_text": "5-8. | | s = 2 | Trivial. |", "friedman_source_image": "8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "095d951ace7a918a", "improvement": -2.136685649301029e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.707507375492412e-07, "max_pair_overlap_depth": -1.170836956105551e-05, "metric_expression": "2.00002", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.000021366856493, "n": 6, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.0, "setup": "squindom", "side": 4.000042733712986, "solution_path": "data/solutions/generated/squindom_006_095d951ace7a918a.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/095d951ace7a918a.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@7", "density": 0.8749756994459561, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 2.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/8.gif", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "5-8. | | s = 2 | Trivial. |", "friedman_source_image": "8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "24268bca52468f25", "improvement": -2.7772640243561852e-05, "max_boundary_excess": -2.0495464614356251e-07, "max_pair_overlap_depth": -1.777420024382249e-07, "metric_expression": "2.00003", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.0000277726402436, "monotone_upper_bound_from_case": "squindom@8", "monotone_upper_bound_from_record": "0b83841ee591689c", "n": 7, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.0, "setup": "squindom", "side": 4.000055545280487, "solution_path": "data/solutions/generated/squindom_007_24268bca52468f25.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/24268bca52468f25.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@7", "density": 0.8749639011598754, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 2.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/8.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "5-8. | | s = 2 | Trivial. |", "friedman_source_image": "8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "50aa0817a0117c40", "improvement": -4.1257093861268856e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4800520663781356e-06, "max_pair_overlap_depth": -1.885899157594295e-05, "metric_expression": "2.00004", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.0000412570938613, "n": 7, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.0, "setup": "squindom", "side": 4.0000825141877225, "solution_path": "data/solutions/generated/squindom_007_50aa0817a0117c40.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/50aa0817a0117c40.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@8", "density": 0.9999722279382355, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 2.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/8.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@8", "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"}, "friedman_reference_text": "5-8. | | s = 2 | Trivial. |", "friedman_source_image": "8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "0b83841ee591689c", "improvement": -2.7772640243561852e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.0495464614356251e-07, "max_pair_overlap_depth": -1.777420024382249e-07, "metric_expression": "2.00003", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.0000277726402436, "n": 8, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.0, "setup": "squindom", "side": 4.000055545280487, "solution_path": "data/solutions/generated/squindom_008_0b83841ee591689c.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/0b83841ee591689c.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@9", "density": 0.7367411181657051, "friedman_analytical_or_proved": true, "friedman_credit": "Evert Stenlund", "friedman_date": "June 2010", "friedman_display_metric_value": 2.471, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/9.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "2 + \u221a2/3 = 2.471+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.471404520791032, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@9", "credit": "Evert Stenlund", "date": "June 2010", "family_updated": "", "metric_expression": "2 + \u221a2/3 = 2.471+", "metric_symbol": "s", "metric_value": 2.471, "reference_text": "9. | | s = 2 + \u221a2/3 = 2.471+ | Found by Evert Stenlund | in June 2010.", "source_image": "9.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/9.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "9. | | s = 2 + \u221a2/3 = 2.471+ | Found by Evert Stenlund | in June 2010.", "friedman_source_image": "9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "2963ae62200cf2c5", "improvement": -2.8257928626818085e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.8492494266174617e-06, "max_pair_overlap_depth": -7.2416104319739105e-06, "metric_expression": "2.47143", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.4714327787196586, "n": 9, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.471404520791032, "setup": "squindom", "side": 4.942865557439317, "solution_path": "data/solutions/generated/squindom_009_2963ae62200cf2c5.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Evert Stenlund", "svg_path": "data/svg/2963ae62200cf2c5.svg", "tolerance": 1e-08, "verified": true}
@@ -416,26 +437,34 @@
{"case": "squindom@12", "density": 0.7147860825866693, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "June 2010", "friedman_display_metric_value": 2.897, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/12.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "6 (1 + \u221a2) / 5 = 2.897+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.897056274847714, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@12", "credit": "Maurizio Morandi", "date": "June 2010", "family_updated": "", "metric_expression": "6 (1 + \u221a2) / 5 = 2.897+", "metric_symbol": "s", "metric_value": 2.897, "reference_text": "12. | | s = 6 (1 + \u221a2) / 5 = 2.897+ | Found by Maurizio Morandi | in June 2010. |", "source_image": "12.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/12.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "12. | | s = 6 (1 + \u221a2) / 5 = 2.897+ | Found by Maurizio Morandi | in June 2010. |", "friedman_source_image": "12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "4c82fd3f736a139f", "improvement": -0.00020446384343131285, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.382389093311346e-06, "max_pair_overlap_depth": -2.837543626332817e-05, "metric_expression": "2.89726", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.8972607386911453, "n": 12, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.897056274847714, "setup": "squindom", "side": 5.794521477382291, "solution_path": "data/solutions/generated/squindom_012_4c82fd3f736a139f.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/4c82fd3f736a139f.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@13", "density": 0.7330751854461901, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_display_metric_value": 2.976, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/13.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "2.976+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.976, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "squindom@13", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "2.976+", "metric_symbol": "s", "metric_value": 2.976, "reference_text": "13. | | s = 2.976+ | Found by David W. Cantrell | in July 2012 (after Maurizio | Morandi in June 2010.)", "source_image": "13.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/13.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "13. | | s = 2.976+ | Found by David W. Cantrell | in July 2012 (after Maurizio | Morandi in June 2010.)", "friedman_source_image": "13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "d738dfc349a8c940", "improvement": -0.0017101333567786092, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.7694174018241284e-06, "max_pair_overlap_depth": -2.048132261660285e-05, "metric_expression": "2.97771", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.9777101333567786, "n": 13, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.976, "setup": "squindom", "side": 5.955420266713557, "solution_path": "data/solutions/generated/squindom_013_d738dfc349a8c940.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/d738dfc349a8c940.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@14", "density": 0.7777474504876777, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 3.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "14-18. | | s = 3 | Trivial. |", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "167b47d5e39e076d", "improvement": -5.849005569213972e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0369016045785884e-06, "max_pair_overlap_depth": -1.5707310774493166e-05, "metric_expression": "3.00006", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.000058490055692, "n": 14, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.0, "setup": "squindom", "side": 6.000116980111384, "solution_path": "data/solutions/generated/squindom_014_167b47d5e39e076d.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/167b47d5e39e076d.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@14", "density": 0.7777539063792901, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 3.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "14-18. | | s = 3 | Trivial. |", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "821200c01d9d607e", "improvement": -4.603875684550118e-05, "max_boundary_excess": -1.0382270816400307e-06, "max_pair_overlap_depth": -3.605764309855264e-06, "metric_expression": "3.00005", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.0000460387568455, "monotone_upper_bound_from_case": "squindom@15", "monotone_upper_bound_from_record": "6d686963e9ef0ece", "n": 14, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.0, "setup": "squindom", "side": 6.000092077513691, "solution_path": "data/solutions/generated/squindom_014_821200c01d9d607e.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/821200c01d9d607e.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@15", "density": 0.8333077568349537, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 3.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@15", "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"}, "friedman_reference_text": "14-18. | | s = 3 | Trivial. |", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "6d686963e9ef0ece", "improvement": -4.603875684550118e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0382270816400307e-06, "max_pair_overlap_depth": -3.605764309855264e-06, "metric_expression": "3.00005", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.0000460387568455, "n": 15, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.0, "setup": "squindom", "side": 6.000092077513691, "solution_path": "data/solutions/generated/squindom_015_6d686963e9ef0ece.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/6d686963e9ef0ece.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@16", "density": 0.888839182217615, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 3.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "14-18. | | s = 3 | Trivial. |", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "0f551306e71d2ff0", "improvement": -8.388352586674586e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4593433725073055e-06, "max_pair_overlap_depth": -5.7555123290642385e-06, "metric_expression": "3.00008", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.0000838835258667, "n": 16, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.0, "setup": "squindom", "side": 6.0001677670517335, "solution_path": "data/solutions/generated/squindom_016_0f551306e71d2ff0.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/0f551306e71d2ff0.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@16", "density": 0.8888532656888313, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 3.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "14-18. | | s = 3 | Trivial. |", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "e980aaa270c37d2c", "improvement": -6.011595701327366e-05, "max_boundary_excess": -8.408701424045262e-07, "max_pair_overlap_depth": -1.079909316636929e-06, "metric_expression": "3.00006", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.0000601159570133, "monotone_upper_bound_from_case": "squindom@18", "monotone_upper_bound_from_record": "a6900e1fd3f925c6", "n": 16, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.0, "setup": "squindom", "side": 6.0001202319140265, "solution_path": "data/solutions/generated/squindom_016_e980aaa270c37d2c.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/e980aaa270c37d2c.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@17", "density": 0.9444002665800334, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 3.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"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"}, "friedman_reference_text": "14-18. | | s = 3 | Trivial. |", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "52c5d2070c49c79c", "improvement": -7.016730512532376e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.650126292496168e-06, "max_pair_overlap_depth": -3.2807804517531736e-07, "metric_expression": "3.00007", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.0000701673051253, "n": 17, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.0, "setup": "squindom", "side": 6.000140334610251, "solution_path": "data/solutions/generated/squindom_017_52c5d2070c49c79c.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/52c5d2070c49c79c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@17", "density": 0.9444065947943833, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 3.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"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"}, "friedman_reference_text": "14-18. | | s = 3 | Trivial. |", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "ccddf4810f297da5", "improvement": -6.011595701327366e-05, "max_boundary_excess": -8.408701424045262e-07, "max_pair_overlap_depth": -1.079909316636929e-06, "metric_expression": "3.00006", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.0000601159570133, "monotone_upper_bound_from_case": "squindom@18", "monotone_upper_bound_from_record": "a6900e1fd3f925c6", "n": 17, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.0, "setup": "squindom", "side": 6.0001202319140265, "solution_path": "data/solutions/generated/squindom_017_ccddf4810f297da5.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/ccddf4810f297da5.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@18", "density": 0.9999599238999353, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 3.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@18", "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"}, "friedman_reference_text": "14-18. | | s = 3 | Trivial. |", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "a6900e1fd3f925c6", "improvement": -6.011595701327366e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.408701424045262e-07, "max_pair_overlap_depth": -1.079909316636929e-06, "metric_expression": "3.00006", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.0000601159570133, "n": 18, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.0, "setup": "squindom", "side": 6.0001202319140265, "solution_path": "data/solutions/generated/squindom_018_a6900e1fd3f925c6.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/a6900e1fd3f925c6.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@19", "density": 0.7988013748197259, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "June 2010", "friedman_display_metric_value": 3.447, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/19.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3 + 1/\u221a5 = 3.447+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.447213595499958, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@19", "credit": "Maurizio Morandi", "date": "June 2010", "family_updated": "", "metric_expression": "3 + 1/\u221a5 = 3.447+", "metric_symbol": "s", "metric_value": 3.447, "reference_text": "19. | | s = 3 + 1/\u221a5 = 3.447+ | Found by Maurizio Morandi | in June 2010.", "source_image": "19.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/19.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "19. | | s = 3 + 1/\u221a5 = 3.447+ | Found by Maurizio Morandi | in June 2010.", "friedman_source_image": "19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "cddf920bff527727", "improvement": -0.0013830451761869256, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.98512899937964e-06, "max_pair_overlap_depth": -1.3782300862574814e-05, "metric_expression": "3.44860", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.448596640676145, "n": 19, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.447213595499958, "setup": "squindom", "side": 6.89719328135229, "solution_path": "data/solutions/generated/squindom_019_cddf920bff527727.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/cddf920bff527727.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@20", "density": 0.8162860196976518, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 3.5, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/21.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "7/2 = 3.5", "friedman_metric_symbol": "s", "friedman_metric_value": 3.5, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "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"}, "friedman_reference_text": "20-21. | | s = 7/2 = 3.5 | Trivial. |", "friedman_source_image": "21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "4a48c58d60a1f36b", "improvement": -8.6848505622239e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.863306484192975e-06, "max_pair_overlap_depth": -2.3696014737728532e-05, "metric_expression": "3.50009", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.5000868485056222, "n": 20, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.5, "setup": "squindom", "side": 7.0001736970112445, "solution_path": "data/solutions/generated/squindom_020_4a48c58d60a1f36b.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/4a48c58d60a1f36b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@20", "density": 0.8162990114455758, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 3.5, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/21.gif", "friedman_metric_expression": "7/2 = 3.5", "friedman_metric_symbol": "s", "friedman_metric_value": 3.5, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "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"}, "friedman_reference_text": "20-21. | | s = 7/2 = 3.5 | Trivial. |", "friedman_source_image": "21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "7351f2aa5e1553d9", "improvement": -5.899570515355279e-05, "max_boundary_excess": -1.8534056853702905e-06, "max_pair_overlap_depth": -1.3464050134803784e-06, "metric_expression": "3.50006", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.5000589957051536, "monotone_upper_bound_from_case": "squindom@21", "monotone_upper_bound_from_record": "f91a28177536f6d3", "n": 20, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.5, "setup": "squindom", "side": 7.000117991410307, "solution_path": "data/solutions/generated/squindom_020_7351f2aa5e1553d9.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/7351f2aa5e1553d9.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@21", "density": 0.8571139620178546, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 3.5, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/21.gif", "friedman_metric_expression": "7/2 = 3.5", "friedman_metric_symbol": "s", "friedman_metric_value": 3.5, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@21", "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"}, "friedman_reference_text": "20-21. | | s = 7/2 = 3.5 | Trivial. |", "friedman_source_image": "21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "f91a28177536f6d3", "improvement": -5.899570515355279e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.8534056853702905e-06, "max_pair_overlap_depth": -1.3464050134803784e-06, "metric_expression": "3.50006", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.5000589957051536, "n": 21, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.5, "setup": "squindom", "side": 7.000117991410307, "solution_path": "data/solutions/generated/squindom_021_f91a28177536f6d3.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/f91a28177536f6d3.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@22", "density": 0.783535190077589, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_display_metric_value": 3.745, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/22.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3.745+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.745, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "squindom@22", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "3.745+", "metric_symbol": "s", "metric_value": 3.745, "reference_text": "22. | | s = 3.745+ | Found by David W. Cantrell | in July 2012 (after Maurizio | Morandi in June 2010.)", "source_image": "22.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/22.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "22. | | s = 3.745+ | Found by David W. Cantrell | in July 2012 (after Maurizio | Morandi in June 2010.)", "friedman_source_image": "22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "d2d5380917cfe7ce", "improvement": -0.0018567500999222553, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.72695736733975e-06, "max_pair_overlap_depth": -9.08031145729904e-06, "metric_expression": "3.74686", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.7468567500999224, "n": 22, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.745, "setup": "squindom", "side": 7.493713500199845, "solution_path": "data/solutions/generated/squindom_022_d2d5380917cfe7ce.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/d2d5380917cfe7ce.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@23", "density": 0.7750857674093591, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "June 2010", "friedman_display_metric_value": 3.849, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/23.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3.849+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.849, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "squindom@23", "credit": "Maurizio Morandi", "date": "June 2010", "family_updated": "", "metric_expression": "3.849+", "metric_symbol": "s", "metric_value": 3.849, "reference_text": "23. | | s = 3.849+ | Found by Maurizio Morandi | in June 2010. |", "source_image": "23.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/23.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "23. | | s = 3.849+ | Found by Maurizio Morandi | in June 2010. |", "friedman_source_image": "23.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "d12d3a9a87ba3b11", "improvement": -0.002891443742406352, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.707858897694649e-06, "max_pair_overlap_depth": -2.3445235071584136e-05, "metric_expression": "3.85189", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.8518914437424066, "n": 23, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.849, "setup": "squindom", "side": 7.703782887484813, "solution_path": "data/solutions/generated/squindom_023_d12d3a9a87ba3b11.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/d12d3a9a87ba3b11.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@24", "density": 0.7961447064751783, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell after Maurizio Morandi", "friedman_date": "August 2012", "friedman_display_metric_value": 3.882, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/24.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3.882+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.882, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "squindom@24", "credit": "David W. Cantrell after Maurizio Morandi", "date": "August 2012", "family_updated": "", "metric_expression": "3.882+", "metric_symbol": "s", "metric_value": 3.882, "reference_text": "24. | | s = 3.882+ | Found by DC after MM | in August 2012. |", "source_image": "24.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/24.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "24. | | s = 3.882+ | Found by DC after MM | in August 2012. |", "friedman_source_image": "24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "f954e5304808181f", "improvement": -0.0003493916461954072, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.7445287290307476e-06, "max_pair_overlap_depth": -1.9183513864096113e-05, "metric_expression": "3.88235", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.8823493916461955, "n": 24, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.882, "setup": "squindom", "side": 7.764698783292391, "solution_path": "data/solutions/generated/squindom_024_f954e5304808181f.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "David W. Cantrell after Maurizio Morandi", "svg_path": "data/svg/f954e5304808181f.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@25", "density": 0.8043920132931294, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell after Maurizio Morandi", "friedman_date": "August 2012", "friedman_display_metric_value": 3.939, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/25.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3.939+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.939, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "squindom@25", "credit": "David W. Cantrell after Maurizio Morandi", "date": "August 2012", "family_updated": "", "metric_expression": "3.939+", "metric_symbol": "s", "metric_value": 3.939, "reference_text": "25. | | s = 3.939+ | Found by DC after MM | in August 2012. |", "source_image": "25.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/25.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "25. | | s = 3.939+ | Found by DC after MM | in August 2012. |", "friedman_source_image": "25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "26273cb13b41bf72", "improvement": -0.003040951197706754, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.269696070480222e-06, "max_pair_overlap_depth": -2.5070959039568547e-05, "metric_expression": "3.94204", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.942040951197707, "n": 25, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.939, "setup": "squindom", "side": 7.884081902395414, "solution_path": "data/solutions/generated/squindom_025_26273cb13b41bf72.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "David W. Cantrell after Maurizio Morandi", "svg_path": "data/svg/26273cb13b41bf72.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@26", "density": 0.8201428647660397, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_display_metric_value": 3.981, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/26.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3.981+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.981, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "squindom@26", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "3.981+", "metric_symbol": "s", "metric_value": 3.981, "reference_text": "26. | | s = 3.981+ | Found by David W. Cantrell | in July 2012 (after Maurizio | Morandi in July 2012.)", "source_image": "26.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/26.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "26. | | s = 3.981+ | Found by David W. Cantrell | in July 2012 (after Maurizio | Morandi in July 2012.)", "friedman_source_image": "26.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "a46204eea3d7f9fe", "improvement": -0.00031848875831208545, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.947984543621445e-06, "max_pair_overlap_depth": -8.516034391559124e-06, "metric_expression": "3.98132", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.981318488758312, "n": 26, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.981, "setup": "squindom", "side": 7.962636977516624, "solution_path": "data/solutions/generated/squindom_026_a46204eea3d7f9fe.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/a46204eea3d7f9fe.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@27", "density": 0.8437137431630357, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 4.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "27-32. | | s = 4 | Trivial. | |", "friedman_source_image": "32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "bc414cdd10a510fa", "improvement": -8.594490193214455e-05, "max_boundary_excess": -4.188605266186585e-06, "max_pair_overlap_depth": -1.644068809536492e-06, "metric_expression": "4.00009", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.000085944901932, "monotone_upper_bound_from_case": "squindom@30", "monotone_upper_bound_from_record": "d67a491f7b4498c0", "n": 27, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.0, "setup": "squindom", "side": 8.000171889803864, "solution_path": "data/solutions/generated/squindom_027_bc414cdd10a510fa.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/bc414cdd10a510fa.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@27", "density": 0.8436994977206457, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 4.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "27-32. | | s = 4 | Trivial. | |", "friedman_source_image": "32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "c68e03cc8f5005db", "improvement": -0.00011971448073744284, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.175631390130775e-07, "max_pair_overlap_depth": -1.4466911184563003e-06, "metric_expression": "4.00012", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.000119714480737, "n": 27, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.0, "setup": "squindom", "side": 8.000239428961475, "solution_path": "data/solutions/generated/squindom_027_c68e03cc8f5005db.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/c68e03cc8f5005db.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@28", "density": 0.8749068629130922, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 4.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@28", "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"}, "friedman_reference_text": "27-32. | | s = 4 | Trivial. | |", "friedman_source_image": "32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "2c8c79126d60cdac", "improvement": -0.00021290176655508475, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.243746303256302e-06, "max_pair_overlap_depth": -1.6386799395987463e-05, "metric_expression": "4.00021", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.000212901766555, "n": 28, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.0, "setup": "squindom", "side": 8.00042580353311, "solution_path": "data/solutions/generated/squindom_028_2c8c79126d60cdac.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/2c8c79126d60cdac.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@28", "density": 0.8749624003172222, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 4.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@28", "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"}, "friedman_reference_text": "27-32. | | s = 4 | Trivial. | |", "friedman_source_image": "32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "a26ece2a03ab477f", "improvement": -8.594490193214455e-05, "max_boundary_excess": -4.188605266186585e-06, "max_pair_overlap_depth": -1.644068809536492e-06, "metric_expression": "4.00009", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.000085944901932, "monotone_upper_bound_from_case": "squindom@30", "monotone_upper_bound_from_record": "d67a491f7b4498c0", "n": 28, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.0, "setup": "squindom", "side": 8.000171889803864, "solution_path": "data/solutions/generated/squindom_028_a26ece2a03ab477f.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/a26ece2a03ab477f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@29", "density": 0.9062110574714087, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 4.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@29", "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"}, "friedman_reference_text": "27-32. | | s = 4 | Trivial. | |", "friedman_source_image": "32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "a006f9c77930af4d", "improvement": -8.594490193214455e-05, "max_boundary_excess": -4.188605266186585e-06, "max_pair_overlap_depth": -1.644068809536492e-06, "metric_expression": "4.00009", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.000085944901932, "monotone_upper_bound_from_case": "squindom@30", "monotone_upper_bound_from_record": "d67a491f7b4498c0", "n": 29, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.0, "setup": "squindom", "side": 8.000171889803864, "solution_path": "data/solutions/generated/squindom_029_a006f9c77930af4d.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/a006f9c77930af4d.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@29", "density": 0.9062096331295707, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 4.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@29", "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"}, "friedman_reference_text": "27-32. | | s = 4 | Trivial. | |", "friedman_source_image": "32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "d2548471aff7d9f5", "improvement": -8.908848335043018e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.8156140130519702e-06, "max_pair_overlap_depth": -2.026030487822794e-06, "metric_expression": "4.00009", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.00008908848335, "n": 29, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.0, "setup": "squindom", "side": 8.0001781769667, "solution_path": "data/solutions/generated/squindom_029_d2548471aff7d9f5.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/d2548471aff7d9f5.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@30", "density": 0.9374597146255952, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 4.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@30", "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"}, "friedman_reference_text": "27-32. | | s = 4 | Trivial. | |", "friedman_source_image": "32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "d67a491f7b4498c0", "improvement": -8.594490193214455e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.188605266186585e-06, "max_pair_overlap_depth": -1.644068809536492e-06, "metric_expression": "4.00009", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.000085944901932, "n": 30, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.0, "setup": "squindom", "side": 8.000171889803864, "solution_path": "data/solutions/generated/squindom_030_d67a491f7b4498c0.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/d67a491f7b4498c0.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@31", "density": 0.9683879031627185, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 4.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@31", "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"}, "friedman_reference_text": "27-32. | | s = 4 | Trivial. | |", "friedman_source_image": "32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "86ec622f54cecf7c", "improvement": -0.000747764390431449, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.872732694063188e-07, "max_pair_overlap_depth": -2.9208615389531944e-06, "metric_expression": "4.00075", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.0007477643904314, "n": 31, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.0, "setup": "squindom", "side": 8.001495528780863, "solution_path": "data/solutions/generated/squindom_031_86ec622f54cecf7c.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/86ec622f54cecf7c.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@32", "density": 0.9989694769831047, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 4.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@32", "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"}, "friedman_reference_text": "27-32. | | s = 4 | Trivial. | |", "friedman_source_image": "32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "2873ac78de186b97", "improvement": -0.0020626403695489515, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.0386280700511747e-06, "max_pair_overlap_depth": -2.476940408779882e-06, "metric_expression": "4.00206", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.002062640369549, "n": 32, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.0, "setup": "squindom", "side": 8.004125280739098, "solution_path": "data/solutions/generated/squindom_032_2873ac78de186b97.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/2873ac78de186b97.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@33", "density": 0.8455857799388354, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "July 2012", "friedman_display_metric_value": 4.414, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/33.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3 + \u221a2 = 4.414+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.414213562373095, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@33", "credit": "Maurizio Morandi", "date": "July 2012", "family_updated": "", "metric_expression": "3 + \u221a2 = 4.414+", "metric_symbol": "s", "metric_value": 4.414, "reference_text": "33. | | s = 3 + \u221a2 = 4.414+ | Found by Maurizio Morandi | in July 2012.", "source_image": "33.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/33.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "33. | | s = 3 + \u221a2 = 4.414+ | Found by Maurizio Morandi | in July 2012.", "friedman_source_image": "33.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "167e5e2a52e9b5fe", "improvement": -0.0031499203989495683, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.856211927670387e-06, "max_pair_overlap_depth": -6.591285420576298e-06, "metric_expression": "4.41736", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.4173634827720445, "n": 33, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.414213562373095, "setup": "squindom", "side": 8.834726965544089, "solution_path": "data/solutions/generated/squindom_033_167e5e2a52e9b5fe.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/167e5e2a52e9b5fe.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@34", "density": 0.8499062715939519, "friedman_analytical_or_proved": true, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_display_metric_value": 4.471, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/34.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "4 + \u221a2/3 = 4.471+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.471404520791031, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@34", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "4 + \u221a2/3 = 4.471+", "metric_symbol": "s", "metric_value": 4.471, "reference_text": "34. | | s = 4 + \u221a2/3 = 4.471+ | Found by David W. Cantrell | in July 2012.", "source_image": "34.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/34.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "34. | | s = 4 + \u221a2/3 = 4.471+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "34.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "b631e5a58c1517e1", "improvement": -0.0009780229400568885, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.603876101403557e-07, "max_pair_overlap_depth": -3.616786977911346e-06, "metric_expression": "4.47238", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.472382543731088, "n": 34, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.471404520791031, "setup": "squindom", "side": 8.944765087462176, "solution_path": "data/solutions/generated/squindom_034_b631e5a58c1517e1.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/b631e5a58c1517e1.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@35", "density": 0.8626972775109039, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 4.5, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/36.gif", "friedman_metric_expression": "9/2 = 4.5", "friedman_metric_symbol": "s", "friedman_metric_value": 4.5, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@35", "credit": "Trivial", "date": "", "family_updated": "", "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/", "title": "Squares in Dominoes"}, "friedman_reference_text": "35-36. | | s = 9/2 = 4.5 | Trivial. |", "friedman_source_image": "36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "7d2266788c6a0b7a", "improvement": -0.003911109790387002, "max_boundary_excess": -9.310647293148122e-07, "max_pair_overlap_depth": -3.943598063660403e-06, "metric_expression": "4.50391", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.503911109790387, "monotone_upper_bound_from_case": "squindom@36", "monotone_upper_bound_from_record": "e0fc61bd01777ea3", "n": 35, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.5, "setup": "squindom", "side": 9.007822219580774, "solution_path": "data/solutions/generated/squindom_035_7d2266788c6a0b7a.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/7d2266788c6a0b7a.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@35", "density": 0.8625827503428114, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 4.5, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/36.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "9/2 = 4.5", "friedman_metric_symbol": "s", "friedman_metric_value": 4.5, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@35", "credit": "Trivial", "date": "", "family_updated": "", "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/", "title": "Squares in Dominoes"}, "friedman_reference_text": "35-36. | | s = 9/2 = 4.5 | Trivial. |", "friedman_source_image": "36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "dcd4cfae3ff101a2", "improvement": -0.004210097373901434, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.9606355010637344e-06, "max_pair_overlap_depth": -1.1537592481902781e-05, "metric_expression": "4.50421", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.504210097373901, "n": 35, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.5, "setup": "squindom", "side": 9.008420194747803, "solution_path": "data/solutions/generated/squindom_035_dcd4cfae3ff101a2.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/dcd4cfae3ff101a2.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@36", "density": 0.8873457711540725, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 4.5, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/36.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "9/2 = 4.5", "friedman_metric_symbol": "s", "friedman_metric_value": 4.5, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@36", "credit": "Trivial", "date": "", "family_updated": "", "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/", "title": "Squares in Dominoes"}, "friedman_reference_text": "35-36. | | s = 9/2 = 4.5 | Trivial. |", "friedman_source_image": "36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "e0fc61bd01777ea3", "improvement": -0.003911109790387002, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.310647293148122e-07, "max_pair_overlap_depth": -3.943598063660403e-06, "metric_expression": "4.50391", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.503911109790387, "n": 36, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.5, "setup": "squindom", "side": 9.007822219580774, "solution_path": "data/solutions/generated/squindom_036_e0fc61bd01777ea3.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/e0fc61bd01777ea3.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@37", "density": 0.8199742411033116, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "August 2012", "friedman_display_metric_value": 4.747, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/37.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "10 / 3 + \u221a2 = 4.747+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.747546895706428, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@37", "credit": "Maurizio Morandi", "date": "August 2012", "family_updated": "", "metric_expression": "10 / 3 + \u221a2 = 4.747+", "metric_symbol": "s", "metric_value": 4.747, "reference_text": "37. | | s = 10 / 3 + \u221a2 = 4.747+ | Found by Maurizio Morandi | in August 2012.", "source_image": "37.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/37.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "37. | | s = 10 / 3 + \u221a2 = 4.747+ | Found by Maurizio Morandi | in August 2012.", "friedman_source_image": "37.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "09a846134d0a3602", "improvement": -0.002367245250098904, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.702594714522945e-06, "max_pair_overlap_depth": -1.6080659807293785e-05, "metric_expression": "4.74991", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.749914140956527, "n": 37, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.747546895706428, "setup": "squindom", "side": 9.499828281913054, "solution_path": "data/solutions/generated/squindom_037_09a846134d0a3602.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/09a846134d0a3602.svg", "tolerance": 1e-08, "verified": true}
@@ -447,11 +476,16 @@
{"case": "squindom@41", "density": 0.8199798328031639, "friedman_analytical_or_proved": false, "friedman_credit": "Sigvart Brendberg", "friedman_date": "June 2023", "friedman_display_metric_value": 4.927, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/41.gif", "friedman_metric_expression": "4.927+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.927, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "squindom@41", "credit": "Sigvart Brendberg", "date": "June 2023", "family_updated": "", "metric_expression": "4.927+", "metric_symbol": "s", "metric_value": 4.927, "reference_text": "41. | | s = 4.927+ | Found by Sigvart Brendberg | in June 2023.", "source_image": "41.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/41.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "41. | | s = 4.927+ | Found by Sigvart Brendberg | in June 2023.", "friedman_source_image": "41.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "cb1462995bd3b2c0", "improvement": -0.07306148649036626, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.465020711057036e-07, "max_pair_overlap_depth": -6.131910444850952e-06, "metric_expression": "5.00006", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.000061486490366, "n": 41, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is outside the benchmark recovery band for the Friedman reference.", "record_type": "verified", "reference_side": 4.927, "setup": "squindom", "side": 10.000122972980732, "solution_path": "data/solutions/generated/squindom_41_cb1462995bd3b2c0.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T13:47:12+00:00", "submitted_by": "Sigvart Brendberg", "svg_path": "data/svg/cb1462995bd3b2c0.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@42", "density": 0.8472590848618945, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "August 2012", "friedman_display_metric_value": 4.989, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/42.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "4.989+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.989, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "squindom@42", "credit": "David W. Cantrell", "date": "August 2012", "family_updated": "", "metric_expression": "4.989+", "metric_symbol": "s", "metric_value": 4.989, "reference_text": "42. | | s = 4.989+ | Found by David W. Cantrell | in August 2012.", "source_image": "42.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/42.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "42. | | s = 4.989+ | Found by David W. Cantrell | in August 2012.", "friedman_source_image": "42.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": false, "id": "7faea8ae450b67a1", "improvement": 0.010465394774865189, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.54667148005916e-06, "max_pair_overlap_depth": -1.5216861767419765e-05, "metric_expression": "4.97853", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.978534605225135, "n": 42, "notes": "Verified coordinate improvement; exceeds the displayed Friedman value by more than 0.15%.", "record_type": "verified", "reference_side": 4.989, "setup": "squindom", "side": 9.95706921045027, "solution_path": "data/solutions/generated/squindom_042_7faea8ae450b67a1.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/7faea8ae450b67a1.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@43", "density": 0.859888117316887, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 5.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@43", "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"}, "friedman_reference_text": "43-50. | | s = 5 | Trivial. |", "friedman_source_image": "50.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "7f9527ccddc9d648", "improvement": -0.00032527209571497906, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.1461540806485566e-08, "max_pair_overlap_depth": -2.005039219099558e-06, "metric_expression": "5.00033", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.000325272095715, "n": 43, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.0, "setup": "squindom", "side": 10.00065054419143, "solution_path": "data/solutions/generated/squindom_043_7f9527ccddc9d648.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/7f9527ccddc9d648.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@43", "density": 0.8599640330943492, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 5.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@43", "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"}, "friedman_reference_text": "43-50. | | s = 5 | Trivial. |", "friedman_source_image": "50.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "83b582ff79edcebf", "improvement": -0.00010455823792376862, "max_boundary_excess": -4.245467354557775e-06, "max_pair_overlap_depth": -3.962242465904708e-06, "metric_expression": "5.00010", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.000104558237924, "monotone_upper_bound_from_case": "squindom@44", "monotone_upper_bound_from_record": "1ffd758f1f058de5", "n": 43, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.0, "setup": "squindom", "side": 10.000209116475848, "solution_path": "data/solutions/generated/squindom_043_83b582ff79edcebf.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/83b582ff79edcebf.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@44", "density": 0.8799631966546829, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 5.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@44", "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"}, "friedman_reference_text": "43-50. | | s = 5 | Trivial. |", "friedman_source_image": "50.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "1ffd758f1f058de5", "improvement": -0.00010455823792376862, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.245467354557775e-06, "max_pair_overlap_depth": -3.962242465904708e-06, "metric_expression": "5.00010", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.000104558237924, "n": 44, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.0, "setup": "squindom", "side": 10.000209116475848, "solution_path": "data/solutions/generated/squindom_044_1ffd758f1f058de5.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/1ffd758f1f058de5.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@45", "density": 0.8999501217119913, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 5.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@45", "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"}, "friedman_reference_text": "43-50. | | s = 5 | Trivial. |", "friedman_source_image": "50.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "d085ed9682ca2968", "improvement": -0.00013855655918781906, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0485935568738114e-06, "max_pair_overlap_depth": -1.361329106774889e-06, "metric_expression": "5.00014", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.000138556559188, "n": 45, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.0, "setup": "squindom", "side": 10.000277113118376, "solution_path": "data/solutions/generated/squindom_045_d085ed9682ca2968.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/d085ed9682ca2968.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@45", "density": 0.8999548660266541, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 5.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@45", "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"}, "friedman_reference_text": "43-50. | | s = 5 | Trivial. |", "friedman_source_image": "50.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "efa26e844da7d857", "improvement": -0.00012537686383318203, "max_boundary_excess": -1.0555756748686917e-06, "max_pair_overlap_depth": -4.3576353303542703e-07, "metric_expression": "5.00013", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.000125376863833, "monotone_upper_bound_from_case": "squindom@48", "monotone_upper_bound_from_record": "0433c6294c4f3fe7", "n": 45, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.0, "setup": "squindom", "side": 10.000250753727666, "solution_path": "data/solutions/generated/squindom_045_efa26e844da7d857.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/efa26e844da7d857.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@46", "density": 0.9198953665582859, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 5.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@46", "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"}, "friedman_reference_text": "43-50. | | s = 5 | Trivial. |", "friedman_source_image": "50.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "3b408e84cd206068", "improvement": -0.00028435426002282327, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1757865436834436e-06, "max_pair_overlap_depth": -7.423155479191479e-07, "metric_expression": "5.00028", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.000284354260023, "n": 46, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.0, "setup": "squindom", "side": 10.000568708520046, "solution_path": "data/solutions/generated/squindom_046_3b408e84cd206068.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/3b408e84cd206068.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@46", "density": 0.9199538630494686, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 5.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@46", "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"}, "friedman_reference_text": "43-50. | | s = 5 | Trivial. |", "friedman_source_image": "50.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "7fb3175fb0fba7a7", "improvement": -0.00012537686383318203, "max_boundary_excess": -1.0555756748686917e-06, "max_pair_overlap_depth": -4.3576353303542703e-07, "metric_expression": "5.00013", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.000125376863833, "monotone_upper_bound_from_case": "squindom@48", "monotone_upper_bound_from_record": "0433c6294c4f3fe7", "n": 46, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.0, "setup": "squindom", "side": 10.000250753727666, "solution_path": "data/solutions/generated/squindom_046_7fb3175fb0fba7a7.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/7fb3175fb0fba7a7.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@47", "density": 0.9398436614252309, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 5.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@47", "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"}, "friedman_reference_text": "43-50. | | s = 5 | Trivial. |", "friedman_source_image": "50.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "2525e331a392c6b9", "improvement": -0.00041584595443833194, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.3105818880431173e-06, "max_pair_overlap_depth": -3.7244704524930228e-06, "metric_expression": "5.00042", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.000415845954438, "n": 47, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.0, "setup": "squindom", "side": 10.000831691908877, "solution_path": "data/solutions/generated/squindom_047_2525e331a392c6b9.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/2525e331a392c6b9.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@47", "density": 0.9399528600722832, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 5.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@47", "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"}, "friedman_reference_text": "43-50. | | s = 5 | Trivial. |", "friedman_source_image": "50.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "a146ed2b5a4b6cf9", "improvement": -0.00012537686383318203, "max_boundary_excess": -1.0555756748686917e-06, "max_pair_overlap_depth": -4.3576353303542703e-07, "metric_expression": "5.00013", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.000125376863833, "monotone_upper_bound_from_case": "squindom@48", "monotone_upper_bound_from_record": "0433c6294c4f3fe7", "n": 47, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.0, "setup": "squindom", "side": 10.000250753727666, "solution_path": "data/solutions/generated/squindom_047_a146ed2b5a4b6cf9.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/a146ed2b5a4b6cf9.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@48", "density": 0.9599518570950978, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 5.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@48", "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"}, "friedman_reference_text": "43-50. | | s = 5 | Trivial. |", "friedman_source_image": "50.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "0433c6294c4f3fe7", "improvement": -0.00012537686383318203, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0555756748686917e-06, "max_pair_overlap_depth": -4.3576353303542703e-07, "metric_expression": "5.00013", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.000125376863833, "n": 48, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.0, "setup": "squindom", "side": 10.000250753727666, "solution_path": "data/solutions/generated/squindom_048_0433c6294c4f3fe7.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/0433c6294c4f3fe7.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@49", "density": 0.9799322342018685, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 5.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "43-50. | | s = 5 | Trivial. |", "friedman_source_image": "50.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "598e94e082ebd7d9", "improvement": -0.00017288089993794387, "max_boundary_excess": -9.912590304139712e-09, "max_pair_overlap_depth": -7.234911549747203e-06, "metric_expression": "5.00017", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.000172880899938, "monotone_upper_bound_from_case": "squindom@50", "monotone_upper_bound_from_record": "df0259815cc5320a", "n": 49, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.0, "setup": "squindom", "side": 10.000345761799876, "solution_path": "data/solutions/generated/squindom_049_598e94e082ebd7d9.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/598e94e082ebd7d9.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@49", "density": 0.9798599360041962, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 5.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "43-50. | | s = 5 | Trivial. |", "friedman_source_image": "50.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "c84a987ae840ebe3", "improvement": -0.0003573444166047324, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.316744310450815e-06, "max_pair_overlap_depth": -1.1644813818767474e-06, "metric_expression": "5.00036", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.000357344416605, "n": 49, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.0, "setup": "squindom", "side": 10.00071468883321, "solution_path": "data/solutions/generated/squindom_049_c84a987ae840ebe3.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/c84a987ae840ebe3.svg", "tolerance": 1e-08, "verified": true}
{"case": "squindom@50", "density": 0.9999308512263965, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 5.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "squindom@50", "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"}, "friedman_reference_text": "43-50. | | s = 5 | Trivial. |", "friedman_source_image": "50.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "df0259815cc5320a", "improvement": -0.00017288089993794387, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.912590304139712e-09, "max_pair_overlap_depth": -7.234911549747203e-06, "metric_expression": "5.00017", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.000172880899938, "n": 50, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.0, "setup": "squindom", "side": 10.000345761799876, "solution_path": "data/solutions/generated/squindom_050_df0259815cc5320a.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/df0259815cc5320a.svg", "tolerance": 1e-08, "verified": true}
{"case": "squinhex@1", "density": 0.6187919623349352, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 0.778, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/1.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "(3+\u221a3)/6 = 0.778+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.7886751345948129, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "squinhex@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "(3+\u221a3)/6 = 0.778+", "metric_symbol": "s", "metric_value": 0.778, "reference_text": "s = (3+\u221a3)/6 = 0.778+ | Trivial. |", "source_image": "1.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/1.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = (3+\u221a3)/6 = 0.778+ | Trivial. |", "friedman_source_image": "1.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "7a4ae27d520be7d9", "improvement": -6.494508079879502e-06, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.500249687614274e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "0.78868", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 0.7886816291028927, "n": 1, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 0.7886751345948129, "setup": "squinhex", "side": 0.7886816291028927, "solution_path": "data/solutions/generated/squinhex_001_7a4ae27d520be7d9.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T03:07:58+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/7a4ae27d520be7d9.svg", "tolerance": 1e-08, "verified": true}
@@ -514,6 +548,7 @@
{"case": "squinoct@14", "density": 0.7369251630765782, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_display_metric_value": 1.9835, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/14.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "1.98350+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.9835, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@14", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.98350+", "metric_symbol": "s", "metric_value": 1.9835, "reference_text": "s = 1.98350+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "14.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/14.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 1.98350+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "14.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "994e0dd57a80d4a3", "improvement": -7.891975572338339e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.49261632156563e-06, "max_pair_overlap_depth": -2.894591689805992e-05, "metric_expression": "1.98358", "metric_recovery_tolerance_abs": 1e-05, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 1.9835789197557234, "n": 14, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 1.9835, "setup": "squinoct", "side": 1.9835789197557234, "solution_path": "data/solutions/generated/squinoct_014_994e0dd57a80d4a3.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/994e0dd57a80d4a3.svg", "tolerance": 1e-08, "verified": true}
{"case": "squinoct@15", "density": 0.7515649396279209, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_display_metric_value": 2.03259, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/15.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "2.03259+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.03259, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@15", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.03259+", "metric_symbol": "s", "metric_value": 2.03259, "reference_text": "s = 2.03259+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "15.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/15.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 2.03259+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "c898d8b3cd2b55db", "improvement": -0.0005137104569299389, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.7983161874182644e-07, "max_pair_overlap_depth": -1.9704810269915995e-05, "metric_expression": "2.03310", "metric_recovery_tolerance_abs": 1e-05, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.03310371045693, "n": 15, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.03259, "setup": "squinoct", "side": 2.03310371045693, "solution_path": "data/solutions/generated/squinoct_015_c898d8b3cd2b55db.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/c898d8b3cd2b55db.svg", "tolerance": 1e-08, "verified": true}
{"case": "squinoct@16", "density": 0.7725175260502497, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 2.07106, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/16.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "5\u221a2 \u2013 5 = 2.07106+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.071067811865475, "friedman_metric_value_source": "exact_expression", "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": 2.07106, "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"}, "friedman_reference_text": "s = 5\u221a2 \u2013 5 = 2.07106+ | Trivial. |", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "3a2e6d99239355fd", "improvement": -4.130460271545289e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.5220970331887e-06, "max_pair_overlap_depth": -8.060079695892952e-07, "metric_expression": "2.07111", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.0711091164681905, "n": 16, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.071067811865475, "setup": "squinoct", "side": 2.0711091164681905, "solution_path": "data/solutions/generated/squinoct_016_3a2e6d99239355fd.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/3a2e6d99239355fd.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinoct@16", "density": 0.7725251391826925, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 2.07106, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/16.gif", "friedman_metric_expression": "5\u221a2 \u2013 5 = 2.07106+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.071067811865475, "friedman_metric_value_source": "exact_expression", "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"}, "friedman_reference_text": "s = 5\u221a2 \u2013 5 = 2.07106+ | Trivial. |", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "4405320aab52a39c", "improvement": -3.109932520750647e-05, "max_boundary_excess": 1.368661750365744e-09, "max_pair_overlap_depth": -1.0480566122961221e-05, "metric_expression": "2.07110", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.0710989111906826, "monotone_upper_bound_from_case": "squinoct@17", "monotone_upper_bound_from_record": "ab660f0510a3dd1c", "n": 16, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.071067811865475, "setup": "squinoct", "side": 2.0710989111906826, "solution_path": "data/solutions/generated/squinoct_016_4405320aab52a39c.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/4405320aab52a39c.svg", "tolerance": 1e-08, "verified": true}
{"case": "squinoct@17", "density": 0.8208079603816107, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 2.07106, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/17.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "5\u221a2 \u2013 5 = 2.07106+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.071067811865475, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "squinoct@17", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5\u221a2 \u2013 5 = 2.07106+", "metric_symbol": "s", "metric_value": 2.07106, "reference_text": "s = 5\u221a2 \u2013 5 = 2.07106+ | Trivial. |", "source_image": "17.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/17.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 5\u221a2 \u2013 5 = 2.07106+ | Trivial. |", "friedman_source_image": "17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "ab660f0510a3dd1c", "improvement": -3.109932520750647e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 1.368661750365744e-09, "max_pair_overlap_depth": -1.0480566122961221e-05, "metric_expression": "2.07110", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.0710989111906826, "n": 17, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.071067811865475, "setup": "squinoct", "side": 2.0710989111906826, "solution_path": "data/solutions/generated/squinoct_017_ab660f0510a3dd1c.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/ab660f0510a3dd1c.svg", "tolerance": 1e-08, "verified": true}
{"case": "squinoct@18", "density": 0.7934296897658486, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_display_metric_value": 2.16778, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/18.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "2.16778+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.16778, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@18", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.16778+", "metric_symbol": "s", "metric_value": 2.16778, "reference_text": "s = 2.16778+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "18.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/18.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 2.16778+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "a7275e2ab7024187", "improvement": 0.00017977070139485463, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.9711774532196686e-06, "max_pair_overlap_depth": -1.4522556675777842e-05, "metric_expression": "2.16760", "metric_recovery_tolerance_abs": 1e-05, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.167600229298605, "n": 18, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.16778, "setup": "squinoct", "side": 2.167600229298605, "solution_path": "data/solutions/generated/squinoct_018_a7275e2ab7024187.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/a7275e2ab7024187.svg", "tolerance": 1e-08, "verified": true}
{"case": "squinoct@19", "density": 0.793469906795261, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_display_metric_value": 2.22823, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/19.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "2.22823+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.22823, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@19", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.22823+", "metric_symbol": "s", "metric_value": 2.22823, "reference_text": "s = 2.22823+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "19.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/19.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 2.22823+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "330906f1db929cdb", "improvement": 0.0012889040224068893, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0543459428412958e-06, "max_pair_overlap_depth": -1.4850274468458657e-05, "metric_expression": "2.22694", "metric_recovery_tolerance_abs": 1e-05, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.226941095977593, "n": 19, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.22823, "setup": "squinoct", "side": 2.226941095977593, "solution_path": "data/solutions/generated/squinoct_019_330906f1db929cdb.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/330906f1db929cdb.svg", "tolerance": 1e-08, "verified": true}
@@ -579,6 +614,7 @@
{"case": "squintri@2", "density": 0.4640775964340414, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 3.154, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts2.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "2 + 2 / \u221a3 = 3.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.1547005383792515, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "squintri@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2 + 2 / \u221a3 = 3.154+", "metric_symbol": "s", "metric_value": 3.154, "reference_text": "s = 2 + 2 / \u221a3 = 3.154+ | Trivial.", "source_image": "ts2.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts2.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 2 + 2 / \u221a3 = 3.154+ | Trivial.", "friedman_source_image": "ts2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "9fcb7fc91bdb6f8d", "improvement": -8.163595671861756e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.2952441585539525e-06, "max_pair_overlap_depth": -3.4919698730116744e-05, "metric_expression": "3.15478", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.15478217433597, "n": 2, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.1547005383792515, "setup": "squintri", "side": 3.15478217433597, "solution_path": "data/solutions/generated/squintri_002_9fcb7fc91bdb6f8d.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/9fcb7fc91bdb6f8d.svg", "tolerance": 1e-08, "verified": true}
{"case": "squintri@3", "density": 0.6632023535454387, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_display_metric_value": 3.232, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts3.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "3 / 2 + \u221a3 = 3.232+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.232050807568877, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "squintri@3", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "3 / 2 + \u221a3 = 3.232+", "metric_symbol": "s", "metric_value": 3.232, "reference_text": "s = 3 / 2 + \u221a3 = 3.232+ | Found by Erich Friedman | in 1997.", "source_image": "ts3.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts3.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 3 / 2 + \u221a3 = 3.232+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "6a0fd09e2fb037d5", "improvement": -6.772906422058966e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.0862631378148764e-06, "max_pair_overlap_depth": -4.0896047962125925e-05, "metric_expression": "3.23212", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.232118536633098, "n": 3, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.232050807568877, "setup": "squintri", "side": 3.232118536633098, "solution_path": "data/solutions/generated/squintri_003_6a0fd09e2fb037d5.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/6a0fd09e2fb037d5.svg", "tolerance": 1e-08, "verified": true}
{"case": "squintri@4", "density": 0.5351297741951115, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_display_metric_value": 4.154, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts4.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "3 + 2 / \u221a3 = 4.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.1547005383792515, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "squintri@4", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "3 + 2 / \u221a3 = 4.154+", "metric_symbol": "s", "metric_value": 4.154, "reference_text": "s = 3 + 2 / \u221a3 = 4.154+ | Found by Erich Friedman | in 1997.", "source_image": "ts4.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts4.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 3 + 2 / \u221a3 = 4.154+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "a14dd3a45e182f53", "improvement": -9.943269259693466e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.3541096500101422e-06, "max_pair_overlap_depth": -2.3979848717270436e-05, "metric_expression": "4.15480", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.154799971071848, "n": 4, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.1547005383792515, "setup": "squintri", "side": 4.154799971071848, "solution_path": "data/solutions/generated/squintri_004_a14dd3a45e182f53.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/a14dd3a45e182f53.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@5", "density": 0.6217428601610049, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_display_metric_value": 4.309, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts6.gif", "friedman_metric_expression": "2 + 4 / \u221a3 = 4.309+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.309401076758503, "friedman_metric_value_source": "exact_expression", "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"}, "friedman_reference_text": "s = 2 + 4 / \u221a3 = 4.309+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "21b5dd023d9a44d0", "improvement": -0.0001226962979972157, "max_boundary_excess": -2.6885485560193324e-06, "max_pair_overlap_depth": -3.4490596049874256e-05, "metric_expression": "4.30952", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.3095237730565, "monotone_upper_bound_from_case": "squintri@6", "monotone_upper_bound_from_record": "cafd65072dcc4c6f", "n": 5, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.309401076758503, "setup": "squintri", "side": 4.3095237730565, "solution_path": "data/solutions/generated/squintri_005_21b5dd023d9a44d0.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/21b5dd023d9a44d0.svg", "tolerance": 1e-08, "verified": true}
{"case": "squintri@5", "density": 0.6217402820027407, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_display_metric_value": 4.309, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts6.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "2 + 4 / \u221a3 = 4.309+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.309401076758503, "friedman_metric_value_source": "exact_expression", "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": 4.309, "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"}, "friedman_reference_text": "s = 2 + 4 / \u221a3 = 4.309+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "b7ec125fb5317da3", "improvement": -0.0001316313976955641, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.818148398440643e-06, "max_pair_overlap_depth": -2.4242157156018074e-05, "metric_expression": "4.30953", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.3095327081561985, "n": 5, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.309401076758503, "setup": "squintri", "side": 4.3095327081561985, "solution_path": "data/solutions/generated/squintri_005_b7ec125fb5317da3.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/b7ec125fb5317da3.svg", "tolerance": 1e-08, "verified": true}
{"case": "squintri@6", "density": 0.7460914321932061, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_display_metric_value": 4.309, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts6.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "2 + 4 / \u221a3 = 4.309+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.309401076758503, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "squintri@6", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "2 + 4 / \u221a3 = 4.309+", "metric_symbol": "s", "metric_value": 4.309, "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"}, "friedman_reference_text": "s = 2 + 4 / \u221a3 = 4.309+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "cafd65072dcc4c6f", "improvement": -0.0001226962979972157, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6885485560193324e-06, "max_pair_overlap_depth": -3.2625800104213454e-05, "metric_expression": "4.30952", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.3095237730565, "n": 6, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.309401076758503, "setup": "squintri", "side": 4.3095237730565, "solution_path": "data/solutions/generated/squintri_006_cafd65072dcc4c6f.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/cafd65072dcc4c6f.svg", "tolerance": 1e-08, "verified": true}
{"case": "squintri@7", "density": 0.6083690864087284, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_display_metric_value": 5.154, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts7.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4 + 2 / \u221a3 = 5.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.1547005383792515, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "squintri@7", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "4 + 2 / \u221a3 = 5.154+", "metric_symbol": "s", "metric_value": 5.154, "reference_text": "s = 4 + 2 / \u221a3 = 5.154+ | Found by Erich Friedman | in 1997.", "source_image": "ts7.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts7.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 4 + 2 / \u221a3 = 5.154+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "b90420d34e1bd6e3", "improvement": -0.0001387881494867571, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.7967605582714725e-06, "max_pair_overlap_depth": -3.132565940933141e-05, "metric_expression": "5.15484", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.154839326528738, "n": 7, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.1547005383792515, "setup": "squintri", "side": 5.154839326528738, "solution_path": "data/solutions/generated/squintri_007_b90420d34e1bd6e3.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/b90420d34e1bd6e3.svg", "tolerance": 1e-08, "verified": true}
@@ -594,6 +630,7 @@
{"case": "squintri@17", "density": 0.7357616143895036, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_display_metric_value": 7.301, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts17.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "7.301+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.301, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "squintri@17", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "7.301+", "metric_symbol": "s", "metric_value": 7.301, "reference_text": "s = 7.301+ | Found by David W. Cantrell | in July 2002.", "source_image": "ts17.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts17.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 7.301+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "ts17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "3a2c987432da5f1d", "improvement": -0.0037537403553287163, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 2.4850519352755694e-10, "max_pair_overlap_depth": -2.8059876787889237e-05, "metric_expression": "7.30475", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 7.304753740355329, "n": 17, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 7.301, "setup": "squintri", "side": 7.304753740355329, "solution_path": "data/solutions/generated/squintri_017_3a2c987432da5f1d.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/3a2c987432da5f1d.svg", "tolerance": 1e-08, "verified": true}
{"case": "squintri@18", "density": 0.7780278532013658, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_display_metric_value": 7.309, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts18.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "5 + 4 / \u221a3 = 7.309+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.309401076758503, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "squintri@18", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "5 + 4 / \u221a3 = 7.309+", "metric_symbol": "s", "metric_value": 7.309, "reference_text": "s = 5 + 4 / \u221a3 = 7.309+ | Found by Erich Friedman | in 1997.", "source_image": "ts18.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts18.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 5 + 4 / \u221a3 = 7.309+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "e729e13f2fcc5856", "improvement": -0.00011056420509358844, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1021248752761939e-06, "max_pair_overlap_depth": -8.579145527987109e-06, "metric_expression": "7.30951", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 7.3095116409635965, "n": 18, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 7.309401076758503, "setup": "squintri", "side": 7.3095116409635965, "solution_path": "data/solutions/generated/squintri_018_e729e13f2fcc5856.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/e729e13f2fcc5856.svg", "tolerance": 1e-08, "verified": true}
{"case": "squintri@19", "density": 0.787537700321716, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_display_metric_value": 7.464, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts19.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4 + 2 \u221a3 = 7.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.464101615137754, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "squintri@19", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "4 + 2 \u221a3 = 7.464+", "metric_symbol": "s", "metric_value": 7.464, "reference_text": "s = 4 + 2 \u221a3 = 7.464+ | Found by Erich Friedman | in 1997.", "source_image": "ts19.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts19.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 4 + 2 \u221a3 = 7.464+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "805d1e50c1db2842", "improvement": -0.00022794589769237206, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.0609380559736765e-06, "max_pair_overlap_depth": -1.836020690809681e-05, "metric_expression": "7.46433", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 7.464329561035447, "n": 19, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 7.464101615137754, "setup": "squintri", "side": 7.464329561035447, "solution_path": "data/solutions/generated/squintri_019_805d1e50c1db2842.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/805d1e50c1db2842.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@20", "density": 0.7956651409816211, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_display_metric_value": 7.618, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts21.gif", "friedman_metric_expression": "3 + 8 / \u221a3 = 7.618+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.618802153517006, "friedman_metric_value_source": "exact_expression", "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"}, "friedman_reference_text": "s = 3 + 8 / \u221a3 = 7.618+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "33507e9f74d78419", "improvement": -0.00022487174892038553, "max_boundary_excess": -2.8035391721895233e-06, "max_pair_overlap_depth": -9.7266150509423e-06, "metric_expression": "7.61903", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 7.619027025265926, "monotone_upper_bound_from_case": "squintri@21", "monotone_upper_bound_from_record": "a2e24601c2557198", "n": 20, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 7.618802153517006, "setup": "squintri", "side": 7.619027025265926, "solution_path": "data/solutions/generated/squintri_020_33507e9f74d78419.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/33507e9f74d78419.svg", "tolerance": 1e-08, "verified": true}
{"case": "squintri@20", "density": 0.795657127614163, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_display_metric_value": 7.618, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts21.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "3 + 8 / \u221a3 = 7.618+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.618802153517006, "friedman_metric_value_source": "exact_expression", "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": 7.618, "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"}, "friedman_reference_text": "s = 3 + 8 / \u221a3 = 7.618+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "f4d21a73470ce742", "improvement": -0.00026323872093936274, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.382631857088383e-07, "max_pair_overlap_depth": -1.1470843414018717e-05, "metric_expression": "7.61907", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 7.619065392237945, "n": 20, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 7.618802153517006, "setup": "squintri", "side": 7.619065392237945, "solution_path": "data/solutions/generated/squintri_020_f4d21a73470ce742.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/f4d21a73470ce742.svg", "tolerance": 1e-08, "verified": true}
{"case": "squintri@21", "density": 0.8354483980307021, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_display_metric_value": 7.618, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts21.gif", "friedman_metric_expression": "3 + 8 / \u221a3 = 7.618+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.618802153517006, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "squintri@21", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "3 + 8 / \u221a3 = 7.618+", "metric_symbol": "s", "metric_value": 7.618, "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"}, "friedman_reference_text": "s = 3 + 8 / \u221a3 = 7.618+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "a2e24601c2557198", "improvement": -0.00022487174892038553, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.8035391721895233e-06, "max_pair_overlap_depth": -9.7266150509423e-06, "metric_expression": "7.61903", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 7.619027025265926, "n": 21, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 7.618802153517006, "setup": "squintri", "side": 7.619027025265926, "solution_path": "data/solutions/generated/squintri_021_a2e24601c2557198.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/a2e24601c2557198.svg", "tolerance": 1e-08, "verified": true}
{"case": "squintri@22", "density": 0.7639711289147475, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_display_metric_value": 8.154, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts22.gif", "friedman_metric_expression": "7 + 2 / \u221a3 = 8.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.154700538379252, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "squintri@22", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "7 + 2 / \u221a3 = 8.154+", "metric_symbol": "s", "metric_value": 8.154, "reference_text": "s = 7 + 2 / \u221a3 = 8.154+ | Found by Erich Friedman | in 1997.", "source_image": "ts22.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts22.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 7 + 2 / \u221a3 = 8.154+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "bbb028380d997350", "improvement": -0.0002728678276717744, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4334644744109685e-07, "max_pair_overlap_depth": -2.16343036094635e-05, "metric_expression": "8.15497", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 8.154973406206924, "n": 22, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 8.154700538379252, "setup": "squintri", "side": 8.154973406206924, "solution_path": "data/solutions/generated/squintri_022_bbb028380d997350.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/bbb028380d997350.svg", "tolerance": 1e-08, "verified": true}
@@ -604,12 +641,15 @@
{"case": "squintri@27", "density": 0.8380226358442442, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_display_metric_value": 8.618, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts27.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4 + 8 / \u221a3 = 8.618+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.618802153517006, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "squintri@27", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "4 + 8 / \u221a3 = 8.618+", "metric_symbol": "s", "metric_value": 8.618, "reference_text": "s = 4 + 8 / \u221a3 = 8.618+ | Found by Erich Friedman | in 1997.", "source_image": "ts27.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts27.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 4 + 8 / \u221a3 = 8.618+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts27.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "3c62cd2126467cec", "improvement": -0.007083352000387677, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.235025861662866e-06, "max_pair_overlap_depth": -2.0721793894562524e-05, "metric_expression": "8.62589", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 8.625885505517394, "n": 27, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 8.618802153517006, "setup": "squintri", "side": 8.625885505517394, "solution_path": "data/solutions/generated/squintri_027_3c62cd2126467cec.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/3c62cd2126467cec.svg", "tolerance": 1e-08, "verified": true}
{"case": "squintri@28", "density": 0.8400113660191094, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_display_metric_value": 8.773, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts28.gif", "friedman_metric_expression": "3 + 10 / \u221a3 = 8.773+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.773502691896258, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "squintri@28", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "3 + 10 / \u221a3 = 8.773+", "metric_symbol": "s", "metric_value": 8.773, "reference_text": "s = 3 + 10 / \u221a3 = 8.773+ | Found by Erich Friedman | in 1997.", "source_image": "ts28.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts28.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 3 + 10 / \u221a3 = 8.773+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "0d20b359435ed501", "improvement": -0.0002647025069464348, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.0145694937175628e-06, "max_pair_overlap_depth": -1.3941973455278145e-05, "metric_expression": "8.77377", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 8.773767394403205, "n": 28, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 8.773502691896258, "setup": "squintri", "side": 8.773767394403205, "solution_path": "data/solutions/generated/squintri_028_0d20b359435ed501.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/0d20b359435ed501.svg", "tolerance": 1e-08, "verified": true}
{"case": "squintri@29", "density": 0.7990846545133872, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_display_metric_value": 9.154, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts29.gif", "friedman_metric_expression": "8 + 2 / \u221a3 = 9.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.154700538379252, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "squintri@29", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "8 + 2 / \u221a3 = 9.154+", "metric_symbol": "s", "metric_value": 9.154, "reference_text": "s = 8 + 2 / \u221a3 = 9.154+ | Found by Erich Friedman | in 1997.", "source_image": "ts29.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts29.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 8 + 2 / \u221a3 = 9.154+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts29.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "3947aab39c0a1a92", "improvement": -0.00017165307172639643, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.2628787428294703e-06, "max_pair_overlap_depth": -1.875431036468811e-05, "metric_expression": "9.15487", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 9.154872191450979, "n": 29, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 9.154700538379252, "setup": "squintri", "side": 9.154872191450979, "solution_path": "data/solutions/generated/squintri_029_3947aab39c0a1a92.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/3947aab39c0a1a92.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@30", "density": 0.7993489079691232, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_display_metric_value": 9.301, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts30.gif", "friedman_metric_expression": "9.301+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.301, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "s = 9.301+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "ts30.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "58fcbf20aa7765b3", "improvement": -0.008837837927465841, "max_boundary_excess": -2.6472234336534766e-06, "max_pair_overlap_depth": -3.090204228461957e-05, "metric_expression": "9.30984", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 9.309837837927466, "monotone_upper_bound_from_case": "squintri@31", "monotone_upper_bound_from_record": "22b5f5f5707858bb", "n": 30, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 9.301, "setup": "squintri", "side": 9.309837837927466, "solution_path": "data/solutions/generated/squintri_030_58fcbf20aa7765b3.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/58fcbf20aa7765b3.svg", "tolerance": 1e-08, "verified": true}
{"case": "squintri@30", "density": 0.7992784153759376, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_display_metric_value": 9.301, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts30.gif", "friedman_metric_expression": "9.301+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.301, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "s = 9.301+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "ts30.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "e0b7505d8b6b1d46", "improvement": -0.009248370808917272, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.5690858985090003e-06, "max_pair_overlap_depth": -2.654303096472077e-05, "metric_expression": "9.31025", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 9.310248370808917, "n": 30, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 9.301, "setup": "squintri", "side": 9.310248370808917, "solution_path": "data/solutions/generated/squintri_030_e0b7505d8b6b1d46.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/e0b7505d8b6b1d46.svg", "tolerance": 1e-08, "verified": true}
{"case": "squintri@31", "density": 0.825993871568094, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_display_metric_value": 9.309, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts31.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "7 + 4 / \u221a3 = 9.309+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.309401076758503, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "squintri@31", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "7 + 4 / \u221a3 = 9.309+", "metric_symbol": "s", "metric_value": 9.309, "reference_text": "s = 7 + 4 / \u221a3 = 9.309+ | Found by Erich Friedman | in 1997.", "source_image": "ts31.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts31.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 7 + 4 / \u221a3 = 9.309+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "22b5f5f5707858bb", "improvement": -0.0004367611689630735, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6472234336534766e-06, "max_pair_overlap_depth": -3.090204228461957e-05, "metric_expression": "9.30984", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 9.309837837927466, "n": 31, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 9.309401076758503, "setup": "squintri", "side": 9.309837837927466, "solution_path": "data/solutions/generated/squintri_031_22b5f5f5707858bb.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/22b5f5f5707858bb.svg", "tolerance": 1e-08, "verified": true}
{"case": "squintri@32", "density": 0.8249717531277396, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_display_metric_value": 9.464, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts32.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "6 + 2 \u221a3 = 9.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.464101615137755, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "squintri@32", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "6 + 2 \u221a3 = 9.464+", "metric_symbol": "s", "metric_value": 9.464, "reference_text": "s = 6 + 2 \u221a3 = 9.464+ | Found by Erich Friedman | in 1997.", "source_image": "ts32.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts32.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 6 + 2 \u221a3 = 9.464+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "a44fbd99145b1ab3", "improvement": -0.0005608790583710999, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.1832282131171085e-07, "max_pair_overlap_depth": -1.699280632205813e-06, "metric_expression": "9.46466", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 9.464662494196126, "n": 32, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 9.464101615137755, "setup": "squintri", "side": 9.464662494196126, "solution_path": "data/solutions/generated/squintri_032_a44fbd99145b1ab3.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/a44fbd99145b1ab3.svg", "tolerance": 1e-08, "verified": true}
{"case": "squintri@33", "density": 0.8228028312642216, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_display_metric_value": 9.618, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts34.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "5 + 8 / \u221a3 = 9.618+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.618802153517006, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "squintri@33", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "5 + 8 / \u221a3 = 9.618+", "metric_symbol": "s", "metric_value": 9.618, "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"}, "friedman_reference_text": "s = 5 + 8 / \u221a3 = 9.618+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts34.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "67de1e08be2d150c", "improvement": -0.005267626054981989, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.226435414550563e-06, "max_pair_overlap_depth": -3.0944328952564604e-06, "metric_expression": "9.62407", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 9.624069779571988, "n": 33, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 9.618802153517006, "setup": "squintri", "side": 9.624069779571988, "solution_path": "data/solutions/generated/squintri_033_67de1e08be2d150c.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/67de1e08be2d150c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@33", "density": 0.8236315271035891, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_display_metric_value": 9.618, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts34.gif", "friedman_metric_expression": "5 + 8 / \u221a3 = 9.618+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.618802153517006, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "squintri@33", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "5 + 8 / \u221a3 = 9.618+", "metric_symbol": "s", "metric_value": 9.618802153517006, "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"}, "friedman_reference_text": "s = 5 + 8 / \u221a3 = 9.618+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts34.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "8d116d694ad5570a", "improvement": -0.0004247846581613146, "max_boundary_excess": -4.159334309452278e-06, "max_pair_overlap_depth": -1.741303470170763e-05, "metric_expression": "9.61923", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 9.619226938175167, "monotone_upper_bound_from_case": "squintri@34", "monotone_upper_bound_from_record": "3661302cdd9d0eb0", "n": 33, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 9.618802153517006, "setup": "squintri", "side": 9.619226938175167, "solution_path": "data/solutions/generated/squintri_033_8d116d694ad5570a.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/8d116d694ad5570a.svg", "tolerance": 1e-08, "verified": true}
{"case": "squintri@34", "density": 0.8485900582279403, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_display_metric_value": 9.618, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts34.gif", "friedman_metric_expression": "5 + 8 / \u221a3 = 9.618+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.618802153517006, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "squintri@34", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "5 + 8 / \u221a3 = 9.618+", "metric_symbol": "s", "metric_value": 9.618, "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"}, "friedman_reference_text": "s = 5 + 8 / \u221a3 = 9.618+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts34.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "3661302cdd9d0eb0", "improvement": -0.0004247846581613146, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.159334309452278e-06, "max_pair_overlap_depth": -1.741303470170763e-05, "metric_expression": "9.61923", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 9.619226938175167, "n": 34, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 9.618802153517006, "setup": "squintri", "side": 9.619226938175167, "solution_path": "data/solutions/generated/squintri_034_3661302cdd9d0eb0.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/3661302cdd9d0eb0.svg", "tolerance": 1e-08, "verified": true}
{"case": "squintri@35", "density": 0.8451911560111607, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_display_metric_value": 9.773, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts36.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4 + 10 / \u221a3 = 9.773+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.773502691896258, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "squintri@35", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "4 + 10 / \u221a3 = 9.773+", "metric_symbol": "s", "metric_value": 9.773, "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"}, "friedman_reference_text": "s = 4 + 10 / \u221a3 = 9.773+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "39d52a823b632afe", "improvement": -0.005762697591846333, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.195518233524095e-06, "max_pair_overlap_depth": -2.612964804571405e-05, "metric_expression": "9.77927", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 9.779265389488105, "n": 35, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 9.773502691896258, "setup": "squintri", "side": 9.779265389488105, "solution_path": "data/solutions/generated/squintri_035_39d52a823b632afe.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/39d52a823b632afe.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@35", "density": 0.8456540660893009, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_display_metric_value": 9.773, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts36.gif", "friedman_metric_expression": "4 + 10 / \u221a3 = 9.773+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.773502691896258, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "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"}, "friedman_reference_text": "s = 4 + 10 / \u221a3 = 9.773+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "454defabaa6b59ca", "improvement": -0.0030857518024500763, "max_boundary_excess": -1.9270562690497914e-06, "max_pair_overlap_depth": -7.45737725771356e-06, "metric_expression": "9.77659", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 9.776588443698708, "monotone_upper_bound_from_case": "squintri@36", "monotone_upper_bound_from_record": "f0473aaa767ebe98", "n": 35, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 9.773502691896258, "setup": "squintri", "side": 9.776588443698708, "solution_path": "data/solutions/generated/squintri_035_454defabaa6b59ca.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/454defabaa6b59ca.svg", "tolerance": 1e-08, "verified": true}
{"case": "squintri@36", "density": 0.8698156108347095, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_display_metric_value": 9.773, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts36.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4 + 10 / \u221a3 = 9.773+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.773502691896258, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "squintri@36", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "4 + 10 / \u221a3 = 9.773+", "metric_symbol": "s", "metric_value": 9.773, "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"}, "friedman_reference_text": "s = 4 + 10 / \u221a3 = 9.773+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "f0473aaa767ebe98", "improvement": -0.0030857518024500763, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.9270562690497914e-06, "max_pair_overlap_depth": -7.45737725771356e-06, "metric_expression": "9.77659", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 9.776588443698708, "n": 36, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 9.773502691896258, "setup": "squintri", "side": 9.776588443698708, "solution_path": "data/solutions/generated/squintri_036_f0473aaa767ebe98.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/f0473aaa767ebe98.svg", "tolerance": 1e-08, "verified": true}
{"case": "squintri@37", "density": 0.8285699034870297, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_display_metric_value": 10.154, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts37.gif", "friedman_metric_expression": "10.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 10.154, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "squintri@37", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "10.154+", "metric_symbol": "s", "metric_value": 10.154, "reference_text": "s = 10.154+ | Found by Emerson Connelly | in April 2026.", "source_image": "ts37.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts37.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 10.154+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "ts37.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "8530be000ac2e9ed", "improvement": -0.0011412138177444575, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.3566887422690367e-06, "max_pair_overlap_depth": -1.2239987045248313e-05, "metric_expression": "10.15514", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 10.155141213817744, "n": 37, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 10.154, "setup": "squintri", "side": 10.155141213817744, "solution_path": "data/solutions/generated/squintri_037_8530be000ac2e9ed.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/8530be000ac2e9ed.svg", "tolerance": 1e-08, "verified": true}
{"case": "squintri@38", "density": 0.8256293316995508, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_display_metric_value": 10.309, "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts38.gif", "friedman_metric_expression": "10.309+", "friedman_metric_symbol": "s", "friedman_metric_value": 10.309, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "squintri@38", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "10.309+", "metric_symbol": "s", "metric_value": 10.309, "reference_text": "s = 10.309+ | Found by Emerson Connelly | in April 2026.", "source_image": "ts38.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts38.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 10.309+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "ts38.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "896b636f9d899bf5", "improvement": -0.0007687369401132571, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.231911901047403e-06, "max_pair_overlap_depth": -1.6045122417374102e-05, "metric_expression": "10.30977", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 10.309768736940113, "n": 38, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 10.309, "setup": "squintri", "side": 10.309768736940113, "solution_path": "data/solutions/generated/squintri_038_896b636f9d899bf5.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/896b636f9d899bf5.svg", "tolerance": 1e-08, "verified": true}
@@ -619,6 +659,7 @@
{"case": "triincir@2", "density": 0.3675414068402789, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 0.866, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc2.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "\u221a3/2 = .866+", "friedman_metric_symbol": "r", "friedman_metric_value": 0.8660254037844386, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "triincir@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a3/2 = .866+", "metric_symbol": "r", "metric_value": 0.866, "reference_text": "r = \u221a3/2 = .866+ | Trivial.", "source_image": "tc2.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc2.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = \u221a3/2 = .866+ | Trivial.", "friedman_source_image": "tc2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "92f5511b689fe041", "improvement": -1.3183335940358809e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.072458078845486e-06, "max_pair_overlap_depth": -1.0890566688626091e-05, "metric_expression": "0.86604", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "r", "metric_value": 0.866038587120379, "n": 2, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 0.8660254037844386, "setup": "triincir", "side": 1.732077174240758, "solution_path": "data/solutions/generated/triincir_002_92f5511b689fe041.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/92f5511b689fe041.svg", "tolerance": 1e-08, "verified": true}
{"case": "triincir@3", "density": 0.45476420302551357, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_display_metric_value": 0.953, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc3.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "3\u221a3 - 3\u221a2 = .953+", "friedman_metric_symbol": "r", "friedman_metric_value": 0.9535117355873467, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "triincir@3", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "3\u221a3 - 3\u221a2 = .953+", "metric_symbol": "r", "metric_value": 0.953, "reference_text": "r = 3\u221a3 - 3\u221a2 = .953+ | Found by Erich Friedman | in 1997.", "source_image": "tc3.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc3.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 3\u221a3 - 3\u221a2 = .953+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "tc3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "6ede8225b5939550", "improvement": -3.6952980157156645e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4231654198736408e-06, "max_pair_overlap_depth": -2.0408752807413166e-05, "metric_expression": "0.95355", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "r", "metric_value": 0.9535486885675039, "n": 3, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 0.9535117355873467, "setup": "triincir", "side": 1.9070973771350077, "solution_path": "data/solutions/generated/triincir_003_6ede8225b5939550.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/6ede8225b5939550.svg", "tolerance": 1e-08, "verified": true}
{"case": "triincir@4", "density": 0.5732690550709103, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_display_metric_value": 0.98, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc4.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": ".980+", "friedman_metric_symbol": "r", "friedman_metric_value": 0.98, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "triincir@4", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": ".980+", "metric_symbol": "r", "metric_value": 0.98, "reference_text": "r = .980+ | Found by Erich Friedman | in 1997.", "source_image": "tc4.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc4.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = .980+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "tc4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "7f9481f33de5ecda", "improvement": -0.0006773112916812485, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.642177417024662e-06, "max_pair_overlap_depth": -2.677155749965807e-05, "metric_expression": "0.98068", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "r", "metric_value": 0.9806773112916812, "n": 4, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 0.98, "setup": "triincir", "side": 1.9613546225833625, "solution_path": "data/solutions/generated/triincir_004_7f9481f33de5ecda.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/7f9481f33de5ecda.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triincir@5", "density": 0.6891363628771783, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 1.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc5.gif", "friedman_metric_expression": "1", "friedman_metric_symbol": "r", "friedman_metric_value": 1.0, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "r = 1 | Trivial.", "friedman_source_image": "tc5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "a3d09cb09f1298d4", "improvement": -1.7961741001881748e-05, "max_boundary_excess": -4.861140867618019e-08, "max_pair_overlap_depth": -1.1418495247272897e-05, "metric_expression": "1.00002", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "r", "metric_value": 1.0000179617410019, "monotone_upper_bound_from_case": "triincir@6", "monotone_upper_bound_from_record": "fe93ba87f23c6c01", "n": 5, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 1.0, "setup": "triincir", "side": 2.0000359234820038, "solution_path": "data/solutions/generated/triincir_005_a3d09cb09f1298d4.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/a3d09cb09f1298d4.svg", "tolerance": 1e-08, "verified": true}
{"case": "triincir@5", "density": 0.6891272508887964, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 1.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc5.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1", "friedman_metric_symbol": "r", "friedman_metric_value": 1.0, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "r = 1 | Trivial.", "friedman_source_image": "tc5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "cf17d061ebfe392b", "improvement": -2.4573090296353683e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.800898255106432e-06, "max_pair_overlap_depth": -9.00554050570603e-06, "metric_expression": "1.00002", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "r", "metric_value": 1.0000245730902964, "n": 5, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 1.0, "setup": "triincir", "side": 2.0000491461805927, "solution_path": "data/solutions/generated/triincir_005_cf17d061ebfe392b.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/cf17d061ebfe392b.svg", "tolerance": 1e-08, "verified": true}
{"case": "triincir@6", "density": 0.826963635452614, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 1.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc6.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1", "friedman_metric_symbol": "r", "friedman_metric_value": 1.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triincir@6", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "r", "metric_value": 1.0, "reference_text": "r = 1 | Trivial.", "source_image": "tc6.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc6.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1 | Trivial.", "friedman_source_image": "tc6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "fe93ba87f23c6c01", "improvement": -1.7961741001881748e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.861140867618019e-08, "max_pair_overlap_depth": -9.037179045745058e-06, "metric_expression": "1.00002", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "r", "metric_value": 1.0000179617410019, "n": 6, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 1.0, "setup": "triincir", "side": 2.0000359234820038, "solution_path": "data/solutions/generated/triincir_006_fe93ba87f23c6c01.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/fe93ba87f23c6c01.svg", "tolerance": 1e-08, "verified": true}
{"case": "triincir@7", "density": 0.7265087780749782, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_display_metric_value": 1.152, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc7.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.152+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.152, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "triincir@7", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "1.152+", "metric_symbol": "r", "metric_value": 1.152, "reference_text": "r = 1.152+ | Found by Erich Friedman | in 1997.", "source_image": "tc7.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc7.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.152+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "tc7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "4d32cb6092e127b9", "improvement": -0.0004019079826915739, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.469968968942808e-06, "max_pair_overlap_depth": -4.662019983989185e-06, "metric_expression": "1.15240", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "r", "metric_value": 1.1524019079826915, "n": 7, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 1.152, "setup": "triincir", "side": 2.304803815965383, "solution_path": "data/solutions/generated/triincir_007_4d32cb6092e127b9.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/4d32cb6092e127b9.svg", "tolerance": 1e-08, "verified": true}
@@ -670,6 +711,7 @@
{"case": "triinhex@2", "density": 0.4444266358884555, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 0.866, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/2.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "\u221a3/2 = .866+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.8660254037844386, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a3/2 = .866+", "metric_symbol": "s", "metric_value": 0.866, "reference_text": "s = \u221a3/2 = .866+ | Trivial.", "source_image": "2.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/2.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = \u221a3/2 = .866+ | Trivial.", "friedman_source_image": "2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "132c3ed1fad59f87", "improvement": -1.735101606104017e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.75428637420658e-06, "max_pair_overlap_depth": -4.36783093215443e-06, "metric_expression": "0.86604", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 0.8660427548004996, "n": 2, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 0.8660254037844386, "setup": "triinhex", "side": 0.8660427548004996, "solution_path": "data/solutions/generated/triinhex_002_132c3ed1fad59f87.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/132c3ed1fad59f87.svg", "tolerance": 1e-08, "verified": true}
{"case": "triinhex@3", "density": 0.5108536175549566, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "April 2005", "friedman_display_metric_value": 0.989, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/3.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "\u221a((6+\u221a33)/12) = .989+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.9892995268765516, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@3", "credit": "Erich Friedman", "date": "April 2005", "family_updated": "", "metric_expression": "\u221a((6+\u221a33)/12) = .989+", "metric_symbol": "s", "metric_value": 0.989, "reference_text": "s = \u221a((6+\u221a33)/12) = .989+ | Found by Erich Friedman | in April 2005.", "source_image": "3.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/3.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = \u221a((6+\u221a33)/12) = .989+ | Found by Erich Friedman | in April 2005.", "friedman_source_image": "3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "3f19c46d9d07a54d", "improvement": -2.0420220901540276e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.001855317125196e-07, "max_pair_overlap_depth": -5.956334362088034e-06, "metric_expression": "0.98932", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 0.9893199470974532, "n": 3, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 0.9892995268765516, "setup": "triinhex", "side": 0.9893199470974532, "solution_path": "data/solutions/generated/triinhex_003_3f19c46d9d07a54d.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/3f19c46d9d07a54d.svg", "tolerance": 1e-08, "verified": true}
{"case": "triinhex@4", "density": 0.6666347131299533, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 1.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/4.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1", "friedman_metric_symbol": "s", "friedman_metric_value": 1.0, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "s = 1 | Trivial.", "friedman_source_image": "4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "9a8f72cbad0da23e", "improvement": -2.396601406218224e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.5130077190296347e-08, "max_pair_overlap_depth": -1.1324383645797624e-05, "metric_expression": "1.00002", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 1.0000239660140622, "n": 4, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 1.0, "setup": "triinhex", "side": 1.0000239660140622, "solution_path": "data/solutions/generated/triinhex_004_9a8f72cbad0da23e.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/9a8f72cbad0da23e.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@4", "density": 0.666641818440701, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 1.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/4.gif", "friedman_metric_expression": "1", "friedman_metric_symbol": "s", "friedman_metric_value": 1.0, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "s = 1 | Trivial.", "friedman_source_image": "4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "d2b5a3b06543cb98", "improvement": -1.863669045065741e-05, "max_boundary_excess": -1.0710555572355552e-06, "max_pair_overlap_depth": -2.545819687190834e-06, "metric_expression": "1.00002", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 1.0000186366904507, "monotone_upper_bound_from_case": "triinhex@5", "monotone_upper_bound_from_record": "0d61e48787b7d5d1", "n": 4, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 1.0, "setup": "triinhex", "side": 1.0000186366904507, "solution_path": "data/solutions/generated/triinhex_004_d2b5a3b06543cb98.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/d2b5a3b06543cb98.svg", "tolerance": 1e-08, "verified": true}
{"case": "triinhex@5", "density": 0.8333022730508762, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 1.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/5.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1", "friedman_metric_symbol": "s", "friedman_metric_value": 1.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@5", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "s", "metric_value": 1.0, "reference_text": "s = 1 | Trivial.", "source_image": "5.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/5.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 1 | Trivial.", "friedman_source_image": "5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "0d61e48787b7d5d1", "improvement": -1.863669045065741e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0710555572355552e-06, "max_pair_overlap_depth": -2.545819687190834e-06, "metric_expression": "1.00002", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 1.0000186366904507, "n": 5, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 1.0, "setup": "triinhex", "side": 1.0000186366904507, "solution_path": "data/solutions/generated/triinhex_005_0d61e48787b7d5d1.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/0d61e48787b7d5d1.svg", "tolerance": 1e-08, "verified": true}
{"case": "triinhex@6", "density": 0.9999558832137269, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 1.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/6.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1", "friedman_metric_symbol": "s", "friedman_metric_value": 1.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@6", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "s", "metric_value": 1.0, "reference_text": "s = 1 | Trivial.", "source_image": "6.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/6.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 1 | Trivial.", "friedman_source_image": "6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "163de15143701be8", "improvement": -2.2059123022444282e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.751347439371281e-07, "max_pair_overlap_depth": -4.053173147537176e-06, "metric_expression": "1.00002", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 1.0000220591230224, "n": 6, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 1.0, "setup": "triinhex", "side": 1.0000220591230224, "solution_path": "data/solutions/generated/triinhex_006_163de15143701be8.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/163de15143701be8.svg", "tolerance": 1e-08, "verified": true}
{"case": "triinhex@7", "density": 0.7149966699512036, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "May 2005", "friedman_display_metric_value": 1.277, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/7.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.277+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.277, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@7", "credit": "David W. Cantrell", "date": "May 2005", "family_updated": "", "metric_expression": "1.277+", "metric_symbol": "s", "metric_value": 1.277, "reference_text": "s = 1.277+ | Found by David W. Cantrell | in May 2005.", "source_image": "7.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/7.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 1.277+ | Found by David W. Cantrell | in May 2005.", "friedman_source_image": "7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "af96c8dfd69688ff", "improvement": -0.0003837447071250999, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.8980834118002716e-06, "max_pair_overlap_depth": -1.0312174508100846e-05, "metric_expression": "1.27738", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 1.277383744707125, "n": 7, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 1.277, "setup": "triinhex", "side": 1.277383744707125, "solution_path": "data/solutions/generated/triinhex_007_af96c8dfd69688ff.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/af96c8dfd69688ff.svg", "tolerance": 1e-08, "verified": true}
@@ -700,6 +742,7 @@
{"case": "triinhex@31", "density": 0.896119877899338, "friedman_analytical_or_proved": true, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_display_metric_value": 2.4, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/31.gif", "friedman_metric_expression": "12/5 = 2.400", "friedman_metric_symbol": "s", "friedman_metric_value": 2.4, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@31", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "12/5 = 2.400", "metric_symbol": "s", "metric_value": 2.4, "reference_text": "s = 12/5 = 2.400 | Found by Emerson Connelly | in April 2026.", "source_image": "31.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/31.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 12/5 = 2.400 | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "cae7087451040120", "improvement": -0.0011658949471424762, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.9409739826535097e-07, "max_pair_overlap_depth": -2.8541100203227643e-06, "metric_expression": "2.40117", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.4011658949471424, "n": 31, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.4, "setup": "triinhex", "side": 2.4011658949471424, "solution_path": "data/solutions/generated/triinhex_31_cae7087451040120.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/cae7087451040120.svg", "tolerance": 1e-08, "verified": true}
{"case": "triinhex@32", "density": 0.8767870241121234, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_display_metric_value": 2.465, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/32.gif", "friedman_metric_expression": "2.465+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.465, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@32", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "2.465+", "metric_symbol": "s", "metric_value": 2.465, "reference_text": "s = 2.465+ | Found by Emerson Connelly | in April 2026.", "source_image": "32.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/32.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 2.465+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "8674581436e5e74d", "improvement": -0.0013363682337788063, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.925122041572899e-07, "max_pair_overlap_depth": -1.2807310418028184e-06, "metric_expression": "2.46634", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.4663363682337787, "n": 32, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.465, "setup": "triinhex", "side": 2.4663363682337787, "solution_path": "data/solutions/generated/triinhex_032_8674581436e5e74d.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/8674581436e5e74d.svg", "tolerance": 1e-08, "verified": true}
{"case": "triinhex@33", "density": 0.8799443949567897, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_display_metric_value": 2.498, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/33.gif", "friedman_metric_expression": "2.498+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.498, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "s = 2.498+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "33.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "5b836a27d2bb958c", "improvement": -0.00207898817969987, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.8634496562140136e-06, "max_pair_overlap_depth": -5.829272115620565e-06, "metric_expression": "2.50008", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.5000789881797, "n": 33, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.498, "setup": "triinhex", "side": 2.5000789881797, "solution_path": "data/solutions/generated/triinhex_033_5b836a27d2bb958c.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/5b836a27d2bb958c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@33", "density": 0.879945195455047, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_display_metric_value": 2.498, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/33.gif", "friedman_metric_expression": "2.498+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.498, "friedman_metric_value_source": "reported_decimal", "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"}, "friedman_reference_text": "s = 2.498+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "33.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "a1101d95622ba18c", "improvement": -0.0020778510013057527, "max_boundary_excess": -1.2753913921730298e-06, "max_pair_overlap_depth": -7.960301433351802e-06, "metric_expression": "2.50008", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.500077851001306, "monotone_upper_bound_from_case": "triinhex@34", "monotone_upper_bound_from_record": "a712744ff3b5bcc9", "n": 33, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.498, "setup": "triinhex", "side": 2.500077851001306, "solution_path": "data/solutions/generated/triinhex_033_a1101d95622ba18c.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/a1101d95622ba18c.svg", "tolerance": 1e-08, "verified": true}
{"case": "triinhex@34", "density": 0.9066102013779274, "friedman_analytical_or_proved": true, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_display_metric_value": 2.5, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/35.gif", "friedman_metric_expression": "5/2 = 2.500", "friedman_metric_symbol": "s", "friedman_metric_value": 2.5, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@34", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "5/2 = 2.500", "metric_symbol": "s", "metric_value": 2.5, "reference_text": "s = 5/2 = 2.500 | Found by Emerson Connelly | in April 2026.", "source_image": "35.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/35.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 5/2 = 2.500 | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "35.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "a712744ff3b5bcc9", "improvement": -7.7851001305973e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.2753913921730298e-06, "max_pair_overlap_depth": -7.960301433351802e-06, "metric_expression": "2.50008", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.500077851001306, "n": 34, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.5, "setup": "triinhex", "side": 2.500077851001306, "solution_path": "data/solutions/generated/triinhex_034_a712744ff3b5bcc9.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/a712744ff3b5bcc9.svg", "tolerance": 1e-08, "verified": true}
{"case": "triinhex@35", "density": 0.93316535041203, "friedman_analytical_or_proved": true, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_display_metric_value": 2.5, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/35.gif", "friedman_metric_expression": "5/2 = 2.500", "friedman_metric_symbol": "s", "friedman_metric_value": 2.5, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@35", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "5/2 = 2.500", "metric_symbol": "s", "metric_value": 2.5, "reference_text": "s = 5/2 = 2.500 | Found by Emerson Connelly | in April 2026.", "source_image": "35.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/35.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 5/2 = 2.500 | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "35.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "ec49778cf450c37b", "improvement": -0.00022500750012577342, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6207066397040256e-06, "max_pair_overlap_depth": -1.5837731757728513e-06, "metric_expression": "2.50023", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.5002250075001258, "n": 35, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.5, "setup": "triinhex", "side": 2.5002250075001258, "solution_path": "data/solutions/generated/triinhex_35_ec49778cf450c37b.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/ec49778cf450c37b.svg", "tolerance": 1e-08, "verified": true}
{"case": "triinhex@36", "density": 0.8942711196026824, "friedman_analytical_or_proved": false, "friedman_credit": "Nathan Roll", "friedman_date": "April 2026", "friedman_display_metric_value": 2.59123, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/36.gif", "friedman_metric_expression": "2.59123+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.59123, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@36", "credit": "Nathan Roll", "date": "April 2026", "family_updated": "", "metric_expression": "2.59123+", "metric_symbol": "s", "metric_value": 2.59123, "reference_text": "s = 2.59123+ | Found by Nathan Roll | in April 2026.", "source_image": "36.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/36.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 2.59123+ | Found by Nathan Roll | in April 2026.", "friedman_source_image": "36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "d30ec5c19b8369f3", "improvement": 0.000983935961939686, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.744185746811731e-07, "max_pair_overlap_depth": -1.519675368655271e-06, "metric_expression": "2.59025", "metric_recovery_tolerance_abs": 1e-05, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.5902460640380602, "n": 36, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.59123, "setup": "triinhex", "side": 2.5902460640380602, "solution_path": "data/solutions/record_hunt/triinhex_36_d30ec5c19b8369f3.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/d30ec5c19b8369f3.svg", "tolerance": 1e-08, "verified": true}
@@ -824,42 +867,44 @@
{"case": "triinsqu@38", "density": 0.87940001283733, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_display_metric_value": 4.327, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t38.gif", "friedman_metric_expression": "4.327+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.327, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@38", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.327+", "metric_symbol": "s", "metric_value": 4.327, "reference_text": "38. | | s = 4.327+ | Found by Emerson Connelly | in April 2026.", "source_image": "t38.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t38.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "38. | | s = 4.327+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "t38.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "f44a4bd75616bc73", "improvement": 0.0013748413573368978, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.0707071410413964e-07, "max_pair_overlap_depth": -1.0916553592199563e-06, "metric_expression": "4.32563", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.325625158642663, "n": 38, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.327, "setup": "triinsqu", "side": 4.325625158642663, "solution_path": "data/solutions/record_hunt/triinsqu_38_f44a4bd75616bc73.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/f44a4bd75616bc73.svg", "tolerance": 1e-08, "verified": true}
{"case": "triinsqu@39", "density": 0.880118042721229, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_display_metric_value": 4.383, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t39.gif", "friedman_metric_expression": "4.383+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.383, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@39", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.383+", "metric_symbol": "s", "metric_value": 4.383, "reference_text": "39. | | s = 4.383+ | Found by Emerson Connelly | in April 2026.", "source_image": "t39.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t39.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "39. | | s = 4.383+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "t39.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "263f7876a25d297b", "improvement": 0.002616247848735931, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.607883972834501e-07, "max_pair_overlap_depth": -1.86255866918561e-06, "metric_expression": "4.38038", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.380383752151264, "n": 39, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.383, "setup": "triinsqu", "side": 4.380383752151264, "solution_path": "data/solutions/record_hunt/triinsqu_39_263f7876a25d297b.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/263f7876a25d297b.svg", "tolerance": 1e-08, "verified": true}
{"case": "triinsqu@40", "density": 0.878868813713771, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_display_metric_value": 4.437, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t40.gif", "friedman_metric_expression": "4.437+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.437, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@40", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.437+", "metric_symbol": "s", "metric_value": 4.437, "reference_text": "40. | | s = 4.437+ | Found by Emerson Connelly | in April 2026.", "source_image": "t40.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t40.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "40. | | s = 4.437+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "t40.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "ccf2360cc8917adc", "improvement": -0.002338759632142029, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.520888573265893e-06, "max_pair_overlap_depth": -1.6780429565876886e-05, "metric_expression": "4.43934", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.439338759632142, "n": 40, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.437, "setup": "triinsqu", "side": 4.439338759632142, "solution_path": "data/solutions/generated/triinsqu_040_ccf2360cc8917adc.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/ccf2360cc8917adc.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@1", "density": 1.0, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 1.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt1.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "1", "friedman_metric_symbol": "s", "friedman_metric_value": 1.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "s", "metric_value": 1.0, "reference_text": "s = 1 | Trivial.", "source_image": "tt1.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt1.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 1 | Trivial.", "friedman_source_image": "tt1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "b09a5ce4a2c03ad3", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 5.551115123125788e-17, "max_pair_overlap_depth": 0.0, "metric_expression": "1.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 1.0, "n": 1, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 1.0, "setup": "triintri", "side": 1.0, "solution_path": "data/solutions/generated/triintri_001_b09a5ce4a2c03ad3.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:08:37+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/b09a5ce4a2c03ad3.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@2", "density": 0.5, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 2.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt4.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "s = 2 | Trivial.", "source_image": "tt4.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt4.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 2 | Trivial.", "friedman_source_image": "tt4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "2d6f654db11412d8", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 1.6653345369377346e-16, "max_pair_overlap_depth": 0.0, "metric_expression": "2.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.0, "n": 2, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.0, "setup": "triintri", "side": 2.0, "solution_path": "data/solutions/generated/triintri_002_2d6f654db11412d8.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:08:37+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/2d6f654db11412d8.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@3", "density": 0.75, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 2.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt4.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@3", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "s = 2 | Trivial.", "source_image": "tt4.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt4.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 2 | Trivial.", "friedman_source_image": "tt4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "76381d42f8f17f41", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 1.6653345369377346e-16, "max_pair_overlap_depth": 3.3306690738754696e-16, "metric_expression": "2.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.0, "n": 3, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.0, "setup": "triintri", "side": 2.0, "solution_path": "data/solutions/generated/triintri_003_76381d42f8f17f41.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:08:37+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/76381d42f8f17f41.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@4", "density": 1.0, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 2.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt4.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@4", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "s = 2 | Trivial.", "source_image": "tt4.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt4.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 2 | Trivial.", "friedman_source_image": "tt4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "be1b0b4b1802d8fa", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 1.6653345369377346e-16, "max_pair_overlap_depth": 3.3306690738754696e-16, "metric_expression": "2.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.0, "n": 4, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.0, "setup": "triintri", "side": 2.0, "solution_path": "data/solutions/generated/triintri_004_be1b0b4b1802d8fa.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:08:37+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/be1b0b4b1802d8fa.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@1", "density": 1.0, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 1.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt1.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "1", "friedman_metric_symbol": "s", "friedman_metric_value": 1.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "s", "metric_value": 1.0, "reference_text": "s = 1 | Trivial.", "source_image": "tt1.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt1.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 1 | Trivial.", "friedman_source_image": "tt1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "b09a5ce4a2c03ad3", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 5.551115123125788e-17, "max_pair_overlap_depth": 0.0, "metric_expression": "1.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 1.0, "n": 1, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 1.0, "setup": "triintri", "side": 1.0, "solution_path": "data/solutions/generated/triintri_001_b09a5ce4a2c03ad3.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:15:00+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/b09a5ce4a2c03ad3.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@2", "density": 0.5, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 2.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt4.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "s = 2 | Trivial.", "source_image": "tt4.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt4.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 2 | Trivial.", "friedman_source_image": "tt4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "2d6f654db11412d8", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 1.6653345369377346e-16, "max_pair_overlap_depth": 0.0, "metric_expression": "2.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.0, "n": 2, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.0, "setup": "triintri", "side": 2.0, "solution_path": "data/solutions/generated/triintri_002_2d6f654db11412d8.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:15:00+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/2d6f654db11412d8.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@3", "density": 0.75, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 2.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt4.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@3", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "s = 2 | Trivial.", "source_image": "tt4.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt4.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 2 | Trivial.", "friedman_source_image": "tt4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "76381d42f8f17f41", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 1.6653345369377346e-16, "max_pair_overlap_depth": 3.3306690738754696e-16, "metric_expression": "2.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.0, "n": 3, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.0, "setup": "triintri", "side": 2.0, "solution_path": "data/solutions/generated/triintri_003_76381d42f8f17f41.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:15:00+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/76381d42f8f17f41.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@4", "density": 1.0, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 2.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt4.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@4", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "s = 2 | Trivial.", "source_image": "tt4.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt4.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 2 | Trivial.", "friedman_source_image": "tt4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "be1b0b4b1802d8fa", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 1.6653345369377346e-16, "max_pair_overlap_depth": 3.3306690738754696e-16, "metric_expression": "2.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.0, "n": 4, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.0, "setup": "triintri", "side": 2.0, "solution_path": "data/solutions/generated/triintri_004_be1b0b4b1802d8fa.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:15:00+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/be1b0b4b1802d8fa.svg", "tolerance": 1e-08, "verified": true}
{"case": "triintri@5", "density": 0.66984024663198, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_display_metric_value": 2.732, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt5.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "1 + \u221a3 = 2.732+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.732050807568877, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@5", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "1 + \u221a3 = 2.732+", "metric_symbol": "s", "metric_value": 2.732, "reference_text": "s = 1 + \u221a3 = 2.732+ | Proved by Erich Friedman | in 1997.", "source_image": "tt5.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt5.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 1 + \u221a3 = 2.732+ | Proved by Erich Friedman | in 1997.", "friedman_source_image": "tt5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "b4ac2c13e986485e", "improvement": -6.675552642354532e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.644640909821311e-06, "max_pair_overlap_depth": -6.265774057961264e-06, "metric_expression": "2.73212", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.7321175630953007, "n": 5, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.732050807568877, "setup": "triintri", "side": 2.7321175630953007, "solution_path": "data/solutions/generated/triintri_005_b4ac2c13e986485e.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/b4ac2c13e986485e.svg", "tolerance": 1e-08, "verified": true}
{"case": "triintri@6", "density": 0.6768918015337515, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "August 2008", "friedman_display_metric_value": 2.977, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt6.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "13/8 + 3\u221a13/8 = 2.977+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.977081728298996, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@6", "credit": "Maurizio Morandi", "date": "August 2008", "family_updated": "", "metric_expression": "13/8 + 3\u221a13/8 = 2.977+", "metric_symbol": "s", "metric_value": 2.977, "reference_text": "s = 13/8 + 3\u221a13/8 = 2.977+ | Found by Maurizio Morandi | in August 2008.", "source_image": "tt6.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt6.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 13/8 + 3\u221a13/8 = 2.977+ | Found by Maurizio Morandi | in August 2008.", "friedman_source_image": "tt6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "df20f3104adbd6a1", "improvement": -0.0001730311672321072, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.3999156171376625e-07, "max_pair_overlap_depth": -1.9147124426771578e-05, "metric_expression": "2.97725", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 2.977254759466228, "n": 6, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 2.977081728298996, "setup": "triintri", "side": 2.977254759466228, "solution_path": "data/solutions/generated/triintri_006_df20f3104adbd6a1.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/df20f3104adbd6a1.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@7", "density": 0.777777777777778, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 3.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt9.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@7", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "s = 3 | Trivial.", "source_image": "tt9.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt9.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 3 | Trivial.", "friedman_source_image": "tt9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "355f218b5ea40614", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 2.9605947323337506e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "3.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.0, "n": 7, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.0, "setup": "triintri", "side": 3.0, "solution_path": "data/solutions/generated/triintri_007_355f218b5ea40614.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:08:37+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/355f218b5ea40614.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@8", "density": 0.8888888888888892, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 3.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt9.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@8", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "s = 3 | Trivial.", "source_image": "tt9.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt9.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 3 | Trivial.", "friedman_source_image": "tt9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "43e3200cf32d77a9", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 2.9605947323337506e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "3.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.0, "n": 8, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.0, "setup": "triintri", "side": 3.0, "solution_path": "data/solutions/generated/triintri_008_43e3200cf32d77a9.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:08:37+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/43e3200cf32d77a9.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@9", "density": 1.0000000000000004, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 3.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt9.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@9", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "s = 3 | Trivial.", "source_image": "tt9.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt9.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 3 | Trivial.", "friedman_source_image": "tt9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "4242c6a66848e1ea", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 2.9605947323337516e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "3.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.0, "n": 9, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.0, "setup": "triintri", "side": 3.0, "solution_path": "data/solutions/generated/triintri_009_4242c6a66848e1ea.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:08:37+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/4242c6a66848e1ea.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@7", "density": 0.777777777777778, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 3.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt9.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@7", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "s = 3 | Trivial.", "source_image": "tt9.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt9.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 3 | Trivial.", "friedman_source_image": "tt9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "355f218b5ea40614", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 2.9605947323337506e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "3.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.0, "n": 7, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.0, "setup": "triintri", "side": 3.0, "solution_path": "data/solutions/generated/triintri_007_355f218b5ea40614.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:15:00+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/355f218b5ea40614.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@8", "density": 0.8888888888888892, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 3.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt9.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@8", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "s = 3 | Trivial.", "source_image": "tt9.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt9.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 3 | Trivial.", "friedman_source_image": "tt9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "43e3200cf32d77a9", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 2.9605947323337506e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "3.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.0, "n": 8, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.0, "setup": "triintri", "side": 3.0, "solution_path": "data/solutions/generated/triintri_008_43e3200cf32d77a9.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:15:00+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/43e3200cf32d77a9.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@9", "density": 1.0000000000000004, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 3.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt9.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@9", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "s = 3 | Trivial.", "source_image": "tt9.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt9.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 3 | Trivial.", "friedman_source_image": "tt9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "4242c6a66848e1ea", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 2.9605947323337516e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "3.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.0, "n": 9, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.0, "setup": "triintri", "side": 3.0, "solution_path": "data/solutions/generated/triintri_009_4242c6a66848e1ea.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:15:00+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/4242c6a66848e1ea.svg", "tolerance": 1e-08, "verified": true}
{"case": "triintri@10", "density": 0.8162770652591925, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_display_metric_value": 3.5, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt10.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "7/2 = 3.500", "friedman_metric_symbol": "s", "friedman_metric_value": 3.5, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@10", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "7/2 = 3.500", "metric_symbol": "s", "metric_value": 3.5, "reference_text": "s = 7/2 = 3.500 | Found by Erich Friedman | in 1997.", "source_image": "tt10.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt10.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 7/2 = 3.500 | Found by Erich Friedman | in 1997.", "friedman_source_image": "tt10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "bcf7fa93627e2e22", "improvement": -0.00010604617003595607, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.918936789115156e-08, "max_pair_overlap_depth": -1.2701382657187565e-05, "metric_expression": "3.50011", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.500106046170036, "n": 10, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.5, "setup": "triintri", "side": 3.500106046170036, "solution_path": "data/solutions/generated/triintri_010_bcf7fa93627e2e22.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/bcf7fa93627e2e22.svg", "tolerance": 1e-08, "verified": true}
{"case": "triintri@11", "density": 0.792568600317471, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_display_metric_value": 3.722, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt11.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "9/4 + 9\u221a21/28 = 3.722+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.7229707590929486, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@11", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "9/4 + 9\u221a21/28 = 3.722+", "metric_symbol": "s", "metric_value": 3.722, "reference_text": "s = 9/4 + 9\u221a21/28 = 3.722+ | Found by Erich Friedman | in 1997.", "source_image": "tt11.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt11.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 9/4 + 9\u221a21/28 = 3.722+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "tt11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "8c37202a929dc807", "improvement": -0.002472140302997694, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.7247990718051686e-06, "max_pair_overlap_depth": -2.4227159337045223e-05, "metric_expression": "3.72544", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.7254428993959463, "n": 11, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.7229707590929486, "setup": "triintri", "side": 3.7254428993959463, "solution_path": "data/solutions/generated/triintri_11_8c37202a929dc807.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/8c37202a929dc807.svg", "tolerance": 1e-08, "verified": true}
{"case": "triintri@12", "density": 0.7960259433336867, "friedman_analytical_or_proved": true, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2007", "friedman_display_metric_value": 3.879, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt12.gif", "friedman_metric_expression": "2 + 2cos(\u03c0/9) = 3.879+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.879385241571817, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@12", "credit": "David W. Cantrell", "date": "July 2007", "family_updated": "", "metric_expression": "2 + 2cos(\u03c0/9) = 3.879+", "metric_symbol": "s", "metric_value": 3.879, "reference_text": "s = 2 + 2cos(\u03c0/9) = 3.879+ | Found by David W. Cantrell | in July 2007.", "source_image": "tt12.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt12.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 2 + 2cos(\u03c0/9) = 3.879+ | Found by David W. Cantrell | in July 2007.", "friedman_source_image": "tt12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "7642b05605706588", "improvement": -0.00325375295575947, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.358547413541538e-08, "max_pair_overlap_depth": -4.590975868801195e-06, "metric_expression": "3.88264", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.8826389945275763, "n": 12, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.879385241571817, "setup": "triintri", "side": 3.8826389945275763, "solution_path": "data/solutions/generated/triintri_12_7642b05605706588.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T07:13:53+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/7642b05605706588.svg", "tolerance": 1e-08, "verified": true}
{"case": "triintri@13", "density": 0.8146483132163195, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "May 2008", "friedman_display_metric_value": 3.992, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt13.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "3.992+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.992, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "triintri@13", "credit": "Maurizio Morandi", "date": "May 2008", "family_updated": "", "metric_expression": "3.992+", "metric_symbol": "s", "metric_value": 3.992, "reference_text": "s = 3.992+ | Found by Maurizio Morandi | in May 2008.", "source_image": "tt13.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt13.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 3.992+ | Found by Maurizio Morandi | in May 2008.", "friedman_source_image": "tt13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "afe3e3dc6de1bc36", "improvement": -0.0027223080586638027, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.48838412272174e-06, "max_pair_overlap_depth": -3.9197094850562664e-05, "metric_expression": "3.99472", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 3.994722308058664, "n": 13, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 3.992, "setup": "triintri", "side": 3.994722308058664, "solution_path": "data/solutions/generated/triintri_013_afe3e3dc6de1bc36.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/afe3e3dc6de1bc36.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@14", "density": 0.875, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 4.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt16.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@14", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "s = 4 | Trivial.", "source_image": "tt16.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt16.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 4 | Trivial.", "friedman_source_image": "tt16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "8676384ad0301449", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 2.775557561562891e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "4.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.0, "n": 14, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.0, "setup": "triintri", "side": 4.0, "solution_path": "data/solutions/generated/triintri_014_8676384ad0301449.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:08:37+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/8676384ad0301449.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@15", "density": 0.9375, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 4.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt16.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@15", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "s = 4 | Trivial.", "source_image": "tt16.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt16.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 4 | Trivial.", "friedman_source_image": "tt16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "31db3ec76cc0421f", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 2.775557561562891e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "4.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.0, "n": 15, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.0, "setup": "triintri", "side": 4.0, "solution_path": "data/solutions/generated/triintri_015_31db3ec76cc0421f.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:08:37+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/31db3ec76cc0421f.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@16", "density": 1.0, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 4.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt16.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@16", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "s = 4 | Trivial.", "source_image": "tt16.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt16.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 4 | Trivial.", "friedman_source_image": "tt16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "2ffa329b602032d6", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 2.775557561562891e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "4.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.0, "n": 16, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.0, "setup": "triintri", "side": 4.0, "solution_path": "data/solutions/generated/triintri_016_2ffa329b602032d6.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:08:37+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/2ffa329b602032d6.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@14", "density": 0.875, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 4.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt16.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@14", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "s = 4 | Trivial.", "source_image": "tt16.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt16.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 4 | Trivial.", "friedman_source_image": "tt16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "8676384ad0301449", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 2.775557561562891e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "4.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.0, "n": 14, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.0, "setup": "triintri", "side": 4.0, "solution_path": "data/solutions/generated/triintri_014_8676384ad0301449.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:15:00+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/8676384ad0301449.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@15", "density": 0.9375, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 4.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt16.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@15", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "s = 4 | Trivial.", "source_image": "tt16.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt16.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 4 | Trivial.", "friedman_source_image": "tt16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "31db3ec76cc0421f", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 2.775557561562891e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "4.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.0, "n": 15, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.0, "setup": "triintri", "side": 4.0, "solution_path": "data/solutions/generated/triintri_015_31db3ec76cc0421f.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:15:00+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/31db3ec76cc0421f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@16", "density": 1.0, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 4.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt16.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@16", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "s = 4 | Trivial.", "source_image": "tt16.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt16.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 4 | Trivial.", "friedman_source_image": "tt16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "2ffa329b602032d6", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 2.775557561562891e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "4.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.0, "n": 16, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.0, "setup": "triintri", "side": 4.0, "solution_path": "data/solutions/generated/triintri_016_2ffa329b602032d6.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:15:00+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/2ffa329b602032d6.svg", "tolerance": 1e-08, "verified": true}
{"case": "triintri@17", "density": 0.8524249732795084, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "May 2008", "friedman_display_metric_value": 4.465, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt17.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "4.465+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.465, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "triintri@17", "credit": "Maurizio Morandi", "date": "May 2008", "family_updated": "", "metric_expression": "4.465+", "metric_symbol": "s", "metric_value": 4.465, "reference_text": "s = 4.465+ | Found by Maurizio Morandi | in May 2008.", "source_image": "tt17.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt17.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 4.465+ | Found by Maurizio Morandi | in May 2008.", "friedman_source_image": "tt17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "ad8aac0e4c06f7d0", "improvement": -0.0007702721912288268, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.8241570152012174e-06, "max_pair_overlap_depth": -7.389272764501964e-06, "metric_expression": "4.46577", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.465770272191229, "n": 17, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.465, "setup": "triintri", "side": 4.465770272191229, "solution_path": "data/solutions/generated/triintri_017_ad8aac0e4c06f7d0.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/ad8aac0e4c06f7d0.svg", "tolerance": 1e-08, "verified": true}
{"case": "triintri@18", "density": 0.8888203671574908, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_display_metric_value": 4.5, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt18.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "9/2 = 4.500", "friedman_metric_symbol": "s", "friedman_metric_value": 4.5, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@18", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "9/2 = 4.500", "metric_symbol": "s", "metric_value": 4.5, "reference_text": "s = 9/2 = 4.500 | Found by Erich Friedman | in 1997.", "source_image": "tt18.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt18.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 9/2 = 4.500 | Found by Erich Friedman | in 1997.", "friedman_source_image": "tt18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "b4cfb14015aca75a", "improvement": -0.00017345566104154386, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.575746594607548e-06, "max_pair_overlap_depth": -5.0550449839370515e-06, "metric_expression": "4.50017", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.5001734556610415, "n": 18, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.5, "setup": "triintri", "side": 4.5001734556610415, "solution_path": "data/solutions/generated/triintri_018_b4cfb14015aca75a.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/b4cfb14015aca75a.svg", "tolerance": 1e-08, "verified": true}
{"case": "triintri@19", "density": 0.872322152073677, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "May 2008", "friedman_display_metric_value": 4.666, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt19.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "14/3 = 4.666+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.666666666666667, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@19", "credit": "Maurizio Morandi", "date": "May 2008", "family_updated": "", "metric_expression": "14/3 = 4.666+", "metric_symbol": "s", "metric_value": 4.666, "reference_text": "s = 14/3 = 4.666+ | Found by Maurizio Morandi | in May 2008.", "source_image": "tt19.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt19.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 14/3 = 4.666+ | Found by Maurizio Morandi | in May 2008.", "friedman_source_image": "tt19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "bf4e89c9b4dd8ae2", "improvement": -0.0003392326096340881, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.331642222161759e-07, "max_pair_overlap_depth": -9.690072424628493e-06, "metric_expression": "4.66701", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.667005899276301, "n": 19, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.666666666666667, "setup": "triintri", "side": 4.667005899276301, "solution_path": "data/solutions/generated/triintri_019_bf4e89c9b4dd8ae2.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/bf4e89c9b4dd8ae2.svg", "tolerance": 1e-08, "verified": true}
{"case": "triintri@20", "density": 0.845304928058783, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "May 2008", "friedman_display_metric_value": 4.861, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt20.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "33/8 + 9\u221a21/56 = 4.861+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.861485379546474, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@20", "credit": "Maurizio Morandi", "date": "May 2008", "family_updated": "", "metric_expression": "33/8 + 9\u221a21/56 = 4.861+", "metric_symbol": "s", "metric_value": 4.861, "reference_text": "s = 33/8 + 9\u221a21/56 = 4.861+ | Found by Maurizio Morandi | in May 2008.", "source_image": "tt20.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt20.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 33/8 + 9\u221a21/56 = 4.861+ | Found by Maurizio Morandi | in May 2008.", "friedman_source_image": "tt20.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "4dff2c76e1273753", "improvement": -0.0026796085219045196, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.619234998604407e-06, "max_pair_overlap_depth": -7.278337082583519e-06, "metric_expression": "4.86416", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.864164988068379, "n": 20, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.861485379546474, "setup": "triintri", "side": 4.864164988068379, "solution_path": "data/solutions/generated/triintri_020_4dff2c76e1273753.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/4dff2c76e1273753.svg", "tolerance": 1e-08, "verified": true}
{"case": "triintri@21", "density": 0.8650098201747314, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "May 2008", "friedman_display_metric_value": 4.923, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt21.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "4.923+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.923, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "triintri@21", "credit": "Maurizio Morandi", "date": "May 2008", "family_updated": "", "metric_expression": "4.923+", "metric_symbol": "s", "metric_value": 4.923, "reference_text": "s = 4.923+ | Found by Maurizio Morandi | in May 2008.", "source_image": "tt21.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt21.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 4.923+ | Found by Maurizio Morandi | in May 2008.", "friedman_source_image": "tt21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "4dc62e9e9421c0aa", "improvement": -0.004187943781052361, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4013824985741632e-06, "max_pair_overlap_depth": -4.2136074186321926e-06, "metric_expression": "4.92719", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 4.927187943781052, "n": 21, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.923, "setup": "triintri", "side": 4.927187943781052, "solution_path": "data/solutions/generated/triintri_021_4dc62e9e9421c0aa.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/4dc62e9e9421c0aa.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@22", "density": 0.8800000000000002, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "May 2008", "friedman_display_metric_value": 4.996, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt22.gif", "friedman_metric_expression": "4.996+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.996, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "triintri@22", "credit": "Maurizio Morandi", "date": "May 2008", "family_updated": "", "metric_expression": "4.996+", "metric_symbol": "s", "metric_value": 4.996, "reference_text": "s = 4.996+ | Found by Maurizio Morandi | in May 2008.", "source_image": "tt22.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt22.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 4.996+ | Found by Maurizio Morandi | in May 2008.", "friedman_source_image": "tt22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "16c9dbca1fb3bede", "improvement": -0.0039999999999995595, "max_boundary_excess": 7.105427357601002e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "5.00000", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.0, "monotone_upper_bound_from_case": "triintri@25", "monotone_upper_bound_from_record": "9c4806b8b37c5d37", "n": 22, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.996, "setup": "triintri", "side": 5.0, "solution_path": "data/solutions/generated/triintri_022_16c9dbca1fb3bede.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/16c9dbca1fb3bede.svg", "tolerance": 1e-08, "verified": true}
{"case": "triintri@22", "density": 0.8799245341930954, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "May 2008", "friedman_display_metric_value": 4.996, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt22.gif", "friedman_metric_expression": "4.996+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.996, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "triintri@22", "credit": "Maurizio Morandi", "date": "May 2008", "family_updated": "", "metric_expression": "4.996+", "metric_symbol": "s", "metric_value": 4.996, "reference_text": "s = 4.996+ | Found by Maurizio Morandi | in May 2008.", "source_image": "tt22.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt22.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 4.996+ | Found by Maurizio Morandi | in May 2008.", "friedman_source_image": "tt22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "2122d833c6fcab70", "improvement": -0.004214405286988132, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.380108740330726e-08, "max_pair_overlap_depth": -8.335028905959163e-07, "metric_expression": "5.00021", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.000214405286989, "n": 22, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 4.996, "setup": "triintri", "side": 5.000214405286989, "solution_path": "data/solutions/generated/triintri_022_2122d833c6fcab70.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/2122d833c6fcab70.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@23", "density": 0.9200000000000002, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 5.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt25.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@23", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "s = 5 | Trivial.", "source_image": "tt25.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt25.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 5 | Trivial.", "friedman_source_image": "tt25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "4f1bdd5d6c07f589", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 7.105427357601002e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "5.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.0, "n": 23, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.0, "setup": "triintri", "side": 5.0, "solution_path": "data/solutions/generated/triintri_023_4f1bdd5d6c07f589.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:08:37+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/4f1bdd5d6c07f589.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@24", "density": 0.9600000000000002, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 5.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt25.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@24", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "s = 5 | Trivial.", "source_image": "tt25.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt25.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 5 | Trivial.", "friedman_source_image": "tt25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "64d77fbf13ae0be7", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 7.105427357601002e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "5.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.0, "n": 24, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.0, "setup": "triintri", "side": 5.0, "solution_path": "data/solutions/generated/triintri_024_64d77fbf13ae0be7.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:08:37+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/64d77fbf13ae0be7.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@25", "density": 1.0000000000000002, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 5.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt25.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@25", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "s = 5 | Trivial.", "source_image": "tt25.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt25.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 5 | Trivial.", "friedman_source_image": "tt25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "9c4806b8b37c5d37", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 7.105427357601002e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "5.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.0, "n": 25, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.0, "setup": "triintri", "side": 5.0, "solution_path": "data/solutions/generated/triintri_025_9c4806b8b37c5d37.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:08:37+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/9c4806b8b37c5d37.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@23", "density": 0.9200000000000002, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 5.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt25.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@23", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "s = 5 | Trivial.", "source_image": "tt25.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt25.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 5 | Trivial.", "friedman_source_image": "tt25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "4f1bdd5d6c07f589", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 7.105427357601002e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "5.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.0, "n": 23, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.0, "setup": "triintri", "side": 5.0, "solution_path": "data/solutions/generated/triintri_023_4f1bdd5d6c07f589.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:15:00+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/4f1bdd5d6c07f589.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@24", "density": 0.9600000000000002, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 5.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt25.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@24", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "s = 5 | Trivial.", "source_image": "tt25.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt25.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 5 | Trivial.", "friedman_source_image": "tt25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "64d77fbf13ae0be7", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 7.105427357601002e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "5.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.0, "n": 24, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.0, "setup": "triintri", "side": 5.0, "solution_path": "data/solutions/generated/triintri_024_64d77fbf13ae0be7.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:15:00+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/64d77fbf13ae0be7.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@25", "density": 1.0000000000000002, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 5.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt25.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@25", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "s = 5 | Trivial.", "source_image": "tt25.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt25.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 5 | Trivial.", "friedman_source_image": "tt25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "9c4806b8b37c5d37", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 7.105427357601002e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "5.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.0, "n": 25, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.0, "setup": "triintri", "side": 5.0, "solution_path": "data/solutions/generated/triintri_025_9c4806b8b37c5d37.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:15:00+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/9c4806b8b37c5d37.svg", "tolerance": 1e-08, "verified": true}
{"case": "triintri@26", "density": 0.8884053356293824, "friedman_analytical_or_proved": false, "friedman_credit": "Joah Frenzley", "friedman_date": "April 2026", "friedman_display_metric_value": 5.406, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt26.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "5.406+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.406, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "triintri@26", "credit": "Joah Frenzley", "date": "April 2026", "family_updated": "", "metric_expression": "5.406+", "metric_symbol": "s", "metric_value": 5.406, "reference_text": "s = 5.406+ | Found by Joah Frenzley | in April 2026.", "source_image": "tt26.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt26.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 5.406+ | Found by Joah Frenzley | in April 2026.", "friedman_source_image": "tt26.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "0af5a3765ecc4828", "improvement": -0.0037985715923722907, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.296466577480461e-06, "max_pair_overlap_depth": -2.0400569136769242e-05, "metric_expression": "5.40980", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.409798571592372, "n": 26, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.406, "setup": "triintri", "side": 5.409798571592372, "solution_path": "data/solutions/generated/triintri_026_0af5a3765ecc4828.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Joah Frenzley", "svg_path": "data/svg/0af5a3765ecc4828.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@27", "density": 0.8925073688875825, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_display_metric_value": 5.5, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt28.gif", "friedman_metric_expression": "11/2 = 5.500", "friedman_metric_symbol": "s", "friedman_metric_value": 5.5, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@27", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "11/2 = 5.500", "metric_symbol": "s", "metric_value": 5.5, "reference_text": "s = 11/2 = 5.500 | Found by Erich Friedman | in 1997.", "source_image": "tt28.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt28.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 11/2 = 5.500 | Found by Erich Friedman | in 1997.", "friedman_source_image": "tt28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "70128a68bb429fb5", "improvement": -0.00016827626559567932, "max_boundary_excess": -2.296352348617567e-06, "max_pair_overlap_depth": -7.147352432124165e-08, "metric_expression": "5.50017", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.500168276265596, "monotone_upper_bound_from_case": "triintri@28", "monotone_upper_bound_from_record": "9562d7d6e075a347", "n": 27, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.5, "setup": "triintri", "side": 5.500168276265596, "solution_path": "data/solutions/generated/triintri_027_70128a68bb429fb5.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:14:49+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/70128a68bb429fb5.svg", "tolerance": 1e-08, "verified": true}
{"case": "triintri@27", "density": 0.8924996192164427, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_display_metric_value": 5.5, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt28.gif", "friedman_metric_expression": "11/2 = 5.500", "friedman_metric_symbol": "s", "friedman_metric_value": 5.5, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@27", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "11/2 = 5.500", "metric_symbol": "s", "metric_value": 5.5, "reference_text": "s = 11/2 = 5.500 | Found by Erich Friedman | in 1997.", "source_image": "tt28.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt28.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 11/2 = 5.500 | Found by Erich Friedman | in 1997.", "friedman_source_image": "tt28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "f1c2be61e829f1e2", "improvement": -0.00019215549305418023, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.8365186496628e-06, "max_pair_overlap_depth": -1.8064988390742442e-06, "metric_expression": "5.50019", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.500192155493054, "n": 27, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.5, "setup": "triintri", "side": 5.500192155493054, "solution_path": "data/solutions/generated/triintri_027_f1c2be61e829f1e2.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/f1c2be61e829f1e2.svg", "tolerance": 1e-08, "verified": true}
{"case": "triintri@28", "density": 0.9255631973649003, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_display_metric_value": 5.5, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt28.gif", "friedman_metric_expression": "11/2 = 5.500", "friedman_metric_symbol": "s", "friedman_metric_value": 5.5, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@28", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "11/2 = 5.500", "metric_symbol": "s", "metric_value": 5.5, "reference_text": "s = 11/2 = 5.500 | Found by Erich Friedman | in 1997.", "source_image": "tt28.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt28.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 11/2 = 5.500 | Found by Erich Friedman | in 1997.", "friedman_source_image": "tt28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "9562d7d6e075a347", "improvement": -0.00016827626559567932, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.296352348617567e-06, "max_pair_overlap_depth": -7.147352432124165e-08, "metric_expression": "5.50017", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.500168276265596, "n": 28, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.5, "setup": "triintri", "side": 5.500168276265596, "solution_path": "data/solutions/generated/triintri_028_9562d7d6e075a347.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/9562d7d6e075a347.svg", "tolerance": 1e-08, "verified": true}
{"case": "triintri@29", "density": 0.9023512759650716, "friedman_analytical_or_proved": true, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2007", "friedman_display_metric_value": 5.666, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt29.gif", "friedman_metric_expression": "17/3 = 5.666+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.666666666666667, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@29", "credit": "David W. Cantrell", "date": "July 2007", "family_updated": "", "metric_expression": "17/3 = 5.666+", "metric_symbol": "s", "metric_value": 5.666, "reference_text": "s = 17/3 = 5.666+ | Found by David W. Cantrell | in July 2007.", "source_image": "tt29.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt29.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 17/3 = 5.666+ | Found by David W. Cantrell | in July 2007.", "friedman_source_image": "tt29.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "38c7a009f9059275", "improvement": -0.0023949920687824644, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.9275914294015247e-06, "max_pair_overlap_depth": -5.088073942505211e-06, "metric_expression": "5.66906", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.669061658735449, "n": 29, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.666666666666667, "setup": "triintri", "side": 5.669061658735449, "solution_path": "data/solutions/generated/triintri_29_38c7a009f9059275.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T13:47:12+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/38c7a009f9059275.svg", "tolerance": 1e-08, "verified": true}
{"case": "triintri@30", "density": 0.9060985977127352, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "May 2008", "friedman_display_metric_value": 5.75, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt30.gif", "friedman_metric_expression": "23/4 = 5.75", "friedman_metric_symbol": "s", "friedman_metric_value": 5.75, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@30", "credit": "Maurizio Morandi", "date": "May 2008", "family_updated": "", "metric_expression": "23/4 = 5.75", "metric_symbol": "s", "metric_value": 5.75, "reference_text": "s = 23/4 = 5.75 | Found by Maurizio Morandi | in May 2008.", "source_image": "tt30.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt30.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 23/4 = 5.75 | Found by Maurizio Morandi | in May 2008.", "friedman_source_image": "tt30.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "149a91e0248df04e", "improvement": -0.004040286099913359, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.2882428961650933e-06, "max_pair_overlap_depth": -6.414942533283785e-06, "metric_expression": "5.75404", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.754040286099913, "n": 30, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.75, "setup": "triintri", "side": 5.754040286099913, "solution_path": "data/solutions/generated/triintri_30_149a91e0248df04e.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/149a91e0248df04e.svg", "tolerance": 1e-08, "verified": true}
{"case": "triintri@31", "density": 0.8852772627280384, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_display_metric_value": 5.912, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt31.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "5.912+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.912, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "triintri@31", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "5.912+", "metric_symbol": "s", "metric_value": 5.912, "reference_text": "s = 5.912+ | Found by Emerson Connelly | in April 2026.", "source_image": "tt31.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt31.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 5.912+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "tt31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "c729ec2b23a32942", "improvement": -0.005539858919105356, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.9513696602846134e-06, "max_pair_overlap_depth": -3.7790737070775293e-06, "metric_expression": "5.91754", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.917539858919105, "n": 31, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.912, "setup": "triintri", "side": 5.917539858919105, "solution_path": "data/solutions/generated/triintri_031_c729ec2b23a32942.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T02:51:36+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/c729ec2b23a32942.svg", "tolerance": 1e-08, "verified": true}
{"case": "triintri@32", "density": 0.9021138463561595, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_display_metric_value": 5.951, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt32.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "5.951+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.951, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "triintri@32", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "5.951+", "metric_symbol": "s", "metric_value": 5.951, "reference_text": "s = 5.951+ | Found by Ian Watson | in April 2026.", "source_image": "tt32.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt32.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 5.951+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "tt32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "fa640909a0850f97", "improvement": -0.004857726462081935, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.7950535548187937e-06, "max_pair_overlap_depth": -4.900081183589755e-06, "metric_expression": "5.95586", "metric_recovery_tolerance_abs": 0.001, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 5.955857726462082, "n": 32, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 5.951, "setup": "triintri", "side": 5.955857726462082, "solution_path": "data/solutions/generated/triintri_032_fa640909a0850f97.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/fa640909a0850f97.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@33", "density": 0.9166666666666669, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 6.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "friedman_metric_expression": "6", "friedman_metric_symbol": "s", "friedman_metric_value": 6.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@33", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "6", "metric_symbol": "s", "metric_value": 6.0, "reference_text": "s = 6 | Trivial. |", "source_image": "tt36.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 6 | Trivial. |", "friedman_source_image": "tt36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "34a5032aebc12aff", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 5.921189464667501e-16, "max_pair_overlap_depth": 6.661338147750939e-16, "metric_expression": "6.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 6.0, "n": 33, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 6.0, "setup": "triintri", "side": 6.0, "solution_path": "data/solutions/generated/triintri_033_34a5032aebc12aff.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:08:37+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/34a5032aebc12aff.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@34", "density": 0.9444444444444448, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 6.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "friedman_metric_expression": "6", "friedman_metric_symbol": "s", "friedman_metric_value": 6.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@34", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "6", "metric_symbol": "s", "metric_value": 6.0, "reference_text": "s = 6 | Trivial. |", "source_image": "tt36.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 6 | Trivial. |", "friedman_source_image": "tt36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "a2ccd0a19cfa1b75", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 5.921189464667501e-16, "max_pair_overlap_depth": 6.661338147750939e-16, "metric_expression": "6.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 6.0, "n": 34, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 6.0, "setup": "triintri", "side": 6.0, "solution_path": "data/solutions/generated/triintri_034_a2ccd0a19cfa1b75.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:08:37+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/a2ccd0a19cfa1b75.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@35", "density": 0.9722222222222225, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 6.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "friedman_metric_expression": "6", "friedman_metric_symbol": "s", "friedman_metric_value": 6.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@35", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "6", "metric_symbol": "s", "metric_value": 6.0, "reference_text": "s = 6 | Trivial. |", "source_image": "tt36.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 6 | Trivial. |", "friedman_source_image": "tt36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "6325f936448cdf94", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 5.921189464667503e-16, "max_pair_overlap_depth": 6.661338147750939e-16, "metric_expression": "6.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 6.0, "n": 35, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 6.0, "setup": "triintri", "side": 6.0, "solution_path": "data/solutions/generated/triintri_035_6325f936448cdf94.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:08:37+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/6325f936448cdf94.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@36", "density": 1.0000000000000004, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 6.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "friedman_metric_expression": "6", "friedman_metric_symbol": "s", "friedman_metric_value": 6.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@36", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "6", "metric_symbol": "s", "metric_value": 6.0, "reference_text": "s = 6 | Trivial. |", "source_image": "tt36.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 6 | Trivial. |", "friedman_source_image": "tt36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "4b90836bf07f8a65", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 6.661338147750939e-16, "max_pair_overlap_depth": 6.661338147750939e-16, "metric_expression": "6.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 6.0, "n": 36, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 6.0, "setup": "triintri", "side": 6.0, "solution_path": "data/solutions/generated/triintri_036_4b90836bf07f8a65.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:08:37+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/4b90836bf07f8a65.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@33", "density": 0.9166666666666669, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 6.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "friedman_metric_expression": "6", "friedman_metric_symbol": "s", "friedman_metric_value": 6.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@33", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "6", "metric_symbol": "s", "metric_value": 6.0, "reference_text": "s = 6 | Trivial. |", "source_image": "tt36.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 6 | Trivial. |", "friedman_source_image": "tt36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "34a5032aebc12aff", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 5.921189464667501e-16, "max_pair_overlap_depth": 6.661338147750939e-16, "metric_expression": "6.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 6.0, "n": 33, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 6.0, "setup": "triintri", "side": 6.0, "solution_path": "data/solutions/generated/triintri_033_34a5032aebc12aff.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:15:00+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/34a5032aebc12aff.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@34", "density": 0.9444444444444448, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 6.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "friedman_metric_expression": "6", "friedman_metric_symbol": "s", "friedman_metric_value": 6.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@34", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "6", "metric_symbol": "s", "metric_value": 6.0, "reference_text": "s = 6 | Trivial. |", "source_image": "tt36.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 6 | Trivial. |", "friedman_source_image": "tt36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "a2ccd0a19cfa1b75", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 5.921189464667501e-16, "max_pair_overlap_depth": 6.661338147750939e-16, "metric_expression": "6.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 6.0, "n": 34, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 6.0, "setup": "triintri", "side": 6.0, "solution_path": "data/solutions/generated/triintri_034_a2ccd0a19cfa1b75.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:15:00+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/a2ccd0a19cfa1b75.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@35", "density": 0.9722222222222225, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 6.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "friedman_metric_expression": "6", "friedman_metric_symbol": "s", "friedman_metric_value": 6.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@35", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "6", "metric_symbol": "s", "metric_value": 6.0, "reference_text": "s = 6 | Trivial. |", "source_image": "tt36.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 6 | Trivial. |", "friedman_source_image": "tt36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "6325f936448cdf94", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 5.921189464667503e-16, "max_pair_overlap_depth": 6.661338147750939e-16, "metric_expression": "6.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 6.0, "n": 35, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 6.0, "setup": "triintri", "side": 6.0, "solution_path": "data/solutions/generated/triintri_035_6325f936448cdf94.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:15:00+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/6325f936448cdf94.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@36", "density": 1.0000000000000004, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_display_metric_value": 6.0, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "friedman_metric_expression": "6", "friedman_metric_symbol": "s", "friedman_metric_value": 6.0, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@36", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "6", "metric_symbol": "s", "metric_value": 6.0, "reference_text": "s = 6 | Trivial. |", "source_image": "tt36.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 6 | Trivial. |", "friedman_source_image": "tt36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "4b90836bf07f8a65", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 6.661338147750939e-16, "max_pair_overlap_depth": 6.661338147750939e-16, "metric_expression": "6.00000", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 6.0, "n": 36, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 6.0, "setup": "triintri", "side": 6.0, "solution_path": "data/solutions/generated/triintri_036_4b90836bf07f8a65.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T18:15:00+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/4b90836bf07f8a65.svg", "tolerance": 1e-08, "verified": true}
{"case": "triintri@37", "density": 0.9223731676430976, "friedman_analytical_or_proved": true, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_display_metric_value": 6.333, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt37.gif", "friedman_metric_expression": "19/3 = 6.333+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.333333333333333, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@37", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "19/3 = 6.333+", "metric_symbol": "s", "metric_value": 6.333, "reference_text": "s = 19/3 = 6.333+ | Found by Ian Watson | in April 2026.", "source_image": "tt37.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt37.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 19/3 = 6.333+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "tt37.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "a824891b660555f5", "improvement": -0.00022145462571288022, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.1058084789851627e-06, "max_pair_overlap_depth": -1.0503324543442716e-06, "metric_expression": "6.33355", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 6.333554787959046, "n": 37, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 6.333333333333333, "setup": "triintri", "side": 6.333554787959046, "solution_path": "data/solutions/generated/triintri_037_a824891b660555f5.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/a824891b660555f5.svg", "tolerance": 1e-08, "verified": true}
{"case": "triintri@38", "density": 0.9184187453437659, "friedman_analytical_or_proved": false, "friedman_credit": "Nathan Roll", "friedman_date": "April 2026", "friedman_display_metric_value": 6.43454, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt38.gif", "friedman_metric_expression": "6.43454+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.43454, "friedman_metric_value_source": "reported_decimal", "friedman_reference": {"analytical_or_proved": false, "case": "triintri@38", "credit": "Nathan Roll", "date": "April 2026", "family_updated": "", "metric_expression": "6.43454+", "metric_symbol": "s", "metric_value": 6.43454, "reference_text": "s = 6.43454+ | Found by Nathan Roll | in April 2026.", "source_image": "tt38.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt38.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 6.43454+ | Found by Nathan Roll | in April 2026.", "friedman_source_image": "tt38.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "b1563d97c893e28d", "improvement": 0.002163914058151306, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.8714774987883973e-07, "max_pair_overlap_depth": -1.6883301962034025e-06, "metric_expression": "6.43238", "metric_recovery_tolerance_abs": 1e-05, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 6.432376085941849, "n": 38, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 6.43454, "setup": "triintri", "side": 6.432376085941849, "solution_path": "data/solutions/record_hunt/triintri_38_b1563d97c893e28d.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/b1563d97c893e28d.svg", "tolerance": 1e-08, "verified": true}
{"case": "triintri@39", "density": 0.922995893638428, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "April 2026", "friedman_display_metric_value": 6.5, "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt40.gif", "friedman_metric_expression": "13/2 = 6.500", "friedman_metric_symbol": "s", "friedman_metric_value": 6.5, "friedman_metric_value_source": "exact_expression", "friedman_reference": {"analytical_or_proved": true, "case": "triintri@39", "credit": "Erich Friedman", "date": "April 2026", "family_updated": "", "metric_expression": "13/2 = 6.500", "metric_symbol": "s", "metric_value": 6.5, "reference_text": "s = 13/2 = 6.500 | Found by Erich Friedman | in April 2026.", "source_image": "tt40.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt40.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 13/2 = 6.500 | Found by Erich Friedman | in April 2026.", "friedman_source_image": "tt40.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "5ff069bd30790b7f", "improvement": -0.0002853099319573005, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.7141701112233934e-06, "max_pair_overlap_depth": -2.524988351737534e-07, "metric_expression": "6.50029", "metric_recovery_tolerance_abs": 1e-08, "metric_recovery_tolerance_rel": 0.001, "metric_symbol": "s", "metric_value": 6.500285309931957, "n": 39, "notes": "Verified coordinate reconstruction credited to the Friedman-listed finder; the exact evaluator metric is within the benchmark recovery band.", "record_type": "verified", "reference_side": 6.5, "setup": "triintri", "side": 6.500285309931957, "solution_path": "data/solutions/generated/triintri_039_5ff069bd30790b7f.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/5ff069bd30790b7f.svg", "tolerance": 1e-08, "verified": true}
diff --git a/data/solutions/generated/cirinhex_006_3e5660c8b6310c57.json b/data/solutions/generated/cirinhex_006_3e5660c8b6310c57.json
new file mode 100644
index 0000000000000000000000000000000000000000..c0278c6a7277de9e480de67374a8a4a9054676fd
--- /dev/null
+++ b/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/"
+}
diff --git a/data/solutions/generated/cirintri_002_befbb923c21388e1.json b/data/solutions/generated/cirintri_002_befbb923c21388e1.json
new file mode 100644
index 0000000000000000000000000000000000000000..c605303bbf2c4c87b6ec4b6da6e1bec9e95ca0e8
--- /dev/null
+++ b/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/"
+}
diff --git a/data/solutions/generated/cirintri_005_64d422ca71110b0d.json b/data/solutions/generated/cirintri_005_64d422ca71110b0d.json
new file mode 100644
index 0000000000000000000000000000000000000000..c14467d5c8711e0d6f77d62f4d67d0fc5cfba04f
--- /dev/null
+++ b/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/"
+}
diff --git a/data/solutions/generated/domindom_007_a480ce1181b75833.json b/data/solutions/generated/domindom_007_a480ce1181b75833.json
new file mode 100644
index 0000000000000000000000000000000000000000..02f043bc562ae65254cdc0d5cba6671d0a249ed4
--- /dev/null
+++ b/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/"
+}
diff --git a/data/solutions/generated/domindom_008_a4f6452b97bb5cc0.json b/data/solutions/generated/domindom_008_a4f6452b97bb5cc0.json
new file mode 100644
index 0000000000000000000000000000000000000000..c3f5a47bcc45f639ef3cda8871a1758c40f4b14f
--- /dev/null
+++ b/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/"
+}
diff --git a/data/solutions/generated/domindom_014_6f795311544960ee.json b/data/solutions/generated/domindom_014_6f795311544960ee.json
new file mode 100644
index 0000000000000000000000000000000000000000..fa8a5a658211ef33f4bdc25838305032e8a5fb86
--- /dev/null
+++ b/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": [
+ {
+ "id": 1,
+ "rotation_radians": -4.6972688014620373e-07,
+ "x": -3.000061323109711,
+ "y": -1.5000301648343772
+ },
+ {
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+ "rotation_radians": 2.2826471184247226e-06,
+ "x": -1.000021531822367,
+ "y": -1.5000330720011108
+ },
+ {
+ "id": 3,
+ "rotation_radians": 1.8921356915879417e-06,
+ "x": 1.0000165752754013,
+ "y": -1.5000326159910415
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+ {
+ "id": 4,
+ "rotation_radians": 7.630129294169187e-07,
+ "x": 3.0000597693379247,
+ "y": -1.500031885010241
+ },
+ {
+ "id": 5,
+ "rotation_radians": -1.790783288515767e-06,
+ "x": -3.0000428974193705,
+ "y": -0.5000096770564719
+ },
+ {
+ "id": 6,
+ "rotation_radians": -2.7351338446214696e-07,
+ "x": -0.9999956531073101,
+ "y": -0.500008129526475
+ },
+ {
+ "id": 7,
+ "rotation_radians": -1.6237729528698304e-06,
+ "x": 1.0000413402407151,
+ "y": -0.5000088554865022
+ },
+ {
+ "id": 8,
+ "rotation_radians": -2.223663485252913e-06,
+ "x": 3.0000715669140843,
+ "y": -0.500009200115029
+ },
+ {
+ "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
+ },
+ {
+ "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
+ },
+ {
+ "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/"
+}
diff --git a/data/solutions/generated/domindom_015_dda9136316a32e15.json b/data/solutions/generated/domindom_015_dda9136316a32e15.json
new file mode 100644
index 0000000000000000000000000000000000000000..7dd810e809f832912fbae3dc6c98089ec6564a6a
--- /dev/null
+++ b/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,
+ "y": -1.5000301648343772
+ },
+ {
+ "id": 2,
+ "rotation_radians": 2.2826471184247226e-06,
+ "x": -1.000021531822367,
+ "y": -1.5000330720011108
+ },
+ {
+ "id": 3,
+ "rotation_radians": 1.8921356915879417e-06,
+ "x": 1.0000165752754013,
+ "y": -1.5000326159910415
+ },
+ {
+ "id": 4,
+ "rotation_radians": 7.630129294169187e-07,
+ "x": 3.0000597693379247,
+ "y": -1.500031885010241
+ },
+ {
+ "id": 5,
+ "rotation_radians": -1.790783288515767e-06,
+ "x": -3.0000428974193705,
+ "y": -0.5000096770564719
+ },
+ {
+ "id": 6,
+ "rotation_radians": -2.7351338446214696e-07,
+ "x": -0.9999956531073101,
+ "y": -0.500008129526475
+ },
+ {
+ "id": 7,
+ "rotation_radians": -1.6237729528698304e-06,
+ "x": 1.0000413402407151,
+ "y": -0.5000088554865022
+ },
+ {
+ "id": 8,
+ "rotation_radians": -2.223663485252913e-06,
+ "x": 3.0000715669140843,
+ "y": -0.500009200115029
+ },
+ {
+ "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
+ },
+ {
+ "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
+ },
+ {
+ "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
+ }
+ ],
+ "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/"
+}
diff --git a/data/solutions/generated/hexincir_006_0d33963cb4bcb842.json b/data/solutions/generated/hexincir_006_0d33963cb4bcb842.json
new file mode 100644
index 0000000000000000000000000000000000000000..a96de755e2aff4eb442f99ec0a3f97c0e9d71ab3
--- /dev/null
+++ b/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
+ },
+ {
+ "id": 2,
+ "rotation_radians": 3.1914712204700044,
+ "x": 0.08633526573246775,
+ "y": -1.7299351830412515
+ },
+ {
+ "id": 3,
+ "rotation_radians": 4.238668164264897,
+ "x": 1.5413350295213695,
+ "y": -0.7901966101691185
+ },
+ {
+ "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/"
+}
diff --git a/data/solutions/generated/hexincir_018_3bf05115c19cc6d5.json b/data/solutions/generated/hexincir_018_3bf05115c19cc6d5.json
new file mode 100644
index 0000000000000000000000000000000000000000..ce69a716f73c365c59666930c941b0a954813322
--- /dev/null
+++ b/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
+ },
+ {
+ "id": 2,
+ "rotation_radians": 2.0984018494288867,
+ "x": -1.6677528334111735,
+ "y": -2.915577274740411
+ },
+ {
+ "id": 3,
+ "rotation_radians": 1.0512116602031218,
+ "x": 0.013891971052751357,
+ "y": -3.464152351237459
+ },
+ {
+ "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/"
+}
diff --git a/data/solutions/generated/hexinhex_002_c8ccd859ecc810e4.json b/data/solutions/generated/hexinhex_002_c8ccd859ecc810e4.json
new file mode 100644
index 0000000000000000000000000000000000000000..d20d02ab6cffead5fb7f0be0576c02ae14526c95
--- /dev/null
+++ b/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/"
+}
diff --git a/data/solutions/generated/hexinhex_006_93e84dfdad8f0036.json b/data/solutions/generated/hexinhex_006_93e84dfdad8f0036.json
new file mode 100644
index 0000000000000000000000000000000000000000..b35fe2443a9213c11e437480569a0cc5710e1966
--- /dev/null
+++ b/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/"
+}
diff --git a/data/solutions/generated/hexinhex_018_2eb7c5be1ce30256.json b/data/solutions/generated/hexinhex_018_2eb7c5be1ce30256.json
new file mode 100644
index 0000000000000000000000000000000000000000..8e92533ef6532751568673b55d9e55710adfa1b6
--- /dev/null
+++ b/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/"
+}
diff --git a/data/solutions/generated/hexinhex_020_197dfdbe3f1f6905.json b/data/solutions/generated/hexinhex_020_197dfdbe3f1f6905.json
new file mode 100644
index 0000000000000000000000000000000000000000..16c709c22f644fc9b686f143912196b3789b04f7
--- /dev/null
+++ b/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
+ },
+ {
+ "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,
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+ {
+ "id": 11,
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+ {
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+ "rotation_radians": 5.977223461539629e-08,
+ "x": 2.500039054190333,
+ "y": -0.8660052621280638
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+ {
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+ "rotation_radians": -1.3478963937982246e-08,
+ "x": -3.500092072845965,
+ "y": 0.8659632371807578
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+ {
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+ "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
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+ {
+ "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/"
+}
diff --git a/data/solutions/generated/hexinhex_029_baea309b7bc15617.json b/data/solutions/generated/hexinhex_029_baea309b7bc15617.json
new file mode 100644
index 0000000000000000000000000000000000000000..0fe732097bb1e22e7b3eae1b77f783f79e2619e7
--- /dev/null
+++ b/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
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+ {
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+ {
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+ "x": 2.500046305729863,
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+ {
+ "id": 8,
+ "rotation_radians": 7.118841366061075e-07,
+ "x": -0.5000034477954115,
+ "y": -2.5981145828228938
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+ {
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+ "x": 2.500040354576674,
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+ {
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+ {
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+ "x": 4.426615534526794,
+ "y": 1.2621780750165221e-05
+ },
+ {
+ "id": 19,
+ "rotation_radians": 8.2150842097607e-07,
+ "x": -3.500057564580012,
+ "y": 0.8660297122558687
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+ {
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+ "rotation_radians": -1.516113667608833e-06,
+ "x": 1.0000039125737086,
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+ {
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+ "rotation_radians": 7.193380867555484e-07,
+ "x": -2.0922316903038887,
+ "y": 3.6238201728422337
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+ {
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+ {
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+ "rotation_radians": 6.036575488920136e-07,
+ "x": -0.4999895478283576,
+ "y": -4.330189492797411
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+ {
+ "id": 24,
+ "rotation_radians": -3.2789721611959128e-06,
+ "x": 2.500037478603552,
+ "y": -4.330182414425288
+ },
+ {
+ "id": 25,
+ "rotation_radians": 1.4954656685310197e-06,
+ "x": -3.5000399678531604,
+ "y": -2.5981301368837437
+ },
+ {
+ "id": 26,
+ "rotation_radians": -4.0079259744672843e-07,
+ "x": 4.000058321751524,
+ "y": 1.7320813891907003
+ },
+ {
+ "id": 27,
+ "rotation_radians": 4.366037913227048e-07,
+ "x": -3.5000607330501903,
+ "y": 2.5981032880488897
+ },
+ {
+ "id": 28,
+ "rotation_radians": 9.76757375278175e-07,
+ "x": -0.5000231083448415,
+ "y": 4.3301883463523865
+ },
+ {
+ "id": 29,
+ "rotation_radians": 9.842621830853817e-07,
+ "x": -5.000068695617427,
+ "y": -7.1268798689748576e-06
+ }
+ ],
+ "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/"
+}
diff --git a/data/solutions/generated/hexinsqu_007_6492fb111d24a3e4.json b/data/solutions/generated/hexinsqu_007_6492fb111d24a3e4.json
new file mode 100644
index 0000000000000000000000000000000000000000..ce5a0889d5d9ee4ca2a4afbf297e4f7f0ef0da13
--- /dev/null
+++ b/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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+ "rotation_radians": 0.8221574350601246,
+ "x": 0.7126124858342775,
+ "y": -0.7084831354689171
+ },
+ {
+ "id": 3,
+ "rotation_radians": 0.7853995123906887,
+ "x": 2.3547411738843413,
+ "y": -0.09476567117498827
+ },
+ {
+ "id": 4,
+ "rotation_radians": -0.26179766104249264,
+ "x": -0.09475265327530596,
+ "y": 2.354754819846214
+ },
+ {
+ "id": 5,
+ "rotation_radians": 0.7854059125449453,
+ "x": 0.09478137113003707,
+ "y": -2.3547267913368533
+ },
+ {
+ "id": 6,
+ "rotation_radians": 0.7854052231788333,
+ "x": -2.3547120325827744,
+ "y": 0.09479659533686603
+ },
+ {
+ "id": 7,
+ "rotation_radians": 1.8326012962028284,
+ "x": -1.1299361893550244,
+ "y": -1.1299358322158408
+ }
+ ],
+ "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/"
+}
diff --git a/data/solutions/generated/hexintri_013_3b4102a5f66851b2.json b/data/solutions/generated/hexintri_013_3b4102a5f66851b2.json
new file mode 100644
index 0000000000000000000000000000000000000000..53bb336cc7198f960ab2f8e7dcc5dc89852a34a9
--- /dev/null
+++ b/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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+ "rotation_radians": -0.0008769437212764101,
+ "x": 1.0000309298712058,
+ "y": -1.7320837124519806
+ },
+ {
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+ "x": 2.5000772782478062,
+ "y": -0.8660374676439979
+ },
+ {
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+ "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
+ }
+ ],
+ "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/"
+}
diff --git a/data/solutions/generated/hexintri_014_aa2a4511d2c644b1.json b/data/solutions/generated/hexintri_014_aa2a4511d2c644b1.json
new file mode 100644
index 0000000000000000000000000000000000000000..42306bfbfd0623e94998f74d18ee042e58b09da0
--- /dev/null
+++ b/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/"
+}
diff --git a/data/solutions/generated/peninpen_005_5517dc706e76853c.json b/data/solutions/generated/peninpen_005_5517dc706e76853c.json
new file mode 100644
index 0000000000000000000000000000000000000000..9b16e3e142805ac3612ce5ec4ad442aa1ad6d02b
--- /dev/null
+++ b/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
+ },
+ {
+ "id": 3,
+ "rotation_radians": 1.2566378945920285,
+ "x": 1.3764035897498028,
+ "y": -3.44180539106247e-06
+ },
+ {
+ "id": 4,
+ "rotation_radians": 5.611088362108696e-06,
+ "x": -1.1135336974442216,
+ "y": 0.8090323048616854
+ },
+ {
+ "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/"
+}
diff --git a/data/solutions/generated/squindom_001_b71122905cd4c5c7.json b/data/solutions/generated/squindom_001_b71122905cd4c5c7.json
new file mode 100644
index 0000000000000000000000000000000000000000..3085b53915fec293358b899475adcb2fb0fd5995
--- /dev/null
+++ b/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/"
+}
diff --git a/data/solutions/generated/squindom_005_30146b12b3cd6b1f.json b/data/solutions/generated/squindom_005_30146b12b3cd6b1f.json
new file mode 100644
index 0000000000000000000000000000000000000000..5568b886610521e3d3e374a6b80f87935e41dbbf
--- /dev/null
+++ b/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,
+ "rotation_radians": -0.7853891503091323,
+ "x": -1.1029545431141596,
+ "y": -0.5000057922707816
+ },
+ {
+ "id": 2,
+ "rotation_radians": -0.785393765849266,
+ "x": 0.12282442070364517,
+ "y": -0.5000057876168164
+ },
+ {
+ "id": 3,
+ "rotation_radians": -0.7853865265288297,
+ "x": 1.4819328261174398,
+ "y": -0.5000046765074337
+ },
+ {
+ "id": 4,
+ "rotation_radians": -0.7853893948892292,
+ "x": -1.48061341767279,
+ "y": 0.5000046125708607
+ },
+ {
+ "id": 5,
+ "rotation_radians": -0.7853955260143537,
+ "x": -0.10290492456776802,
+ "y": 0.5000058081885114
+ }
+ ],
+ "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/"
+}
diff --git a/data/solutions/generated/squindom_007_24268bca52468f25.json b/data/solutions/generated/squindom_007_24268bca52468f25.json
new file mode 100644
index 0000000000000000000000000000000000000000..663f51581aab54497f860a400a5e2b8c8910fa1f
--- /dev/null
+++ b/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,
+ "rotation_radians": 0.7854025982248328,
+ "x": 1.5000149119527248,
+ "y": -0.5000097739490859
+ },
+ {
+ "id": 2,
+ "rotation_radians": -0.7853937944765449,
+ "x": -0.5000178582951299,
+ "y": -0.5000087167870138
+ },
+ {
+ "id": 3,
+ "rotation_radians": 3.9269947363494615,
+ "x": -1.5000251801567914,
+ "y": -0.5000060956347121
+ },
+ {
+ "id": 4,
+ "rotation_radians": 5.497791251488889,
+ "x": 0.5000184314486544,
+ "y": 0.5000087826722156
+ },
+ {
+ "id": 5,
+ "rotation_radians": 2.3561991235548816,
+ "x": -1.5000141550241783,
+ "y": 0.5000096521148194
+ },
+ {
+ "id": 6,
+ "rotation_radians": 2.356198903146586,
+ "x": -0.49999402074017957,
+ "y": 0.5000089601704955
+ },
+ {
+ "id": 7,
+ "rotation_radians": 0.7854026230935394,
+ "x": 1.5000253378425243,
+ "y": 0.5000061105664929
+ }
+ ],
+ "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/"
+}
diff --git a/data/solutions/generated/squindom_014_821200c01d9d607e.json b/data/solutions/generated/squindom_014_821200c01d9d607e.json
new file mode 100644
index 0000000000000000000000000000000000000000..f6d58b53a7d84ca170ea012b7d62792099f8ab67
--- /dev/null
+++ b/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,
+ "rotation_radians": -0.7853996463353461,
+ "x": -2.500038636355437,
+ "y": -1.0000145856688325
+ },
+ {
+ "id": 2,
+ "rotation_radians": -0.7854009136127643,
+ "x": -0.5000076742870805,
+ "y": -1.0000141285015212
+ },
+ {
+ "id": 3,
+ "rotation_radians": -0.7854020318133327,
+ "x": 1.5000228966002565,
+ "y": -1.0000154649740374
+ },
+ {
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+ "problem": "14 equal items for Squares in Dominoes",
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+ "source_image": "https://erich-friedman.github.io/packing/squindom/18.gif",
+ "source_page": "https://erich-friedman.github.io/packing/squindom/"
+}
diff --git a/data/solutions/generated/squindom_016_e980aaa270c37d2c.json b/data/solutions/generated/squindom_016_e980aaa270c37d2c.json
new file mode 100644
index 0000000000000000000000000000000000000000..b62fff0d4896e62228be3c62aad3d43b1e09d122
--- /dev/null
+++ b/data/solutions/generated/squindom_016_e980aaa270c37d2c.json
@@ -0,0 +1,158 @@
+{
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+ "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": {
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+ "sides": 4,
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+ "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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+ "monotone_upper_bound_from_case": "squindom@18",
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+ "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"
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+ "source_image": "https://erich-friedman.github.io/packing/squindom/18.gif",
+ "source_page": "https://erich-friedman.github.io/packing/squindom/"
+}
diff --git a/data/solutions/generated/squindom_017_ccddf4810f297da5.json b/data/solutions/generated/squindom_017_ccddf4810f297da5.json
new file mode 100644
index 0000000000000000000000000000000000000000..7415ec95372cf96b7e0f60edb61595f22b8714dc
--- /dev/null
+++ b/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,
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+ "metadata": {
+ "friedman_reference": {
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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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+ "generator": "monotone_subset_upper_bound",
+ "metric_recovery_tolerance_rel": 0.001,
+ "monotone_upper_bound_from_case": "squindom@18",
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+ "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",
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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/"
+}
diff --git a/data/solutions/generated/squindom_020_7351f2aa5e1553d9.json b/data/solutions/generated/squindom_020_7351f2aa5e1553d9.json
new file mode 100644
index 0000000000000000000000000000000000000000..1fb19b2278e49bf81eaa9067c728380f2878e701
--- /dev/null
+++ b/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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+ "case": "squindom@20",
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+ "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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+ "generator": "monotone_subset_upper_bound",
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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",
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+}
diff --git a/data/solutions/generated/squindom_027_bc414cdd10a510fa.json b/data/solutions/generated/squindom_027_bc414cdd10a510fa.json
new file mode 100644
index 0000000000000000000000000000000000000000..c11d010e5546a35142e5aea7b60c91341989dc32
--- /dev/null
+++ b/data/solutions/generated/squindom_027_bc414cdd10a510fa.json
@@ -0,0 +1,224 @@
+{
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+ "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"
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+ "sides": 4,
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+ "metric_expression": "4",
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diff --git a/data/solutions/generated/squindom_028_a26ece2a03ab477f.json b/data/solutions/generated/squindom_028_a26ece2a03ab477f.json
new file mode 100644
index 0000000000000000000000000000000000000000..9e0e069be519ea4c8862aedd3c8a1a6fbfff026e
--- /dev/null
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diff --git a/data/solutions/generated/squindom_029_a006f9c77930af4d.json b/data/solutions/generated/squindom_029_a006f9c77930af4d.json
new file mode 100644
index 0000000000000000000000000000000000000000..ef7e732f4ace1700335ccabc3dc24030acb85b28
--- /dev/null
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index 0000000000000000000000000000000000000000..3709d5907d21f42a5f92f89ad99409a75e7e1011
--- /dev/null
+++ b/data/solutions/generated/squindom_035_7d2266788c6a0b7a.json
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diff --git a/data/solutions/generated/squindom_043_83b582ff79edcebf.json b/data/solutions/generated/squindom_043_83b582ff79edcebf.json
new file mode 100644
index 0000000000000000000000000000000000000000..3ceb7fa95dad75597cac07bfcc1f72ce94051474
--- /dev/null
+++ b/data/solutions/generated/squindom_043_83b582ff79edcebf.json
@@ -0,0 +1,320 @@
+{
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+ "reference_text": "43-50. | | s = 5 | Trivial. |",
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+ "source_page": "https://erich-friedman.github.io/packing/squindom/",
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diff --git a/data/solutions/generated/squindom_045_efa26e844da7d857.json b/data/solutions/generated/squindom_045_efa26e844da7d857.json
new file mode 100644
index 0000000000000000000000000000000000000000..34b3b3cf137367774b59e4cd290fb8de88a4e85e
--- /dev/null
+++ b/data/solutions/generated/squindom_045_efa26e844da7d857.json
@@ -0,0 +1,332 @@
+{
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+ "case": "squindom@45",
+ "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": {
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+ "case": "squindom@45",
+ "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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+ "generator": "monotone_subset_upper_bound",
+ "metric_recovery_tolerance_rel": 0.001,
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+ "monotone_upper_bound_from_record": "0433c6294c4f3fe7",
+ "notes": "Monotone upper-bound coordinate JSON derived by deleting items from squindom@48; 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/",
+ "submitted_at": "2026-05-01T18:14:49+00:00",
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diff --git a/data/solutions/generated/squindom_046_7fb3175fb0fba7a7.json b/data/solutions/generated/squindom_046_7fb3175fb0fba7a7.json
new file mode 100644
index 0000000000000000000000000000000000000000..940363c98977f7d2ba83528ac09477fd5ee89ef7
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diff --git a/data/solutions/generated/squindom_047_a146ed2b5a4b6cf9.json b/data/solutions/generated/squindom_047_a146ed2b5a4b6cf9.json
new file mode 100644
index 0000000000000000000000000000000000000000..4fa09e162588eab7abb5e3e2ae2b3d10f96cf381
--- /dev/null
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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/"
+}
diff --git a/data/solutions/generated/squindom_049_598e94e082ebd7d9.json b/data/solutions/generated/squindom_049_598e94e082ebd7d9.json
new file mode 100644
index 0000000000000000000000000000000000000000..1e65a4feaa0297f899c3c709b383800f76bf8a6d
--- /dev/null
+++ b/data/solutions/generated/squindom_049_598e94e082ebd7d9.json
@@ -0,0 +1,356 @@
+{
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+ "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"
+ },
+ "item": {
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+ "side_length": 1.0,
+ "sides": 4,
+ "type": "regular_polygon"
+ },
+ "metadata": {
+ "friedman_reference": {
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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"
+ },
+ "generator": "monotone_subset_upper_bound",
+ "metric_recovery_tolerance_rel": 0.001,
+ "monotone_upper_bound_from_case": "squindom@50",
+ "monotone_upper_bound_from_record": "df0259815cc5320a",
+ "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/",
+ "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/50.gif",
+ "source_page": "https://erich-friedman.github.io/packing/squindom/"
+}
diff --git a/data/solutions/generated/squinoct_016_4405320aab52a39c.json b/data/solutions/generated/squinoct_016_4405320aab52a39c.json
new file mode 100644
index 0000000000000000000000000000000000000000..31bf4068dcd8daf8df239e09e169c219adb97c1a
--- /dev/null
+++ b/data/solutions/generated/squinoct_016_4405320aab52a39c.json
@@ -0,0 +1,160 @@
+{
+ "case": "squinoct@16",
+ "container": {
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+ "orientation_radians": 0.0,
+ "side_length": 2.0710989111906826,
+ "sides": 8,
+ "type": "regular_polygon"
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+ "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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+ "metadata": {
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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/",
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+ "rotation_radians": -0.3926972955850281,
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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/"
+}
diff --git a/data/solutions/generated/squintri_005_21b5dd023d9a44d0.json b/data/solutions/generated/squintri_005_21b5dd023d9a44d0.json
new file mode 100644
index 0000000000000000000000000000000000000000..2a49cc874521dd02ba49af603e87510dfefe715a
--- /dev/null
+++ b/data/solutions/generated/squintri_005_21b5dd023d9a44d0.json
@@ -0,0 +1,94 @@
+{
+ "case": "squintri@5",
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+ "side_length": 4.3095237730565,
+ "sides": 3,
+ "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"
+ },
+ "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"
+ },
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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/"
+}
diff --git a/data/solutions/generated/squintri_020_33507e9f74d78419.json b/data/solutions/generated/squintri_020_33507e9f74d78419.json
new file mode 100644
index 0000000000000000000000000000000000000000..dab77b93ac893ff04abfc7a54be73638d7e0fe24
--- /dev/null
+++ b/data/solutions/generated/squintri_020_33507e9f74d78419.json
@@ -0,0 +1,184 @@
+{
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+ "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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+ "metadata": {
+ "friedman_reference": {
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+ "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"
+ },
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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/"
+}
diff --git a/data/solutions/generated/squintri_030_58fcbf20aa7765b3.json b/data/solutions/generated/squintri_030_58fcbf20aa7765b3.json
new file mode 100644
index 0000000000000000000000000000000000000000..b780264465660ab4ff811a881fa9491e00cc8e3f
--- /dev/null
+++ b/data/solutions/generated/squintri_030_58fcbf20aa7765b3.json
@@ -0,0 +1,244 @@
+{
+ "case": "squintri@30",
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+ "side_length": 9.309837837927466,
+ "sides": 3,
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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,
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+ },
+ "metadata": {
+ "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"
+ },
+ "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,
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+ "source_page": "https://erich-friedman.github.io/packing/squintri/"
+}
diff --git a/data/solutions/generated/squintri_033_8d116d694ad5570a.json b/data/solutions/generated/squintri_033_8d116d694ad5570a.json
new file mode 100644
index 0000000000000000000000000000000000000000..4e26d2c4aac65e97ef95c6cc51a0a8b8c55157a2
--- /dev/null
+++ b/data/solutions/generated/squintri_033_8d116d694ad5570a.json
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+{
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+ "source_page": "https://erich-friedman.github.io/packing/squintri/",
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+ "source_page": "https://erich-friedman.github.io/packing/squintri/",
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diff --git a/data/solutions/generated/squintri_035_454defabaa6b59ca.json b/data/solutions/generated/squintri_035_454defabaa6b59ca.json
new file mode 100644
index 0000000000000000000000000000000000000000..11156f3f3175a8bf8334f7026ddd1e19bcee159c
--- /dev/null
+++ b/data/solutions/generated/squintri_035_454defabaa6b59ca.json
@@ -0,0 +1,274 @@
+{
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+ "metric_expression": "4 + 10 / \u221a3 = 9.773+",
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+ "reference_text": "s = 4 + 10 / \u221a3 = 9.773+ | Found by Erich Friedman | in 1997.",
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+ "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts36.gif",
+ "source_page": "https://erich-friedman.github.io/packing/squintri/",
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+ "metric_expression": "4 + 10 / \u221a3 = 9.773+",
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+ "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/",
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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": 9.773502691896258,
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+ "source_page": "https://erich-friedman.github.io/packing/squintri/"
+}
diff --git a/data/solutions/generated/triincir_005_a3d09cb09f1298d4.json b/data/solutions/generated/triincir_005_a3d09cb09f1298d4.json
new file mode 100644
index 0000000000000000000000000000000000000000..b0ad311b53c6889e6cf391f2bc7192ae38c15eea
--- /dev/null
+++ b/data/solutions/generated/triincir_005_a3d09cb09f1298d4.json
@@ -0,0 +1,91 @@
+{
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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": {
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+ "sides": 3,
+ "type": "regular_polygon"
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+ "metadata": {
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+ "case": "triincir@5",
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+ "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"
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+ "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/"
+}
diff --git a/data/solutions/generated/triinhex_004_d2b5a3b06543cb98.json b/data/solutions/generated/triinhex_004_d2b5a3b06543cb98.json
new file mode 100644
index 0000000000000000000000000000000000000000..cf684efd03ff05c0a482e5041acd2ee9b1d7d8eb
--- /dev/null
+++ b/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/"
+}
diff --git a/data/solutions/generated/triinhex_033_a1101d95622ba18c.json b/data/solutions/generated/triinhex_033_a1101d95622ba18c.json
new file mode 100644
index 0000000000000000000000000000000000000000..fb2981e57f4ce26a2bf1d4c719b0dc5a74f04e4c
--- /dev/null
+++ b/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/"
+}
diff --git a/data/solutions/generated/triintri_001_b09a5ce4a2c03ad3.json b/data/solutions/generated/triintri_001_b09a5ce4a2c03ad3.json
index 02547c9dd2439364787816a8fcc8af004837a441..0deb9edb58256b17cb922687b66029aba1d78b15 100644
--- a/data/solutions/generated/triintri_001_b09a5ce4a2c03ad3.json
+++ b/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": [
diff --git a/data/solutions/generated/triintri_002_2d6f654db11412d8.json b/data/solutions/generated/triintri_002_2d6f654db11412d8.json
index 9c5b56e8b9dc841df2a6dc27bdc2f8a97f8921fa..944aa672fec347628e1952c79c75086004a91428 100644
--- a/data/solutions/generated/triintri_002_2d6f654db11412d8.json
+++ b/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": [
diff --git a/data/solutions/generated/triintri_003_76381d42f8f17f41.json b/data/solutions/generated/triintri_003_76381d42f8f17f41.json
index 8baeec11e6672b44209322ef0dc5cf814e9e32e7..921b08454c37c6dc70bd687c219ea6b2e4481538 100644
--- a/data/solutions/generated/triintri_003_76381d42f8f17f41.json
+++ b/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": [
diff --git a/data/solutions/generated/triintri_004_be1b0b4b1802d8fa.json b/data/solutions/generated/triintri_004_be1b0b4b1802d8fa.json
index 155406b8db87dd36ef5077db793afe5e6433ce73..1a90c63ed7bbe5ca9ad8939aba22143ca5ac6b30 100644
--- a/data/solutions/generated/triintri_004_be1b0b4b1802d8fa.json
+++ b/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": [
diff --git a/data/solutions/generated/triintri_007_355f218b5ea40614.json b/data/solutions/generated/triintri_007_355f218b5ea40614.json
index 4d86c2fb5b805dd89362bc85dcef4702d6f5a600..ac188f39c959298f9d2424a8550ad386509f5f4b 100644
--- a/data/solutions/generated/triintri_007_355f218b5ea40614.json
+++ b/data/solutions/generated/triintri_007_355f218b5ea40614.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": 3.0,
"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": [
diff --git a/data/solutions/generated/triintri_008_43e3200cf32d77a9.json b/data/solutions/generated/triintri_008_43e3200cf32d77a9.json
index 65416af63d964bdc6038ce46c3d12d7d80f7b316..3722de84f5086b7d14d55de03dfd7583363e018d 100644
--- a/data/solutions/generated/triintri_008_43e3200cf32d77a9.json
+++ b/data/solutions/generated/triintri_008_43e3200cf32d77a9.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": 3.0,
"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": [
diff --git a/data/solutions/generated/triintri_009_4242c6a66848e1ea.json b/data/solutions/generated/triintri_009_4242c6a66848e1ea.json
index 30a471c35f72ffb492a79110840047ac38e6cf97..d4458badff377b5bd7d65aa87577585f09123d52 100644
--- a/data/solutions/generated/triintri_009_4242c6a66848e1ea.json
+++ b/data/solutions/generated/triintri_009_4242c6a66848e1ea.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": 3.0,
"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": [
diff --git a/data/solutions/generated/triintri_014_8676384ad0301449.json b/data/solutions/generated/triintri_014_8676384ad0301449.json
index 23bbf3f964c00aa7eda38b445300e336bee491ae..62887d696b728cfac04214abe25bff003e3488af 100644
--- a/data/solutions/generated/triintri_014_8676384ad0301449.json
+++ b/data/solutions/generated/triintri_014_8676384ad0301449.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": 4.0,
"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": [
diff --git a/data/solutions/generated/triintri_015_31db3ec76cc0421f.json b/data/solutions/generated/triintri_015_31db3ec76cc0421f.json
index 897323ab6de3aea22fcbb58b22d868a3208d8ab2..efc90fe897046cc754dce7191713a16caa44e617 100644
--- a/data/solutions/generated/triintri_015_31db3ec76cc0421f.json
+++ b/data/solutions/generated/triintri_015_31db3ec76cc0421f.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": 4.0,
"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": [
diff --git a/data/solutions/generated/triintri_016_2ffa329b602032d6.json b/data/solutions/generated/triintri_016_2ffa329b602032d6.json
index 94a72b828658618d8e29a754135f924190f34095..e3f6f3a70c858a0a5621f58358ea653c9ffc25b7 100644
--- a/data/solutions/generated/triintri_016_2ffa329b602032d6.json
+++ b/data/solutions/generated/triintri_016_2ffa329b602032d6.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": 4.0,
"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": [
diff --git a/data/solutions/generated/triintri_022_16c9dbca1fb3bede.json b/data/solutions/generated/triintri_022_16c9dbca1fb3bede.json
new file mode 100644
index 0000000000000000000000000000000000000000..5130886422b1c65ee8f01c79154c0c432ad08c90
--- /dev/null
+++ b/data/solutions/generated/triintri_022_16c9dbca1fb3bede.json
@@ -0,0 +1,196 @@
+{
+ "case": "triintri@22",
+ "container": {
+ "circumradius": 2.886751345948129,
+ "orientation_radians": 0.0,
+ "side_length": 5.0,
+ "sides": 3,
+ "type": "regular_polygon"
+ },
+ "friedman_reference": {
+ "analytical_or_proved": false,
+ "case": "triintri@22",
+ "credit": "Maurizio Morandi",
+ "date": "May 2008",
+ "family_updated": "",
+ "metric_expression": "4.996+",
+ "metric_symbol": "s",
+ "metric_value": 4.996,
+ "reference_text": "s = 4.996+ | Found by Maurizio Morandi | in May 2008.",
+ "source_image": "tt22.gif",
+ "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt22.gif",
+ "source_page": "https://erich-friedman.github.io/packing/triintri/",
+ "title": "Triangles in Triangles"
+ },
+ "item": {
+ "circumradius": 0.5773502691896258,
+ "side_length": 1.0,
+ "sides": 3,
+ "type": "regular_polygon"
+ },
+ "metadata": {
+ "friedman_reference": {
+ "analytical_or_proved": false,
+ "case": "triintri@22",
+ "credit": "Maurizio Morandi",
+ "date": "May 2008",
+ "family_updated": "",
+ "metric_expression": "4.996+",
+ "metric_symbol": "s",
+ "metric_value": 4.996,
+ "reference_text": "s = 4.996+ | Found by Maurizio Morandi | in May 2008.",
+ "source_image": "tt22.gif",
+ "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt22.gif",
+ "source_page": "https://erich-friedman.github.io/packing/triintri/",
+ "title": "Triangles in Triangles"
+ },
+ "generator": "monotone_subset_upper_bound",
+ "metric_recovery_tolerance_rel": 0.001,
+ "monotone_upper_bound_from_case": "triintri@25",
+ "monotone_upper_bound_from_record": "9c4806b8b37c5d37",
+ "notes": "Monotone upper-bound coordinate JSON derived by deleting items from triintri@25; 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": 4.996,
+ "source_url": "https://erich-friedman.github.io/packing/triintri/",
+ "submitted_at": "2026-05-01T18:14:49+00:00",
+ "submitted_by": "Maurizio Morandi"
+ },
+ "placements": [
+ {
+ "id": 1,
+ "rotation_radians": -1.2819751242557092e-16,
+ "x": 2.3094010767585034,
+ "y": 7.401486830834377e-17
+ },
+ {
+ "id": 2,
+ "rotation_radians": 1.0471975511965974,
+ "x": 1.7320508075688776,
+ "y": 1.4802973661668753e-16
+ },
+ {
+ "id": 3,
+ "rotation_radians": -9.61481343191782e-17,
+ "x": 1.4433756729740645,
+ "y": -0.49999999999999983
+ },
+ {
+ "id": 4,
+ "rotation_radians": 1.0471975511965972,
+ "x": 0.8660254037844387,
+ "y": -0.4999999999999997
+ },
+ {
+ "id": 5,
+ "rotation_radians": -1.9229626863835636e-16,
+ "x": 0.5773502691896254,
+ "y": -0.9999999999999997
+ },
+ {
+ "id": 6,
+ "rotation_radians": 1.0471975511965972,
+ "x": -3.7007434154171886e-16,
+ "y": -0.9999999999999996
+ },
+ {
+ "id": 7,
+ "rotation_radians": -3.845925372767128e-16,
+ "x": -0.28867513459481353,
+ "y": -1.4999999999999993
+ },
+ {
+ "id": 8,
+ "rotation_radians": 1.0471975511965974,
+ "x": -0.8660254037844393,
+ "y": -1.4999999999999993
+ },
+ {
+ "id": 9,
+ "rotation_radians": -3.845925372767128e-16,
+ "x": -1.1547005383792524,
+ "y": -1.9999999999999993
+ },
+ {
+ "id": 10,
+ "rotation_radians": 0.0,
+ "x": 1.4433756729740648,
+ "y": 0.5000000000000001
+ },
+ {
+ "id": 11,
+ "rotation_radians": 1.0471975511965974,
+ "x": 0.8660254037844389,
+ "y": 0.5000000000000003
+ },
+ {
+ "id": 12,
+ "rotation_radians": -1.2819751242557082e-16,
+ "x": 0.5773502691896258,
+ "y": 2.9605947323337506e-16
+ },
+ {
+ "id": 13,
+ "rotation_radians": 1.0471975511965974,
+ "x": -3.700743415417188e-17,
+ "y": 3.7007434154171886e-16
+ },
+ {
+ "id": 14,
+ "rotation_radians": -1.9229626863835633e-16,
+ "x": -0.2886751345948131,
+ "y": -0.49999999999999956
+ },
+ {
+ "id": 15,
+ "rotation_radians": 1.0471975511965974,
+ "x": -0.8660254037844389,
+ "y": -0.4999999999999994
+ },
+ {
+ "id": 16,
+ "rotation_radians": 0.0,
+ "x": -1.1547005383792521,
+ "y": -0.9999999999999994
+ },
+ {
+ "id": 17,
+ "rotation_radians": 0.0,
+ "x": 0.5773502691896262,
+ "y": 1.0000000000000002
+ },
+ {
+ "id": 18,
+ "rotation_radians": 1.0471975511965974,
+ "x": 3.7007434154171886e-16,
+ "y": 1.0000000000000004
+ },
+ {
+ "id": 19,
+ "rotation_radians": -1.922962686383564e-16,
+ "x": -0.2886751345948128,
+ "y": 0.5000000000000004
+ },
+ {
+ "id": 20,
+ "rotation_radians": 1.0471975511965974,
+ "x": -0.8660254037844385,
+ "y": 0.5000000000000006
+ },
+ {
+ "id": 21,
+ "rotation_radians": -2.563950248511418e-16,
+ "x": -1.1547005383792517,
+ "y": 5.921189464667501e-16
+ },
+ {
+ "id": 22,
+ "rotation_radians": -3.8459253727671266e-16,
+ "x": -0.28867513459481237,
+ "y": 1.5000000000000007
+ }
+ ],
+ "problem": "22 equal items for Triangles in Triangles",
+ "setup": "triintri",
+ "source_image": "https://erich-friedman.github.io/packing/triintri/tt22.gif",
+ "source_page": "https://erich-friedman.github.io/packing/triintri/"
+}
diff --git a/data/solutions/generated/triintri_023_4f1bdd5d6c07f589.json b/data/solutions/generated/triintri_023_4f1bdd5d6c07f589.json
index 2ea52b4a9160762d6abd32c690a19a685200d638..d6aa31e86507266f168c695c0042fc52e593201f 100644
--- a/data/solutions/generated/triintri_023_4f1bdd5d6c07f589.json
+++ b/data/solutions/generated/triintri_023_4f1bdd5d6c07f589.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": 5.0,
"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": [
diff --git a/data/solutions/generated/triintri_024_64d77fbf13ae0be7.json b/data/solutions/generated/triintri_024_64d77fbf13ae0be7.json
index 1118173a3f008d5d2dcc4fe8c277cecbf9980efe..71891cdf021b4a69bfc9ddd6f4d0cfafb1adf27e 100644
--- a/data/solutions/generated/triintri_024_64d77fbf13ae0be7.json
+++ b/data/solutions/generated/triintri_024_64d77fbf13ae0be7.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": 5.0,
"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": [
diff --git a/data/solutions/generated/triintri_025_9c4806b8b37c5d37.json b/data/solutions/generated/triintri_025_9c4806b8b37c5d37.json
index 9b5f673f25302eab07cbd4ca292b780ed7ea2b40..76f1464fc1e4c78612c7e85b4651efa83807cd15 100644
--- a/data/solutions/generated/triintri_025_9c4806b8b37c5d37.json
+++ b/data/solutions/generated/triintri_025_9c4806b8b37c5d37.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": 5.0,
"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": [
diff --git a/data/solutions/generated/triintri_027_70128a68bb429fb5.json b/data/solutions/generated/triintri_027_70128a68bb429fb5.json
new file mode 100644
index 0000000000000000000000000000000000000000..edf99d6b1d25ee6fe28afd24978bae15a9c7ee60
--- /dev/null
+++ b/data/solutions/generated/triintri_027_70128a68bb429fb5.json
@@ -0,0 +1,226 @@
+{
+ "case": "triintri@27",
+ "container": {
+ "circumradius": 3.175523634890182,
+ "orientation_radians": 0.0,
+ "side_length": 5.500168276265596,
+ "sides": 3,
+ "type": "regular_polygon"
+ },
+ "friedman_reference": {
+ "analytical_or_proved": true,
+ "case": "triintri@27",
+ "credit": "Erich Friedman",
+ "date": "1997",
+ "family_updated": "",
+ "metric_expression": "11/2 = 5.500",
+ "metric_symbol": "s",
+ "metric_value": 5.5,
+ "reference_text": "s = 11/2 = 5.500 | Found by Erich Friedman | in 1997.",
+ "source_image": "tt28.gif",
+ "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt28.gif",
+ "source_page": "https://erich-friedman.github.io/packing/triintri/",
+ "title": "Triangles in Triangles"
+ },
+ "item": {
+ "circumradius": 0.5773502691896258,
+ "side_length": 1.0,
+ "sides": 3,
+ "type": "regular_polygon"
+ },
+ "metadata": {
+ "friedman_reference": {
+ "analytical_or_proved": true,
+ "case": "triintri@27",
+ "credit": "Erich Friedman",
+ "date": "1997",
+ "family_updated": "",
+ "metric_expression": "11/2 = 5.500",
+ "metric_symbol": "s",
+ "metric_value": 5.5,
+ "reference_text": "s = 11/2 = 5.500 | Found by Erich Friedman | in 1997.",
+ "source_image": "tt28.gif",
+ "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt28.gif",
+ "source_page": "https://erich-friedman.github.io/packing/triintri/",
+ "title": "Triangles in Triangles"
+ },
+ "generator": "monotone_subset_upper_bound",
+ "metric_recovery_tolerance_rel": 0.001,
+ "monotone_upper_bound_from_case": "triintri@28",
+ "monotone_upper_bound_from_record": "9562d7d6e075a347",
+ "notes": "Monotone upper-bound coordinate JSON derived by deleting items from triintri@28; 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": 5.5,
+ "source_url": "https://erich-friedman.github.io/packing/triintri/",
+ "submitted_at": "2026-05-01T18:14:49+00:00",
+ "submitted_by": "Erich Friedman"
+ },
+ "placements": [
+ {
+ "id": 1,
+ "rotation_radians": 2.314758340428289e-05,
+ "x": -4.2531865809583077e-07,
+ "y": -3.1175523115254195e-06
+ },
+ {
+ "id": 2,
+ "rotation_radians": -3.1415671720576253,
+ "x": -0.5773658132173131,
+ "y": -8.432805377512893e-06
+ },
+ {
+ "id": 3,
+ "rotation_radians": -5.1505474047317785e-06,
+ "x": -0.4330356688884721,
+ "y": -0.750026519900571
+ },
+ {
+ "id": 4,
+ "rotation_radians": -3.1415678858963654,
+ "x": 0.28868865974605556,
+ "y": -0.5000135691584721
+ },
+ {
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+ "rotation_radians": 1.2493640579083933e-05,
+ "x": 0.8660552602829118,
+ "y": -9.906916513191095e-07
+ },
+ {
+ "id": 6,
+ "rotation_radians": -3.141565947061667,
+ "x": 0.28867733340651325,
+ "y": 0.5000143957074368
+ },
+ {
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+ "rotation_radians": 1.0891380966307694e-05,
+ "x": -0.4330255044215008,
+ "y": 0.7500252026220282
+ },
+ {
+ "id": 8,
+ "rotation_radians": -3.1415826894045353,
+ "x": -1.0103943943222498,
+ "y": 0.7500197985571919
+ },
+ {
+ "id": 9,
+ "rotation_radians": 2.3291115865861655e-05,
+ "x": -1.2990684891201176,
+ "y": 0.24998731078360928
+ },
+ {
+ "id": 10,
+ "rotation_radians": -3.141595852057269,
+ "x": -0.1443424011762592,
+ "y": -1.2500361922480143
+ },
+ {
+ "id": 11,
+ "rotation_radians": 1.7744455777302309e-06,
+ "x": 0.4330300778657851,
+ "y": -1.2500346059800018
+ },
+ {
+ "id": 12,
+ "rotation_radians": -3.1415817375664807,
+ "x": 1.1547346082716248,
+ "y": 0.5000175042909349
+ },
+ {
+ "id": 13,
+ "rotation_radians": 1.7783349492600918e-05,
+ "x": 0.8660342392539183,
+ "y": 1.000032839011964
+ },
+ {
+ "id": 14,
+ "rotation_radians": 9.380927490717909e-06,
+ "x": -1.2990791545419709,
+ "y": 1.2500387862374776
+ },
+ {
+ "id": 15,
+ "rotation_radians": -3.141586016672962,
+ "x": -1.0103925124793443,
+ "y": 1.7500533495566808
+ },
+ {
+ "id": 16,
+ "rotation_radians": -3.1415977643783144,
+ "x": -1.0103923499853305,
+ "y": -1.7500561539273904
+ },
+ {
+ "id": 17,
+ "rotation_radians": -3.973786663127966e-06,
+ "x": -0.4330234273864937,
+ "y": -1.750053954596603
+ },
+ {
+ "id": 18,
+ "rotation_radians": -3.1415850794381335,
+ "x": 2.020786436238312,
+ "y": -1.3296589983097e-06
+ },
+ {
+ "id": 19,
+ "rotation_radians": 7.519099540054636e-06,
+ "x": 1.7321056145463598,
+ "y": 0.5000160361111167
+ },
+ {
+ "id": 20,
+ "rotation_radians": -3.1415965701193587,
+ "x": -1.0104008997921337,
+ "y": -0.7500266899467389
+ },
+ {
+ "id": 21,
+ "rotation_radians": -6.344902384175991e-06,
+ "x": -1.299081214052549,
+ "y": -1.2500441976408654
+ },
+ {
+ "id": 22,
+ "rotation_radians": 3.22297910597702e-06,
+ "x": -1.2990785253239168,
+ "y": 2.2500713843273457
+ },
+ {
+ "id": 23,
+ "rotation_radians": -3.1415818576827457,
+ "x": 1.154736529587972,
+ "y": -0.5000169378527385
+ },
+ {
+ "id": 24,
+ "rotation_radians": 8.763585979686832e-06,
+ "x": 1.732104258918298,
+ "y": -0.5000169436704749
+ },
+ {
+ "id": 25,
+ "rotation_radians": -3.141581954097244,
+ "x": -0.14433947205272424,
+ "y": 1.2500388832162375
+ },
+ {
+ "id": 26,
+ "rotation_radians": 7.579213112610313e-06,
+ "x": -0.43302227584468417,
+ "y": 1.7500533444599347
+ },
+ {
+ "id": 27,
+ "rotation_radians": -2.1550190193066363e-06,
+ "x": -1.2990783250927131,
+ "y": -2.2500728964714147
+ }
+ ],
+ "problem": "27 equal items for Triangles in Triangles",
+ "setup": "triintri",
+ "source_image": "https://erich-friedman.github.io/packing/triintri/tt28.gif",
+ "source_page": "https://erich-friedman.github.io/packing/triintri/"
+}
diff --git a/data/solutions/generated/triintri_033_34a5032aebc12aff.json b/data/solutions/generated/triintri_033_34a5032aebc12aff.json
index 91990b50eb928a19ce598502c6c973ce7e9cc9f9..72b655e80918d5574e6810b172f5dc0a4e1930b6 100644
--- a/data/solutions/generated/triintri_033_34a5032aebc12aff.json
+++ b/data/solutions/generated/triintri_033_34a5032aebc12aff.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": 6.0,
"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": [
diff --git a/data/solutions/generated/triintri_034_a2ccd0a19cfa1b75.json b/data/solutions/generated/triintri_034_a2ccd0a19cfa1b75.json
index eb00562e077e142f93663b93cf6a68fd4f23cda6..c32741692e9a97da40e19aff0494eaa292ff9110 100644
--- a/data/solutions/generated/triintri_034_a2ccd0a19cfa1b75.json
+++ b/data/solutions/generated/triintri_034_a2ccd0a19cfa1b75.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": 6.0,
"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": [
diff --git a/data/solutions/generated/triintri_035_6325f936448cdf94.json b/data/solutions/generated/triintri_035_6325f936448cdf94.json
index 38462af5d18ca5f185f4f6bfc75c1f83eca5cd8f..d0ad2db4efe4fd17e37f121daa497bc48db96f29 100644
--- a/data/solutions/generated/triintri_035_6325f936448cdf94.json
+++ b/data/solutions/generated/triintri_035_6325f936448cdf94.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": 6.0,
"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"
},
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diff --git a/data/solutions/generated/triintri_036_4b90836bf07f8a65.json b/data/solutions/generated/triintri_036_4b90836bf07f8a65.json
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"reference_side": 6.0,
"source_url": "https://erich-friedman.github.io/packing/triintri/",
- "submitted_at": "2026-05-01T18:08:37+00:00",
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},
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