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          "mean_abs_psi4": 0.34868761013915145,
          "abs_mean_psi4": 0.20951603919037617,
          "mean_abs_psi6": 0.4278785836995793,
          "abs_mean_psi6": 0.020253188127467486,
          "mean_abs_psi8": 0.34867396110059545,
          "abs_mean_psi8": 0.20945747062332193,
          "mean_abs_psi12": 0.9999998376606561,
          "abs_mean_psi12": 0.9999997777442562
        },
        "inner_six_radii": [
          0.9999163164364883,
          0.9999319010538906,
          0.9999358531555248,
          0.999938942147848,
          0.999951439161693,
          0.9999584781342381
        ],
        "outer_twelve_radii": [
          1.9316627072094963,
          1.931709541782771,
          1.931710192150029,
          1.9317104106392615,
          1.9317106501364658,
          1.9317186394103334,
          1.93172144492094,
          1.9317231899338025,
          1.9317280747590166,
          1.9317467493040124,
          1.9317516059462099,
          1.9317610043492806
        ],
        "motif_note": "Center plus inner hexagon plus outer dodecagon: the full 19-point motif has sixfold, not twelvefold, symmetry."
      }
    ],
    "template_check": {
      "template_centers": 349,
      "remaining_disks": 5702,
      "initial_disks": 6400,
      "removed_disks": 698,
      "density": 0.94
    },
    "limits": [
      "Small independent calculations use the same harmonic-repulsion family and pressure, with L-BFGS/Newton instead of original FIRE.",
      "A positive Hessian after removing translations is local mechanical stability, not a global free-energy minimum.",
      "The hard-like model produces an approximant or local motifs, not a complete aperiodic DDQC.",
      "The public high-quality TLS example embeds ideal-tiling information in chosen vacancy locations."
    ]
  },
  "orientation_comparison": [
    {
      "structure": "square",
      "local_abs_psi4": 1.0,
      "local_abs_psi6": 1.83697019872103e-16,
      "local_abs_psi12": 1.0
    },
    {
      "structure": "hexagonal",
      "local_abs_psi4": 3.0348352473391116e-16,
      "local_abs_psi6": 1.0,
      "local_abs_psi12": 1.0
    },
    {
      "structure": "periodic_sigma",
      "local_abs_psi4": 0.20003381617675312,
      "local_abs_psi6": 0.19999999999999996,
      "local_abs_psi12": 0.9999999914232547
    },
    {
      "structure": "experimental_patch",
      "local_abs_psi4": 0.25115107314755125,
      "local_abs_psi6": 0.36951140415719796,
      "local_abs_psi12": 0.9403873480602027
    }
  ],
  "geometry": {
    "at_half_size": {
      "projected": 0.7071067811865476,
      "additive": 0.75,
      "nonadditivity": -0.05719095841793653
    },
    "tiling_geometry": [
      {
        "structure": "sigma_approximant",
        "triangle_square_count_ratio": 2,
        "area_per_vertex": 0.9330127018922193,
        "hard_circle_fraction": 0.8417872144769329
      },
      {
        "structure": "ideal_dodecagonal",
        "triangle_square_count_ratio": 2.3094010767585034,
        "area_per_vertex": 0.9282032302755092,
        "hard_circle_fraction": 0.8461489227572786
      }
    ],
    "limits": "Ideal tile-area bookkeeping, not a free-energy calculation or proof that the densest candidate is always selected."
  },
  "inputs_sha256": {
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    "gao-coordinate-for-fig4.xlsx": "eb51b0859472a77648de96cee06b367d77c2c669b131eaef6edce99c6314fb61",
    "gao-coordinate-for-fig2.xlsx": "f3ad8d1551cb401ccadca0df3c10b1a28d0f09c59883c0d226e089338851b77a",
    "gao-coordinate-for-fig1.xlsx": "189574e86f02758c88cee4c6385b60b7fd5cb71e5b2ce497c29fd168aacc5bbd",
    "plati-source-4.xlsx": "517c1e03652ee8af7f77dcd31d975d0c8bbc6e15e6afab1f991ec55839c8545b",
    "mechanical-code/01_template_config.dat": "4c66896dc97cb8ebc8340136abfd0ad5e9d0e55a59ecd8f35d71f8b19e503d59",
    "mechanical-code/02_removed_config.dat": "b4388c6ab59f2fdce9749c4b5695e655c6546666e7a7f8193ebacaa884cc4dc9"
  }
}