{
  "date": "2026-10-09",
  "status": "passed",
  "script_sha256": "7bfae84d552e136ed36b24fa52203f8c56f7720fa9d169ded7f5e79304739e9f",
  "source": {
    "authors": "Emmanuel Jeandel and Michael Rao",
    "title": "An aperiodic set of 11 Wang tiles",
    "url": "https://arxiv.org/html/1506.06492v4#S4.F4",
    "figure": 4,
    "theorem": 3,
    "extraction": "Math labels and their positions in the authors' Figure 4 SVG; left-to-right order."
  },
  "tuple_order": [
    "west",
    "east",
    "south",
    "north"
  ],
  "color_amplitude_map": [
    1,
    2,
    3,
    4
  ],
  "palette": [
    {
      "id": 1,
      "colors": [
        0,
        0,
        1,
        0
      ],
      "edge_amplitudes": [
        1,
        1,
        2,
        1
      ],
      "cosine_coefficients": [
        "5/4",
        "0",
        "1/4",
        "-1/2",
        "-3/4",
        "0",
        "-1/4"
      ]
    },
    {
      "id": 2,
      "colors": [
        0,
        3,
        2,
        1
      ],
      "edge_amplitudes": [
        1,
        4,
        3,
        2
      ],
      "cosine_coefficients": [
        "5/2",
        "-3/4",
        "1/4",
        "-5/4",
        "-5/4",
        "3/4",
        "-1/4"
      ]
    },
    {
      "id": 3,
      "colors": [
        1,
        0,
        2,
        2
      ],
      "edge_amplitudes": [
        2,
        1,
        3,
        3
      ],
      "cosine_coefficients": [
        "9/4",
        "1/4",
        "0",
        "-3/4",
        "-3/2",
        "-1/4",
        "0"
      ]
    },
    {
      "id": 4,
      "colors": [
        1,
        1,
        0,
        2
      ],
      "edge_amplitudes": [
        2,
        2,
        1,
        3
      ],
      "cosine_coefficients": [
        "2",
        "0",
        "-1/2",
        "-1",
        "-1",
        "0",
        "1/2"
      ]
    },
    {
      "id": 5,
      "colors": [
        1,
        3,
        2,
        3
      ],
      "edge_amplitudes": [
        2,
        4,
        3,
        4
      ],
      "cosine_coefficients": [
        "13/4",
        "-1/2",
        "-1/4",
        "-3/2",
        "-7/4",
        "1/2",
        "1/4"
      ]
    },
    {
      "id": 6,
      "colors": [
        3,
        0,
        1,
        1
      ],
      "edge_amplitudes": [
        4,
        1,
        2,
        2
      ],
      "cosine_coefficients": [
        "9/4",
        "3/4",
        "0",
        "-5/4",
        "-1",
        "-3/4",
        "0"
      ]
    },
    {
      "id": 7,
      "colors": [
        3,
        1,
        1,
        1
      ],
      "edge_amplitudes": [
        4,
        2,
        2,
        2
      ],
      "cosine_coefficients": [
        "5/2",
        "1/2",
        "0",
        "-3/2",
        "-1",
        "-1/2",
        "0"
      ]
    },
    {
      "id": 8,
      "colors": [
        3,
        1,
        2,
        2
      ],
      "edge_amplitudes": [
        4,
        2,
        3,
        3
      ],
      "cosine_coefficients": [
        "3",
        "1/2",
        "0",
        "-3/2",
        "-3/2",
        "-1/2",
        "0"
      ]
    },
    {
      "id": 9,
      "colors": [
        3,
        3,
        3,
        1
      ],
      "edge_amplitudes": [
        4,
        4,
        4,
        2
      ],
      "cosine_coefficients": [
        "7/2",
        "0",
        "1/2",
        "-2",
        "-3/2",
        "0",
        "-1/2"
      ]
    },
    {
      "id": 10,
      "colors": [
        2,
        2,
        1,
        0
      ],
      "edge_amplitudes": [
        3,
        3,
        2,
        1
      ],
      "cosine_coefficients": [
        "9/4",
        "0",
        "1/4",
        "-3/2",
        "-3/4",
        "0",
        "-1/4"
      ]
    },
    {
      "id": 11,
      "colors": [
        2,
        2,
        0,
        2
      ],
      "edge_amplitudes": [
        3,
        3,
        1,
        3
      ],
      "cosine_coefficients": [
        "5/2",
        "0",
        "-1/2",
        "-3/2",
        "-1",
        "0",
        "1/2"
      ]
    }
  ],
  "Neumann_cosine_modes": [
    {
      "indices": [
        0,
        0
      ],
      "normalized_eigenvalue": 0
    },
    {
      "indices": [
        1,
        0
      ],
      "normalized_eigenvalue": 1
    },
    {
      "indices": [
        0,
        1
      ],
      "normalized_eigenvalue": 1
    },
    {
      "indices": [
        0,
        2
      ],
      "normalized_eigenvalue": 4
    },
    {
      "indices": [
        2,
        0
      ],
      "normalized_eigenvalue": 4
    },
    {
      "indices": [
        1,
        2
      ],
      "normalized_eigenvalue": 5
    },
    {
      "indices": [
        2,
        1
      ],
      "normalized_eigenvalue": 5
    }
  ],
  "number_of_Neumann_modes": 7,
  "normalized_eigenvalues_used": [
    0,
    1,
    4,
    5
  ],
  "edge_carrier_space_dimension": 4,
  "horizontal_adjacency": [
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      0,
      0,
      0,
      0,
      0,
      0,
      0
    ],
    [
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    [
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    [
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    [
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      0
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    [
      0,
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      1,
      0,
      0,
      0,
      0,
      0,
      0
    ],
    [
      0,
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      1,
      1,
      0,
      0,
      0,
      0,
      0,
      0
    ],
    [
      0,
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      0,
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      1,
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      0
    ],
    [
      0,
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      0,
      0,
      0,
      0,
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      1,
      1
    ],
    [
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      1,
      1
    ]
  ],
  "vertical_adjacency": [
    [
      0,
      0,
      0,
      1,
      0,
      0,
      0,
      0,
      0,
      0,
      1
    ],
    [
      1,
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      1,
      1,
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      0,
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    ],
    [
      0,
      1,
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      1,
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      0,
      1,
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      0,
      0
    ],
    [
      0,
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      1,
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    ],
    [
      0,
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    ],
    [
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    ],
    [
      1,
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      0,
      1,
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    ],
    [
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      1,
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      0,
      0
    ],
    [
      1,
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      1,
      0,
      0,
      1,
      0
    ],
    [
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    [
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      0,
      0
    ]
  ],
  "directed_value_compatible_pairs": {
    "horizontal": 31,
    "vertical": 37
  },
  "projected_outward_flux_matrix": [
    [
      1,
      -1,
      -2,
      -2
    ],
    [
      -1,
      1,
      -2,
      -2
    ],
    [
      -2,
      -2,
      1,
      -1
    ],
    [
      -2,
      -2,
      -1,
      1
    ]
  ],
  "boundary_basis_norm_squared": "1/2",
  "maximum_sample_trace_error": 2.220446049250313e-16,
  "maximum_sample_cosine_expansion_error": 8.881784197001252e-16,
  "maximum_sample_Neumann_normal_derivative": 3.4626072486992214e-15,
  "single_frequency_alternative": {
    "normalized_Helmholtz_value": 1,
    "basis_dimension": 4,
    "maximum_sample_trace_error": 4.898587196589413e-16,
    "fixed_operator": "Trace in F and P_F outward_flux = B trace; common self-adjoint nonlocal operator. Lowest eigenvalue pi^2 has multiplicity four by the boundary-form proof in the Notes.",
    "flux_matching": "Impossible for the strictly positive palette; this alternative has interface sources."
  },
  "placement_model": "One unit square per integer grid cell; translations only; fixed amplitude palette; equality of scalar traces.",
  "aperiodicity": "Exact bijection to the cited Wang set. Infinite existence and absence of nonzero grid periods use Jeandel--Rao's theorem, not these finite pair checks.",
  "flux_matching": "The total static field is continuously matched and every square has exactly zero outward normal derivative. Each summand is a homogeneous Neumann eigenfunction.",
  "limitations": "The total static field combines four eigenvalues; it is not a single Helmholtz eigenfunction, and separate frequencies generally do not match. The finite coefficient palette is essential. Reflections of states are excluded. No tiling/search engine or wave evolution is constructed.",
  "atlas": {
    "file": "boundary-mode-palette.png",
    "sha256": "77f8e4cc1f167597495844653909dd2af7e6bc5217815b435336f871c278bd58",
    "sample_grid_per_field": [
      181,
      181
    ],
    "shared_range": [
      0,
      7.06246079595714
    ],
    "meaning": "Sampled exact formulas. No independent normalization or FEM solve."
  }
}