Complete Fibers, Positive-Mass Conditional Laws and Sufficiency

([0, 1, 2, 3], [0, 1, 0, 1], [[0, [0, 2]], [1, [1, 3]]], Yes, total_statistic_fibers_partition_sample_space)

Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([0, 1, 2, 3], [0, 1, 0, 1], [[0, [0, 2]], [1, [1, 3]]], Yes, total_statistic_fibers_partition_sample_space).

([10, 11, 12, 13], [0, 1, 0, 1], [[0, [10, 12]], [1, [11, 13]]], Yes, total_statistic_fibers_partition_sample_space)

Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([10, 11, 12, 13], [0, 1, 0, 1], [[0, [10, 12]], [1, [11, 13]]], Yes, total_statistic_fibers_partition_sample_space).

([20, 21, 22, 23], [0, 1, 0, 1], [[0, [20, 22]], [1, [21, 23]]], Yes, total_statistic_fibers_partition_sample_space)

Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([20, 21, 22, 23], [0, 1, 0, 1], [[0, [20, 22]], [1, [21, 23]]], Yes, total_statistic_fibers_partition_sample_space).

([30, 31, 32, 33], [0, 1, 0, 1], [[0, [30, 32]], [1, [31, 33]]], Yes, total_statistic_fibers_partition_sample_space)

Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([30, 31, 32, 33], [0, 1, 0, 1], [[0, [30, 32]], [1, [31, 33]]], Yes, total_statistic_fibers_partition_sample_space).

([40, 41, 42, 43], [0, 1, 0, 1], [[0, [40, 42]], [1, [41, 43]]], Yes, total_statistic_fibers_partition_sample_space)

Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([40, 41, 42, 43], [0, 1, 0, 1], [[0, [40, 42]], [1, [41, 43]]], Yes, total_statistic_fibers_partition_sample_space).

([50, 51, 52, 53], [0, 1, 0, 1], [[0, [50, 52]], [1, [51, 53]]], Yes, total_statistic_fibers_partition_sample_space)

Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([50, 51, 52, 53], [0, 1, 0, 1], [[0, [50, 52]], [1, [51, 53]]], Yes, total_statistic_fibers_partition_sample_space).

([60, 61, 62, 63], [0, 1, 0, 1], [[0, [60, 62]], [1, [61, 63]]], Yes, total_statistic_fibers_partition_sample_space)

Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([60, 61, 62, 63], [0, 1, 0, 1], [[0, [60, 62]], [1, [61, 63]]], Yes, total_statistic_fibers_partition_sample_space).

([70, 71, 72, 73], [0, 1, 0, 1], [[0, [70, 72]], [1, [71, 73]]], Yes, total_statistic_fibers_partition_sample_space)

Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([70, 71, 72, 73], [0, 1, 0, 1], [[0, [70, 72]], [1, [71, 73]]], Yes, total_statistic_fibers_partition_sample_space).

([80, 81, 82, 83], [0, 1, 0, 1], [[0, [80, 82]], [1, [81, 83]]], Yes, total_statistic_fibers_partition_sample_space)

Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([80, 81, 82, 83], [0, 1, 0, 1], [[0, [80, 82]], [1, [81, 83]]], Yes, total_statistic_fibers_partition_sample_space).

([90, 91, 92, 93], [0, 1, 0, 1], [[0, [90, 92]], [1, [91, 93]]], Yes, total_statistic_fibers_partition_sample_space)

Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([90, 91, 92, 93], [0, 1, 0, 1], [[0, [90, 92]], [1, [91, 93]]], Yes, total_statistic_fibers_partition_sample_space).

([100, 101, 102, 103], [0, 1, 0, 1], [[0, [100, 102]], [1, [101, 103]]], Yes, total_statistic_fibers_partition_sample_space)

Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([100, 101, 102, 103], [0, 1, 0, 1], [[0, [100, 102]], [1, [101, 103]]], Yes, total_statistic_fibers_partition_sample_space).

([110, 111, 112, 113], [0, 1, 0, 1], [[0, [110, 112]], [1, [111, 113]]], Yes, total_statistic_fibers_partition_sample_space)

Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([110, 111, 112, 113], [0, 1, 0, 1], [[0, [110, 112]], [1, [111, 113]]], Yes, total_statistic_fibers_partition_sample_space).

([120, 121, 122, 123], [0, 1, 0, 1], [[0, [120, 122]], [1, [121, 123]]], Yes, total_statistic_fibers_partition_sample_space)

Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([120, 121, 122, 123], [0, 1, 0, 1], [[0, [120, 122]], [1, [121, 123]]], Yes, total_statistic_fibers_partition_sample_space).

([130, 131, 132, 133], [0, 1, 0, 1], [[0, [130, 132]], [1, [131, 133]]], Yes, total_statistic_fibers_partition_sample_space)

Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([130, 131, 132, 133], [0, 1, 0, 1], [[0, [130, 132]], [1, [131, 133]]], Yes, total_statistic_fibers_partition_sample_space).

([140, 141, 142, 143], [0, 1, 0, 1], [[0, [140, 142]], [1, [141, 143]]], Yes, total_statistic_fibers_partition_sample_space)

Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([140, 141, 142, 143], [0, 1, 0, 1], [[0, [140, 142]], [1, [141, 143]]], Yes, total_statistic_fibers_partition_sample_space).

([150, 151, 152, 153], [0, 1, 0, 1], [[0, [150, 152]], [1, [151, 153]]], Yes, total_statistic_fibers_partition_sample_space)

Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([150, 151, 152, 153], [0, 1, 0, 1], [[0, [150, 152]], [1, [151, 153]]], Yes, total_statistic_fibers_partition_sample_space).

([160, 161, 162, 163], [0, 1, 0, 1], [[0, [160, 162]], [1, [161, 163]]], Yes, total_statistic_fibers_partition_sample_space)

Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([160, 161, 162, 163], [0, 1, 0, 1], [[0, [160, 162]], [1, [161, 163]]], Yes, total_statistic_fibers_partition_sample_space).

([170, 171, 172, 173], [0, 1, 0, 1], [[0, [170, 172]], [1, [171, 173]]], Yes, total_statistic_fibers_partition_sample_space)

Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([170, 171, 172, 173], [0, 1, 0, 1], [[0, [170, 172]], [1, [171, 173]]], Yes, total_statistic_fibers_partition_sample_space).

([180, 181, 182, 183], [0, 1, 0, 1], [[0, [180, 182]], [1, [181, 183]]], Yes, total_statistic_fibers_partition_sample_space)

Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([180, 181, 182, 183], [0, 1, 0, 1], [[0, [180, 182]], [1, [181, 183]]], Yes, total_statistic_fibers_partition_sample_space).

([190, 191, 192, 193], [0, 1, 0, 1], [[0, [190, 192]], [1, [191, 193]]], Yes, total_statistic_fibers_partition_sample_space)

Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([190, 191, 192, 193], [0, 1, 0, 1], [[0, [190, 192]], [1, [191, 193]]], Yes, total_statistic_fibers_partition_sample_space).

([200, 201, 202, 203], [0, 1, 0, 1], [[0, [200, 202]], [1, [201, 203]]], Yes, total_statistic_fibers_partition_sample_space)

Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([200, 201, 202, 203], [0, 1, 0, 1], [[0, [200, 202]], [1, [201, 203]]], Yes, total_statistic_fibers_partition_sample_space).

([210, 211, 212, 213], [0, 1, 0, 1], [[0, [210, 212]], [1, [211, 213]]], Yes, total_statistic_fibers_partition_sample_space)

Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([210, 211, 212, 213], [0, 1, 0, 1], [[0, [210, 212]], [1, [211, 213]]], Yes, total_statistic_fibers_partition_sample_space).

([220, 221, 222, 223], [0, 1, 0, 1], [[0, [220, 222]], [1, [221, 223]]], Yes, total_statistic_fibers_partition_sample_space)

Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([220, 221, 222, 223], [0, 1, 0, 1], [[0, [220, 222]], [1, [221, 223]]], Yes, total_statistic_fibers_partition_sample_space).

([230, 231, 232, 233], [0, 1, 0, 1], [[0, [230, 232]], [1, [231, 233]]], Yes, total_statistic_fibers_partition_sample_space)

Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([230, 231, 232, 233], [0, 1, 0, 1], [[0, [230, 232]], [1, [231, 233]]], Yes, total_statistic_fibers_partition_sample_space).

([240, 241, 242, 243], [0, 1, 0, 1], [[0, [240, 242]], [1, [241, 243]]], Yes, total_statistic_fibers_partition_sample_space)

Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([240, 241, 242, 243], [0, 1, 0, 1], [[0, [240, 242]], [1, [241, 243]]], Yes, total_statistic_fibers_partition_sample_space).

([[a0, 0, 4/5, [[0, 1/2], [2, 1/2]]], [a0, 1, 1/5, [[1, 1/2], [3, 1/2]]], [b0, 0, 1/5, [[0, 1/2], [2, 1/2]]], [b0, 1, 4/5, [[1, 1/2], [3, 1/2]]]], zero_mass_cells_remain_undefined)

For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a0, 0, 4/5, [[0, 1/2], [2, 1/2]]], [a0, 1, 1/5, [[1, 1/2], [3, 1/2]]], [b0, 0, 1/5, [[0, 1/2], [2, 1/2]]], [b0, 1, 4/5, [[1, 1/2], [3, 1/2]]]], zero_mass_cells_remain_undefined).

([[a1, 0, 4/5, [[10, 1/2], [12, 1/2]]], [a1, 1, 1/5, [[11, 1/2], [13, 1/2]]], [b1, 0, 1/5, [[10, 1/2], [12, 1/2]]], [b1, 1, 4/5, [[11, 1/2], [13, 1/2]]]], zero_mass_cells_remain_undefined)

For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a1, 0, 4/5, [[10, 1/2], [12, 1/2]]], [a1, 1, 1/5, [[11, 1/2], [13, 1/2]]], [b1, 0, 1/5, [[10, 1/2], [12, 1/2]]], [b1, 1, 4/5, [[11, 1/2], [13, 1/2]]]], zero_mass_cells_remain_undefined).

([[a2, 0, 4/5, [[20, 1/2], [22, 1/2]]], [a2, 1, 1/5, [[21, 1/2], [23, 1/2]]], [b2, 0, 1/5, [[20, 1/2], [22, 1/2]]], [b2, 1, 4/5, [[21, 1/2], [23, 1/2]]]], zero_mass_cells_remain_undefined)

For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a2, 0, 4/5, [[20, 1/2], [22, 1/2]]], [a2, 1, 1/5, [[21, 1/2], [23, 1/2]]], [b2, 0, 1/5, [[20, 1/2], [22, 1/2]]], [b2, 1, 4/5, [[21, 1/2], [23, 1/2]]]], zero_mass_cells_remain_undefined).

([[a3, 0, 4/5, [[30, 1/2], [32, 1/2]]], [a3, 1, 1/5, [[31, 1/2], [33, 1/2]]], [b3, 0, 1/5, [[30, 1/2], [32, 1/2]]], [b3, 1, 4/5, [[31, 1/2], [33, 1/2]]]], zero_mass_cells_remain_undefined)

For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a3, 0, 4/5, [[30, 1/2], [32, 1/2]]], [a3, 1, 1/5, [[31, 1/2], [33, 1/2]]], [b3, 0, 1/5, [[30, 1/2], [32, 1/2]]], [b3, 1, 4/5, [[31, 1/2], [33, 1/2]]]], zero_mass_cells_remain_undefined).

([[a4, 0, 4/5, [[40, 1/2], [42, 1/2]]], [a4, 1, 1/5, [[41, 1/2], [43, 1/2]]], [b4, 0, 1/5, [[40, 1/2], [42, 1/2]]], [b4, 1, 4/5, [[41, 1/2], [43, 1/2]]]], zero_mass_cells_remain_undefined)

For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a4, 0, 4/5, [[40, 1/2], [42, 1/2]]], [a4, 1, 1/5, [[41, 1/2], [43, 1/2]]], [b4, 0, 1/5, [[40, 1/2], [42, 1/2]]], [b4, 1, 4/5, [[41, 1/2], [43, 1/2]]]], zero_mass_cells_remain_undefined).

([[a5, 0, 4/5, [[50, 1/2], [52, 1/2]]], [a5, 1, 1/5, [[51, 1/2], [53, 1/2]]], [b5, 0, 1/5, [[50, 1/2], [52, 1/2]]], [b5, 1, 4/5, [[51, 1/2], [53, 1/2]]]], zero_mass_cells_remain_undefined)

For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a5, 0, 4/5, [[50, 1/2], [52, 1/2]]], [a5, 1, 1/5, [[51, 1/2], [53, 1/2]]], [b5, 0, 1/5, [[50, 1/2], [52, 1/2]]], [b5, 1, 4/5, [[51, 1/2], [53, 1/2]]]], zero_mass_cells_remain_undefined).

([[a6, 0, 4/5, [[60, 1/2], [62, 1/2]]], [a6, 1, 1/5, [[61, 1/2], [63, 1/2]]], [b6, 0, 1/5, [[60, 1/2], [62, 1/2]]], [b6, 1, 4/5, [[61, 1/2], [63, 1/2]]]], zero_mass_cells_remain_undefined)

For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a6, 0, 4/5, [[60, 1/2], [62, 1/2]]], [a6, 1, 1/5, [[61, 1/2], [63, 1/2]]], [b6, 0, 1/5, [[60, 1/2], [62, 1/2]]], [b6, 1, 4/5, [[61, 1/2], [63, 1/2]]]], zero_mass_cells_remain_undefined).

([[a7, 0, 4/5, [[70, 1/2], [72, 1/2]]], [a7, 1, 1/5, [[71, 1/2], [73, 1/2]]], [b7, 0, 1/5, [[70, 1/2], [72, 1/2]]], [b7, 1, 4/5, [[71, 1/2], [73, 1/2]]]], zero_mass_cells_remain_undefined)

For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a7, 0, 4/5, [[70, 1/2], [72, 1/2]]], [a7, 1, 1/5, [[71, 1/2], [73, 1/2]]], [b7, 0, 1/5, [[70, 1/2], [72, 1/2]]], [b7, 1, 4/5, [[71, 1/2], [73, 1/2]]]], zero_mass_cells_remain_undefined).

([[a8, 0, 4/5, [[80, 1/2], [82, 1/2]]], [a8, 1, 1/5, [[81, 1/2], [83, 1/2]]], [b8, 0, 1/5, [[80, 1/2], [82, 1/2]]], [b8, 1, 4/5, [[81, 1/2], [83, 1/2]]]], zero_mass_cells_remain_undefined)

For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a8, 0, 4/5, [[80, 1/2], [82, 1/2]]], [a8, 1, 1/5, [[81, 1/2], [83, 1/2]]], [b8, 0, 1/5, [[80, 1/2], [82, 1/2]]], [b8, 1, 4/5, [[81, 1/2], [83, 1/2]]]], zero_mass_cells_remain_undefined).

([[a9, 0, 4/5, [[90, 1/2], [92, 1/2]]], [a9, 1, 1/5, [[91, 1/2], [93, 1/2]]], [b9, 0, 1/5, [[90, 1/2], [92, 1/2]]], [b9, 1, 4/5, [[91, 1/2], [93, 1/2]]]], zero_mass_cells_remain_undefined)

For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a9, 0, 4/5, [[90, 1/2], [92, 1/2]]], [a9, 1, 1/5, [[91, 1/2], [93, 1/2]]], [b9, 0, 1/5, [[90, 1/2], [92, 1/2]]], [b9, 1, 4/5, [[91, 1/2], [93, 1/2]]]], zero_mass_cells_remain_undefined).

([[a10, 0, 4/5, [[100, 1/2], [102, 1/2]]], [a10, 1, 1/5, [[101, 1/2], [103, 1/2]]], [b10, 0, 1/5, [[100, 1/2], [102, 1/2]]], [b10, 1, 4/5, [[101, 1/2], [103, 1/2]]]], zero_mass_cells_remain_undefined)

For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a10, 0, 4/5, [[100, 1/2], [102, 1/2]]], [a10, 1, 1/5, [[101, 1/2], [103, 1/2]]], [b10, 0, 1/5, [[100, 1/2], [102, 1/2]]], [b10, 1, 4/5, [[101, 1/2], [103, 1/2]]]], zero_mass_cells_remain_undefined).

([[a11, 0, 4/5, [[110, 1/2], [112, 1/2]]], [a11, 1, 1/5, [[111, 1/2], [113, 1/2]]], [b11, 0, 1/5, [[110, 1/2], [112, 1/2]]], [b11, 1, 4/5, [[111, 1/2], [113, 1/2]]]], zero_mass_cells_remain_undefined)

For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a11, 0, 4/5, [[110, 1/2], [112, 1/2]]], [a11, 1, 1/5, [[111, 1/2], [113, 1/2]]], [b11, 0, 1/5, [[110, 1/2], [112, 1/2]]], [b11, 1, 4/5, [[111, 1/2], [113, 1/2]]]], zero_mass_cells_remain_undefined).

([[a12, 0, 4/5, [[120, 1/2], [122, 1/2]]], [a12, 1, 1/5, [[121, 1/2], [123, 1/2]]], [b12, 0, 1/5, [[120, 1/2], [122, 1/2]]], [b12, 1, 4/5, [[121, 1/2], [123, 1/2]]]], zero_mass_cells_remain_undefined)

For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a12, 0, 4/5, [[120, 1/2], [122, 1/2]]], [a12, 1, 1/5, [[121, 1/2], [123, 1/2]]], [b12, 0, 1/5, [[120, 1/2], [122, 1/2]]], [b12, 1, 4/5, [[121, 1/2], [123, 1/2]]]], zero_mass_cells_remain_undefined).

([[a13, 0, 4/5, [[130, 1/2], [132, 1/2]]], [a13, 1, 1/5, [[131, 1/2], [133, 1/2]]], [b13, 0, 1/5, [[130, 1/2], [132, 1/2]]], [b13, 1, 4/5, [[131, 1/2], [133, 1/2]]]], zero_mass_cells_remain_undefined)

For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a13, 0, 4/5, [[130, 1/2], [132, 1/2]]], [a13, 1, 1/5, [[131, 1/2], [133, 1/2]]], [b13, 0, 1/5, [[130, 1/2], [132, 1/2]]], [b13, 1, 4/5, [[131, 1/2], [133, 1/2]]]], zero_mass_cells_remain_undefined).

([[a14, 0, 4/5, [[140, 1/2], [142, 1/2]]], [a14, 1, 1/5, [[141, 1/2], [143, 1/2]]], [b14, 0, 1/5, [[140, 1/2], [142, 1/2]]], [b14, 1, 4/5, [[141, 1/2], [143, 1/2]]]], zero_mass_cells_remain_undefined)

For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a14, 0, 4/5, [[140, 1/2], [142, 1/2]]], [a14, 1, 1/5, [[141, 1/2], [143, 1/2]]], [b14, 0, 1/5, [[140, 1/2], [142, 1/2]]], [b14, 1, 4/5, [[141, 1/2], [143, 1/2]]]], zero_mass_cells_remain_undefined).

([[a15, 0, 4/5, [[150, 1/2], [152, 1/2]]], [a15, 1, 1/5, [[151, 1/2], [153, 1/2]]], [b15, 0, 1/5, [[150, 1/2], [152, 1/2]]], [b15, 1, 4/5, [[151, 1/2], [153, 1/2]]]], zero_mass_cells_remain_undefined)

For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a15, 0, 4/5, [[150, 1/2], [152, 1/2]]], [a15, 1, 1/5, [[151, 1/2], [153, 1/2]]], [b15, 0, 1/5, [[150, 1/2], [152, 1/2]]], [b15, 1, 4/5, [[151, 1/2], [153, 1/2]]]], zero_mass_cells_remain_undefined).

([[a16, 0, 4/5, [[160, 1/2], [162, 1/2]]], [a16, 1, 1/5, [[161, 1/2], [163, 1/2]]], [b16, 0, 1/5, [[160, 1/2], [162, 1/2]]], [b16, 1, 4/5, [[161, 1/2], [163, 1/2]]]], zero_mass_cells_remain_undefined)

For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a16, 0, 4/5, [[160, 1/2], [162, 1/2]]], [a16, 1, 1/5, [[161, 1/2], [163, 1/2]]], [b16, 0, 1/5, [[160, 1/2], [162, 1/2]]], [b16, 1, 4/5, [[161, 1/2], [163, 1/2]]]], zero_mass_cells_remain_undefined).

([[a17, 0, 4/5, [[170, 1/2], [172, 1/2]]], [a17, 1, 1/5, [[171, 1/2], [173, 1/2]]], [b17, 0, 1/5, [[170, 1/2], [172, 1/2]]], [b17, 1, 4/5, [[171, 1/2], [173, 1/2]]]], zero_mass_cells_remain_undefined)

For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a17, 0, 4/5, [[170, 1/2], [172, 1/2]]], [a17, 1, 1/5, [[171, 1/2], [173, 1/2]]], [b17, 0, 1/5, [[170, 1/2], [172, 1/2]]], [b17, 1, 4/5, [[171, 1/2], [173, 1/2]]]], zero_mass_cells_remain_undefined).

([[a18, 0, 4/5, [[180, 1/2], [182, 1/2]]], [a18, 1, 1/5, [[181, 1/2], [183, 1/2]]], [b18, 0, 1/5, [[180, 1/2], [182, 1/2]]], [b18, 1, 4/5, [[181, 1/2], [183, 1/2]]]], zero_mass_cells_remain_undefined)

For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a18, 0, 4/5, [[180, 1/2], [182, 1/2]]], [a18, 1, 1/5, [[181, 1/2], [183, 1/2]]], [b18, 0, 1/5, [[180, 1/2], [182, 1/2]]], [b18, 1, 4/5, [[181, 1/2], [183, 1/2]]]], zero_mass_cells_remain_undefined).

([[a19, 0, 4/5, [[190, 1/2], [192, 1/2]]], [a19, 1, 1/5, [[191, 1/2], [193, 1/2]]], [b19, 0, 1/5, [[190, 1/2], [192, 1/2]]], [b19, 1, 4/5, [[191, 1/2], [193, 1/2]]]], zero_mass_cells_remain_undefined)

For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a19, 0, 4/5, [[190, 1/2], [192, 1/2]]], [a19, 1, 1/5, [[191, 1/2], [193, 1/2]]], [b19, 0, 1/5, [[190, 1/2], [192, 1/2]]], [b19, 1, 4/5, [[191, 1/2], [193, 1/2]]]], zero_mass_cells_remain_undefined).

([[a20, 0, 4/5, [[200, 1/2], [202, 1/2]]], [a20, 1, 1/5, [[201, 1/2], [203, 1/2]]], [b20, 0, 1/5, [[200, 1/2], [202, 1/2]]], [b20, 1, 4/5, [[201, 1/2], [203, 1/2]]]], zero_mass_cells_remain_undefined)

For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a20, 0, 4/5, [[200, 1/2], [202, 1/2]]], [a20, 1, 1/5, [[201, 1/2], [203, 1/2]]], [b20, 0, 1/5, [[200, 1/2], [202, 1/2]]], [b20, 1, 4/5, [[201, 1/2], [203, 1/2]]]], zero_mass_cells_remain_undefined).

([[a21, 0, 4/5, [[210, 1/2], [212, 1/2]]], [a21, 1, 1/5, [[211, 1/2], [213, 1/2]]], [b21, 0, 1/5, [[210, 1/2], [212, 1/2]]], [b21, 1, 4/5, [[211, 1/2], [213, 1/2]]]], zero_mass_cells_remain_undefined)

For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a21, 0, 4/5, [[210, 1/2], [212, 1/2]]], [a21, 1, 1/5, [[211, 1/2], [213, 1/2]]], [b21, 0, 1/5, [[210, 1/2], [212, 1/2]]], [b21, 1, 4/5, [[211, 1/2], [213, 1/2]]]], zero_mass_cells_remain_undefined).

([[a22, 0, 4/5, [[220, 1/2], [222, 1/2]]], [a22, 1, 1/5, [[221, 1/2], [223, 1/2]]], [b22, 0, 1/5, [[220, 1/2], [222, 1/2]]], [b22, 1, 4/5, [[221, 1/2], [223, 1/2]]]], zero_mass_cells_remain_undefined)

For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a22, 0, 4/5, [[220, 1/2], [222, 1/2]]], [a22, 1, 1/5, [[221, 1/2], [223, 1/2]]], [b22, 0, 1/5, [[220, 1/2], [222, 1/2]]], [b22, 1, 4/5, [[221, 1/2], [223, 1/2]]]], zero_mass_cells_remain_undefined).

([[a23, 0, 4/5, [[230, 1/2], [232, 1/2]]], [a23, 1, 1/5, [[231, 1/2], [233, 1/2]]], [b23, 0, 1/5, [[230, 1/2], [232, 1/2]]], [b23, 1, 4/5, [[231, 1/2], [233, 1/2]]]], zero_mass_cells_remain_undefined)

For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a23, 0, 4/5, [[230, 1/2], [232, 1/2]]], [a23, 1, 1/5, [[231, 1/2], [233, 1/2]]], [b23, 0, 1/5, [[230, 1/2], [232, 1/2]]], [b23, 1, 4/5, [[231, 1/2], [233, 1/2]]]], zero_mass_cells_remain_undefined).

([[a24, 0, 4/5, [[240, 1/2], [242, 1/2]]], [a24, 1, 1/5, [[241, 1/2], [243, 1/2]]], [b24, 0, 1/5, [[240, 1/2], [242, 1/2]]], [b24, 1, 4/5, [[241, 1/2], [243, 1/2]]]], zero_mass_cells_remain_undefined)

For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a24, 0, 4/5, [[240, 1/2], [242, 1/2]]], [a24, 1, 1/5, [[241, 1/2], [243, 1/2]]], [b24, 0, 1/5, [[240, 1/2], [242, 1/2]]], [b24, 1, 4/5, [[241, 1/2], [243, 1/2]]]], zero_mass_cells_remain_undefined).

([[0, [[a0, [[0, 1/2], [2, 1/2]]], [b0, [[0, 1/2], [2, 1/2]]]], Yes], [1, [[a0, [[1, 1/2], [3, 1/2]]], [b0, [[1, 1/2], [3, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)

For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a0, [[0, 1/2], [2, 1/2]]], [b0, [[0, 1/2], [2, 1/2]]]], Yes], [1, [[a0, [[1, 1/2], [3, 1/2]]], [b0, [[1, 1/2], [3, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell).

([[0, [[a1, [[10, 1/2], [12, 1/2]]], [b1, [[10, 1/2], [12, 1/2]]]], Yes], [1, [[a1, [[11, 1/2], [13, 1/2]]], [b1, [[11, 1/2], [13, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)

For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a1, [[10, 1/2], [12, 1/2]]], [b1, [[10, 1/2], [12, 1/2]]]], Yes], [1, [[a1, [[11, 1/2], [13, 1/2]]], [b1, [[11, 1/2], [13, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell).

([[0, [[a2, [[20, 1/2], [22, 1/2]]], [b2, [[20, 1/2], [22, 1/2]]]], Yes], [1, [[a2, [[21, 1/2], [23, 1/2]]], [b2, [[21, 1/2], [23, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)

For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a2, [[20, 1/2], [22, 1/2]]], [b2, [[20, 1/2], [22, 1/2]]]], Yes], [1, [[a2, [[21, 1/2], [23, 1/2]]], [b2, [[21, 1/2], [23, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell).

([[0, [[a3, [[30, 1/2], [32, 1/2]]], [b3, [[30, 1/2], [32, 1/2]]]], Yes], [1, [[a3, [[31, 1/2], [33, 1/2]]], [b3, [[31, 1/2], [33, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)

For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a3, [[30, 1/2], [32, 1/2]]], [b3, [[30, 1/2], [32, 1/2]]]], Yes], [1, [[a3, [[31, 1/2], [33, 1/2]]], [b3, [[31, 1/2], [33, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell).

([[0, [[a4, [[40, 1/2], [42, 1/2]]], [b4, [[40, 1/2], [42, 1/2]]]], Yes], [1, [[a4, [[41, 1/2], [43, 1/2]]], [b4, [[41, 1/2], [43, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)

For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a4, [[40, 1/2], [42, 1/2]]], [b4, [[40, 1/2], [42, 1/2]]]], Yes], [1, [[a4, [[41, 1/2], [43, 1/2]]], [b4, [[41, 1/2], [43, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell).

([[0, [[a5, [[50, 1/2], [52, 1/2]]], [b5, [[50, 1/2], [52, 1/2]]]], Yes], [1, [[a5, [[51, 1/2], [53, 1/2]]], [b5, [[51, 1/2], [53, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)

For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a5, [[50, 1/2], [52, 1/2]]], [b5, [[50, 1/2], [52, 1/2]]]], Yes], [1, [[a5, [[51, 1/2], [53, 1/2]]], [b5, [[51, 1/2], [53, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell).

([[0, [[a6, [[60, 1/2], [62, 1/2]]], [b6, [[60, 1/2], [62, 1/2]]]], Yes], [1, [[a6, [[61, 1/2], [63, 1/2]]], [b6, [[61, 1/2], [63, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)

For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a6, [[60, 1/2], [62, 1/2]]], [b6, [[60, 1/2], [62, 1/2]]]], Yes], [1, [[a6, [[61, 1/2], [63, 1/2]]], [b6, [[61, 1/2], [63, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell).

([[0, [[a7, [[70, 1/2], [72, 1/2]]], [b7, [[70, 1/2], [72, 1/2]]]], Yes], [1, [[a7, [[71, 1/2], [73, 1/2]]], [b7, [[71, 1/2], [73, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)

For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a7, [[70, 1/2], [72, 1/2]]], [b7, [[70, 1/2], [72, 1/2]]]], Yes], [1, [[a7, [[71, 1/2], [73, 1/2]]], [b7, [[71, 1/2], [73, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell).

([[0, [[a8, [[80, 1/2], [82, 1/2]]], [b8, [[80, 1/2], [82, 1/2]]]], Yes], [1, [[a8, [[81, 1/2], [83, 1/2]]], [b8, [[81, 1/2], [83, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)

For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a8, [[80, 1/2], [82, 1/2]]], [b8, [[80, 1/2], [82, 1/2]]]], Yes], [1, [[a8, [[81, 1/2], [83, 1/2]]], [b8, [[81, 1/2], [83, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell).

([[0, [[a9, [[90, 1/2], [92, 1/2]]], [b9, [[90, 1/2], [92, 1/2]]]], Yes], [1, [[a9, [[91, 1/2], [93, 1/2]]], [b9, [[91, 1/2], [93, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)

For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a9, [[90, 1/2], [92, 1/2]]], [b9, [[90, 1/2], [92, 1/2]]]], Yes], [1, [[a9, [[91, 1/2], [93, 1/2]]], [b9, [[91, 1/2], [93, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell).

([[0, [[a10, [[100, 1/2], [102, 1/2]]], [b10, [[100, 1/2], [102, 1/2]]]], Yes], [1, [[a10, [[101, 1/2], [103, 1/2]]], [b10, [[101, 1/2], [103, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)

For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a10, [[100, 1/2], [102, 1/2]]], [b10, [[100, 1/2], [102, 1/2]]]], Yes], [1, [[a10, [[101, 1/2], [103, 1/2]]], [b10, [[101, 1/2], [103, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell).

([[0, [[a11, [[110, 1/2], [112, 1/2]]], [b11, [[110, 1/2], [112, 1/2]]]], Yes], [1, [[a11, [[111, 1/2], [113, 1/2]]], [b11, [[111, 1/2], [113, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)

For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a11, [[110, 1/2], [112, 1/2]]], [b11, [[110, 1/2], [112, 1/2]]]], Yes], [1, [[a11, [[111, 1/2], [113, 1/2]]], [b11, [[111, 1/2], [113, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell).

([[0, [[a12, [[120, 1/2], [122, 1/2]]], [b12, [[120, 1/2], [122, 1/2]]]], Yes], [1, [[a12, [[121, 1/2], [123, 1/2]]], [b12, [[121, 1/2], [123, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)

For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a12, [[120, 1/2], [122, 1/2]]], [b12, [[120, 1/2], [122, 1/2]]]], Yes], [1, [[a12, [[121, 1/2], [123, 1/2]]], [b12, [[121, 1/2], [123, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell).

([[0, [[a13, [[130, 1/2], [132, 1/2]]], [b13, [[130, 1/2], [132, 1/2]]]], Yes], [1, [[a13, [[131, 1/2], [133, 1/2]]], [b13, [[131, 1/2], [133, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)

For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a13, [[130, 1/2], [132, 1/2]]], [b13, [[130, 1/2], [132, 1/2]]]], Yes], [1, [[a13, [[131, 1/2], [133, 1/2]]], [b13, [[131, 1/2], [133, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell).

([[0, [[a14, [[140, 1/2], [142, 1/2]]], [b14, [[140, 1/2], [142, 1/2]]]], Yes], [1, [[a14, [[141, 1/2], [143, 1/2]]], [b14, [[141, 1/2], [143, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)

For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a14, [[140, 1/2], [142, 1/2]]], [b14, [[140, 1/2], [142, 1/2]]]], Yes], [1, [[a14, [[141, 1/2], [143, 1/2]]], [b14, [[141, 1/2], [143, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell).

([[0, [[a15, [[150, 1/2], [152, 1/2]]], [b15, [[150, 1/2], [152, 1/2]]]], Yes], [1, [[a15, [[151, 1/2], [153, 1/2]]], [b15, [[151, 1/2], [153, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)

For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a15, [[150, 1/2], [152, 1/2]]], [b15, [[150, 1/2], [152, 1/2]]]], Yes], [1, [[a15, [[151, 1/2], [153, 1/2]]], [b15, [[151, 1/2], [153, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell).

([[0, [[a16, [[160, 1/2], [162, 1/2]]], [b16, [[160, 1/2], [162, 1/2]]]], Yes], [1, [[a16, [[161, 1/2], [163, 1/2]]], [b16, [[161, 1/2], [163, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)

For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a16, [[160, 1/2], [162, 1/2]]], [b16, [[160, 1/2], [162, 1/2]]]], Yes], [1, [[a16, [[161, 1/2], [163, 1/2]]], [b16, [[161, 1/2], [163, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell).

([[0, [[a17, [[170, 1/2], [172, 1/2]]], [b17, [[170, 1/2], [172, 1/2]]]], Yes], [1, [[a17, [[171, 1/2], [173, 1/2]]], [b17, [[171, 1/2], [173, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)

For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a17, [[170, 1/2], [172, 1/2]]], [b17, [[170, 1/2], [172, 1/2]]]], Yes], [1, [[a17, [[171, 1/2], [173, 1/2]]], [b17, [[171, 1/2], [173, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell).

([[0, [[a18, [[180, 1/2], [182, 1/2]]], [b18, [[180, 1/2], [182, 1/2]]]], Yes], [1, [[a18, [[181, 1/2], [183, 1/2]]], [b18, [[181, 1/2], [183, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)

For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a18, [[180, 1/2], [182, 1/2]]], [b18, [[180, 1/2], [182, 1/2]]]], Yes], [1, [[a18, [[181, 1/2], [183, 1/2]]], [b18, [[181, 1/2], [183, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell).

([[0, [[a19, [[190, 1/2], [192, 1/2]]], [b19, [[190, 1/2], [192, 1/2]]]], Yes], [1, [[a19, [[191, 1/2], [193, 1/2]]], [b19, [[191, 1/2], [193, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)

For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a19, [[190, 1/2], [192, 1/2]]], [b19, [[190, 1/2], [192, 1/2]]]], Yes], [1, [[a19, [[191, 1/2], [193, 1/2]]], [b19, [[191, 1/2], [193, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell).

([[0, [[a20, [[200, 1/2], [202, 1/2]]], [b20, [[200, 1/2], [202, 1/2]]]], Yes], [1, [[a20, [[201, 1/2], [203, 1/2]]], [b20, [[201, 1/2], [203, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)

For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a20, [[200, 1/2], [202, 1/2]]], [b20, [[200, 1/2], [202, 1/2]]]], Yes], [1, [[a20, [[201, 1/2], [203, 1/2]]], [b20, [[201, 1/2], [203, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell).

([[0, [[a21, [[210, 1/2], [212, 1/2]]], [b21, [[210, 1/2], [212, 1/2]]]], Yes], [1, [[a21, [[211, 1/2], [213, 1/2]]], [b21, [[211, 1/2], [213, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)

For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a21, [[210, 1/2], [212, 1/2]]], [b21, [[210, 1/2], [212, 1/2]]]], Yes], [1, [[a21, [[211, 1/2], [213, 1/2]]], [b21, [[211, 1/2], [213, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell).

([[0, [[a22, [[220, 1/2], [222, 1/2]]], [b22, [[220, 1/2], [222, 1/2]]]], Yes], [1, [[a22, [[221, 1/2], [223, 1/2]]], [b22, [[221, 1/2], [223, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)

For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a22, [[220, 1/2], [222, 1/2]]], [b22, [[220, 1/2], [222, 1/2]]]], Yes], [1, [[a22, [[221, 1/2], [223, 1/2]]], [b22, [[221, 1/2], [223, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell).

([[0, [[a23, [[230, 1/2], [232, 1/2]]], [b23, [[230, 1/2], [232, 1/2]]]], Yes], [1, [[a23, [[231, 1/2], [233, 1/2]]], [b23, [[231, 1/2], [233, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)

For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a23, [[230, 1/2], [232, 1/2]]], [b23, [[230, 1/2], [232, 1/2]]]], Yes], [1, [[a23, [[231, 1/2], [233, 1/2]]], [b23, [[231, 1/2], [233, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell).

([[0, [[a24, [[240, 1/2], [242, 1/2]]], [b24, [[240, 1/2], [242, 1/2]]]], Yes], [1, [[a24, [[241, 1/2], [243, 1/2]]], [b24, [[241, 1/2], [243, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)

For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a24, [[240, 1/2], [242, 1/2]]], [b24, [[240, 1/2], [242, 1/2]]]], Yes], [1, [[a24, [[241, 1/2], [243, 1/2]]], [b24, [[241, 1/2], [243, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell).

(nonsufficient, [[0, [[a0, [[0, 3/4], [2, 1/4]]], [b0, [[0, 1/2], [2, 1/2]]]], No], [1, [[a0, [[1, 1/2], [3, 1/2]]], [b0, [[1, 1/2], [3, 1/2]]]], Yes]], No, [], [0, [[a0, [[0, 3/4], [2, 1/4]]], [b0, [[0, 1/2], [2, 1/2]]]], No], No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)

Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (nonsufficient, [[0, [[a0, [[0, 3/4], [2, 1/4]]], [b0, [[0, 1/2], [2, 1/2]]]], No], [1, [[a0, [[1, 1/2], [3, 1/2]]], [b0, [[1, 1/2], [3, 1/2]]]], Yes]], No, [], [0, [[a0, [[0, 3/4], [2, 1/4]]], [b0, [[0, 1/2], [2, 1/2]]]], No], No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim).

(zero_mass, [[0, [[a1, [[10, 1/2], [12, 1/2]]]], Yes], [1, [[b1, [[11, 1/2], [13, 1/2]]]], Yes]], Yes, [[a1, 1, 0, null], [b1, 0, 0, null]], null, No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)

Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (zero_mass, [[0, [[a1, [[10, 1/2], [12, 1/2]]]], Yes], [1, [[b1, [[11, 1/2], [13, 1/2]]]], Yes]], Yes, [[a1, 1, 0, null], [b1, 0, 0, null]], null, No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim).

(identity, [[0, [[a2, [[20, 1]]], [b2, [[20, 1]]]], Yes], [1, [[a2, [[21, 1]]], [b2, [[21, 1]]]], Yes], [2, [[a2, [[22, 1]]], [b2, [[22, 1]]]], Yes], [3, [[a2, [[23, 1]]], [b2, [[23, 1]]]], Yes]], Yes, [], null, Yes, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)

Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (identity, [[0, [[a2, [[20, 1]]], [b2, [[20, 1]]]], Yes], [1, [[a2, [[21, 1]]], [b2, [[21, 1]]]], Yes], [2, [[a2, [[22, 1]]], [b2, [[22, 1]]]], Yes], [3, [[a2, [[23, 1]]], [b2, [[23, 1]]]], Yes]], Yes, [], null, Yes, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim).

(constant, [[0, [[a3, [[30, 2/5], [31, 1/10], [32, 2/5], [33, 1/10]]], [b3, [[30, 1/10], [31, 2/5], [32, 1/10], [33, 2/5]]]], No]], No, [], null, No, Yes, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)

Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (constant, [[0, [[a3, [[30, 2/5], [31, 1/10], [32, 2/5], [33, 1/10]]], [b3, [[30, 1/10], [31, 2/5], [32, 1/10], [33, 2/5]]]], No]], No, [], null, No, Yes, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim).

(nonsufficient, [[0, [[a4, [[40, 3/4], [42, 1/4]]], [b4, [[40, 1/2], [42, 1/2]]]], No], [1, [[a4, [[41, 1/2], [43, 1/2]]], [b4, [[41, 1/2], [43, 1/2]]]], Yes]], No, [], [0, [[a4, [[40, 3/4], [42, 1/4]]], [b4, [[40, 1/2], [42, 1/2]]]], No], No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)

Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (nonsufficient, [[0, [[a4, [[40, 3/4], [42, 1/4]]], [b4, [[40, 1/2], [42, 1/2]]]], No], [1, [[a4, [[41, 1/2], [43, 1/2]]], [b4, [[41, 1/2], [43, 1/2]]]], Yes]], No, [], [0, [[a4, [[40, 3/4], [42, 1/4]]], [b4, [[40, 1/2], [42, 1/2]]]], No], No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim).

(zero_mass, [[0, [[a5, [[50, 1/2], [52, 1/2]]]], Yes], [1, [[b5, [[51, 1/2], [53, 1/2]]]], Yes]], Yes, [[a5, 1, 0, null], [b5, 0, 0, null]], null, No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)

Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (zero_mass, [[0, [[a5, [[50, 1/2], [52, 1/2]]]], Yes], [1, [[b5, [[51, 1/2], [53, 1/2]]]], Yes]], Yes, [[a5, 1, 0, null], [b5, 0, 0, null]], null, No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim).

(identity, [[0, [[a6, [[60, 1]]], [b6, [[60, 1]]]], Yes], [1, [[a6, [[61, 1]]], [b6, [[61, 1]]]], Yes], [2, [[a6, [[62, 1]]], [b6, [[62, 1]]]], Yes], [3, [[a6, [[63, 1]]], [b6, [[63, 1]]]], Yes]], Yes, [], null, Yes, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)

Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (identity, [[0, [[a6, [[60, 1]]], [b6, [[60, 1]]]], Yes], [1, [[a6, [[61, 1]]], [b6, [[61, 1]]]], Yes], [2, [[a6, [[62, 1]]], [b6, [[62, 1]]]], Yes], [3, [[a6, [[63, 1]]], [b6, [[63, 1]]]], Yes]], Yes, [], null, Yes, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim).

(constant, [[0, [[a7, [[70, 2/5], [71, 1/10], [72, 2/5], [73, 1/10]]], [b7, [[70, 1/10], [71, 2/5], [72, 1/10], [73, 2/5]]]], No]], No, [], null, No, Yes, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)

Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (constant, [[0, [[a7, [[70, 2/5], [71, 1/10], [72, 2/5], [73, 1/10]]], [b7, [[70, 1/10], [71, 2/5], [72, 1/10], [73, 2/5]]]], No]], No, [], null, No, Yes, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim).

(nonsufficient, [[0, [[a8, [[80, 3/4], [82, 1/4]]], [b8, [[80, 1/2], [82, 1/2]]]], No], [1, [[a8, [[81, 1/2], [83, 1/2]]], [b8, [[81, 1/2], [83, 1/2]]]], Yes]], No, [], [0, [[a8, [[80, 3/4], [82, 1/4]]], [b8, [[80, 1/2], [82, 1/2]]]], No], No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)

Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (nonsufficient, [[0, [[a8, [[80, 3/4], [82, 1/4]]], [b8, [[80, 1/2], [82, 1/2]]]], No], [1, [[a8, [[81, 1/2], [83, 1/2]]], [b8, [[81, 1/2], [83, 1/2]]]], Yes]], No, [], [0, [[a8, [[80, 3/4], [82, 1/4]]], [b8, [[80, 1/2], [82, 1/2]]]], No], No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim).

(zero_mass, [[0, [[a9, [[90, 1/2], [92, 1/2]]]], Yes], [1, [[b9, [[91, 1/2], [93, 1/2]]]], Yes]], Yes, [[a9, 1, 0, null], [b9, 0, 0, null]], null, No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)

Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (zero_mass, [[0, [[a9, [[90, 1/2], [92, 1/2]]]], Yes], [1, [[b9, [[91, 1/2], [93, 1/2]]]], Yes]], Yes, [[a9, 1, 0, null], [b9, 0, 0, null]], null, No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim).

(identity, [[0, [[a10, [[100, 1]]], [b10, [[100, 1]]]], Yes], [1, [[a10, [[101, 1]]], [b10, [[101, 1]]]], Yes], [2, [[a10, [[102, 1]]], [b10, [[102, 1]]]], Yes], [3, [[a10, [[103, 1]]], [b10, [[103, 1]]]], Yes]], Yes, [], null, Yes, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)

Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (identity, [[0, [[a10, [[100, 1]]], [b10, [[100, 1]]]], Yes], [1, [[a10, [[101, 1]]], [b10, [[101, 1]]]], Yes], [2, [[a10, [[102, 1]]], [b10, [[102, 1]]]], Yes], [3, [[a10, [[103, 1]]], [b10, [[103, 1]]]], Yes]], Yes, [], null, Yes, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim).

(constant, [[0, [[a11, [[110, 2/5], [111, 1/10], [112, 2/5], [113, 1/10]]], [b11, [[110, 1/10], [111, 2/5], [112, 1/10], [113, 2/5]]]], No]], No, [], null, No, Yes, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)

Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (constant, [[0, [[a11, [[110, 2/5], [111, 1/10], [112, 2/5], [113, 1/10]]], [b11, [[110, 1/10], [111, 2/5], [112, 1/10], [113, 2/5]]]], No]], No, [], null, No, Yes, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim).

(nonsufficient, [[0, [[a12, [[120, 3/4], [122, 1/4]]], [b12, [[120, 1/2], [122, 1/2]]]], No], [1, [[a12, [[121, 1/2], [123, 1/2]]], [b12, [[121, 1/2], [123, 1/2]]]], Yes]], No, [], [0, [[a12, [[120, 3/4], [122, 1/4]]], [b12, [[120, 1/2], [122, 1/2]]]], No], No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)

Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (nonsufficient, [[0, [[a12, [[120, 3/4], [122, 1/4]]], [b12, [[120, 1/2], [122, 1/2]]]], No], [1, [[a12, [[121, 1/2], [123, 1/2]]], [b12, [[121, 1/2], [123, 1/2]]]], Yes]], No, [], [0, [[a12, [[120, 3/4], [122, 1/4]]], [b12, [[120, 1/2], [122, 1/2]]]], No], No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim).

(zero_mass, [[0, [[a13, [[130, 1/2], [132, 1/2]]]], Yes], [1, [[b13, [[131, 1/2], [133, 1/2]]]], Yes]], Yes, [[a13, 1, 0, null], [b13, 0, 0, null]], null, No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)

Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (zero_mass, [[0, [[a13, [[130, 1/2], [132, 1/2]]]], Yes], [1, [[b13, [[131, 1/2], [133, 1/2]]]], Yes]], Yes, [[a13, 1, 0, null], [b13, 0, 0, null]], null, No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim).

(identity, [[0, [[a14, [[140, 1]]], [b14, [[140, 1]]]], Yes], [1, [[a14, [[141, 1]]], [b14, [[141, 1]]]], Yes], [2, [[a14, [[142, 1]]], [b14, [[142, 1]]]], Yes], [3, [[a14, [[143, 1]]], [b14, [[143, 1]]]], Yes]], Yes, [], null, Yes, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)

Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (identity, [[0, [[a14, [[140, 1]]], [b14, [[140, 1]]]], Yes], [1, [[a14, [[141, 1]]], [b14, [[141, 1]]]], Yes], [2, [[a14, [[142, 1]]], [b14, [[142, 1]]]], Yes], [3, [[a14, [[143, 1]]], [b14, [[143, 1]]]], Yes]], Yes, [], null, Yes, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim).

(constant, [[0, [[a15, [[150, 2/5], [151, 1/10], [152, 2/5], [153, 1/10]]], [b15, [[150, 1/10], [151, 2/5], [152, 1/10], [153, 2/5]]]], No]], No, [], null, No, Yes, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)

Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (constant, [[0, [[a15, [[150, 2/5], [151, 1/10], [152, 2/5], [153, 1/10]]], [b15, [[150, 1/10], [151, 2/5], [152, 1/10], [153, 2/5]]]], No]], No, [], null, No, Yes, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim).

(nonsufficient, [[0, [[a16, [[160, 3/4], [162, 1/4]]], [b16, [[160, 1/2], [162, 1/2]]]], No], [1, [[a16, [[161, 1/2], [163, 1/2]]], [b16, [[161, 1/2], [163, 1/2]]]], Yes]], No, [], [0, [[a16, [[160, 3/4], [162, 1/4]]], [b16, [[160, 1/2], [162, 1/2]]]], No], No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)

Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (nonsufficient, [[0, [[a16, [[160, 3/4], [162, 1/4]]], [b16, [[160, 1/2], [162, 1/2]]]], No], [1, [[a16, [[161, 1/2], [163, 1/2]]], [b16, [[161, 1/2], [163, 1/2]]]], Yes]], No, [], [0, [[a16, [[160, 3/4], [162, 1/4]]], [b16, [[160, 1/2], [162, 1/2]]]], No], No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim).

(zero_mass, [[0, [[a17, [[170, 1/2], [172, 1/2]]]], Yes], [1, [[b17, [[171, 1/2], [173, 1/2]]]], Yes]], Yes, [[a17, 1, 0, null], [b17, 0, 0, null]], null, No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)

Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (zero_mass, [[0, [[a17, [[170, 1/2], [172, 1/2]]]], Yes], [1, [[b17, [[171, 1/2], [173, 1/2]]]], Yes]], Yes, [[a17, 1, 0, null], [b17, 0, 0, null]], null, No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim).

(identity, [[0, [[a18, [[180, 1]]], [b18, [[180, 1]]]], Yes], [1, [[a18, [[181, 1]]], [b18, [[181, 1]]]], Yes], [2, [[a18, [[182, 1]]], [b18, [[182, 1]]]], Yes], [3, [[a18, [[183, 1]]], [b18, [[183, 1]]]], Yes]], Yes, [], null, Yes, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)

Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (identity, [[0, [[a18, [[180, 1]]], [b18, [[180, 1]]]], Yes], [1, [[a18, [[181, 1]]], [b18, [[181, 1]]]], Yes], [2, [[a18, [[182, 1]]], [b18, [[182, 1]]]], Yes], [3, [[a18, [[183, 1]]], [b18, [[183, 1]]]], Yes]], Yes, [], null, Yes, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim).

(constant, [[0, [[a19, [[190, 2/5], [191, 1/10], [192, 2/5], [193, 1/10]]], [b19, [[190, 1/10], [191, 2/5], [192, 1/10], [193, 2/5]]]], No]], No, [], null, No, Yes, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)

Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (constant, [[0, [[a19, [[190, 2/5], [191, 1/10], [192, 2/5], [193, 1/10]]], [b19, [[190, 1/10], [191, 2/5], [192, 1/10], [193, 2/5]]]], No]], No, [], null, No, Yes, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim).

(nonsufficient, [[0, [[a20, [[200, 3/4], [202, 1/4]]], [b20, [[200, 1/2], [202, 1/2]]]], No], [1, [[a20, [[201, 1/2], [203, 1/2]]], [b20, [[201, 1/2], [203, 1/2]]]], Yes]], No, [], [0, [[a20, [[200, 3/4], [202, 1/4]]], [b20, [[200, 1/2], [202, 1/2]]]], No], No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)

Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (nonsufficient, [[0, [[a20, [[200, 3/4], [202, 1/4]]], [b20, [[200, 1/2], [202, 1/2]]]], No], [1, [[a20, [[201, 1/2], [203, 1/2]]], [b20, [[201, 1/2], [203, 1/2]]]], Yes]], No, [], [0, [[a20, [[200, 3/4], [202, 1/4]]], [b20, [[200, 1/2], [202, 1/2]]]], No], No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim).

(zero_mass, [[0, [[a21, [[210, 1/2], [212, 1/2]]]], Yes], [1, [[b21, [[211, 1/2], [213, 1/2]]]], Yes]], Yes, [[a21, 1, 0, null], [b21, 0, 0, null]], null, No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)

Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (zero_mass, [[0, [[a21, [[210, 1/2], [212, 1/2]]]], Yes], [1, [[b21, [[211, 1/2], [213, 1/2]]]], Yes]], Yes, [[a21, 1, 0, null], [b21, 0, 0, null]], null, No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim).

(identity, [[0, [[a22, [[220, 1]]], [b22, [[220, 1]]]], Yes], [1, [[a22, [[221, 1]]], [b22, [[221, 1]]]], Yes], [2, [[a22, [[222, 1]]], [b22, [[222, 1]]]], Yes], [3, [[a22, [[223, 1]]], [b22, [[223, 1]]]], Yes]], Yes, [], null, Yes, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)

Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (identity, [[0, [[a22, [[220, 1]]], [b22, [[220, 1]]]], Yes], [1, [[a22, [[221, 1]]], [b22, [[221, 1]]]], Yes], [2, [[a22, [[222, 1]]], [b22, [[222, 1]]]], Yes], [3, [[a22, [[223, 1]]], [b22, [[223, 1]]]], Yes]], Yes, [], null, Yes, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim).

(constant, [[0, [[a23, [[230, 2/5], [231, 1/10], [232, 2/5], [233, 1/10]]], [b23, [[230, 1/10], [231, 2/5], [232, 1/10], [233, 2/5]]]], No]], No, [], null, No, Yes, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)

Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (constant, [[0, [[a23, [[230, 2/5], [231, 1/10], [232, 2/5], [233, 1/10]]], [b23, [[230, 1/10], [231, 2/5], [232, 1/10], [233, 2/5]]]], No]], No, [], null, No, Yes, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim).

(nonsufficient, [[0, [[a24, [[240, 3/4], [242, 1/4]]], [b24, [[240, 1/2], [242, 1/2]]]], No], [1, [[a24, [[241, 1/2], [243, 1/2]]], [b24, [[241, 1/2], [243, 1/2]]]], Yes]], No, [], [0, [[a24, [[240, 3/4], [242, 1/4]]], [b24, [[240, 1/2], [242, 1/2]]]], No], No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)

Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (nonsufficient, [[0, [[a24, [[240, 3/4], [242, 1/4]]], [b24, [[240, 1/2], [242, 1/2]]]], No], [1, [[a24, [[241, 1/2], [243, 1/2]]], [b24, [[241, 1/2], [243, 1/2]]]], Yes]], No, [], [0, [[a24, [[240, 3/4], [242, 1/4]]], [b24, [[240, 1/2], [242, 1/2]]]], No], No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim).