# Complete Fibers, Positive-Mass Conditional Laws and Sufficiency

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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations · 2026-10-15 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. A Total Statistic Partitions the Complete Sample Carrier into Fibers

Use complete theta labels=[a0, b0], samples=[0, 1, 2, 3], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).

View answer and explanation([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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0001/)

Sufficient statistic in a finite model · qf-mod47-suff-fibers · candidate





## 2. A Total Statistic Partitions the Complete Sample Carrier into Fibers

Use complete theta labels=[a1, b1], samples=[10, 11, 12, 13], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).

View answer and explanation([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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0002/)

Sufficient statistic in a finite model · qf-mod47-suff-fibers · candidate





## 3. A Total Statistic Partitions the Complete Sample Carrier into Fibers

Use complete theta labels=[a2, b2], samples=[20, 21, 22, 23], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).

View answer and explanation([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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0003/)

Sufficient statistic in a finite model · qf-mod47-suff-fibers · candidate





## 4. A Total Statistic Partitions the Complete Sample Carrier into Fibers

Use complete theta labels=[a3, b3], samples=[30, 31, 32, 33], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).

View answer and explanation([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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0004/)

Sufficient statistic in a finite model · qf-mod47-suff-fibers · candidate





## 5. A Total Statistic Partitions the Complete Sample Carrier into Fibers

Use complete theta labels=[a4, b4], samples=[40, 41, 42, 43], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).

View answer and explanation([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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0005/)

Sufficient statistic in a finite model · qf-mod47-suff-fibers · candidate





## 6. A Total Statistic Partitions the Complete Sample Carrier into Fibers

Use complete theta labels=[a5, b5], samples=[50, 51, 52, 53], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).

View answer and explanation([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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0006/)

Sufficient statistic in a finite model · qf-mod47-suff-fibers · candidate





## 7. A Total Statistic Partitions the Complete Sample Carrier into Fibers

Use complete theta labels=[a6, b6], samples=[60, 61, 62, 63], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).

View answer and explanation([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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0007/)

Sufficient statistic in a finite model · qf-mod47-suff-fibers · candidate





## 8. A Total Statistic Partitions the Complete Sample Carrier into Fibers

Use complete theta labels=[a7, b7], samples=[70, 71, 72, 73], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).

View answer and explanation([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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0008/)

Sufficient statistic in a finite model · qf-mod47-suff-fibers · candidate





## 9. A Total Statistic Partitions the Complete Sample Carrier into Fibers

Use complete theta labels=[a8, b8], samples=[80, 81, 82, 83], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).

View answer and explanation([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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0009/)

Sufficient statistic in a finite model · qf-mod47-suff-fibers · candidate





## 10. A Total Statistic Partitions the Complete Sample Carrier into Fibers

Use complete theta labels=[a9, b9], samples=[90, 91, 92, 93], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).

View answer and explanation([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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0010/)

Sufficient statistic in a finite model · qf-mod47-suff-fibers · candidate





## 11. A Total Statistic Partitions the Complete Sample Carrier into Fibers

Use complete theta labels=[a10, b10], samples=[100, 101, 102, 103], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).

View answer and explanation([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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0011/)

Sufficient statistic in a finite model · qf-mod47-suff-fibers · candidate





## 12. A Total Statistic Partitions the Complete Sample Carrier into Fibers

Use complete theta labels=[a11, b11], samples=[110, 111, 112, 113], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).

View answer and explanation([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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0012/)

Sufficient statistic in a finite model · qf-mod47-suff-fibers · candidate





## 13. A Total Statistic Partitions the Complete Sample Carrier into Fibers

Use complete theta labels=[a12, b12], samples=[120, 121, 122, 123], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).

View answer and explanation([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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0013/)

Sufficient statistic in a finite model · qf-mod47-suff-fibers · candidate





## 14. A Total Statistic Partitions the Complete Sample Carrier into Fibers

Use complete theta labels=[a13, b13], samples=[130, 131, 132, 133], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).

View answer and explanation([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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0014/)

Sufficient statistic in a finite model · qf-mod47-suff-fibers · candidate





## 15. A Total Statistic Partitions the Complete Sample Carrier into Fibers

Use complete theta labels=[a14, b14], samples=[140, 141, 142, 143], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).

View answer and explanation([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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0015/)

Sufficient statistic in a finite model · qf-mod47-suff-fibers · candidate





## 16. A Total Statistic Partitions the Complete Sample Carrier into Fibers

Use complete theta labels=[a15, b15], samples=[150, 151, 152, 153], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).

View answer and explanation([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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0016/)

Sufficient statistic in a finite model · qf-mod47-suff-fibers · candidate





## 17. A Total Statistic Partitions the Complete Sample Carrier into Fibers

Use complete theta labels=[a16, b16], samples=[160, 161, 162, 163], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).

View answer and explanation([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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0017/)

Sufficient statistic in a finite model · qf-mod47-suff-fibers · candidate





## 18. A Total Statistic Partitions the Complete Sample Carrier into Fibers

Use complete theta labels=[a17, b17], samples=[170, 171, 172, 173], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).

View answer and explanation([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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0018/)

Sufficient statistic in a finite model · qf-mod47-suff-fibers · candidate





## 19. A Total Statistic Partitions the Complete Sample Carrier into Fibers

Use complete theta labels=[a18, b18], samples=[180, 181, 182, 183], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).

View answer and explanation([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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0019/)

Sufficient statistic in a finite model · qf-mod47-suff-fibers · candidate





## 20. A Total Statistic Partitions the Complete Sample Carrier into Fibers

Use complete theta labels=[a19, b19], samples=[190, 191, 192, 193], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).

View answer and explanation([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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0020/)

Sufficient statistic in a finite model · qf-mod47-suff-fibers · candidate





## 21. A Total Statistic Partitions the Complete Sample Carrier into Fibers

Use complete theta labels=[a20, b20], samples=[200, 201, 202, 203], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).

View answer and explanation([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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0021/)

Sufficient statistic in a finite model · qf-mod47-suff-fibers · candidate





## 22. A Total Statistic Partitions the Complete Sample Carrier into Fibers

Use complete theta labels=[a21, b21], samples=[210, 211, 212, 213], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).

View answer and explanation([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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0022/)

Sufficient statistic in a finite model · qf-mod47-suff-fibers · candidate





## 23. A Total Statistic Partitions the Complete Sample Carrier into Fibers

Use complete theta labels=[a22, b22], samples=[220, 221, 222, 223], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).

View answer and explanation([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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0023/)

Sufficient statistic in a finite model · qf-mod47-suff-fibers · candidate





## 24. A Total Statistic Partitions the Complete Sample Carrier into Fibers

Use complete theta labels=[a23, b23], samples=[230, 231, 232, 233], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).

View answer and explanation([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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0024/)

Sufficient statistic in a finite model · qf-mod47-suff-fibers · candidate





## 25. A Total Statistic Partitions the Complete Sample Carrier into Fibers

Use complete theta labels=[a24, b24], samples=[240, 241, 242, 243], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).

View answer and explanation([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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0025/)

Sufficient statistic in a finite model · qf-mod47-suff-fibers · candidate





## 26. Conditional Laws Are Defined Only on Positive-Mass Statistic Cells

Use complete theta labels=[a0, b0], samples=[0, 1, 2, 3], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).

A zero-mass conditional law is undefined and encoded as null, not an empty table, zero distribution or a value obtained by division by zero.

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0026/)

Sufficient statistic in a finite model · qf-mod47-suff-conditional-table · candidate





## 27. Conditional Laws Are Defined Only on Positive-Mass Statistic Cells

Use complete theta labels=[a1, b1], samples=[10, 11, 12, 13], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).

A zero-mass conditional law is undefined and encoded as null, not an empty table, zero distribution or a value obtained by division by zero.

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0027/)

Sufficient statistic in a finite model · qf-mod47-suff-conditional-table · candidate





## 28. Conditional Laws Are Defined Only on Positive-Mass Statistic Cells

Use complete theta labels=[a2, b2], samples=[20, 21, 22, 23], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).

A zero-mass conditional law is undefined and encoded as null, not an empty table, zero distribution or a value obtained by division by zero.

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0028/)

Sufficient statistic in a finite model · qf-mod47-suff-conditional-table · candidate





## 29. Conditional Laws Are Defined Only on Positive-Mass Statistic Cells

Use complete theta labels=[a3, b3], samples=[30, 31, 32, 33], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).

A zero-mass conditional law is undefined and encoded as null, not an empty table, zero distribution or a value obtained by division by zero.

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0029/)

Sufficient statistic in a finite model · qf-mod47-suff-conditional-table · candidate





## 30. Conditional Laws Are Defined Only on Positive-Mass Statistic Cells

Use complete theta labels=[a4, b4], samples=[40, 41, 42, 43], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).

A zero-mass conditional law is undefined and encoded as null, not an empty table, zero distribution or a value obtained by division by zero.

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0030/)

Sufficient statistic in a finite model · qf-mod47-suff-conditional-table · candidate





## 31. Conditional Laws Are Defined Only on Positive-Mass Statistic Cells

Use complete theta labels=[a5, b5], samples=[50, 51, 52, 53], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).

A zero-mass conditional law is undefined and encoded as null, not an empty table, zero distribution or a value obtained by division by zero.

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0031/)

Sufficient statistic in a finite model · qf-mod47-suff-conditional-table · candidate





## 32. Conditional Laws Are Defined Only on Positive-Mass Statistic Cells

Use complete theta labels=[a6, b6], samples=[60, 61, 62, 63], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).

A zero-mass conditional law is undefined and encoded as null, not an empty table, zero distribution or a value obtained by division by zero.

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0032/)

Sufficient statistic in a finite model · qf-mod47-suff-conditional-table · candidate





## 33. Conditional Laws Are Defined Only on Positive-Mass Statistic Cells

Use complete theta labels=[a7, b7], samples=[70, 71, 72, 73], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).

A zero-mass conditional law is undefined and encoded as null, not an empty table, zero distribution or a value obtained by division by zero.

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0033/)

Sufficient statistic in a finite model · qf-mod47-suff-conditional-table · candidate





## 34. Conditional Laws Are Defined Only on Positive-Mass Statistic Cells

Use complete theta labels=[a8, b8], samples=[80, 81, 82, 83], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).

A zero-mass conditional law is undefined and encoded as null, not an empty table, zero distribution or a value obtained by division by zero.

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0034/)

Sufficient statistic in a finite model · qf-mod47-suff-conditional-table · candidate





## 35. Conditional Laws Are Defined Only on Positive-Mass Statistic Cells

Use complete theta labels=[a9, b9], samples=[90, 91, 92, 93], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).

A zero-mass conditional law is undefined and encoded as null, not an empty table, zero distribution or a value obtained by division by zero.

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0035/)

Sufficient statistic in a finite model · qf-mod47-suff-conditional-table · candidate





## 36. Conditional Laws Are Defined Only on Positive-Mass Statistic Cells

Use complete theta labels=[a10, b10], samples=[100, 101, 102, 103], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).

A zero-mass conditional law is undefined and encoded as null, not an empty table, zero distribution or a value obtained by division by zero.

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0036/)

Sufficient statistic in a finite model · qf-mod47-suff-conditional-table · candidate





## 37. Conditional Laws Are Defined Only on Positive-Mass Statistic Cells

Use complete theta labels=[a11, b11], samples=[110, 111, 112, 113], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).

A zero-mass conditional law is undefined and encoded as null, not an empty table, zero distribution or a value obtained by division by zero.

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0037/)

Sufficient statistic in a finite model · qf-mod47-suff-conditional-table · candidate





## 38. Conditional Laws Are Defined Only on Positive-Mass Statistic Cells

Use complete theta labels=[a12, b12], samples=[120, 121, 122, 123], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).

A zero-mass conditional law is undefined and encoded as null, not an empty table, zero distribution or a value obtained by division by zero.

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0038/)

Sufficient statistic in a finite model · qf-mod47-suff-conditional-table · candidate





## 39. Conditional Laws Are Defined Only on Positive-Mass Statistic Cells

Use complete theta labels=[a13, b13], samples=[130, 131, 132, 133], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).

A zero-mass conditional law is undefined and encoded as null, not an empty table, zero distribution or a value obtained by division by zero.

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0039/)

Sufficient statistic in a finite model · qf-mod47-suff-conditional-table · candidate





## 40. Conditional Laws Are Defined Only on Positive-Mass Statistic Cells

Use complete theta labels=[a14, b14], samples=[140, 141, 142, 143], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).

A zero-mass conditional law is undefined and encoded as null, not an empty table, zero distribution or a value obtained by division by zero.

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0040/)

Sufficient statistic in a finite model · qf-mod47-suff-conditional-table · candidate





## 41. Conditional Laws Are Defined Only on Positive-Mass Statistic Cells

Use complete theta labels=[a15, b15], samples=[150, 151, 152, 153], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).

A zero-mass conditional law is undefined and encoded as null, not an empty table, zero distribution or a value obtained by division by zero.

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0041/)

Sufficient statistic in a finite model · qf-mod47-suff-conditional-table · candidate





## 42. Conditional Laws Are Defined Only on Positive-Mass Statistic Cells

Use complete theta labels=[a16, b16], samples=[160, 161, 162, 163], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).

A zero-mass conditional law is undefined and encoded as null, not an empty table, zero distribution or a value obtained by division by zero.

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0042/)

Sufficient statistic in a finite model · qf-mod47-suff-conditional-table · candidate





## 43. Conditional Laws Are Defined Only on Positive-Mass Statistic Cells

Use complete theta labels=[a17, b17], samples=[170, 171, 172, 173], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).

A zero-mass conditional law is undefined and encoded as null, not an empty table, zero distribution or a value obtained by division by zero.

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0043/)

Sufficient statistic in a finite model · qf-mod47-suff-conditional-table · candidate





## 44. Conditional Laws Are Defined Only on Positive-Mass Statistic Cells

Use complete theta labels=[a18, b18], samples=[180, 181, 182, 183], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).

A zero-mass conditional law is undefined and encoded as null, not an empty table, zero distribution or a value obtained by division by zero.

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0044/)

Sufficient statistic in a finite model · qf-mod47-suff-conditional-table · candidate





## 45. Conditional Laws Are Defined Only on Positive-Mass Statistic Cells

Use complete theta labels=[a19, b19], samples=[190, 191, 192, 193], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).

A zero-mass conditional law is undefined and encoded as null, not an empty table, zero distribution or a value obtained by division by zero.

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0045/)

Sufficient statistic in a finite model · qf-mod47-suff-conditional-table · candidate





## 46. Conditional Laws Are Defined Only on Positive-Mass Statistic Cells

Use complete theta labels=[a20, b20], samples=[200, 201, 202, 203], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).

A zero-mass conditional law is undefined and encoded as null, not an empty table, zero distribution or a value obtained by division by zero.

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0046/)

Sufficient statistic in a finite model · qf-mod47-suff-conditional-table · candidate





## 47. Conditional Laws Are Defined Only on Positive-Mass Statistic Cells

Use complete theta labels=[a21, b21], samples=[210, 211, 212, 213], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).

A zero-mass conditional law is undefined and encoded as null, not an empty table, zero distribution or a value obtained by division by zero.

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0047/)

Sufficient statistic in a finite model · qf-mod47-suff-conditional-table · candidate





## 48. Conditional Laws Are Defined Only on Positive-Mass Statistic Cells

Use complete theta labels=[a22, b22], samples=[220, 221, 222, 223], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).

A zero-mass conditional law is undefined and encoded as null, not an empty table, zero distribution or a value obtained by division by zero.

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0048/)

Sufficient statistic in a finite model · qf-mod47-suff-conditional-table · candidate





## 49. Conditional Laws Are Defined Only on Positive-Mass Statistic Cells

Use complete theta labels=[a23, b23], samples=[230, 231, 232, 233], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).

A zero-mass conditional law is undefined and encoded as null, not an empty table, zero distribution or a value obtained by division by zero.

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0049/)

Sufficient statistic in a finite model · qf-mod47-suff-conditional-table · candidate





## 50. Conditional Laws Are Defined Only on Positive-Mass Statistic Cells

Use complete theta labels=[a24, b24], samples=[240, 241, 242, 243], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).

A zero-mass conditional law is undefined and encoded as null, not an empty table, zero distribution or a value obtained by division by zero.

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0050/)

Sufficient statistic in a finite model · qf-mod47-suff-conditional-table · candidate





## 51. Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber

Use complete theta labels=[a0, b0], samples=[0, 1, 2, 3], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0051/)

Sufficient statistic in a finite model · qf-mod47-suff-invariance · candidate





## 52. Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber

Use complete theta labels=[a1, b1], samples=[10, 11, 12, 13], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0052/)

Sufficient statistic in a finite model · qf-mod47-suff-invariance · candidate





## 53. Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber

Use complete theta labels=[a2, b2], samples=[20, 21, 22, 23], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0053/)

Sufficient statistic in a finite model · qf-mod47-suff-invariance · candidate





## 54. Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber

Use complete theta labels=[a3, b3], samples=[30, 31, 32, 33], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0054/)

Sufficient statistic in a finite model · qf-mod47-suff-invariance · candidate





## 55. Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber

Use complete theta labels=[a4, b4], samples=[40, 41, 42, 43], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0055/)

Sufficient statistic in a finite model · qf-mod47-suff-invariance · candidate





## 56. Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber

Use complete theta labels=[a5, b5], samples=[50, 51, 52, 53], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0056/)

Sufficient statistic in a finite model · qf-mod47-suff-invariance · candidate





## 57. Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber

Use complete theta labels=[a6, b6], samples=[60, 61, 62, 63], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0057/)

Sufficient statistic in a finite model · qf-mod47-suff-invariance · candidate





## 58. Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber

Use complete theta labels=[a7, b7], samples=[70, 71, 72, 73], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0058/)

Sufficient statistic in a finite model · qf-mod47-suff-invariance · candidate





## 59. Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber

Use complete theta labels=[a8, b8], samples=[80, 81, 82, 83], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0059/)

Sufficient statistic in a finite model · qf-mod47-suff-invariance · candidate





## 60. Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber

Use complete theta labels=[a9, b9], samples=[90, 91, 92, 93], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0060/)

Sufficient statistic in a finite model · qf-mod47-suff-invariance · candidate





## 61. Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber

Use complete theta labels=[a10, b10], samples=[100, 101, 102, 103], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0061/)

Sufficient statistic in a finite model · qf-mod47-suff-invariance · candidate





## 62. Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber

Use complete theta labels=[a11, b11], samples=[110, 111, 112, 113], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0062/)

Sufficient statistic in a finite model · qf-mod47-suff-invariance · candidate





## 63. Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber

Use complete theta labels=[a12, b12], samples=[120, 121, 122, 123], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0063/)

Sufficient statistic in a finite model · qf-mod47-suff-invariance · candidate





## 64. Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber

Use complete theta labels=[a13, b13], samples=[130, 131, 132, 133], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0064/)

Sufficient statistic in a finite model · qf-mod47-suff-invariance · candidate





## 65. Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber

Use complete theta labels=[a14, b14], samples=[140, 141, 142, 143], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0065/)

Sufficient statistic in a finite model · qf-mod47-suff-invariance · candidate





## 66. Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber

Use complete theta labels=[a15, b15], samples=[150, 151, 152, 153], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0066/)

Sufficient statistic in a finite model · qf-mod47-suff-invariance · candidate





## 67. Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber

Use complete theta labels=[a16, b16], samples=[160, 161, 162, 163], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0067/)

Sufficient statistic in a finite model · qf-mod47-suff-invariance · candidate





## 68. Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber

Use complete theta labels=[a17, b17], samples=[170, 171, 172, 173], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0068/)

Sufficient statistic in a finite model · qf-mod47-suff-invariance · candidate





## 69. Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber

Use complete theta labels=[a18, b18], samples=[180, 181, 182, 183], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0069/)

Sufficient statistic in a finite model · qf-mod47-suff-invariance · candidate





## 70. Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber

Use complete theta labels=[a19, b19], samples=[190, 191, 192, 193], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0070/)

Sufficient statistic in a finite model · qf-mod47-suff-invariance · candidate





## 71. Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber

Use complete theta labels=[a20, b20], samples=[200, 201, 202, 203], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0071/)

Sufficient statistic in a finite model · qf-mod47-suff-invariance · candidate





## 72. Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber

Use complete theta labels=[a21, b21], samples=[210, 211, 212, 213], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0072/)

Sufficient statistic in a finite model · qf-mod47-suff-invariance · candidate





## 73. Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber

Use complete theta labels=[a22, b22], samples=[220, 221, 222, 223], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0073/)

Sufficient statistic in a finite model · qf-mod47-suff-invariance · candidate





## 74. Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber

Use complete theta labels=[a23, b23], samples=[230, 231, 232, 233], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0074/)

Sufficient statistic in a finite model · qf-mod47-suff-invariance · candidate





## 75. Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber

Use complete theta labels=[a24, b24], samples=[240, 241, 242, 243], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).

View answer and explanation([[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0075/)

Sufficient statistic in a finite model · qf-mod47-suff-invariance · candidate





## 76. Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks

Use complete theta labels=[a0, b0], samples=[0, 1, 2, 3], aligned exact probability rows=[[[3, 5], [1, 10], [1, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=nonsufficient. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).

Compare positive-mass theta cells within each fiber; zero-mass cells stay null. constant_equivalence_if_applicable tests sufficiency versus equal laws only in constant mode; other modes retain a source true placeholder, not a new theorem.

View answer and explanation(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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0076/)

Sufficient statistic in a finite model · qf-mod47-suff-nonsufficient-zero-boundary · candidate





## 77. Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks

Use complete theta labels=[a1, b1], samples=[10, 11, 12, 13], aligned exact probability rows=[[[1, 2], 0, [1, 2], 0], [0, [1, 2], 0, [1, 2]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=zero_mass. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).

Compare positive-mass theta cells within each fiber; zero-mass cells stay null. constant_equivalence_if_applicable tests sufficiency versus equal laws only in constant mode; other modes retain a source true placeholder, not a new theorem.

View answer and explanation(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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0077/)

Sufficient statistic in a finite model · qf-mod47-suff-nonsufficient-zero-boundary · candidate





## 78. Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks

Use complete theta labels=[a2, b2], samples=[20, 21, 22, 23], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 2, 3]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=identity. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).

Compare positive-mass theta cells within each fiber; zero-mass cells stay null. constant_equivalence_if_applicable tests sufficiency versus equal laws only in constant mode; other modes retain a source true placeholder, not a new theorem.

View answer and explanation(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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0078/)

Sufficient statistic in a finite model · qf-mod47-suff-nonsufficient-zero-boundary · candidate





## 79. Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks

Use complete theta labels=[a3, b3], samples=[30, 31, 32, 33], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 0, 0, 0]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=constant. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).

Compare positive-mass theta cells within each fiber; zero-mass cells stay null. constant_equivalence_if_applicable tests sufficiency versus equal laws only in constant mode; other modes retain a source true placeholder, not a new theorem.

View answer and explanation(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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0079/)

Sufficient statistic in a finite model · qf-mod47-suff-nonsufficient-zero-boundary · candidate





## 80. Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks

Use complete theta labels=[a4, b4], samples=[40, 41, 42, 43], aligned exact probability rows=[[[3, 5], [1, 10], [1, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=nonsufficient. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).

Compare positive-mass theta cells within each fiber; zero-mass cells stay null. constant_equivalence_if_applicable tests sufficiency versus equal laws only in constant mode; other modes retain a source true placeholder, not a new theorem.

View answer and explanation(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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0080/)

Sufficient statistic in a finite model · qf-mod47-suff-nonsufficient-zero-boundary · candidate





## 81. Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks

Use complete theta labels=[a5, b5], samples=[50, 51, 52, 53], aligned exact probability rows=[[[1, 2], 0, [1, 2], 0], [0, [1, 2], 0, [1, 2]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=zero_mass. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).

Compare positive-mass theta cells within each fiber; zero-mass cells stay null. constant_equivalence_if_applicable tests sufficiency versus equal laws only in constant mode; other modes retain a source true placeholder, not a new theorem.

View answer and explanation(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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0081/)

Sufficient statistic in a finite model · qf-mod47-suff-nonsufficient-zero-boundary · candidate





## 82. Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks

Use complete theta labels=[a6, b6], samples=[60, 61, 62, 63], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 2, 3]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=identity. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).

Compare positive-mass theta cells within each fiber; zero-mass cells stay null. constant_equivalence_if_applicable tests sufficiency versus equal laws only in constant mode; other modes retain a source true placeholder, not a new theorem.

View answer and explanation(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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0082/)

Sufficient statistic in a finite model · qf-mod47-suff-nonsufficient-zero-boundary · candidate





## 83. Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks

Use complete theta labels=[a7, b7], samples=[70, 71, 72, 73], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 0, 0, 0]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=constant. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).

Compare positive-mass theta cells within each fiber; zero-mass cells stay null. constant_equivalence_if_applicable tests sufficiency versus equal laws only in constant mode; other modes retain a source true placeholder, not a new theorem.

View answer and explanation(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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0083/)

Sufficient statistic in a finite model · qf-mod47-suff-nonsufficient-zero-boundary · candidate





## 84. Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks

Use complete theta labels=[a8, b8], samples=[80, 81, 82, 83], aligned exact probability rows=[[[3, 5], [1, 10], [1, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=nonsufficient. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).

Compare positive-mass theta cells within each fiber; zero-mass cells stay null. constant_equivalence_if_applicable tests sufficiency versus equal laws only in constant mode; other modes retain a source true placeholder, not a new theorem.

View answer and explanation(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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0084/)

Sufficient statistic in a finite model · qf-mod47-suff-nonsufficient-zero-boundary · candidate





## 85. Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks

Use complete theta labels=[a9, b9], samples=[90, 91, 92, 93], aligned exact probability rows=[[[1, 2], 0, [1, 2], 0], [0, [1, 2], 0, [1, 2]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=zero_mass. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).

Compare positive-mass theta cells within each fiber; zero-mass cells stay null. constant_equivalence_if_applicable tests sufficiency versus equal laws only in constant mode; other modes retain a source true placeholder, not a new theorem.

View answer and explanation(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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0085/)

Sufficient statistic in a finite model · qf-mod47-suff-nonsufficient-zero-boundary · candidate





## 86. Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks

Use complete theta labels=[a10, b10], samples=[100, 101, 102, 103], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 2, 3]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=identity. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).

Compare positive-mass theta cells within each fiber; zero-mass cells stay null. constant_equivalence_if_applicable tests sufficiency versus equal laws only in constant mode; other modes retain a source true placeholder, not a new theorem.

View answer and explanation(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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0086/)

Sufficient statistic in a finite model · qf-mod47-suff-nonsufficient-zero-boundary · candidate





## 87. Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks

Use complete theta labels=[a11, b11], samples=[110, 111, 112, 113], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 0, 0, 0]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=constant. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).

Compare positive-mass theta cells within each fiber; zero-mass cells stay null. constant_equivalence_if_applicable tests sufficiency versus equal laws only in constant mode; other modes retain a source true placeholder, not a new theorem.

View answer and explanation(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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0087/)

Sufficient statistic in a finite model · qf-mod47-suff-nonsufficient-zero-boundary · candidate





## 88. Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks

Use complete theta labels=[a12, b12], samples=[120, 121, 122, 123], aligned exact probability rows=[[[3, 5], [1, 10], [1, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=nonsufficient. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).

Compare positive-mass theta cells within each fiber; zero-mass cells stay null. constant_equivalence_if_applicable tests sufficiency versus equal laws only in constant mode; other modes retain a source true placeholder, not a new theorem.

View answer and explanation(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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0088/)

Sufficient statistic in a finite model · qf-mod47-suff-nonsufficient-zero-boundary · candidate





## 89. Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks

Use complete theta labels=[a13, b13], samples=[130, 131, 132, 133], aligned exact probability rows=[[[1, 2], 0, [1, 2], 0], [0, [1, 2], 0, [1, 2]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=zero_mass. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).

Compare positive-mass theta cells within each fiber; zero-mass cells stay null. constant_equivalence_if_applicable tests sufficiency versus equal laws only in constant mode; other modes retain a source true placeholder, not a new theorem.

View answer and explanation(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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0089/)

Sufficient statistic in a finite model · qf-mod47-suff-nonsufficient-zero-boundary · candidate





## 90. Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks

Use complete theta labels=[a14, b14], samples=[140, 141, 142, 143], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 2, 3]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=identity. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).

Compare positive-mass theta cells within each fiber; zero-mass cells stay null. constant_equivalence_if_applicable tests sufficiency versus equal laws only in constant mode; other modes retain a source true placeholder, not a new theorem.

View answer and explanation(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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0090/)

Sufficient statistic in a finite model · qf-mod47-suff-nonsufficient-zero-boundary · candidate





## 91. Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks

Use complete theta labels=[a15, b15], samples=[150, 151, 152, 153], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 0, 0, 0]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=constant. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).

Compare positive-mass theta cells within each fiber; zero-mass cells stay null. constant_equivalence_if_applicable tests sufficiency versus equal laws only in constant mode; other modes retain a source true placeholder, not a new theorem.

View answer and explanation(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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0091/)

Sufficient statistic in a finite model · qf-mod47-suff-nonsufficient-zero-boundary · candidate





## 92. Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks

Use complete theta labels=[a16, b16], samples=[160, 161, 162, 163], aligned exact probability rows=[[[3, 5], [1, 10], [1, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=nonsufficient. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).

Compare positive-mass theta cells within each fiber; zero-mass cells stay null. constant_equivalence_if_applicable tests sufficiency versus equal laws only in constant mode; other modes retain a source true placeholder, not a new theorem.

View answer and explanation(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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0092/)

Sufficient statistic in a finite model · qf-mod47-suff-nonsufficient-zero-boundary · candidate





## 93. Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks

Use complete theta labels=[a17, b17], samples=[170, 171, 172, 173], aligned exact probability rows=[[[1, 2], 0, [1, 2], 0], [0, [1, 2], 0, [1, 2]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=zero_mass. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).

Compare positive-mass theta cells within each fiber; zero-mass cells stay null. constant_equivalence_if_applicable tests sufficiency versus equal laws only in constant mode; other modes retain a source true placeholder, not a new theorem.

View answer and explanation(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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0093/)

Sufficient statistic in a finite model · qf-mod47-suff-nonsufficient-zero-boundary · candidate





## 94. Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks

Use complete theta labels=[a18, b18], samples=[180, 181, 182, 183], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 2, 3]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=identity. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).

Compare positive-mass theta cells within each fiber; zero-mass cells stay null. constant_equivalence_if_applicable tests sufficiency versus equal laws only in constant mode; other modes retain a source true placeholder, not a new theorem.

View answer and explanation(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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0094/)

Sufficient statistic in a finite model · qf-mod47-suff-nonsufficient-zero-boundary · candidate





## 95. Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks

Use complete theta labels=[a19, b19], samples=[190, 191, 192, 193], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 0, 0, 0]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=constant. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).

Compare positive-mass theta cells within each fiber; zero-mass cells stay null. constant_equivalence_if_applicable tests sufficiency versus equal laws only in constant mode; other modes retain a source true placeholder, not a new theorem.

View answer and explanation(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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0095/)

Sufficient statistic in a finite model · qf-mod47-suff-nonsufficient-zero-boundary · candidate





## 96. Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks

Use complete theta labels=[a20, b20], samples=[200, 201, 202, 203], aligned exact probability rows=[[[3, 5], [1, 10], [1, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=nonsufficient. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).

Compare positive-mass theta cells within each fiber; zero-mass cells stay null. constant_equivalence_if_applicable tests sufficiency versus equal laws only in constant mode; other modes retain a source true placeholder, not a new theorem.

View answer and explanation(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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0096/)

Sufficient statistic in a finite model · qf-mod47-suff-nonsufficient-zero-boundary · candidate





## 97. Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks

Use complete theta labels=[a21, b21], samples=[210, 211, 212, 213], aligned exact probability rows=[[[1, 2], 0, [1, 2], 0], [0, [1, 2], 0, [1, 2]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=zero_mass. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).

Compare positive-mass theta cells within each fiber; zero-mass cells stay null. constant_equivalence_if_applicable tests sufficiency versus equal laws only in constant mode; other modes retain a source true placeholder, not a new theorem.

View answer and explanation(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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0097/)

Sufficient statistic in a finite model · qf-mod47-suff-nonsufficient-zero-boundary · candidate





## 98. Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks

Use complete theta labels=[a22, b22], samples=[220, 221, 222, 223], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 2, 3]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=identity. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).

Compare positive-mass theta cells within each fiber; zero-mass cells stay null. constant_equivalence_if_applicable tests sufficiency versus equal laws only in constant mode; other modes retain a source true placeholder, not a new theorem.

View answer and explanation(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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0098/)

Sufficient statistic in a finite model · qf-mod47-suff-nonsufficient-zero-boundary · candidate





## 99. Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks

Use complete theta labels=[a23, b23], samples=[230, 231, 232, 233], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 0, 0, 0]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=constant. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).

Compare positive-mass theta cells within each fiber; zero-mass cells stay null. constant_equivalence_if_applicable tests sufficiency versus equal laws only in constant mode; other modes retain a source true placeholder, not a new theorem.

View answer and explanation(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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0099/)

Sufficient statistic in a finite model · qf-mod47-suff-nonsufficient-zero-boundary · candidate





## 100. Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks

Use complete theta labels=[a24, b24], samples=[240, 241, 242, 243], aligned exact probability rows=[[[3, 5], [1, 10], [1, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=nonsufficient. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).

Compare positive-mass theta cells within each fiber; zero-mass cells stay null. constant_equivalence_if_applicable tests sufficiency versus equal laws only in constant mode; other modes retain a source true placeholder, not a new theorem.

View answer and explanation(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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0100/)

Sufficient statistic in a finite model · qf-mod47-suff-nonsufficient-zero-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations/validation-report.json)