# Bernoulli Cylinders, Full Marginals, Finite Factorization and Infinite-IID Evidence

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batch-20261014-advance-20261029-r06-iid-sequence-foundations · 2026-10-14 · 100 questions

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



## 1. Exact Bernoulli Cylinder Probability under a Declared Product Law

Use the explicitly declared Bernoulli product-law model with exact p=1/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[0, 0, 0, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law).

View answer and explanation([0, 0, 0, 0], 0, 4, 1/29, 614656/707281, symbolic_product_law)

Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([0, 0, 0, 0], 0, 4, 1/29, 614656/707281, symbolic_product_law).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0001/)

Independent identically distributed sequence · qf-mod29-iid-bernoulli-cylinder · candidate





## 2. Exact Bernoulli Cylinder Probability under a Declared Product Law

Use the explicitly declared Bernoulli product-law model with exact p=2/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[1, 0, 0, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law).

View answer and explanation([1, 0, 0, 0], 1, 3, 2/29, 39366/707281, symbolic_product_law)

Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([1, 0, 0, 0], 1, 3, 2/29, 39366/707281, symbolic_product_law).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0002/)

Independent identically distributed sequence · qf-mod29-iid-bernoulli-cylinder · candidate





## 3. Exact Bernoulli Cylinder Probability under a Declared Product Law

Use the explicitly declared Bernoulli product-law model with exact p=3/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[0, 1, 0, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law).

View answer and explanation([0, 1, 0, 0], 1, 3, 3/29, 52728/707281, symbolic_product_law)

Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([0, 1, 0, 0], 1, 3, 3/29, 52728/707281, symbolic_product_law).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0003/)

Independent identically distributed sequence · qf-mod29-iid-bernoulli-cylinder · candidate





## 4. Exact Bernoulli Cylinder Probability under a Declared Product Law

Use the explicitly declared Bernoulli product-law model with exact p=4/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[1, 1, 0, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law).

View answer and explanation([1, 1, 0, 0], 2, 2, 4/29, 10000/707281, symbolic_product_law)

Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([1, 1, 0, 0], 2, 2, 4/29, 10000/707281, symbolic_product_law).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0004/)

Independent identically distributed sequence · qf-mod29-iid-bernoulli-cylinder · candidate





## 5. Exact Bernoulli Cylinder Probability under a Declared Product Law

Use the explicitly declared Bernoulli product-law model with exact p=5/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[0, 0, 1, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law).

View answer and explanation([0, 0, 1, 0], 1, 3, 5/29, 69120/707281, symbolic_product_law)

Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([0, 0, 1, 0], 1, 3, 5/29, 69120/707281, symbolic_product_law).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0005/)

Independent identically distributed sequence · qf-mod29-iid-bernoulli-cylinder · candidate





## 6. Exact Bernoulli Cylinder Probability under a Declared Product Law

Use the explicitly declared Bernoulli product-law model with exact p=6/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[1, 0, 1, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law).

View answer and explanation([1, 0, 1, 0], 2, 2, 6/29, 19044/707281, symbolic_product_law)

Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([1, 0, 1, 0], 2, 2, 6/29, 19044/707281, symbolic_product_law).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0006/)

Independent identically distributed sequence · qf-mod29-iid-bernoulli-cylinder · candidate





## 7. Exact Bernoulli Cylinder Probability under a Declared Product Law

Use the explicitly declared Bernoulli product-law model with exact p=7/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[0, 1, 1, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law).

View answer and explanation([0, 1, 1, 0], 2, 2, 7/29, 23716/707281, symbolic_product_law)

Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([0, 1, 1, 0], 2, 2, 7/29, 23716/707281, symbolic_product_law).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0007/)

Independent identically distributed sequence · qf-mod29-iid-bernoulli-cylinder · candidate





## 8. Exact Bernoulli Cylinder Probability under a Declared Product Law

Use the explicitly declared Bernoulli product-law model with exact p=8/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[1, 1, 1, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law).

View answer and explanation([1, 1, 1, 0], 3, 1, 8/29, 10752/707281, symbolic_product_law)

Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([1, 1, 1, 0], 3, 1, 8/29, 10752/707281, symbolic_product_law).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0008/)

Independent identically distributed sequence · qf-mod29-iid-bernoulli-cylinder · candidate





## 9. Exact Bernoulli Cylinder Probability under a Declared Product Law

Use the explicitly declared Bernoulli product-law model with exact p=9/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[0, 0, 0, 1]. Give (word,ones,zeros,p,probability,symbolic_product_law).

View answer and explanation([0, 0, 0, 1], 1, 3, 9/29, 72000/707281, symbolic_product_law)

Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([0, 0, 0, 1], 1, 3, 9/29, 72000/707281, symbolic_product_law).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0009/)

Independent identically distributed sequence · qf-mod29-iid-bernoulli-cylinder · candidate





## 10. Exact Bernoulli Cylinder Probability under a Declared Product Law

Use the explicitly declared Bernoulli product-law model with exact p=10/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[1, 0, 0, 1]. Give (word,ones,zeros,p,probability,symbolic_product_law).

View answer and explanation([1, 0, 0, 1], 2, 2, 10/29, 36100/707281, symbolic_product_law)

Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([1, 0, 0, 1], 2, 2, 10/29, 36100/707281, symbolic_product_law).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0010/)

Independent identically distributed sequence · qf-mod29-iid-bernoulli-cylinder · candidate





## 11. Exact Bernoulli Cylinder Probability under a Declared Product Law

Use the explicitly declared Bernoulli product-law model with exact p=11/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[0, 1, 0, 1]. Give (word,ones,zeros,p,probability,symbolic_product_law).

View answer and explanation([0, 1, 0, 1], 2, 2, 11/29, 39204/707281, symbolic_product_law)

Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([0, 1, 0, 1], 2, 2, 11/29, 39204/707281, symbolic_product_law).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0011/)

Independent identically distributed sequence · qf-mod29-iid-bernoulli-cylinder · candidate





## 12. Exact Bernoulli Cylinder Probability under a Declared Product Law

Use the explicitly declared Bernoulli product-law model with exact p=12/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[1, 1, 0, 1]. Give (word,ones,zeros,p,probability,symbolic_product_law).

View answer and explanation([1, 1, 0, 1], 3, 1, 12/29, 29376/707281, symbolic_product_law)

Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([1, 1, 0, 1], 3, 1, 12/29, 29376/707281, symbolic_product_law).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0012/)

Independent identically distributed sequence · qf-mod29-iid-bernoulli-cylinder · candidate





## 13. Exact Bernoulli Cylinder Probability under a Declared Product Law

Use the explicitly declared Bernoulli product-law model with exact p=13/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[0, 0, 1, 1]. Give (word,ones,zeros,p,probability,symbolic_product_law).

View answer and explanation([0, 0, 1, 1], 2, 2, 13/29, 43264/707281, symbolic_product_law)

Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([0, 0, 1, 1], 2, 2, 13/29, 43264/707281, symbolic_product_law).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0013/)

Independent identically distributed sequence · qf-mod29-iid-bernoulli-cylinder · candidate





## 14. Exact Bernoulli Cylinder Probability under a Declared Product Law

Use the explicitly declared Bernoulli product-law model with exact p=14/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[1, 0, 1, 1]. Give (word,ones,zeros,p,probability,symbolic_product_law).

View answer and explanation([1, 0, 1, 1], 3, 1, 14/29, 41160/707281, symbolic_product_law)

Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([1, 0, 1, 1], 3, 1, 14/29, 41160/707281, symbolic_product_law).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0014/)

Independent identically distributed sequence · qf-mod29-iid-bernoulli-cylinder · candidate





## 15. Exact Bernoulli Cylinder Probability under a Declared Product Law

Use the explicitly declared Bernoulli product-law model with exact p=15/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[0, 1, 1, 1]. Give (word,ones,zeros,p,probability,symbolic_product_law).

View answer and explanation([0, 1, 1, 1], 3, 1, 15/29, 47250/707281, symbolic_product_law)

Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([0, 1, 1, 1], 3, 1, 15/29, 47250/707281, symbolic_product_law).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0015/)

Independent identically distributed sequence · qf-mod29-iid-bernoulli-cylinder · candidate





## 16. Exact Bernoulli Cylinder Probability under a Declared Product Law

Use the explicitly declared Bernoulli product-law model with exact p=16/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[1, 1, 1, 1]. Give (word,ones,zeros,p,probability,symbolic_product_law).

View answer and explanation([1, 1, 1, 1], 4, 0, 16/29, 65536/707281, symbolic_product_law)

Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([1, 1, 1, 1], 4, 0, 16/29, 65536/707281, symbolic_product_law).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0016/)

Independent identically distributed sequence · qf-mod29-iid-bernoulli-cylinder · candidate





## 17. Exact Bernoulli Cylinder Probability under a Declared Product Law

Use the explicitly declared Bernoulli product-law model with exact p=17/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[0, 0, 0, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law).

View answer and explanation([0, 0, 0, 0], 0, 4, 17/29, 20736/707281, symbolic_product_law)

Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([0, 0, 0, 0], 0, 4, 17/29, 20736/707281, symbolic_product_law).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0017/)

Independent identically distributed sequence · qf-mod29-iid-bernoulli-cylinder · candidate





## 18. Exact Bernoulli Cylinder Probability under a Declared Product Law

Use the explicitly declared Bernoulli product-law model with exact p=18/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[1, 0, 0, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law).

View answer and explanation([1, 0, 0, 0], 1, 3, 18/29, 23958/707281, symbolic_product_law)

Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([1, 0, 0, 0], 1, 3, 18/29, 23958/707281, symbolic_product_law).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0018/)

Independent identically distributed sequence · qf-mod29-iid-bernoulli-cylinder · candidate





## 19. Exact Bernoulli Cylinder Probability under a Declared Product Law

Use the explicitly declared Bernoulli product-law model with exact p=19/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[0, 1, 0, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law).

View answer and explanation([0, 1, 0, 0], 1, 3, 19/29, 19000/707281, symbolic_product_law)

Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([0, 1, 0, 0], 1, 3, 19/29, 19000/707281, symbolic_product_law).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0019/)

Independent identically distributed sequence · qf-mod29-iid-bernoulli-cylinder · candidate





## 20. Exact Bernoulli Cylinder Probability under a Declared Product Law

Use the explicitly declared Bernoulli product-law model with exact p=20/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[1, 1, 0, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law).

View answer and explanation([1, 1, 0, 0], 2, 2, 20/29, 32400/707281, symbolic_product_law)

Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([1, 1, 0, 0], 2, 2, 20/29, 32400/707281, symbolic_product_law).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0020/)

Independent identically distributed sequence · qf-mod29-iid-bernoulli-cylinder · candidate





## 21. Exact Bernoulli Cylinder Probability under a Declared Product Law

Use the explicitly declared Bernoulli product-law model with exact p=21/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[0, 0, 1, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law).

View answer and explanation([0, 0, 1, 0], 1, 3, 21/29, 10752/707281, symbolic_product_law)

Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([0, 0, 1, 0], 1, 3, 21/29, 10752/707281, symbolic_product_law).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0021/)

Independent identically distributed sequence · qf-mod29-iid-bernoulli-cylinder · candidate





## 22. Exact Bernoulli Cylinder Probability under a Declared Product Law

Use the explicitly declared Bernoulli product-law model with exact p=22/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[1, 0, 1, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law).

View answer and explanation([1, 0, 1, 0], 2, 2, 22/29, 23716/707281, symbolic_product_law)

Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([1, 0, 1, 0], 2, 2, 22/29, 23716/707281, symbolic_product_law).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0022/)

Independent identically distributed sequence · qf-mod29-iid-bernoulli-cylinder · candidate





## 23. Exact Bernoulli Cylinder Probability under a Declared Product Law

Use the explicitly declared Bernoulli product-law model with exact p=23/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[0, 1, 1, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law).

View answer and explanation([0, 1, 1, 0], 2, 2, 23/29, 19044/707281, symbolic_product_law)

Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([0, 1, 1, 0], 2, 2, 23/29, 19044/707281, symbolic_product_law).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0023/)

Independent identically distributed sequence · qf-mod29-iid-bernoulli-cylinder · candidate





## 24. Exact Bernoulli Cylinder Probability under a Declared Product Law

Use the explicitly declared Bernoulli product-law model with exact p=24/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[1, 1, 1, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law).

View answer and explanation([1, 1, 1, 0], 3, 1, 24/29, 69120/707281, symbolic_product_law)

Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([1, 1, 1, 0], 3, 1, 24/29, 69120/707281, symbolic_product_law).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0024/)

Independent identically distributed sequence · qf-mod29-iid-bernoulli-cylinder · candidate





## 25. Exact Bernoulli Cylinder Probability under a Declared Product Law

Use the explicitly declared Bernoulli product-law model with exact p=25/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[0, 0, 0, 1]. Give (word,ones,zeros,p,probability,symbolic_product_law).

View answer and explanation([0, 0, 0, 1], 1, 3, 25/29, 1600/707281, symbolic_product_law)

Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([0, 0, 0, 1], 1, 3, 25/29, 1600/707281, symbolic_product_law).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0025/)

Independent identically distributed sequence · qf-mod29-iid-bernoulli-cylinder · candidate





## 26. Identical Distribution Means the Complete Marginal, Not Just Its Mean

Use the explicitly declared Bernoulli product-law model with exact p=1/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=0, right_index=17. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).

View answer and explanation(0, 17, [0, 28/29], [1, 1/29], Yes, full_Bernoulli_marginal_not_mean_only)

Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (0, 17, [0, 28/29], [1, 1/29], Yes, full_Bernoulli_marginal_not_mean_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0026/)

Independent identically distributed sequence · qf-mod29-iid-identical-marginal · candidate





## 27. Identical Distribution Means the Complete Marginal, Not Just Its Mean

Use the explicitly declared Bernoulli product-law model with exact p=2/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=1, right_index=18. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).

View answer and explanation(1, 18, [0, 27/29], [1, 2/29], Yes, full_Bernoulli_marginal_not_mean_only)

Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (1, 18, [0, 27/29], [1, 2/29], Yes, full_Bernoulli_marginal_not_mean_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0027/)

Independent identically distributed sequence · qf-mod29-iid-identical-marginal · candidate





## 28. Identical Distribution Means the Complete Marginal, Not Just Its Mean

Use the explicitly declared Bernoulli product-law model with exact p=3/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=2, right_index=19. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).

View answer and explanation(2, 19, [0, 26/29], [1, 3/29], Yes, full_Bernoulli_marginal_not_mean_only)

Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (2, 19, [0, 26/29], [1, 3/29], Yes, full_Bernoulli_marginal_not_mean_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0028/)

Independent identically distributed sequence · qf-mod29-iid-identical-marginal · candidate





## 29. Identical Distribution Means the Complete Marginal, Not Just Its Mean

Use the explicitly declared Bernoulli product-law model with exact p=4/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=3, right_index=20. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).

View answer and explanation(3, 20, [0, 25/29], [1, 4/29], Yes, full_Bernoulli_marginal_not_mean_only)

Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (3, 20, [0, 25/29], [1, 4/29], Yes, full_Bernoulli_marginal_not_mean_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0029/)

Independent identically distributed sequence · qf-mod29-iid-identical-marginal · candidate





## 30. Identical Distribution Means the Complete Marginal, Not Just Its Mean

Use the explicitly declared Bernoulli product-law model with exact p=5/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=4, right_index=21. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).

View answer and explanation(4, 21, [0, 24/29], [1, 5/29], Yes, full_Bernoulli_marginal_not_mean_only)

Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (4, 21, [0, 24/29], [1, 5/29], Yes, full_Bernoulli_marginal_not_mean_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0030/)

Independent identically distributed sequence · qf-mod29-iid-identical-marginal · candidate





## 31. Identical Distribution Means the Complete Marginal, Not Just Its Mean

Use the explicitly declared Bernoulli product-law model with exact p=6/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=5, right_index=22. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).

View answer and explanation(5, 22, [0, 23/29], [1, 6/29], Yes, full_Bernoulli_marginal_not_mean_only)

Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (5, 22, [0, 23/29], [1, 6/29], Yes, full_Bernoulli_marginal_not_mean_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0031/)

Independent identically distributed sequence · qf-mod29-iid-identical-marginal · candidate





## 32. Identical Distribution Means the Complete Marginal, Not Just Its Mean

Use the explicitly declared Bernoulli product-law model with exact p=7/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=6, right_index=23. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).

View answer and explanation(6, 23, [0, 22/29], [1, 7/29], Yes, full_Bernoulli_marginal_not_mean_only)

Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (6, 23, [0, 22/29], [1, 7/29], Yes, full_Bernoulli_marginal_not_mean_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0032/)

Independent identically distributed sequence · qf-mod29-iid-identical-marginal · candidate





## 33. Identical Distribution Means the Complete Marginal, Not Just Its Mean

Use the explicitly declared Bernoulli product-law model with exact p=8/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=7, right_index=24. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).

View answer and explanation(7, 24, [0, 21/29], [1, 8/29], Yes, full_Bernoulli_marginal_not_mean_only)

Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (7, 24, [0, 21/29], [1, 8/29], Yes, full_Bernoulli_marginal_not_mean_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0033/)

Independent identically distributed sequence · qf-mod29-iid-identical-marginal · candidate





## 34. Identical Distribution Means the Complete Marginal, Not Just Its Mean

Use the explicitly declared Bernoulli product-law model with exact p=9/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=8, right_index=25. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).

View answer and explanation(8, 25, [0, 20/29], [1, 9/29], Yes, full_Bernoulli_marginal_not_mean_only)

Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (8, 25, [0, 20/29], [1, 9/29], Yes, full_Bernoulli_marginal_not_mean_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0034/)

Independent identically distributed sequence · qf-mod29-iid-identical-marginal · candidate





## 35. Identical Distribution Means the Complete Marginal, Not Just Its Mean

Use the explicitly declared Bernoulli product-law model with exact p=10/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=9, right_index=26. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).

View answer and explanation(9, 26, [0, 19/29], [1, 10/29], Yes, full_Bernoulli_marginal_not_mean_only)

Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (9, 26, [0, 19/29], [1, 10/29], Yes, full_Bernoulli_marginal_not_mean_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0035/)

Independent identically distributed sequence · qf-mod29-iid-identical-marginal · candidate





## 36. Identical Distribution Means the Complete Marginal, Not Just Its Mean

Use the explicitly declared Bernoulli product-law model with exact p=11/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=10, right_index=27. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).

View answer and explanation(10, 27, [0, 18/29], [1, 11/29], Yes, full_Bernoulli_marginal_not_mean_only)

Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (10, 27, [0, 18/29], [1, 11/29], Yes, full_Bernoulli_marginal_not_mean_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0036/)

Independent identically distributed sequence · qf-mod29-iid-identical-marginal · candidate





## 37. Identical Distribution Means the Complete Marginal, Not Just Its Mean

Use the explicitly declared Bernoulli product-law model with exact p=12/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=11, right_index=28. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).

View answer and explanation(11, 28, [0, 17/29], [1, 12/29], Yes, full_Bernoulli_marginal_not_mean_only)

Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (11, 28, [0, 17/29], [1, 12/29], Yes, full_Bernoulli_marginal_not_mean_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0037/)

Independent identically distributed sequence · qf-mod29-iid-identical-marginal · candidate





## 38. Identical Distribution Means the Complete Marginal, Not Just Its Mean

Use the explicitly declared Bernoulli product-law model with exact p=13/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=12, right_index=29. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).

View answer and explanation(12, 29, [0, 16/29], [1, 13/29], Yes, full_Bernoulli_marginal_not_mean_only)

Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (12, 29, [0, 16/29], [1, 13/29], Yes, full_Bernoulli_marginal_not_mean_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0038/)

Independent identically distributed sequence · qf-mod29-iid-identical-marginal · candidate





## 39. Identical Distribution Means the Complete Marginal, Not Just Its Mean

Use the explicitly declared Bernoulli product-law model with exact p=14/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=13, right_index=30. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).

View answer and explanation(13, 30, [0, 15/29], [1, 14/29], Yes, full_Bernoulli_marginal_not_mean_only)

Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (13, 30, [0, 15/29], [1, 14/29], Yes, full_Bernoulli_marginal_not_mean_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0039/)

Independent identically distributed sequence · qf-mod29-iid-identical-marginal · candidate





## 40. Identical Distribution Means the Complete Marginal, Not Just Its Mean

Use the explicitly declared Bernoulli product-law model with exact p=15/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=14, right_index=31. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).

View answer and explanation(14, 31, [0, 14/29], [1, 15/29], Yes, full_Bernoulli_marginal_not_mean_only)

Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (14, 31, [0, 14/29], [1, 15/29], Yes, full_Bernoulli_marginal_not_mean_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0040/)

Independent identically distributed sequence · qf-mod29-iid-identical-marginal · candidate





## 41. Identical Distribution Means the Complete Marginal, Not Just Its Mean

Use the explicitly declared Bernoulli product-law model with exact p=16/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=15, right_index=32. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).

View answer and explanation(15, 32, [0, 13/29], [1, 16/29], Yes, full_Bernoulli_marginal_not_mean_only)

Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (15, 32, [0, 13/29], [1, 16/29], Yes, full_Bernoulli_marginal_not_mean_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0041/)

Independent identically distributed sequence · qf-mod29-iid-identical-marginal · candidate





## 42. Identical Distribution Means the Complete Marginal, Not Just Its Mean

Use the explicitly declared Bernoulli product-law model with exact p=17/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=16, right_index=33. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).

View answer and explanation(16, 33, [0, 12/29], [1, 17/29], Yes, full_Bernoulli_marginal_not_mean_only)

Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (16, 33, [0, 12/29], [1, 17/29], Yes, full_Bernoulli_marginal_not_mean_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0042/)

Independent identically distributed sequence · qf-mod29-iid-identical-marginal · candidate





## 43. Identical Distribution Means the Complete Marginal, Not Just Its Mean

Use the explicitly declared Bernoulli product-law model with exact p=18/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=17, right_index=34. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).

View answer and explanation(17, 34, [0, 11/29], [1, 18/29], Yes, full_Bernoulli_marginal_not_mean_only)

Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (17, 34, [0, 11/29], [1, 18/29], Yes, full_Bernoulli_marginal_not_mean_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0043/)

Independent identically distributed sequence · qf-mod29-iid-identical-marginal · candidate





## 44. Identical Distribution Means the Complete Marginal, Not Just Its Mean

Use the explicitly declared Bernoulli product-law model with exact p=19/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=18, right_index=35. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).

View answer and explanation(18, 35, [0, 10/29], [1, 19/29], Yes, full_Bernoulli_marginal_not_mean_only)

Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (18, 35, [0, 10/29], [1, 19/29], Yes, full_Bernoulli_marginal_not_mean_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0044/)

Independent identically distributed sequence · qf-mod29-iid-identical-marginal · candidate





## 45. Identical Distribution Means the Complete Marginal, Not Just Its Mean

Use the explicitly declared Bernoulli product-law model with exact p=20/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=19, right_index=36. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).

View answer and explanation(19, 36, [0, 9/29], [1, 20/29], Yes, full_Bernoulli_marginal_not_mean_only)

Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (19, 36, [0, 9/29], [1, 20/29], Yes, full_Bernoulli_marginal_not_mean_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0045/)

Independent identically distributed sequence · qf-mod29-iid-identical-marginal · candidate





## 46. Identical Distribution Means the Complete Marginal, Not Just Its Mean

Use the explicitly declared Bernoulli product-law model with exact p=21/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=20, right_index=37. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).

View answer and explanation(20, 37, [0, 8/29], [1, 21/29], Yes, full_Bernoulli_marginal_not_mean_only)

Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (20, 37, [0, 8/29], [1, 21/29], Yes, full_Bernoulli_marginal_not_mean_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0046/)

Independent identically distributed sequence · qf-mod29-iid-identical-marginal · candidate





## 47. Identical Distribution Means the Complete Marginal, Not Just Its Mean

Use the explicitly declared Bernoulli product-law model with exact p=22/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=21, right_index=38. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).

View answer and explanation(21, 38, [0, 7/29], [1, 22/29], Yes, full_Bernoulli_marginal_not_mean_only)

Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (21, 38, [0, 7/29], [1, 22/29], Yes, full_Bernoulli_marginal_not_mean_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0047/)

Independent identically distributed sequence · qf-mod29-iid-identical-marginal · candidate





## 48. Identical Distribution Means the Complete Marginal, Not Just Its Mean

Use the explicitly declared Bernoulli product-law model with exact p=23/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=22, right_index=39. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).

View answer and explanation(22, 39, [0, 6/29], [1, 23/29], Yes, full_Bernoulli_marginal_not_mean_only)

Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (22, 39, [0, 6/29], [1, 23/29], Yes, full_Bernoulli_marginal_not_mean_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0048/)

Independent identically distributed sequence · qf-mod29-iid-identical-marginal · candidate





## 49. Identical Distribution Means the Complete Marginal, Not Just Its Mean

Use the explicitly declared Bernoulli product-law model with exact p=24/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=23, right_index=40. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).

View answer and explanation(23, 40, [0, 5/29], [1, 24/29], Yes, full_Bernoulli_marginal_not_mean_only)

Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (23, 40, [0, 5/29], [1, 24/29], Yes, full_Bernoulli_marginal_not_mean_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0049/)

Independent identically distributed sequence · qf-mod29-iid-identical-marginal · candidate





## 50. Identical Distribution Means the Complete Marginal, Not Just Its Mean

Use the explicitly declared Bernoulli product-law model with exact p=25/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=24, right_index=41. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).

View answer and explanation(24, 41, [0, 4/29], [1, 25/29], Yes, full_Bernoulli_marginal_not_mean_only)

Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (24, 41, [0, 4/29], [1, 25/29], Yes, full_Bernoulli_marginal_not_mean_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0050/)

Independent identically distributed sequence · qf-mod29-iid-identical-marginal · candidate





## 51. Multiply Every Ordered Atom Factor in the Finite Word

Use the explicitly declared Bernoulli product-law model with exact p=1/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[0, 0, 0, 0]. Give (word,atom_factors,product,pass?).

View answer and explanation([0, 0, 0, 0], [28/29, 28/29, 28/29, 28/29], 614656/707281, Yes)

Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([0, 0, 0, 0], [28/29, 28/29, 28/29, 28/29], 614656/707281, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0051/)

Independent identically distributed sequence · qf-mod29-iid-finite-factorization · candidate





## 52. Multiply Every Ordered Atom Factor in the Finite Word

Use the explicitly declared Bernoulli product-law model with exact p=2/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[1, 0, 0, 0]. Give (word,atom_factors,product,pass?).

View answer and explanation([1, 0, 0, 0], [2/29, 27/29, 27/29, 27/29], 39366/707281, Yes)

Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([1, 0, 0, 0], [2/29, 27/29, 27/29, 27/29], 39366/707281, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0052/)

Independent identically distributed sequence · qf-mod29-iid-finite-factorization · candidate





## 53. Multiply Every Ordered Atom Factor in the Finite Word

Use the explicitly declared Bernoulli product-law model with exact p=3/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[0, 1, 0, 0]. Give (word,atom_factors,product,pass?).

View answer and explanation([0, 1, 0, 0], [26/29, 3/29, 26/29, 26/29], 52728/707281, Yes)

Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([0, 1, 0, 0], [26/29, 3/29, 26/29, 26/29], 52728/707281, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0053/)

Independent identically distributed sequence · qf-mod29-iid-finite-factorization · candidate





## 54. Multiply Every Ordered Atom Factor in the Finite Word

Use the explicitly declared Bernoulli product-law model with exact p=4/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[1, 1, 0, 0]. Give (word,atom_factors,product,pass?).

View answer and explanation([1, 1, 0, 0], [4/29, 4/29, 25/29, 25/29], 10000/707281, Yes)

Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([1, 1, 0, 0], [4/29, 4/29, 25/29, 25/29], 10000/707281, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0054/)

Independent identically distributed sequence · qf-mod29-iid-finite-factorization · candidate





## 55. Multiply Every Ordered Atom Factor in the Finite Word

Use the explicitly declared Bernoulli product-law model with exact p=5/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[0, 0, 1, 0]. Give (word,atom_factors,product,pass?).

View answer and explanation([0, 0, 1, 0], [24/29, 24/29, 5/29, 24/29], 69120/707281, Yes)

Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([0, 0, 1, 0], [24/29, 24/29, 5/29, 24/29], 69120/707281, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0055/)

Independent identically distributed sequence · qf-mod29-iid-finite-factorization · candidate





## 56. Multiply Every Ordered Atom Factor in the Finite Word

Use the explicitly declared Bernoulli product-law model with exact p=6/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[1, 0, 1, 0]. Give (word,atom_factors,product,pass?).

View answer and explanation([1, 0, 1, 0], [6/29, 23/29, 6/29, 23/29], 19044/707281, Yes)

Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([1, 0, 1, 0], [6/29, 23/29, 6/29, 23/29], 19044/707281, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0056/)

Independent identically distributed sequence · qf-mod29-iid-finite-factorization · candidate





## 57. Multiply Every Ordered Atom Factor in the Finite Word

Use the explicitly declared Bernoulli product-law model with exact p=7/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[0, 1, 1, 0]. Give (word,atom_factors,product,pass?).

View answer and explanation([0, 1, 1, 0], [22/29, 7/29, 7/29, 22/29], 23716/707281, Yes)

Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([0, 1, 1, 0], [22/29, 7/29, 7/29, 22/29], 23716/707281, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0057/)

Independent identically distributed sequence · qf-mod29-iid-finite-factorization · candidate





## 58. Multiply Every Ordered Atom Factor in the Finite Word

Use the explicitly declared Bernoulli product-law model with exact p=8/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[1, 1, 1, 0]. Give (word,atom_factors,product,pass?).

View answer and explanation([1, 1, 1, 0], [8/29, 8/29, 8/29, 21/29], 10752/707281, Yes)

Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([1, 1, 1, 0], [8/29, 8/29, 8/29, 21/29], 10752/707281, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0058/)

Independent identically distributed sequence · qf-mod29-iid-finite-factorization · candidate





## 59. Multiply Every Ordered Atom Factor in the Finite Word

Use the explicitly declared Bernoulli product-law model with exact p=9/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[0, 0, 0, 1]. Give (word,atom_factors,product,pass?).

View answer and explanation([0, 0, 0, 1], [20/29, 20/29, 20/29, 9/29], 72000/707281, Yes)

Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([0, 0, 0, 1], [20/29, 20/29, 20/29, 9/29], 72000/707281, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0059/)

Independent identically distributed sequence · qf-mod29-iid-finite-factorization · candidate





## 60. Multiply Every Ordered Atom Factor in the Finite Word

Use the explicitly declared Bernoulli product-law model with exact p=10/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[1, 0, 0, 1]. Give (word,atom_factors,product,pass?).

View answer and explanation([1, 0, 0, 1], [10/29, 19/29, 19/29, 10/29], 36100/707281, Yes)

Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([1, 0, 0, 1], [10/29, 19/29, 19/29, 10/29], 36100/707281, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0060/)

Independent identically distributed sequence · qf-mod29-iid-finite-factorization · candidate





## 61. Multiply Every Ordered Atom Factor in the Finite Word

Use the explicitly declared Bernoulli product-law model with exact p=11/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[0, 1, 0, 1]. Give (word,atom_factors,product,pass?).

View answer and explanation([0, 1, 0, 1], [18/29, 11/29, 18/29, 11/29], 39204/707281, Yes)

Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([0, 1, 0, 1], [18/29, 11/29, 18/29, 11/29], 39204/707281, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0061/)

Independent identically distributed sequence · qf-mod29-iid-finite-factorization · candidate





## 62. Multiply Every Ordered Atom Factor in the Finite Word

Use the explicitly declared Bernoulli product-law model with exact p=12/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[1, 1, 0, 1]. Give (word,atom_factors,product,pass?).

View answer and explanation([1, 1, 0, 1], [12/29, 12/29, 17/29, 12/29], 29376/707281, Yes)

Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([1, 1, 0, 1], [12/29, 12/29, 17/29, 12/29], 29376/707281, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0062/)

Independent identically distributed sequence · qf-mod29-iid-finite-factorization · candidate





## 63. Multiply Every Ordered Atom Factor in the Finite Word

Use the explicitly declared Bernoulli product-law model with exact p=13/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[0, 0, 1, 1]. Give (word,atom_factors,product,pass?).

View answer and explanation([0, 0, 1, 1], [16/29, 16/29, 13/29, 13/29], 43264/707281, Yes)

Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([0, 0, 1, 1], [16/29, 16/29, 13/29, 13/29], 43264/707281, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0063/)

Independent identically distributed sequence · qf-mod29-iid-finite-factorization · candidate





## 64. Multiply Every Ordered Atom Factor in the Finite Word

Use the explicitly declared Bernoulli product-law model with exact p=14/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[1, 0, 1, 1]. Give (word,atom_factors,product,pass?).

View answer and explanation([1, 0, 1, 1], [14/29, 15/29, 14/29, 14/29], 41160/707281, Yes)

Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([1, 0, 1, 1], [14/29, 15/29, 14/29, 14/29], 41160/707281, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0064/)

Independent identically distributed sequence · qf-mod29-iid-finite-factorization · candidate





## 65. Multiply Every Ordered Atom Factor in the Finite Word

Use the explicitly declared Bernoulli product-law model with exact p=15/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[0, 1, 1, 1]. Give (word,atom_factors,product,pass?).

View answer and explanation([0, 1, 1, 1], [14/29, 15/29, 15/29, 15/29], 47250/707281, Yes)

Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([0, 1, 1, 1], [14/29, 15/29, 15/29, 15/29], 47250/707281, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0065/)

Independent identically distributed sequence · qf-mod29-iid-finite-factorization · candidate





## 66. Multiply Every Ordered Atom Factor in the Finite Word

Use the explicitly declared Bernoulli product-law model with exact p=16/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[1, 1, 1, 1]. Give (word,atom_factors,product,pass?).

View answer and explanation([1, 1, 1, 1], [16/29, 16/29, 16/29, 16/29], 65536/707281, Yes)

Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([1, 1, 1, 1], [16/29, 16/29, 16/29, 16/29], 65536/707281, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0066/)

Independent identically distributed sequence · qf-mod29-iid-finite-factorization · candidate





## 67. Multiply Every Ordered Atom Factor in the Finite Word

Use the explicitly declared Bernoulli product-law model with exact p=17/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[0, 0, 0, 0]. Give (word,atom_factors,product,pass?).

View answer and explanation([0, 0, 0, 0], [12/29, 12/29, 12/29, 12/29], 20736/707281, Yes)

Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([0, 0, 0, 0], [12/29, 12/29, 12/29, 12/29], 20736/707281, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0067/)

Independent identically distributed sequence · qf-mod29-iid-finite-factorization · candidate





## 68. Multiply Every Ordered Atom Factor in the Finite Word

Use the explicitly declared Bernoulli product-law model with exact p=18/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[1, 0, 0, 0]. Give (word,atom_factors,product,pass?).

View answer and explanation([1, 0, 0, 0], [18/29, 11/29, 11/29, 11/29], 23958/707281, Yes)

Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([1, 0, 0, 0], [18/29, 11/29, 11/29, 11/29], 23958/707281, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0068/)

Independent identically distributed sequence · qf-mod29-iid-finite-factorization · candidate





## 69. Multiply Every Ordered Atom Factor in the Finite Word

Use the explicitly declared Bernoulli product-law model with exact p=19/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[0, 1, 0, 0]. Give (word,atom_factors,product,pass?).

View answer and explanation([0, 1, 0, 0], [10/29, 19/29, 10/29, 10/29], 19000/707281, Yes)

Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([0, 1, 0, 0], [10/29, 19/29, 10/29, 10/29], 19000/707281, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0069/)

Independent identically distributed sequence · qf-mod29-iid-finite-factorization · candidate





## 70. Multiply Every Ordered Atom Factor in the Finite Word

Use the explicitly declared Bernoulli product-law model with exact p=20/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[1, 1, 0, 0]. Give (word,atom_factors,product,pass?).

View answer and explanation([1, 1, 0, 0], [20/29, 20/29, 9/29, 9/29], 32400/707281, Yes)

Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([1, 1, 0, 0], [20/29, 20/29, 9/29, 9/29], 32400/707281, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0070/)

Independent identically distributed sequence · qf-mod29-iid-finite-factorization · candidate





## 71. Multiply Every Ordered Atom Factor in the Finite Word

Use the explicitly declared Bernoulli product-law model with exact p=21/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[0, 0, 1, 0]. Give (word,atom_factors,product,pass?).

View answer and explanation([0, 0, 1, 0], [8/29, 8/29, 21/29, 8/29], 10752/707281, Yes)

Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([0, 0, 1, 0], [8/29, 8/29, 21/29, 8/29], 10752/707281, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0071/)

Independent identically distributed sequence · qf-mod29-iid-finite-factorization · candidate





## 72. Multiply Every Ordered Atom Factor in the Finite Word

Use the explicitly declared Bernoulli product-law model with exact p=22/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[1, 0, 1, 0]. Give (word,atom_factors,product,pass?).

View answer and explanation([1, 0, 1, 0], [22/29, 7/29, 22/29, 7/29], 23716/707281, Yes)

Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([1, 0, 1, 0], [22/29, 7/29, 22/29, 7/29], 23716/707281, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0072/)

Independent identically distributed sequence · qf-mod29-iid-finite-factorization · candidate





## 73. Multiply Every Ordered Atom Factor in the Finite Word

Use the explicitly declared Bernoulli product-law model with exact p=23/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[0, 1, 1, 0]. Give (word,atom_factors,product,pass?).

View answer and explanation([0, 1, 1, 0], [6/29, 23/29, 23/29, 6/29], 19044/707281, Yes)

Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([0, 1, 1, 0], [6/29, 23/29, 23/29, 6/29], 19044/707281, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0073/)

Independent identically distributed sequence · qf-mod29-iid-finite-factorization · candidate





## 74. Multiply Every Ordered Atom Factor in the Finite Word

Use the explicitly declared Bernoulli product-law model with exact p=24/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[1, 1, 1, 0]. Give (word,atom_factors,product,pass?).

View answer and explanation([1, 1, 1, 0], [24/29, 24/29, 24/29, 5/29], 69120/707281, Yes)

Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([1, 1, 1, 0], [24/29, 24/29, 24/29, 5/29], 69120/707281, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0074/)

Independent identically distributed sequence · qf-mod29-iid-finite-factorization · candidate





## 75. Multiply Every Ordered Atom Factor in the Finite Word

Use the explicitly declared Bernoulli product-law model with exact p=25/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[0, 0, 0, 1]. Give (word,atom_factors,product,pass?).

View answer and explanation([0, 0, 0, 1], [4/29, 4/29, 4/29, 25/29], 1600/707281, Yes)

Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([0, 0, 0, 1], [4/29, 4/29, 4/29, 25/29], 1600/707281, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0075/)

Independent identically distributed sequence · qf-mod29-iid-finite-factorization · candidate





## 76. A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence

Use the explicitly declared Bernoulli product-law model with exact p=1/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=finite_prefix_only, checked_prefix=1. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.

View answer and explanation(1/29, finite_prefix_only, 1, No, finite_prefix_only_not_infinite_IID)

Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (1/29, finite_prefix_only, 1, No, finite_prefix_only_not_infinite_IID).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0076/)

Independent identically distributed sequence · qf-mod29-iid-evidence-mode · candidate





## 77. A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence

Use the explicitly declared Bernoulli product-law model with exact p=2/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=2. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.

View answer and explanation(2/29, symbolic_product_law, 2, Yes, infinite_IID_certificate)

Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (2/29, symbolic_product_law, 2, Yes, infinite_IID_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0077/)

Independent identically distributed sequence · qf-mod29-iid-evidence-mode · candidate





## 78. A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence

Use the explicitly declared Bernoulli product-law model with exact p=3/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=3. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.

View answer and explanation(3/29, symbolic_product_law, 3, Yes, infinite_IID_certificate)

Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (3/29, symbolic_product_law, 3, Yes, infinite_IID_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0078/)

Independent identically distributed sequence · qf-mod29-iid-evidence-mode · candidate





## 79. A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence

Use the explicitly declared Bernoulli product-law model with exact p=4/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=4. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.

View answer and explanation(4/29, symbolic_product_law, 4, Yes, infinite_IID_certificate)

Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (4/29, symbolic_product_law, 4, Yes, infinite_IID_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0079/)

Independent identically distributed sequence · qf-mod29-iid-evidence-mode · candidate





## 80. A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence

Use the explicitly declared Bernoulli product-law model with exact p=5/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=finite_prefix_only, checked_prefix=5. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.

View answer and explanation(5/29, finite_prefix_only, 5, No, finite_prefix_only_not_infinite_IID)

Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (5/29, finite_prefix_only, 5, No, finite_prefix_only_not_infinite_IID).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0080/)

Independent identically distributed sequence · qf-mod29-iid-evidence-mode · candidate





## 81. A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence

Use the explicitly declared Bernoulli product-law model with exact p=6/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=6. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.

View answer and explanation(6/29, symbolic_product_law, 6, Yes, infinite_IID_certificate)

Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (6/29, symbolic_product_law, 6, Yes, infinite_IID_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0081/)

Independent identically distributed sequence · qf-mod29-iid-evidence-mode · candidate





## 82. A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence

Use the explicitly declared Bernoulli product-law model with exact p=7/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=7. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.

View answer and explanation(7/29, symbolic_product_law, 7, Yes, infinite_IID_certificate)

Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (7/29, symbolic_product_law, 7, Yes, infinite_IID_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0082/)

Independent identically distributed sequence · qf-mod29-iid-evidence-mode · candidate





## 83. A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence

Use the explicitly declared Bernoulli product-law model with exact p=8/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=8. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.

View answer and explanation(8/29, symbolic_product_law, 8, Yes, infinite_IID_certificate)

Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (8/29, symbolic_product_law, 8, Yes, infinite_IID_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0083/)

Independent identically distributed sequence · qf-mod29-iid-evidence-mode · candidate





## 84. A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence

Use the explicitly declared Bernoulli product-law model with exact p=9/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=finite_prefix_only, checked_prefix=9. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.

View answer and explanation(9/29, finite_prefix_only, 9, No, finite_prefix_only_not_infinite_IID)

Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (9/29, finite_prefix_only, 9, No, finite_prefix_only_not_infinite_IID).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0084/)

Independent identically distributed sequence · qf-mod29-iid-evidence-mode · candidate





## 85. A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence

Use the explicitly declared Bernoulli product-law model with exact p=10/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=10. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.

View answer and explanation(10/29, symbolic_product_law, 10, Yes, infinite_IID_certificate)

Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (10/29, symbolic_product_law, 10, Yes, infinite_IID_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0085/)

Independent identically distributed sequence · qf-mod29-iid-evidence-mode · candidate





## 86. A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence

Use the explicitly declared Bernoulli product-law model with exact p=11/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=11. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.

View answer and explanation(11/29, symbolic_product_law, 11, Yes, infinite_IID_certificate)

Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (11/29, symbolic_product_law, 11, Yes, infinite_IID_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0086/)

Independent identically distributed sequence · qf-mod29-iid-evidence-mode · candidate





## 87. A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence

Use the explicitly declared Bernoulli product-law model with exact p=12/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=12. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.

View answer and explanation(12/29, symbolic_product_law, 12, Yes, infinite_IID_certificate)

Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (12/29, symbolic_product_law, 12, Yes, infinite_IID_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0087/)

Independent identically distributed sequence · qf-mod29-iid-evidence-mode · candidate





## 88. A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence

Use the explicitly declared Bernoulli product-law model with exact p=13/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=finite_prefix_only, checked_prefix=13. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.

View answer and explanation(13/29, finite_prefix_only, 13, No, finite_prefix_only_not_infinite_IID)

Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (13/29, finite_prefix_only, 13, No, finite_prefix_only_not_infinite_IID).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0088/)

Independent identically distributed sequence · qf-mod29-iid-evidence-mode · candidate





## 89. A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence

Use the explicitly declared Bernoulli product-law model with exact p=14/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=14. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.

View answer and explanation(14/29, symbolic_product_law, 14, Yes, infinite_IID_certificate)

Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (14/29, symbolic_product_law, 14, Yes, infinite_IID_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0089/)

Independent identically distributed sequence · qf-mod29-iid-evidence-mode · candidate





## 90. A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence

Use the explicitly declared Bernoulli product-law model with exact p=15/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=15. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.

View answer and explanation(15/29, symbolic_product_law, 15, Yes, infinite_IID_certificate)

Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (15/29, symbolic_product_law, 15, Yes, infinite_IID_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0090/)

Independent identically distributed sequence · qf-mod29-iid-evidence-mode · candidate





## 91. A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence

Use the explicitly declared Bernoulli product-law model with exact p=16/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=16. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.

View answer and explanation(16/29, symbolic_product_law, 16, Yes, infinite_IID_certificate)

Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (16/29, symbolic_product_law, 16, Yes, infinite_IID_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0091/)

Independent identically distributed sequence · qf-mod29-iid-evidence-mode · candidate





## 92. A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence

Use the explicitly declared Bernoulli product-law model with exact p=17/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=finite_prefix_only, checked_prefix=17. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.

View answer and explanation(17/29, finite_prefix_only, 17, No, finite_prefix_only_not_infinite_IID)

Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (17/29, finite_prefix_only, 17, No, finite_prefix_only_not_infinite_IID).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0092/)

Independent identically distributed sequence · qf-mod29-iid-evidence-mode · candidate





## 93. A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence

Use the explicitly declared Bernoulli product-law model with exact p=18/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=18. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.

View answer and explanation(18/29, symbolic_product_law, 18, Yes, infinite_IID_certificate)

Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (18/29, symbolic_product_law, 18, Yes, infinite_IID_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0093/)

Independent identically distributed sequence · qf-mod29-iid-evidence-mode · candidate





## 94. A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence

Use the explicitly declared Bernoulli product-law model with exact p=19/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=19. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.

View answer and explanation(19/29, symbolic_product_law, 19, Yes, infinite_IID_certificate)

Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (19/29, symbolic_product_law, 19, Yes, infinite_IID_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0094/)

Independent identically distributed sequence · qf-mod29-iid-evidence-mode · candidate





## 95. A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence

Use the explicitly declared Bernoulli product-law model with exact p=20/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=20. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.

View answer and explanation(20/29, symbolic_product_law, 20, Yes, infinite_IID_certificate)

Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (20/29, symbolic_product_law, 20, Yes, infinite_IID_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0095/)

Independent identically distributed sequence · qf-mod29-iid-evidence-mode · candidate





## 96. A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence

Use the explicitly declared Bernoulli product-law model with exact p=21/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=finite_prefix_only, checked_prefix=21. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.

View answer and explanation(21/29, finite_prefix_only, 21, No, finite_prefix_only_not_infinite_IID)

Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (21/29, finite_prefix_only, 21, No, finite_prefix_only_not_infinite_IID).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0096/)

Independent identically distributed sequence · qf-mod29-iid-evidence-mode · candidate





## 97. A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence

Use the explicitly declared Bernoulli product-law model with exact p=22/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=22. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.

View answer and explanation(22/29, symbolic_product_law, 22, Yes, infinite_IID_certificate)

Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (22/29, symbolic_product_law, 22, Yes, infinite_IID_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0097/)

Independent identically distributed sequence · qf-mod29-iid-evidence-mode · candidate





## 98. A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence

Use the explicitly declared Bernoulli product-law model with exact p=23/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=23. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.

View answer and explanation(23/29, symbolic_product_law, 23, Yes, infinite_IID_certificate)

Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (23/29, symbolic_product_law, 23, Yes, infinite_IID_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0098/)

Independent identically distributed sequence · qf-mod29-iid-evidence-mode · candidate





## 99. A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence

Use the explicitly declared Bernoulli product-law model with exact p=24/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=24. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.

View answer and explanation(24/29, symbolic_product_law, 24, Yes, infinite_IID_certificate)

Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (24/29, symbolic_product_law, 24, Yes, infinite_IID_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0099/)

Independent identically distributed sequence · qf-mod29-iid-evidence-mode · candidate





## 100. A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence

Use the explicitly declared Bernoulli product-law model with exact p=25/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=finite_prefix_only, checked_prefix=25. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.

View answer and explanation(25/29, finite_prefix_only, 25, No, finite_prefix_only_not_infinite_IID)

Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (25/29, finite_prefix_only, 25, No, finite_prefix_only_not_infinite_IID).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r06-iid-sequence-0100/)

Independent identically distributed sequence · qf-mod29-iid-evidence-mode · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261029-r06-iid-sequence-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261029-r06-iid-sequence-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261029-r06-iid-sequence-foundations/validation-report.json)