Bernoulli Cylinders, Full Marginals, Finite Factorization and Infinite-IID Evidence
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Independent identically distributed sequence · qf-mod29-iid-evidence-mode · candidate
Canonical question JSONL · English JSONL · Validation report