# Question 14452: Multiply Every Ordered Atom Factor in the Finite Word [Return to catalogue page 73](https://ocme.evemisslab.com/en/questions/page/73/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261029-r06-iid-sequence-foundations/#q-20261014-advance-20261029-r06-iid-sequence-0052) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0027-14452 · q-20261014-advance-20261029-r06-iid-sequence-0052 ## Original question and givens 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?). - G001 — 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?). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-iid-sequence](https://ocme.evemisslab.com/en/mko/mko-iid-sequence/) ## Step-by-step derivation S001 · calculationBefore Bernoulli zero-outcome mass := sub(1, 2/29) After Bernoulli zero-outcome mass = 27/29 Conditions: none Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001 S002 · impliesBefore raw Bernoulli parameter=[2, 29] After normalized p=2/29;mass at0=27/29;mass at1=2/29;symbolic product-law model explicitly declared Conditions: none Retain the complete{0,1} marginal and the explicitly declared product-law premise.Infinite IID is not inferred from a data set,equal means or pairwise checks. Step identifiers and dependenciesRule: Bernoulli_complete_marginal_and_product_law_premise · Uses: G001, S001 S003 · impliesBefore original ordered finite binary word=[1, 0, 0, 0];p=2/29;one-minus-p=27/29 After cylinder atom factorization uses the same declared symbolic product law Conditions: none Preserve word order,use p for1 and1−p for0.Factorization is a model premise,not a test proving independence of an unknown dependent law.Retain empty words and p=0 or1 boundaries. Step identifiers and dependenciesRule: IID_ordered_word_and_given_model_not_unknown_law_test · Uses: G001, S002 S004 · impliesBefore word position=0;original bit=1 After aligned atom factor=2/29 Conditions: none Choose the correct marginal factor at every original position,retaining zero factors without any conditional-probability division. Step identifiers and dependenciesRule: IID_each_original_bit_factor · Uses: G001, S003 S005 · calculationBefore ordered factor prefix product0 := mul(1, 2/29) After ordered factor prefix product0 = 2/29 Conditions: none Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore word position=1;original bit=0 After aligned atom factor=27/29 Conditions: none Choose the correct marginal factor at every original position,retaining zero factors without any conditional-probability division. Step identifiers and dependenciesRule: IID_each_original_bit_factor · Uses: G001, S005 S007 · calculationBefore ordered factor prefix product1 := mul(2/29, 27/29) After ordered factor prefix product1 = 54/841 Conditions: none Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore word position=2;original bit=0 After aligned atom factor=27/29 Conditions: none Choose the correct marginal factor at every original position,retaining zero factors without any conditional-probability division. Step identifiers and dependenciesRule: IID_each_original_bit_factor · Uses: G001, S007 S009 · calculationBefore ordered factor prefix product2 := mul(54/841, 27/29) After ordered factor prefix product2 = 1458/24389 Conditions: none Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore word position=3;original bit=0 After aligned atom factor=27/29 Conditions: none Choose the correct marginal factor at every original position,retaining zero factors without any conditional-probability division. Step identifiers and dependenciesRule: IID_each_original_bit_factor · Uses: G001, S009 S011 · calculationBefore ordered factor prefix product3 := mul(1458/24389, 27/29) After ordered factor prefix product3 = 39366/707281 Conditions: none Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · impliesBefore ordered atom factors=[2/29, 27/29, 27/29, 27/29] After finite product=39366/707281;model factorization check=Yes Conditions: none Multiply every factor in original order,starting an empty list at identity1.Pass checks this finite calculation within the declared product model;it does not extrapolate arbitrary infinite IID from finite samples. Step identifiers and dependenciesRule: IID_finite_factorization_scope_not_new_infinite_proof · Uses: G001, S011 ## Reverse checks - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Recompute ordered factor prefix product3 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Recompute ordered factor prefix product2 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Recompute ordered factor prefix product1 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Recompute ordered factor prefix product0 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Recompute Bernoulli zero-outcome mass with exact rational arithmetic, not rounded decimals. ## Result ([1, 0, 0, 0], [2/29, 27/29, 27/29, 27/29], 39366/707281, Yes) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: e06eca07455c01f3a42fa3cef36873d47baae5657f4153b65da1d52b8c1ea252 - candidate record SHA-256: 922d11adb4ac2291190693d533c90da03a7b58ddec8349a28c15dd3c3636805a - semantic review receipt SHA-256: e49bf5eaa968c09ebac4ad538ce58251671ed922c957cee1471cfc7b68934bfe - question-first witness SHA-256: a7976057c26051fab92342fc95d48af224466c0e1e4ab6ae4c8811c02772e6da - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261029-r06-iid-sequence-0052.json)