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R0027-14471 · q-20261014-advance-20261029-r06-iid-sequence-0071
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?).
G001 — 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?).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.
Before
Bernoulli zero-outcome mass := sub(1, 21/29)
After
Bernoulli zero-outcome mass = 8/29
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001
Before
raw Bernoulli parameter=[21, 29]
After
normalized p=21/29;mass at0=8/29;mass at1=21/29;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.
Rule: Bernoulli_complete_marginal_and_product_law_premise · Uses: G001, S001
Before
original ordered finite binary word=[0, 0, 1, 0];p=21/29;one-minus-p=8/29
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.
Rule: IID_ordered_word_and_given_model_not_unknown_law_test · Uses: G001, S002
Before
word position=0;original bit=0
After
aligned atom factor=8/29
Conditions: none
Choose the correct marginal factor at every original position,retaining zero factors without any conditional-probability division.
Rule: IID_each_original_bit_factor · Uses: G001, S003
Before
ordered factor prefix product0 := mul(1, 8/29)
After
ordered factor prefix product0 = 8/29
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
word position=1;original bit=0
After
aligned atom factor=8/29
Conditions: none
Choose the correct marginal factor at every original position,retaining zero factors without any conditional-probability division.
Rule: IID_each_original_bit_factor · Uses: G001, S005
Before
ordered factor prefix product1 := mul(8/29, 8/29)
After
ordered factor prefix product1 = 64/841
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
word position=2;original bit=1
After
aligned atom factor=21/29
Conditions: none
Choose the correct marginal factor at every original position,retaining zero factors without any conditional-probability division.
Rule: IID_each_original_bit_factor · Uses: G001, S007
Before
ordered factor prefix product2 := mul(64/841, 21/29)
After
ordered factor prefix product2 = 1344/24389
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
word position=3;original bit=0
After
aligned atom factor=8/29
Conditions: none
Choose the correct marginal factor at every original position,retaining zero factors without any conditional-probability division.
Rule: IID_each_original_bit_factor · Uses: G001, S009
Before
ordered factor prefix product3 := mul(1344/24389, 8/29)
After
ordered factor prefix product3 = 10752/707281
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
ordered atom factors=[8/29, 8/29, 21/29, 8/29]
After
finite product=10752/707281;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.
Rule: IID_finite_factorization_scope_not_new_infinite_proof · Uses: G001, S011
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.([0, 0, 1, 0], [8/29, 8/29, 21/29, 8/29], 10752/707281, Yes)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
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