Question 14475: Multiply Every Ordered Atom Factor in the Finite Word

Original question and givens

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?).

• G001 — 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?).

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

Step-by-step derivation

Bernoulli zero-outcome mass := sub(1, 25/29)

Bernoulli zero-outcome mass = 4/29

raw Bernoulli parameter=[25, 29]

normalized p=25/29;mass at0=4/29;mass at1=25/29;symbolic product-law model explicitly declared

original ordered finite binary word=[0, 0, 0, 1];p=25/29;one-minus-p=4/29

cylinder atom factorization uses the same declared symbolic product law

word position=0;original bit=0

aligned atom factor=4/29

ordered factor prefix product0 := mul(1, 4/29)

ordered factor prefix product0 = 4/29

word position=1;original bit=0

aligned atom factor=4/29

ordered factor prefix product1 := mul(4/29, 4/29)

ordered factor prefix product1 = 16/841

word position=2;original bit=0

aligned atom factor=4/29

ordered factor prefix product2 := mul(16/841, 4/29)

ordered factor prefix product2 = 64/24389

word position=3;original bit=1

aligned atom factor=25/29

ordered factor prefix product3 := mul(64/24389, 25/29)

ordered factor prefix product3 = 1600/707281

ordered atom factors=[4/29, 4/29, 4/29, 25/29]

finite product=1600/707281;model factorization check=Yes

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

([0, 0, 0, 1], [4/29, 4/29, 4/29, 25/29], 1600/707281, 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.