Question 14439: A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence

Original question and givens

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.

• G001 — 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.

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, 14/29)

Bernoulli zero-outcome mass = 15/29

raw Bernoulli parameter=[14, 29]

normalized p=14/29;mass at0=15/29;mass at1=14/29;symbolic product-law model explicitly declared

p=14/29;selected evidence=symbolic_product_law;checked finite prefix length=14

sufficient infinite-IID evidence=Yes

Reverse checks

• 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

(14/29, symbolic_product_law, 14, Yes, infinite_IID_certificate)

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.