# Question 14443: A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence [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-0093) 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. R0027-14443 · q-20261014-advance-20261029-r06-iid-sequence-0093 ## Original question and givens 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. - G001 — 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. ## 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, 18/29) After Bernoulli zero-outcome mass = 11/29 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=[18, 29] After normalized p=18/29;mass at0=11/29;mass at1=18/29;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 p=18/29;selected evidence=symbolic_product_law;checked finite prefix length=18 After sufficient infinite-IID evidence=Yes Conditions: none The declared product model assigns the identical-marginal product to every finite-index cylinder,providing this symbolic certificate.A fixed prefix,however long,does not control all future subfamilies.No rejects evidence sufficiency,not existence of the Bernoulli product model,and supplies no LLN/CLT. Step identifiers and dependenciesRule: IID_infinite_cylinders_not_fixed_prefix · Uses: G001, S002 ## 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 (18/29, symbolic_product_law, 18, 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: b06647bdd93b8550a6cb8e521527c293cbb3d1462adef2a35625cc6c43d4d455 - candidate record SHA-256: 7625ae546961db9974d0c5d3e2088461ccef607a58077d6ea1a9a7e366135aab - 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-0093.json)