# Question 27215: Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles [Return to catalogue page 137](https://ocme.evemisslab.com/en/questions/page/137/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations/#q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0090) 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. R0053-27215 · q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0090 ## Original question and givens Use mode=finite_histogram and complete model={"values":[2,3],"probabilities":[[1,4],[3,4]],"process_certificate":"infinite_iid","n":6}. Give (mode, mean, variance, n, evidence_scope, limit_proof_claim, boundary). - G001 — Use mode=finite_histogram and complete model={"values":[2,3],"probabilities":[[1,4],[3,4]],"process_certificate":"infinite_iid","n":6}. Give (mode, mean, variance, n, evidence_scope, limit_proof_claim, boundary). ## 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-central-limit-theorem-finite-variance-iid](https://ocme.evemisslab.com/en/mko/mko-central-limit-theorem-finite-variance-iid/) ## Step-by-step derivation S001 · impliesBefore source roles = [clt-application-boundary] After derived = [[mode, mean, variance, n, evidence_scope, limit_proof_claim, boundary]] Conditions: none List the complete source field order and mode-specific roles without JavaScript or answer-shape inference;No,undefined and infinity retain their distinct meanings. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · impliesBefore source roles = [infinite_iid] After derived = [ASSUME infinite sequence is IID with this common marginal] Conditions: none infinite_iid is an explicit infinite-process premise token,not a construction or finite verification of infinite independence.Subsequent reasoning is conditional on that premise. Step identifiers and dependenciesRule: CLT_infinite_iid_is_EXACT_GIVEN_premise_not_constructed_process · Uses: G001, S001 S003 · calculationBefore CLT ORIGINAL conditional marginal ORIGINAL law mass sum0 := sum(0, 1/4) After CLT ORIGINAL conditional marginal ORIGINAL law mass sum0 = 1/4 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore CLT ORIGINAL conditional marginal ORIGINAL law mass sum1 := sum(1/4, 3/4) After CLT ORIGINAL conditional marginal ORIGINAL law mass sum1 = 1 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore source roles = [CLT ORIGINAL conditional marginal, [2, 3], [[1, 4], [3, 4]]] After derived = [[[2, 1/4], [3, 3/4]], 1] Conditions: none Pair ORIGINAL values/probabilities before numeric sorting,retaining zero-mass labels and checking total mass1.This qualifies a finite marginal,not an infinite IID process inferred from samples. Step identifiers and dependenciesRule: CLT_ORIGINAL_value_probability_pairs_NUMERIC_order_normalized · Uses: G001, S004 S006 · calculationBefore CLT ORIGINAL conditional marginal ORIGINAL value times probability0 := mul(2, 1/4) After CLT ORIGINAL conditional marginal ORIGINAL value times probability0 = 1/2 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore CLT ORIGINAL conditional marginal ORIGINAL ascending mean sum0 := sum(0, 1/2) After CLT ORIGINAL conditional marginal ORIGINAL ascending mean sum0 = 1/2 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore CLT ORIGINAL conditional marginal exact value square0 := mul(2, 2) After CLT ORIGINAL conditional marginal exact value square0 = 4 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore CLT ORIGINAL conditional marginal ORIGINAL square times probability0 := mul(4, 1/4) After CLT ORIGINAL conditional marginal ORIGINAL square times probability0 = 1 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore CLT ORIGINAL conditional marginal ORIGINAL ascending second moment sum0 := sum(0, 1) After CLT ORIGINAL conditional marginal ORIGINAL ascending second moment sum0 = 1 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore CLT ORIGINAL conditional marginal ORIGINAL value times probability1 := mul(3, 3/4) After CLT ORIGINAL conditional marginal ORIGINAL value times probability1 = 9/4 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore CLT ORIGINAL conditional marginal ORIGINAL ascending mean sum1 := sum(1/2, 9/4) After CLT ORIGINAL conditional marginal ORIGINAL ascending mean sum1 = 11/4 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore CLT ORIGINAL conditional marginal exact value square1 := mul(3, 3) After CLT ORIGINAL conditional marginal exact value square1 = 9 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore CLT ORIGINAL conditional marginal ORIGINAL square times probability1 := mul(9, 3/4) After CLT ORIGINAL conditional marginal ORIGINAL square times probability1 = 27/4 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore CLT ORIGINAL conditional marginal ORIGINAL ascending second moment sum1 := sum(1, 27/4) After CLT ORIGINAL conditional marginal ORIGINAL ascending second moment sum1 = 31/4 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore CLT ORIGINAL conditional marginal ORIGINAL mean square := mul(11/4, 11/4) After CLT ORIGINAL conditional marginal ORIGINAL mean square = 121/16 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore CLT ORIGINAL conditional marginal ORIGINAL second minus mean square := sub(31/4, 121/16) After CLT ORIGINAL conditional marginal ORIGINAL second minus mean square = 3/16 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · impliesBefore source roles = [CLT ORIGINAL conditional marginal, [[2, 1/4], [3, 3/4]]] After derived = [11/4, 31/4, 3/16] Conditions: none Use the SAME complete law for E[X],E[X²] and Var(X)=E[X²]-E[X]².Keep value² as an exact teaching intermediate and apply the source profile to its product with probability,not an extra rejection for a large zero-mass value square.Finite moments prove neither independence,an infinite process nor a distributional limit. Step identifiers and dependenciesRule: CLT_ALL_finite_marginal_moments_not_a_process_or_limit_proof · Uses: G001, S017 S019 · impliesBefore source roles = [finite_histogram, 11/4, 3/16, 6] After derived = [finite_histogram, 11/4, 3/16, 6, one_finite_histogram, No, finite_histogram_does_not_prove_distributional_limit] Conditions: none The denominator must be sigma*sqrt(n),since Var(S_n)=n*sigma².The general formula n*sigma differs,but coincides at n=1;No does not assert numerical inequality in every instance.One finite histogram,even if normal-looking,does not prove a distributional limit. Step identifiers and dependenciesRule: CLT_original_wrong_scale_or_finite_evidence_inference_boundary · Uses: G001, S018 ## Reverse checks - S019 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct. - S018 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct. - S017 · premise_and_result_audit — Recompute CLT ORIGINAL conditional marginal ORIGINAL second minus mean square with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute CLT ORIGINAL conditional marginal ORIGINAL mean square with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute CLT ORIGINAL conditional marginal ORIGINAL ascending second moment sum1 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute CLT ORIGINAL conditional marginal ORIGINAL square times probability1 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute CLT ORIGINAL conditional marginal exact value square1 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute CLT ORIGINAL conditional marginal ORIGINAL ascending mean sum1 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute CLT ORIGINAL conditional marginal ORIGINAL value times probability1 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute CLT ORIGINAL conditional marginal ORIGINAL ascending second moment sum0 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute CLT ORIGINAL conditional marginal ORIGINAL square times probability0 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute CLT ORIGINAL conditional marginal exact value square0 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute CLT ORIGINAL conditional marginal ORIGINAL ascending mean sum0 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute CLT ORIGINAL conditional marginal ORIGINAL value times probability0 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct. - S004 · premise_and_result_audit — Recompute CLT ORIGINAL conditional marginal ORIGINAL law mass sum1 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute CLT ORIGINAL conditional marginal ORIGINAL law mass sum0 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct. - S001 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct. ## Result (finite_histogram, 11/4, 3/16, 6, one_finite_histogram, No, finite_histogram_does_not_prove_distributional_limit) 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: 693a2ed264ebf50bbabf342d1f474f21b638d9745fbf127524d0701b3529c676 - candidate record SHA-256: 99e99713478dcb43cf8f1072c0bd1c5877dda96cb23d612db9ab44d7687c23bd - semantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62 - question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0090.json)