# Question 27283: Complete Fixed-n Sum Moments Do Not Prove a Distributional Limit [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-0033) 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. R0053-27283 · q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0033 ## Original question and givens Use the aligned finite law values=[0, 1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use sample size n=2 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope). - G001 — Use the aligned finite law values=[0, 1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use sample size n=2 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope). ## 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-standardized-sum-moments] After derived = [[n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope]] 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 common marginal ORIGINAL law mass sum0 := sum(0, 1/2) After CLT ORIGINAL common marginal ORIGINAL law mass 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, S002 S004 · calculationBefore CLT ORIGINAL common marginal ORIGINAL law mass sum1 := sum(1/2, 1/2) After CLT ORIGINAL common 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 common marginal, [0, 1], [[1, 2], [1, 2]]] After derived = [[[0, 1/2], [1, 1/2]], 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 common marginal ORIGINAL value times probability0 := mul(0, 1/2) After CLT ORIGINAL common marginal ORIGINAL value times probability0 = 0 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 common marginal ORIGINAL ascending mean sum0 := sum(0, 0) After CLT ORIGINAL common marginal ORIGINAL ascending mean sum0 = 0 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 common marginal exact value square0 := mul(0, 0) After CLT ORIGINAL common marginal exact value square0 = 0 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 common marginal ORIGINAL square times probability0 := mul(0, 1/2) After CLT ORIGINAL common marginal ORIGINAL square times probability0 = 0 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 common marginal ORIGINAL ascending second moment sum0 := sum(0, 0) After CLT ORIGINAL common marginal ORIGINAL ascending second moment sum0 = 0 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 common marginal ORIGINAL value times probability1 := mul(1, 1/2) After CLT ORIGINAL common marginal ORIGINAL value times probability1 = 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, S010 S012 · calculationBefore CLT ORIGINAL common marginal ORIGINAL ascending mean sum1 := sum(0, 1/2) After CLT ORIGINAL common marginal ORIGINAL ascending mean sum1 = 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, S011 S013 · calculationBefore CLT ORIGINAL common marginal exact value square1 := mul(1, 1) After CLT ORIGINAL common marginal exact value square1 = 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, S012 S014 · calculationBefore CLT ORIGINAL common marginal ORIGINAL square times probability1 := mul(1, 1/2) After CLT ORIGINAL common marginal ORIGINAL square times probability1 = 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, S013 S015 · calculationBefore CLT ORIGINAL common marginal ORIGINAL ascending second moment sum1 := sum(0, 1/2) After CLT ORIGINAL common marginal ORIGINAL ascending second moment sum1 = 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, S014 S016 · calculationBefore CLT ORIGINAL common marginal ORIGINAL mean square := mul(1/2, 1/2) After CLT ORIGINAL common marginal ORIGINAL mean square = 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, S015 S017 · calculationBefore CLT ORIGINAL common marginal ORIGINAL second minus mean square := sub(1/2, 1/4) After CLT ORIGINAL common marginal ORIGINAL second minus mean square = 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, S016 S018 · impliesBefore source roles = [CLT ORIGINAL common marginal, [[0, 1/2], [1, 1/2]]] After derived = [1/2, 1/2, 1/4] 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 · calculationBefore CLT joint ORIGINAL word0 sum := sum(0, 0) After CLT joint ORIGINAL word0 sum = 0 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, S018 S020 · calculationBefore CLT joint ORIGINAL word0 probability factor0 := mul(1, 1/2) After CLT joint ORIGINAL word0 probability factor0 = 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, S019 S021 · calculationBefore CLT joint ORIGINAL word0 probability factor1 := mul(1/2, 1/2) After CLT joint ORIGINAL word0 probability factor1 = 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, S020 S022 · calculationBefore CLT joint ORIGINAL sum0 bucket word0 := sum(0, 1/4) After CLT joint ORIGINAL sum0 bucket word0 = 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, S021 S023 · impliesBefore source roles = [2, 0, [0, 0], [1/2, 1/2]] After derived = [0, 1/4, 0, 1/4] Conditions: none The declared IID premise gives each finite word the product of coordinate probabilities.Enumerate EVERY word in source order and add it to its actual sum fibre,retaining zero-mass words/buckets.Do not guess the distribution from samples or moments. Step identifiers and dependenciesRule: CLT_EVERY_joint_word_original_coordinate_products_AND_full_sum_fibre · Uses: G001, S022 S024 · calculationBefore CLT joint ORIGINAL word1 sum := sum(0, 1) After CLT joint ORIGINAL word1 sum = 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, S023 S025 · calculationBefore CLT joint ORIGINAL word1 probability factor0 := mul(1, 1/2) After CLT joint ORIGINAL word1 probability factor0 = 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, S024 S026 · calculationBefore CLT joint ORIGINAL word1 probability factor1 := mul(1/2, 1/2) After CLT joint ORIGINAL word1 probability factor1 = 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, S025 S027 · calculationBefore CLT joint ORIGINAL sum1 bucket word1 := sum(0, 1/4) After CLT joint ORIGINAL sum1 bucket word1 = 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, S026 S028 · impliesBefore source roles = [2, 1, [0, 1], [1/2, 1/2]] After derived = [1, 1/4, 0, 1/4] Conditions: none The declared IID premise gives each finite word the product of coordinate probabilities.Enumerate EVERY word in source order and add it to its actual sum fibre,retaining zero-mass words/buckets.Do not guess the distribution from samples or moments. Step identifiers and dependenciesRule: CLT_EVERY_joint_word_original_coordinate_products_AND_full_sum_fibre · Uses: G001, S027 S029 · calculationBefore CLT joint ORIGINAL word2 sum := sum(1, 0) After CLT joint ORIGINAL word2 sum = 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, S028 S030 · calculationBefore CLT joint ORIGINAL word2 probability factor0 := mul(1, 1/2) After CLT joint ORIGINAL word2 probability factor0 = 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, S029 S031 · calculationBefore CLT joint ORIGINAL word2 probability factor1 := mul(1/2, 1/2) After CLT joint ORIGINAL word2 probability factor1 = 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, S030 S032 · calculationBefore CLT joint ORIGINAL sum1 bucket word2 := sum(1/4, 1/4) After CLT joint ORIGINAL sum1 bucket word2 = 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, S031 S033 · impliesBefore source roles = [2, 2, [1, 0], [1/2, 1/2]] After derived = [1, 1/4, 1/4, 1/2] Conditions: none The declared IID premise gives each finite word the product of coordinate probabilities.Enumerate EVERY word in source order and add it to its actual sum fibre,retaining zero-mass words/buckets.Do not guess the distribution from samples or moments. Step identifiers and dependenciesRule: CLT_EVERY_joint_word_original_coordinate_products_AND_full_sum_fibre · Uses: G001, S032 S034 · calculationBefore CLT joint ORIGINAL word3 sum := sum(1, 1) After CLT joint ORIGINAL word3 sum = 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, S033 S035 · calculationBefore CLT joint ORIGINAL word3 probability factor0 := mul(1, 1/2) After CLT joint ORIGINAL word3 probability factor0 = 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, S034 S036 · calculationBefore CLT joint ORIGINAL word3 probability factor1 := mul(1/2, 1/2) After CLT joint ORIGINAL word3 probability factor1 = 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, S035 S037 · calculationBefore CLT joint ORIGINAL sum2 bucket word3 := sum(0, 1/4) After CLT joint ORIGINAL sum2 bucket word3 = 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, S036 S038 · impliesBefore source roles = [2, 3, [1, 1], [1/2, 1/2]] After derived = [2, 1/4, 0, 1/4] Conditions: none The declared IID premise gives each finite word the product of coordinate probabilities.Enumerate EVERY word in source order and add it to its actual sum fibre,retaining zero-mass words/buckets.Do not guess the distribution from samples or moments. Step identifiers and dependenciesRule: CLT_EVERY_joint_word_original_coordinate_products_AND_full_sum_fibre · Uses: G001, S037 S039 · calculationBefore CLT SUM ORIGINAL total distribution mass0 := sum(0, 1/4) After CLT SUM ORIGINAL total distribution mass0 = 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, S038 S040 · calculationBefore CLT SUM ORIGINAL total distribution mass1 := sum(1/4, 1/2) After CLT SUM ORIGINAL total distribution mass1 = 3/4 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, S039 S041 · calculationBefore CLT SUM ORIGINAL total distribution mass2 := sum(3/4, 1/4) After CLT SUM ORIGINAL total distribution mass2 = 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, S040 S042 · calculationBefore CLT SUM ORIGINAL value times probability0 := mul(0, 1/4) After CLT SUM ORIGINAL value times probability0 = 0 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, S041 S043 · calculationBefore CLT SUM ORIGINAL ascending mean sum0 := sum(0, 0) After CLT SUM ORIGINAL ascending mean sum0 = 0 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, S042 S044 · calculationBefore CLT SUM exact value square0 := mul(0, 0) After CLT SUM exact value square0 = 0 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, S043 S045 · calculationBefore CLT SUM ORIGINAL square times probability0 := mul(0, 1/4) After CLT SUM ORIGINAL square times probability0 = 0 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, S044 S046 · calculationBefore CLT SUM ORIGINAL ascending second moment sum0 := sum(0, 0) After CLT SUM ORIGINAL ascending second moment sum0 = 0 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, S045 S047 · calculationBefore CLT SUM ORIGINAL value times probability1 := mul(1, 1/2) After CLT SUM ORIGINAL value times probability1 = 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, S046 S048 · calculationBefore CLT SUM ORIGINAL ascending mean sum1 := sum(0, 1/2) After CLT SUM ORIGINAL ascending mean sum1 = 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, S047 S049 · calculationBefore CLT SUM exact value square1 := mul(1, 1) After CLT SUM exact value square1 = 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, S048 S050 · calculationBefore CLT SUM ORIGINAL square times probability1 := mul(1, 1/2) After CLT SUM ORIGINAL square times probability1 = 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, S049 S051 · calculationBefore CLT SUM ORIGINAL ascending second moment sum1 := sum(0, 1/2) After CLT SUM ORIGINAL ascending second moment sum1 = 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, S050 S052 · calculationBefore CLT SUM ORIGINAL value times probability2 := mul(2, 1/4) After CLT SUM ORIGINAL value times probability2 = 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, S051 S053 · calculationBefore CLT SUM ORIGINAL ascending mean sum2 := sum(1/2, 1/2) After CLT SUM ORIGINAL ascending mean sum2 = 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, S052 S054 · calculationBefore CLT SUM exact value square2 := mul(2, 2) After CLT SUM exact value square2 = 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, S053 S055 · calculationBefore CLT SUM ORIGINAL square times probability2 := mul(4, 1/4) After CLT SUM ORIGINAL square times probability2 = 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, S054 S056 · calculationBefore CLT SUM ORIGINAL ascending second moment sum2 := sum(1/2, 1) After CLT SUM ORIGINAL ascending second moment sum2 = 3/2 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, S055 S057 · calculationBefore CLT SUM ORIGINAL mean square := mul(1, 1) After CLT SUM ORIGINAL mean square = 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, S056 S058 · calculationBefore CLT SUM ORIGINAL second minus mean square := sub(3/2, 1) After CLT SUM ORIGINAL second minus mean square = 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, S057 S059 · calculationBefore CLT ORIGINAL n times common mean := mul(2, 1/2) After CLT ORIGINAL n times common mean = 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, S058 S060 · calculationBefore CLT ORIGINAL n times common variance := mul(2, 1/4) After CLT ORIGINAL n times common variance = 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, S059 S061 · impliesBefore source roles = [2, 1/2, 1/4, [(0, 1/4), (1, 1/2), (2, 1/4)], 1, 1/2] After derived = [1, 1, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n))] Conditions: none sigma²>0 and n>0 make the denominator nonzero.E[Z]=(nmu-nmu)/(sigma sqrt(n))=0 and Var(Z)=n sigma²/(sigma² n)=1.These exact moments belong to this complete FIXED-n law;they do not make it normal or prove the general CLT. Step identifiers and dependenciesRule: CLT_standardization_EXACT_0_1_moments_NOT_normality_or_limit · Uses: G001, S060 ## Reverse checks - S061 · 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. - S060 · premise_and_result_audit — Recompute CLT ORIGINAL n times common variance with exact rational arithmetic, not rounded decimals. - S059 · premise_and_result_audit — Recompute CLT ORIGINAL n times common mean with exact rational arithmetic, not rounded decimals. - S058 · premise_and_result_audit — Recompute CLT SUM ORIGINAL second minus mean square with exact rational arithmetic, not rounded decimals. - S057 · premise_and_result_audit — Recompute CLT SUM ORIGINAL mean square with exact rational arithmetic, not rounded decimals. - S056 · premise_and_result_audit — Recompute CLT SUM ORIGINAL ascending second moment sum2 with exact rational arithmetic, not rounded decimals. - S055 · premise_and_result_audit — Recompute CLT SUM ORIGINAL square times probability2 with exact rational arithmetic, not rounded decimals. - S054 · premise_and_result_audit — Recompute CLT SUM exact value square2 with exact rational arithmetic, not rounded decimals. - S053 · premise_and_result_audit — Recompute CLT SUM ORIGINAL ascending mean sum2 with exact rational arithmetic, not rounded decimals. - S052 · premise_and_result_audit — Recompute CLT SUM ORIGINAL value times probability2 with exact rational arithmetic, not rounded decimals. - S051 · premise_and_result_audit — Recompute CLT SUM ORIGINAL ascending second moment sum1 with exact rational arithmetic, not rounded decimals. - S050 · premise_and_result_audit — Recompute CLT SUM ORIGINAL square times probability1 with exact rational arithmetic, not rounded decimals. - S049 · premise_and_result_audit — Recompute CLT SUM exact value square1 with exact rational arithmetic, not rounded decimals. - S048 · premise_and_result_audit — Recompute CLT SUM ORIGINAL ascending mean sum1 with exact rational arithmetic, not rounded decimals. - S047 · premise_and_result_audit — Recompute CLT SUM ORIGINAL value times probability1 with exact rational arithmetic, not rounded decimals. - S046 · premise_and_result_audit — Recompute CLT SUM ORIGINAL ascending second moment sum0 with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Recompute CLT SUM ORIGINAL square times probability0 with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Recompute CLT SUM exact value square0 with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Recompute CLT SUM ORIGINAL ascending mean sum0 with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute CLT SUM ORIGINAL value times probability0 with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute CLT SUM ORIGINAL total distribution mass2 with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute CLT SUM ORIGINAL total distribution mass1 with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute CLT SUM ORIGINAL total distribution mass0 with exact rational arithmetic, not rounded decimals. - S038 · 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. - S037 · premise_and_result_audit — Recompute CLT joint ORIGINAL sum2 bucket word3 with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute CLT joint ORIGINAL word3 probability factor1 with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute CLT joint ORIGINAL word3 probability factor0 with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute CLT joint ORIGINAL word3 sum with exact rational arithmetic, not rounded decimals. - S033 · 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. - S032 · premise_and_result_audit — Recompute CLT joint ORIGINAL sum1 bucket word2 with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute CLT joint ORIGINAL word2 probability factor1 with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute CLT joint ORIGINAL word2 probability factor0 with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute CLT joint ORIGINAL word2 sum with exact rational arithmetic, not rounded decimals. - S028 · 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. - S027 · premise_and_result_audit — Recompute CLT joint ORIGINAL sum1 bucket word1 with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute CLT joint ORIGINAL word1 probability factor1 with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute CLT joint ORIGINAL word1 probability factor0 with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute CLT joint ORIGINAL word1 sum with exact rational arithmetic, not rounded decimals. - S023 · 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. - S022 · premise_and_result_audit — Recompute CLT joint ORIGINAL sum0 bucket word0 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute CLT joint ORIGINAL word0 probability factor1 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute CLT joint ORIGINAL word0 probability factor0 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute CLT joint ORIGINAL word0 sum with exact rational arithmetic, not rounded decimals. - 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 common marginal ORIGINAL second minus mean square with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute CLT ORIGINAL common marginal ORIGINAL mean square with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute CLT ORIGINAL common marginal ORIGINAL ascending second moment sum1 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute CLT ORIGINAL common marginal ORIGINAL square times probability1 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute CLT ORIGINAL common marginal exact value square1 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute CLT ORIGINAL common marginal ORIGINAL ascending mean sum1 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute CLT ORIGINAL common marginal ORIGINAL value times probability1 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute CLT ORIGINAL common marginal ORIGINAL ascending second moment sum0 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute CLT ORIGINAL common marginal ORIGINAL square times probability0 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute CLT ORIGINAL common marginal exact value square0 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute CLT ORIGINAL common marginal ORIGINAL ascending mean sum0 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute CLT ORIGINAL common 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 common marginal ORIGINAL law mass sum1 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute CLT ORIGINAL common 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 (2, [[0, 1/4], [1, 1/2], [2, 1/4]], 1, 1, 1/2, 1, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence) 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: 8adecfe3df466f59ab307b7c7bc764b89851d605aa8537a9069b09844c7c3919 - candidate record SHA-256: 5d5d27afb22e951158793692a501a279a312cbaa3f04d7f15ceef001b36b1de6 - 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-0033.json)