# Question 27295: 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-0045) 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. R0053-27295 · q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0045 ## Original question and givens Use the aligned finite law values=[2, 3], 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=4 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=[2, 3], 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=4 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, [2, 3], [[1, 2], [1, 2]]] After derived = [[[2, 1/2], [3, 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(2, 1/2) After CLT ORIGINAL common marginal ORIGINAL value 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, S005 S007 · calculationBefore CLT ORIGINAL common marginal ORIGINAL ascending mean sum0 := sum(0, 1) After CLT ORIGINAL common marginal ORIGINAL ascending mean 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, S006 S008 · calculationBefore CLT ORIGINAL common marginal exact value square0 := mul(2, 2) After CLT ORIGINAL common 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 common marginal ORIGINAL square times probability0 := mul(4, 1/2) After CLT ORIGINAL common marginal ORIGINAL square times probability0 = 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, S008 S010 · calculationBefore CLT ORIGINAL common marginal ORIGINAL ascending second moment sum0 := sum(0, 2) After CLT ORIGINAL common marginal ORIGINAL ascending second moment sum0 = 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, S009 S011 · calculationBefore CLT ORIGINAL common marginal ORIGINAL value times probability1 := mul(3, 1/2) After CLT ORIGINAL common marginal ORIGINAL value times probability1 = 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, S010 S012 · calculationBefore CLT ORIGINAL common marginal ORIGINAL ascending mean sum1 := sum(1, 3/2) After CLT ORIGINAL common marginal ORIGINAL ascending mean sum1 = 5/2 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(3, 3) After CLT ORIGINAL common 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 common marginal ORIGINAL square times probability1 := mul(9, 1/2) After CLT ORIGINAL common marginal ORIGINAL square times probability1 = 9/2 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(2, 9/2) After CLT ORIGINAL common marginal ORIGINAL ascending second moment sum1 = 13/2 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(5/2, 5/2) After CLT ORIGINAL common marginal ORIGINAL mean square = 25/4 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(13/2, 25/4) 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, [[2, 1/2], [3, 1/2]]] After derived = [5/2, 13/2, 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(2, 2, 2, 2) After CLT joint ORIGINAL word0 sum = 8 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 word0 probability factor2 := mul(1/4, 1/2) After CLT joint ORIGINAL word0 probability factor2 = 1/8 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 · calculationBefore CLT joint ORIGINAL word0 probability factor3 := mul(1/8, 1/2) After CLT joint ORIGINAL word0 probability factor3 = 1/16 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, S022 S024 · calculationBefore CLT joint ORIGINAL sum8 bucket word0 := sum(0, 1/16) After CLT joint ORIGINAL sum8 bucket word0 = 1/16 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, S023 S025 · impliesBefore source roles = [4, 0, [2, 2, 2, 2], [1/2, 1/2, 1/2, 1/2]] After derived = [8, 1/16, 0, 1/16] 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, S024 S026 · calculationBefore CLT joint ORIGINAL word1 sum := sum(2, 2, 2, 3) After CLT joint ORIGINAL word1 sum = 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, S025 S027 · 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, S026 S028 · 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, S027 S029 · calculationBefore CLT joint ORIGINAL word1 probability factor2 := mul(1/4, 1/2) After CLT joint ORIGINAL word1 probability factor2 = 1/8 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 word1 probability factor3 := mul(1/8, 1/2) After CLT joint ORIGINAL word1 probability factor3 = 1/16 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, S029 S031 · calculationBefore CLT joint ORIGINAL sum9 bucket word1 := sum(0, 1/16) After CLT joint ORIGINAL sum9 bucket word1 = 1/16 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, S030 S032 · impliesBefore source roles = [4, 1, [2, 2, 2, 3], [1/2, 1/2, 1/2, 1/2]] After derived = [9, 1/16, 0, 1/16] 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, S031 S033 · calculationBefore CLT joint ORIGINAL word2 sum := sum(2, 2, 3, 2) After CLT joint ORIGINAL word2 sum = 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, S032 S034 · 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, S033 S035 · 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, S034 S036 · calculationBefore CLT joint ORIGINAL word2 probability factor2 := mul(1/4, 1/2) After CLT joint ORIGINAL word2 probability factor2 = 1/8 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 word2 probability factor3 := mul(1/8, 1/2) After CLT joint ORIGINAL word2 probability factor3 = 1/16 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, S036 S038 · calculationBefore CLT joint ORIGINAL sum9 bucket word2 := sum(1/16, 1/16) After CLT joint ORIGINAL sum9 bucket word2 = 1/8 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, S037 S039 · impliesBefore source roles = [4, 2, [2, 2, 3, 2], [1/2, 1/2, 1/2, 1/2]] After derived = [9, 1/16, 1/16, 1/8] 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, S038 S040 · calculationBefore CLT joint ORIGINAL word3 sum := sum(2, 2, 3, 3) After CLT joint ORIGINAL word3 sum = 10 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 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, S040 S042 · 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, S041 S043 · calculationBefore CLT joint ORIGINAL word3 probability factor2 := mul(1/4, 1/2) After CLT joint ORIGINAL word3 probability factor2 = 1/8 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 joint ORIGINAL word3 probability factor3 := mul(1/8, 1/2) After CLT joint ORIGINAL word3 probability factor3 = 1/16 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, S043 S045 · calculationBefore CLT joint ORIGINAL sum10 bucket word3 := sum(0, 1/16) After CLT joint ORIGINAL sum10 bucket word3 = 1/16 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, S044 S046 · impliesBefore source roles = [4, 3, [2, 2, 3, 3], [1/2, 1/2, 1/2, 1/2]] After derived = [10, 1/16, 0, 1/16] 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, S045 S047 · calculationBefore CLT joint ORIGINAL word4 sum := sum(2, 3, 2, 2) After CLT joint ORIGINAL word4 sum = 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, S046 S048 · calculationBefore CLT joint ORIGINAL word4 probability factor0 := mul(1, 1/2) After CLT joint ORIGINAL word4 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, S047 S049 · calculationBefore CLT joint ORIGINAL word4 probability factor1 := mul(1/2, 1/2) After CLT joint ORIGINAL word4 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, S048 S050 · calculationBefore CLT joint ORIGINAL word4 probability factor2 := mul(1/4, 1/2) After CLT joint ORIGINAL word4 probability factor2 = 1/8 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 joint ORIGINAL word4 probability factor3 := mul(1/8, 1/2) After CLT joint ORIGINAL word4 probability factor3 = 1/16 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, S050 S052 · calculationBefore CLT joint ORIGINAL sum9 bucket word4 := sum(1/8, 1/16) After CLT joint ORIGINAL sum9 bucket word4 = 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, S051 S053 · impliesBefore source roles = [4, 4, [2, 3, 2, 2], [1/2, 1/2, 1/2, 1/2]] After derived = [9, 1/16, 1/8, 3/16] 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, S052 S054 · calculationBefore CLT joint ORIGINAL word5 sum := sum(2, 3, 2, 3) After CLT joint ORIGINAL word5 sum = 10 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 joint ORIGINAL word5 probability factor0 := mul(1, 1/2) After CLT joint ORIGINAL word5 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, S054 S056 · calculationBefore CLT joint ORIGINAL word5 probability factor1 := mul(1/2, 1/2) After CLT joint ORIGINAL word5 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, S055 S057 · calculationBefore CLT joint ORIGINAL word5 probability factor2 := mul(1/4, 1/2) After CLT joint ORIGINAL word5 probability factor2 = 1/8 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 joint ORIGINAL word5 probability factor3 := mul(1/8, 1/2) After CLT joint ORIGINAL word5 probability factor3 = 1/16 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, S057 S059 · calculationBefore CLT joint ORIGINAL sum10 bucket word5 := sum(1/16, 1/16) After CLT joint ORIGINAL sum10 bucket word5 = 1/8 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 · impliesBefore source roles = [4, 5, [2, 3, 2, 3], [1/2, 1/2, 1/2, 1/2]] After derived = [10, 1/16, 1/16, 1/8] 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, S059 S061 · calculationBefore CLT joint ORIGINAL word6 sum := sum(2, 3, 3, 2) After CLT joint ORIGINAL word6 sum = 10 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, S060 S062 · calculationBefore CLT joint ORIGINAL word6 probability factor0 := mul(1, 1/2) After CLT joint ORIGINAL word6 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, S061 S063 · calculationBefore CLT joint ORIGINAL word6 probability factor1 := mul(1/2, 1/2) After CLT joint ORIGINAL word6 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, S062 S064 · calculationBefore CLT joint ORIGINAL word6 probability factor2 := mul(1/4, 1/2) After CLT joint ORIGINAL word6 probability factor2 = 1/8 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, S063 S065 · calculationBefore CLT joint ORIGINAL word6 probability factor3 := mul(1/8, 1/2) After CLT joint ORIGINAL word6 probability factor3 = 1/16 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, S064 S066 · calculationBefore CLT joint ORIGINAL sum10 bucket word6 := sum(1/8, 1/16) After CLT joint ORIGINAL sum10 bucket word6 = 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, S065 S067 · impliesBefore source roles = [4, 6, [2, 3, 3, 2], [1/2, 1/2, 1/2, 1/2]] After derived = [10, 1/16, 1/8, 3/16] 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, S066 S068 · calculationBefore CLT joint ORIGINAL word7 sum := sum(2, 3, 3, 3) After CLT joint ORIGINAL word7 sum = 11 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, S067 S069 · calculationBefore CLT joint ORIGINAL word7 probability factor0 := mul(1, 1/2) After CLT joint ORIGINAL word7 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, S068 S070 · calculationBefore CLT joint ORIGINAL word7 probability factor1 := mul(1/2, 1/2) After CLT joint ORIGINAL word7 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, S069 S071 · calculationBefore CLT joint ORIGINAL word7 probability factor2 := mul(1/4, 1/2) After CLT joint ORIGINAL word7 probability factor2 = 1/8 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, S070 S072 · calculationBefore CLT joint ORIGINAL word7 probability factor3 := mul(1/8, 1/2) After CLT joint ORIGINAL word7 probability factor3 = 1/16 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, S071 S073 · calculationBefore CLT joint ORIGINAL sum11 bucket word7 := sum(0, 1/16) After CLT joint ORIGINAL sum11 bucket word7 = 1/16 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, S072 S074 · impliesBefore source roles = [4, 7, [2, 3, 3, 3], [1/2, 1/2, 1/2, 1/2]] After derived = [11, 1/16, 0, 1/16] 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, S073 S075 · calculationBefore CLT joint ORIGINAL word8 sum := sum(3, 2, 2, 2) After CLT joint ORIGINAL word8 sum = 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, S074 S076 · calculationBefore CLT joint ORIGINAL word8 probability factor0 := mul(1, 1/2) After CLT joint ORIGINAL word8 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, S075 S077 · calculationBefore CLT joint ORIGINAL word8 probability factor1 := mul(1/2, 1/2) After CLT joint ORIGINAL word8 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, S076 S078 · calculationBefore CLT joint ORIGINAL word8 probability factor2 := mul(1/4, 1/2) After CLT joint ORIGINAL word8 probability factor2 = 1/8 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, S077 S079 · calculationBefore CLT joint ORIGINAL word8 probability factor3 := mul(1/8, 1/2) After CLT joint ORIGINAL word8 probability factor3 = 1/16 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, S078 S080 · calculationBefore CLT joint ORIGINAL sum9 bucket word8 := sum(3/16, 1/16) After CLT joint ORIGINAL sum9 bucket word8 = 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, S079 S081 · impliesBefore source roles = [4, 8, [3, 2, 2, 2], [1/2, 1/2, 1/2, 1/2]] After derived = [9, 1/16, 3/16, 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, S080 S082 · calculationBefore CLT joint ORIGINAL word9 sum := sum(3, 2, 2, 3) After CLT joint ORIGINAL word9 sum = 10 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, S081 S083 · calculationBefore CLT joint ORIGINAL word9 probability factor0 := mul(1, 1/2) After CLT joint ORIGINAL word9 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, S082 S084 · calculationBefore CLT joint ORIGINAL word9 probability factor1 := mul(1/2, 1/2) After CLT joint ORIGINAL word9 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, S083 S085 · calculationBefore CLT joint ORIGINAL word9 probability factor2 := mul(1/4, 1/2) After CLT joint ORIGINAL word9 probability factor2 = 1/8 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, S084 S086 · calculationBefore CLT joint ORIGINAL word9 probability factor3 := mul(1/8, 1/2) After CLT joint ORIGINAL word9 probability factor3 = 1/16 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, S085 S087 · calculationBefore CLT joint ORIGINAL sum10 bucket word9 := sum(3/16, 1/16) After CLT joint ORIGINAL sum10 bucket word9 = 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, S086 S088 · impliesBefore source roles = [4, 9, [3, 2, 2, 3], [1/2, 1/2, 1/2, 1/2]] After derived = [10, 1/16, 3/16, 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, S087 S089 · calculationBefore CLT joint ORIGINAL word10 sum := sum(3, 2, 3, 2) After CLT joint ORIGINAL word10 sum = 10 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, S088 S090 · calculationBefore CLT joint ORIGINAL word10 probability factor0 := mul(1, 1/2) After CLT joint ORIGINAL word10 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, S089 S091 · calculationBefore CLT joint ORIGINAL word10 probability factor1 := mul(1/2, 1/2) After CLT joint ORIGINAL word10 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, S090 S092 · calculationBefore CLT joint ORIGINAL word10 probability factor2 := mul(1/4, 1/2) After CLT joint ORIGINAL word10 probability factor2 = 1/8 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, S091 S093 · calculationBefore CLT joint ORIGINAL word10 probability factor3 := mul(1/8, 1/2) After CLT joint ORIGINAL word10 probability factor3 = 1/16 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, S092 S094 · calculationBefore CLT joint ORIGINAL sum10 bucket word10 := sum(1/4, 1/16) After CLT joint ORIGINAL sum10 bucket word10 = 5/16 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, S093 S095 · impliesBefore source roles = [4, 10, [3, 2, 3, 2], [1/2, 1/2, 1/2, 1/2]] After derived = [10, 1/16, 1/4, 5/16] 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, S094 S096 · calculationBefore CLT joint ORIGINAL word11 sum := sum(3, 2, 3, 3) After CLT joint ORIGINAL word11 sum = 11 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, S095 S097 · calculationBefore CLT joint ORIGINAL word11 probability factor0 := mul(1, 1/2) After CLT joint ORIGINAL word11 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, S096 S098 · calculationBefore CLT joint ORIGINAL word11 probability factor1 := mul(1/2, 1/2) After CLT joint ORIGINAL word11 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, S097 S099 · calculationBefore CLT joint ORIGINAL word11 probability factor2 := mul(1/4, 1/2) After CLT joint ORIGINAL word11 probability factor2 = 1/8 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, S098 S100 · calculationBefore CLT joint ORIGINAL word11 probability factor3 := mul(1/8, 1/2) After CLT joint ORIGINAL word11 probability factor3 = 1/16 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, S099 S101 · calculationBefore CLT joint ORIGINAL sum11 bucket word11 := sum(1/16, 1/16) After CLT joint ORIGINAL sum11 bucket word11 = 1/8 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, S100 S102 · impliesBefore source roles = [4, 11, [3, 2, 3, 3], [1/2, 1/2, 1/2, 1/2]] After derived = [11, 1/16, 1/16, 1/8] 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, S101 S103 · calculationBefore CLT joint ORIGINAL word12 sum := sum(3, 3, 2, 2) After CLT joint ORIGINAL word12 sum = 10 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, S102 S104 · calculationBefore CLT joint ORIGINAL word12 probability factor0 := mul(1, 1/2) After CLT joint ORIGINAL word12 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, S103 S105 · calculationBefore CLT joint ORIGINAL word12 probability factor1 := mul(1/2, 1/2) After CLT joint ORIGINAL word12 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, S104 S106 · calculationBefore CLT joint ORIGINAL word12 probability factor2 := mul(1/4, 1/2) After CLT joint ORIGINAL word12 probability factor2 = 1/8 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, S105 S107 · calculationBefore CLT joint ORIGINAL word12 probability factor3 := mul(1/8, 1/2) After CLT joint ORIGINAL word12 probability factor3 = 1/16 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, S106 S108 · calculationBefore CLT joint ORIGINAL sum10 bucket word12 := sum(5/16, 1/16) After CLT joint ORIGINAL sum10 bucket word12 = 3/8 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, S107 S109 · impliesBefore source roles = [4, 12, [3, 3, 2, 2], [1/2, 1/2, 1/2, 1/2]] After derived = [10, 1/16, 5/16, 3/8] 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, S108 S110 · calculationBefore CLT joint ORIGINAL word13 sum := sum(3, 3, 2, 3) After CLT joint ORIGINAL word13 sum = 11 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, S109 S111 · calculationBefore CLT joint ORIGINAL word13 probability factor0 := mul(1, 1/2) After CLT joint ORIGINAL word13 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, S110 S112 · calculationBefore CLT joint ORIGINAL word13 probability factor1 := mul(1/2, 1/2) After CLT joint ORIGINAL word13 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, S111 S113 · calculationBefore CLT joint ORIGINAL word13 probability factor2 := mul(1/4, 1/2) After CLT joint ORIGINAL word13 probability factor2 = 1/8 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, S112 S114 · calculationBefore CLT joint ORIGINAL word13 probability factor3 := mul(1/8, 1/2) After CLT joint ORIGINAL word13 probability factor3 = 1/16 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, S113 S115 · calculationBefore CLT joint ORIGINAL sum11 bucket word13 := sum(1/8, 1/16) After CLT joint ORIGINAL sum11 bucket word13 = 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, S114 S116 · impliesBefore source roles = [4, 13, [3, 3, 2, 3], [1/2, 1/2, 1/2, 1/2]] After derived = [11, 1/16, 1/8, 3/16] 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, S115 S117 · calculationBefore CLT joint ORIGINAL word14 sum := sum(3, 3, 3, 2) After CLT joint ORIGINAL word14 sum = 11 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, S116 S118 · calculationBefore CLT joint ORIGINAL word14 probability factor0 := mul(1, 1/2) After CLT joint ORIGINAL word14 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, S117 S119 · calculationBefore CLT joint ORIGINAL word14 probability factor1 := mul(1/2, 1/2) After CLT joint ORIGINAL word14 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, S118 S120 · calculationBefore CLT joint ORIGINAL word14 probability factor2 := mul(1/4, 1/2) After CLT joint ORIGINAL word14 probability factor2 = 1/8 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, S119 S121 · calculationBefore CLT joint ORIGINAL word14 probability factor3 := mul(1/8, 1/2) After CLT joint ORIGINAL word14 probability factor3 = 1/16 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, S120 S122 · calculationBefore CLT joint ORIGINAL sum11 bucket word14 := sum(3/16, 1/16) After CLT joint ORIGINAL sum11 bucket word14 = 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, S121 S123 · impliesBefore source roles = [4, 14, [3, 3, 3, 2], [1/2, 1/2, 1/2, 1/2]] After derived = [11, 1/16, 3/16, 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, S122 S124 · calculationBefore CLT joint ORIGINAL word15 sum := sum(3, 3, 3, 3) After CLT joint ORIGINAL word15 sum = 12 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, S123 S125 · calculationBefore CLT joint ORIGINAL word15 probability factor0 := mul(1, 1/2) After CLT joint ORIGINAL word15 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, S124 S126 · calculationBefore CLT joint ORIGINAL word15 probability factor1 := mul(1/2, 1/2) After CLT joint ORIGINAL word15 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, S125 S127 · calculationBefore CLT joint ORIGINAL word15 probability factor2 := mul(1/4, 1/2) After CLT joint ORIGINAL word15 probability factor2 = 1/8 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, S126 S128 · calculationBefore CLT joint ORIGINAL word15 probability factor3 := mul(1/8, 1/2) After CLT joint ORIGINAL word15 probability factor3 = 1/16 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, S127 S129 · calculationBefore CLT joint ORIGINAL sum12 bucket word15 := sum(0, 1/16) After CLT joint ORIGINAL sum12 bucket word15 = 1/16 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, S128 S130 · impliesBefore source roles = [4, 15, [3, 3, 3, 3], [1/2, 1/2, 1/2, 1/2]] After derived = [12, 1/16, 0, 1/16] 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, S129 S131 · calculationBefore CLT SUM ORIGINAL total distribution mass0 := sum(0, 1/16) After CLT SUM ORIGINAL total distribution mass0 = 1/16 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, S130 S132 · calculationBefore CLT SUM ORIGINAL total distribution mass1 := sum(1/16, 1/4) After CLT SUM ORIGINAL total distribution mass1 = 5/16 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, S131 S133 · calculationBefore CLT SUM ORIGINAL total distribution mass2 := sum(5/16, 3/8) After CLT SUM ORIGINAL total distribution mass2 = 11/16 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, S132 S134 · calculationBefore CLT SUM ORIGINAL total distribution mass3 := sum(11/16, 1/4) After CLT SUM ORIGINAL total distribution mass3 = 15/16 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, S133 S135 · calculationBefore CLT SUM ORIGINAL total distribution mass4 := sum(15/16, 1/16) After CLT SUM ORIGINAL total distribution mass4 = 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, S134 S136 · calculationBefore CLT SUM ORIGINAL value times probability0 := mul(8, 1/16) After CLT SUM 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, S135 S137 · calculationBefore CLT SUM ORIGINAL ascending mean sum0 := sum(0, 1/2) After CLT SUM 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, S136 S138 · calculationBefore CLT SUM exact value square0 := mul(8, 8) After CLT SUM exact value square0 = 64 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, S137 S139 · calculationBefore CLT SUM ORIGINAL square times probability0 := mul(64, 1/16) After CLT SUM ORIGINAL square times probability0 = 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, S138 S140 · calculationBefore CLT SUM ORIGINAL ascending second moment sum0 := sum(0, 4) After CLT SUM ORIGINAL ascending second moment sum0 = 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, S139 S141 · calculationBefore CLT SUM ORIGINAL value times probability1 := mul(9, 1/4) After CLT SUM 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, S140 S142 · calculationBefore CLT SUM ORIGINAL ascending mean sum1 := sum(1/2, 9/4) After CLT SUM 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, S141 S143 · calculationBefore CLT SUM exact value square1 := mul(9, 9) After CLT SUM exact value square1 = 81 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, S142 S144 · calculationBefore CLT SUM ORIGINAL square times probability1 := mul(81, 1/4) After CLT SUM ORIGINAL square times probability1 = 81/4 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, S143 S145 · calculationBefore CLT SUM ORIGINAL ascending second moment sum1 := sum(4, 81/4) After CLT SUM ORIGINAL ascending second moment sum1 = 97/4 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, S144 S146 · calculationBefore CLT SUM ORIGINAL value times probability2 := mul(10, 3/8) After CLT SUM ORIGINAL value times probability2 = 15/4 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, S145 S147 · calculationBefore CLT SUM ORIGINAL ascending mean sum2 := sum(11/4, 15/4) After CLT SUM ORIGINAL ascending mean sum2 = 13/2 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, S146 S148 · calculationBefore CLT SUM exact value square2 := mul(10, 10) After CLT SUM exact value square2 = 100 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, S147 S149 · calculationBefore CLT SUM ORIGINAL square times probability2 := mul(100, 3/8) After CLT SUM ORIGINAL square times probability2 = 75/2 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, S148 S150 · calculationBefore CLT SUM ORIGINAL ascending second moment sum2 := sum(97/4, 75/2) After CLT SUM ORIGINAL ascending second moment sum2 = 247/4 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, S149 S151 · calculationBefore CLT SUM ORIGINAL value times probability3 := mul(11, 1/4) After CLT SUM ORIGINAL value times probability3 = 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, S150 S152 · calculationBefore CLT SUM ORIGINAL ascending mean sum3 := sum(13/2, 11/4) After CLT SUM ORIGINAL ascending mean sum3 = 37/4 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, S151 S153 · calculationBefore CLT SUM exact value square3 := mul(11, 11) After CLT SUM exact value square3 = 121 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, S152 S154 · calculationBefore CLT SUM ORIGINAL square times probability3 := mul(121, 1/4) After CLT SUM ORIGINAL square times probability3 = 121/4 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, S153 S155 · calculationBefore CLT SUM ORIGINAL ascending second moment sum3 := sum(247/4, 121/4) After CLT SUM ORIGINAL ascending second moment sum3 = 92 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, S154 S156 · calculationBefore CLT SUM ORIGINAL value times probability4 := mul(12, 1/16) After CLT SUM ORIGINAL value times probability4 = 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, S155 S157 · calculationBefore CLT SUM ORIGINAL ascending mean sum4 := sum(37/4, 3/4) After CLT SUM ORIGINAL ascending mean sum4 = 10 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, S156 S158 · calculationBefore CLT SUM exact value square4 := mul(12, 12) After CLT SUM exact value square4 = 144 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, S157 S159 · calculationBefore CLT SUM ORIGINAL square times probability4 := mul(144, 1/16) After CLT SUM ORIGINAL square times probability4 = 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, S158 S160 · calculationBefore CLT SUM ORIGINAL ascending second moment sum4 := sum(92, 9) After CLT SUM ORIGINAL ascending second moment sum4 = 101 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, S159 S161 · calculationBefore CLT SUM ORIGINAL mean square := mul(10, 10) After CLT SUM ORIGINAL mean square = 100 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, S160 S162 · calculationBefore CLT SUM ORIGINAL second minus mean square := sub(101, 100) After CLT SUM ORIGINAL second minus 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, S161 S163 · calculationBefore CLT ORIGINAL n times common mean := mul(4, 5/2) After CLT ORIGINAL n times common mean = 10 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, S162 S164 · calculationBefore CLT ORIGINAL n times common variance := mul(4, 1/4) After CLT ORIGINAL n times common variance = 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, S163 S165 · impliesBefore source roles = [4, 5/2, 1/4, [(8, 1/16), (9, 1/4), (10, 3/8), (11, 1/4), (12, 1/16)], 10, 1] After derived = [1, 10, 1, 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, S164 ## Reverse checks - S165 · 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. - S164 · premise_and_result_audit — Recompute CLT ORIGINAL n times common variance with exact rational arithmetic, not rounded decimals. - S163 · premise_and_result_audit — Recompute CLT ORIGINAL n times common mean with exact rational arithmetic, not rounded decimals. - S162 · premise_and_result_audit — Recompute CLT SUM ORIGINAL second minus mean square with exact rational arithmetic, not rounded decimals. - S161 · premise_and_result_audit — Recompute CLT SUM ORIGINAL mean square with exact rational arithmetic, not rounded decimals. - S160 · premise_and_result_audit — Recompute CLT SUM ORIGINAL ascending second moment sum4 with exact rational arithmetic, not rounded decimals. - S159 · premise_and_result_audit — Recompute CLT SUM ORIGINAL square times probability4 with exact rational arithmetic, not rounded decimals. - S158 · premise_and_result_audit — Recompute CLT SUM exact value square4 with exact rational arithmetic, not rounded decimals. - S157 · premise_and_result_audit — Recompute CLT SUM ORIGINAL ascending mean sum4 with exact rational arithmetic, not rounded decimals. - S156 · premise_and_result_audit — Recompute CLT SUM ORIGINAL value times probability4 with exact rational arithmetic, not rounded decimals. - S155 · premise_and_result_audit — Recompute CLT SUM ORIGINAL ascending second moment sum3 with exact rational arithmetic, not rounded decimals. - S154 · premise_and_result_audit — Recompute CLT SUM ORIGINAL square times probability3 with exact rational arithmetic, not rounded decimals. - S153 · premise_and_result_audit — Recompute CLT SUM exact value square3 with exact rational arithmetic, not rounded decimals. - S152 · premise_and_result_audit — Recompute CLT SUM ORIGINAL ascending mean sum3 with exact rational arithmetic, not rounded decimals. - S151 · premise_and_result_audit — Recompute CLT SUM ORIGINAL value times probability3 with exact rational arithmetic, not rounded decimals. - S150 · premise_and_result_audit — Recompute CLT SUM ORIGINAL ascending second moment sum2 with exact rational arithmetic, not rounded decimals. - S149 · premise_and_result_audit — Recompute CLT SUM ORIGINAL square times probability2 with exact rational arithmetic, not rounded decimals. - S148 · premise_and_result_audit — Recompute CLT SUM exact value square2 with exact rational arithmetic, not rounded decimals. - S147 · premise_and_result_audit — Recompute CLT SUM ORIGINAL ascending mean sum2 with exact rational arithmetic, not rounded decimals. - S146 · premise_and_result_audit — Recompute CLT SUM ORIGINAL value times probability2 with exact rational arithmetic, not rounded decimals. - S145 · premise_and_result_audit — Recompute CLT SUM ORIGINAL ascending second moment sum1 with exact rational arithmetic, not rounded decimals. - S144 · premise_and_result_audit — Recompute CLT SUM ORIGINAL square times probability1 with exact rational arithmetic, not rounded decimals. - S143 · premise_and_result_audit — Recompute CLT SUM exact value square1 with exact rational arithmetic, not rounded decimals. - S142 · premise_and_result_audit — Recompute CLT SUM ORIGINAL ascending mean sum1 with exact rational arithmetic, not rounded decimals. - S141 · premise_and_result_audit — Recompute CLT SUM ORIGINAL value times probability1 with exact rational arithmetic, not rounded decimals. - S140 · premise_and_result_audit — Recompute CLT SUM ORIGINAL ascending second moment sum0 with exact rational arithmetic, not rounded decimals. - S139 · premise_and_result_audit — Recompute CLT SUM ORIGINAL square times probability0 with exact rational arithmetic, not rounded decimals. - S138 · premise_and_result_audit — Recompute CLT SUM exact value square0 with exact rational arithmetic, not rounded decimals. - S137 · premise_and_result_audit — Recompute CLT SUM ORIGINAL ascending mean sum0 with exact rational arithmetic, not rounded decimals. - S136 · premise_and_result_audit — Recompute CLT SUM ORIGINAL value times probability0 with exact rational arithmetic, not rounded decimals. - S135 · premise_and_result_audit — Recompute CLT SUM ORIGINAL total distribution mass4 with exact rational arithmetic, not rounded decimals. - S134 · premise_and_result_audit — Recompute CLT SUM ORIGINAL total distribution mass3 with exact rational arithmetic, not rounded decimals. - S133 · premise_and_result_audit — Recompute CLT SUM ORIGINAL total distribution mass2 with exact rational arithmetic, not rounded decimals. - S132 · premise_and_result_audit — Recompute CLT SUM ORIGINAL total distribution mass1 with exact rational arithmetic, not rounded decimals. - S131 · premise_and_result_audit — Recompute CLT SUM ORIGINAL total distribution mass0 with exact rational arithmetic, not rounded decimals. - S130 · 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. - S129 · premise_and_result_audit — Recompute CLT joint ORIGINAL sum12 bucket word15 with exact rational arithmetic, not rounded decimals. - S128 · premise_and_result_audit — Recompute CLT joint ORIGINAL word15 probability factor3 with exact rational arithmetic, not rounded decimals. - S127 · premise_and_result_audit — Recompute CLT joint ORIGINAL word15 probability factor2 with exact rational arithmetic, not rounded decimals. - S126 · premise_and_result_audit — Recompute CLT joint ORIGINAL word15 probability factor1 with exact rational arithmetic, not rounded decimals. - S125 · premise_and_result_audit — Recompute CLT joint ORIGINAL word15 probability factor0 with exact rational arithmetic, not rounded decimals. - S124 · premise_and_result_audit — Recompute CLT joint ORIGINAL word15 sum with exact rational arithmetic, not rounded decimals. - S123 · 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. - S122 · premise_and_result_audit — Recompute CLT joint ORIGINAL sum11 bucket word14 with exact rational arithmetic, not rounded decimals. - S121 · premise_and_result_audit — Recompute CLT joint ORIGINAL word14 probability factor3 with exact rational arithmetic, not rounded decimals. - S120 · premise_and_result_audit — Recompute CLT joint ORIGINAL word14 probability factor2 with exact rational arithmetic, not rounded decimals. - S119 · premise_and_result_audit — Recompute CLT joint ORIGINAL word14 probability factor1 with exact rational arithmetic, not rounded decimals. - S118 · premise_and_result_audit — Recompute CLT joint ORIGINAL word14 probability factor0 with exact rational arithmetic, not rounded decimals. - S117 · premise_and_result_audit — Recompute CLT joint ORIGINAL word14 sum with exact rational arithmetic, not rounded decimals. - S116 · 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. - S115 · premise_and_result_audit — Recompute CLT joint ORIGINAL sum11 bucket word13 with exact rational arithmetic, not rounded decimals. - S114 · premise_and_result_audit — Recompute CLT joint ORIGINAL word13 probability factor3 with exact rational arithmetic, not rounded decimals. - S113 · premise_and_result_audit — Recompute CLT joint ORIGINAL word13 probability factor2 with exact rational arithmetic, not rounded decimals. - S112 · premise_and_result_audit — Recompute CLT joint ORIGINAL word13 probability factor1 with exact rational arithmetic, not rounded decimals. - S111 · premise_and_result_audit — Recompute CLT joint ORIGINAL word13 probability factor0 with exact rational arithmetic, not rounded decimals. - S110 · premise_and_result_audit — Recompute CLT joint ORIGINAL word13 sum with exact rational arithmetic, not rounded decimals. - S109 · 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. - S108 · premise_and_result_audit — Recompute CLT joint ORIGINAL sum10 bucket word12 with exact rational arithmetic, not rounded decimals. - S107 · premise_and_result_audit — Recompute CLT joint ORIGINAL word12 probability factor3 with exact rational arithmetic, not rounded decimals. - S106 · premise_and_result_audit — Recompute CLT joint ORIGINAL word12 probability factor2 with exact rational arithmetic, not rounded decimals. - S105 · premise_and_result_audit — Recompute CLT joint ORIGINAL word12 probability factor1 with exact rational arithmetic, not rounded decimals. - S104 · premise_and_result_audit — Recompute CLT joint ORIGINAL word12 probability factor0 with exact rational arithmetic, not rounded decimals. - S103 · premise_and_result_audit — Recompute CLT joint ORIGINAL word12 sum with exact rational arithmetic, not rounded decimals. - S102 · 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. - S101 · premise_and_result_audit — Recompute CLT joint ORIGINAL sum11 bucket word11 with exact rational arithmetic, not rounded decimals. - S100 · premise_and_result_audit — Recompute CLT joint ORIGINAL word11 probability factor3 with exact rational arithmetic, not rounded decimals. - S099 · premise_and_result_audit — Recompute CLT joint ORIGINAL word11 probability factor2 with exact rational arithmetic, not rounded decimals. - S098 · premise_and_result_audit — Recompute CLT joint ORIGINAL word11 probability factor1 with exact rational arithmetic, not rounded decimals. - S097 · premise_and_result_audit — Recompute CLT joint ORIGINAL word11 probability factor0 with exact rational arithmetic, not rounded decimals. - S096 · premise_and_result_audit — Recompute CLT joint ORIGINAL word11 sum with exact rational arithmetic, not rounded decimals. - S095 · 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. - S094 · premise_and_result_audit — Recompute CLT joint ORIGINAL sum10 bucket word10 with exact rational arithmetic, not rounded decimals. - S093 · premise_and_result_audit — Recompute CLT joint ORIGINAL word10 probability factor3 with exact rational arithmetic, not rounded decimals. - S092 · premise_and_result_audit — Recompute CLT joint ORIGINAL word10 probability factor2 with exact rational arithmetic, not rounded decimals. - S091 · premise_and_result_audit — Recompute CLT joint ORIGINAL word10 probability factor1 with exact rational arithmetic, not rounded decimals. - S090 · premise_and_result_audit — Recompute CLT joint ORIGINAL word10 probability factor0 with exact rational arithmetic, not rounded decimals. - S089 · premise_and_result_audit — Recompute CLT joint ORIGINAL word10 sum with exact rational arithmetic, not rounded decimals. - S088 · 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. - S087 · premise_and_result_audit — Recompute CLT joint ORIGINAL sum10 bucket word9 with exact rational arithmetic, not rounded decimals. - S086 · premise_and_result_audit — Recompute CLT joint ORIGINAL word9 probability factor3 with exact rational arithmetic, not rounded decimals. - S085 · premise_and_result_audit — Recompute CLT joint ORIGINAL word9 probability factor2 with exact rational arithmetic, not rounded decimals. - S084 · premise_and_result_audit — Recompute CLT joint ORIGINAL word9 probability factor1 with exact rational arithmetic, not rounded decimals. - S083 · premise_and_result_audit — Recompute CLT joint ORIGINAL word9 probability factor0 with exact rational arithmetic, not rounded decimals. - S082 · premise_and_result_audit — Recompute CLT joint ORIGINAL word9 sum with exact rational arithmetic, not rounded decimals. - S081 · 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. - S080 · premise_and_result_audit — Recompute CLT joint ORIGINAL sum9 bucket word8 with exact rational arithmetic, not rounded decimals. - S079 · premise_and_result_audit — Recompute CLT joint ORIGINAL word8 probability factor3 with exact rational arithmetic, not rounded decimals. - S078 · premise_and_result_audit — Recompute CLT joint ORIGINAL word8 probability factor2 with exact rational arithmetic, not rounded decimals. - S077 · premise_and_result_audit — Recompute CLT joint ORIGINAL word8 probability factor1 with exact rational arithmetic, not rounded decimals. - S076 · premise_and_result_audit — Recompute CLT joint ORIGINAL word8 probability factor0 with exact rational arithmetic, not rounded decimals. - S075 · premise_and_result_audit — Recompute CLT joint ORIGINAL word8 sum with exact rational arithmetic, not rounded decimals. - S074 · 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. - S073 · premise_and_result_audit — Recompute CLT joint ORIGINAL sum11 bucket word7 with exact rational arithmetic, not rounded decimals. - S072 · premise_and_result_audit — Recompute CLT joint ORIGINAL word7 probability factor3 with exact rational arithmetic, not rounded decimals. - S071 · premise_and_result_audit — Recompute CLT joint ORIGINAL word7 probability factor2 with exact rational arithmetic, not rounded decimals. - S070 · premise_and_result_audit — Recompute CLT joint ORIGINAL word7 probability factor1 with exact rational arithmetic, not rounded decimals. - S069 · premise_and_result_audit — Recompute CLT joint ORIGINAL word7 probability factor0 with exact rational arithmetic, not rounded decimals. - S068 · premise_and_result_audit — Recompute CLT joint ORIGINAL word7 sum with exact rational arithmetic, not rounded decimals. - S067 · 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. - S066 · premise_and_result_audit — Recompute CLT joint ORIGINAL sum10 bucket word6 with exact rational arithmetic, not rounded decimals. - S065 · premise_and_result_audit — Recompute CLT joint ORIGINAL word6 probability factor3 with exact rational arithmetic, not rounded decimals. - S064 · premise_and_result_audit — Recompute CLT joint ORIGINAL word6 probability factor2 with exact rational arithmetic, not rounded decimals. - S063 · premise_and_result_audit — Recompute CLT joint ORIGINAL word6 probability factor1 with exact rational arithmetic, not rounded decimals. - S062 · premise_and_result_audit — Recompute CLT joint ORIGINAL word6 probability factor0 with exact rational arithmetic, not rounded decimals. - S061 · premise_and_result_audit — Recompute CLT joint ORIGINAL word6 sum with exact rational arithmetic, not rounded decimals. - S060 · 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. - S059 · premise_and_result_audit — Recompute CLT joint ORIGINAL sum10 bucket word5 with exact rational arithmetic, not rounded decimals. - S058 · premise_and_result_audit — Recompute CLT joint ORIGINAL word5 probability factor3 with exact rational arithmetic, not rounded decimals. - S057 · premise_and_result_audit — Recompute CLT joint ORIGINAL word5 probability factor2 with exact rational arithmetic, not rounded decimals. - S056 · premise_and_result_audit — Recompute CLT joint ORIGINAL word5 probability factor1 with exact rational arithmetic, not rounded decimals. - S055 · premise_and_result_audit — Recompute CLT joint ORIGINAL word5 probability factor0 with exact rational arithmetic, not rounded decimals. - S054 · premise_and_result_audit — Recompute CLT joint ORIGINAL word5 sum with exact rational arithmetic, not rounded decimals. - S053 · 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. - S052 · premise_and_result_audit — Recompute CLT joint ORIGINAL sum9 bucket word4 with exact rational arithmetic, not rounded decimals. - S051 · premise_and_result_audit — Recompute CLT joint ORIGINAL word4 probability factor3 with exact rational arithmetic, not rounded decimals. - S050 · premise_and_result_audit — Recompute CLT joint ORIGINAL word4 probability factor2 with exact rational arithmetic, not rounded decimals. - S049 · premise_and_result_audit — Recompute CLT joint ORIGINAL word4 probability factor1 with exact rational arithmetic, not rounded decimals. - S048 · premise_and_result_audit — Recompute CLT joint ORIGINAL word4 probability factor0 with exact rational arithmetic, not rounded decimals. - S047 · premise_and_result_audit — Recompute CLT joint ORIGINAL word4 sum with exact rational arithmetic, not rounded decimals. - S046 · 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. - S045 · premise_and_result_audit — Recompute CLT joint ORIGINAL sum10 bucket word3 with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Recompute CLT joint ORIGINAL word3 probability factor3 with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Recompute CLT joint ORIGINAL word3 probability factor2 with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute CLT joint ORIGINAL word3 probability factor1 with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute CLT joint ORIGINAL word3 probability factor0 with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute CLT joint ORIGINAL word3 sum with exact rational arithmetic, not rounded decimals. - S039 · 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. - S038 · premise_and_result_audit — Recompute CLT joint ORIGINAL sum9 bucket word2 with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute CLT joint ORIGINAL word2 probability factor3 with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute CLT joint ORIGINAL word2 probability factor2 with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute CLT joint ORIGINAL word2 probability factor1 with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute CLT joint ORIGINAL word2 probability factor0 with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute CLT joint ORIGINAL word2 sum with exact rational arithmetic, not rounded decimals. - S032 · 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. - S031 · premise_and_result_audit — Recompute CLT joint ORIGINAL sum9 bucket word1 with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute CLT joint ORIGINAL word1 probability factor3 with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute CLT joint ORIGINAL word1 probability factor2 with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute CLT joint ORIGINAL word1 probability factor1 with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute CLT joint ORIGINAL word1 probability factor0 with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute CLT joint ORIGINAL word1 sum with exact rational arithmetic, not rounded decimals. - S025 · 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. - S024 · premise_and_result_audit — Recompute CLT joint ORIGINAL sum8 bucket word0 with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute CLT joint ORIGINAL word0 probability factor3 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute CLT joint ORIGINAL word0 probability factor2 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 (4, [[8, 1/16], [9, 1/4], [10, 3/8], [11, 1/4], [12, 1/16]], 1, 10, 1, 10, 1, 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: ab91acb4d9dc927d49780dfb4d6265fdd9ad33573dc0e69521da443775348411 - candidate record SHA-256: 41e6614c83f1c752dfdb7e54b63d726548b6f15ba19336304504e6f676aaa3d3 - 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-0045.json)