Question 27281: Complete Fixed-n Sum Moments Do Not Prove a Distributional Limit

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0053-27281 · q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0031

Original question and givens

Use the aligned finite law values=[-2, -1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use sample size n=2 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).

G001 — Use the aligned finite law values=[-2, -1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use sample size n=2 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

mko-central-limit-theorem-finite-variance-iid

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · implies

Before

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 dependencies

Rule: CLT_infinite_iid_is_EXACT_GIVEN_premise_not_constructed_process · Uses: G001, S001

S003 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · implies

Before

source roles = [CLT ORIGINAL common marginal, [-2, -1], [[1, 2], [1, 2]]]

After

derived = [[[-2, 1/2], [-1, 1/2]], 1]

Conditions: none

Pair ORIGINAL values/probabilities before numeric sorting,retaining zero-mass labels and checking total mass1.This qualifies a finite marginal,not an infinite IID process inferred from samples.

Step identifiers and dependencies

Rule: CLT_ORIGINAL_value_probability_pairs_NUMERIC_order_normalized · Uses: G001, S004

S006 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

CLT ORIGINAL common marginal ORIGINAL value times probability1 := mul(-1, 1/2)

After

CLT ORIGINAL common marginal ORIGINAL value times probability1 = -1/2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

CLT ORIGINAL common marginal ORIGINAL ascending mean sum1 := sum(-1, -1/2)

After

CLT ORIGINAL common marginal ORIGINAL ascending mean sum1 = -3/2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

CLT ORIGINAL common marginal exact value square1 := mul(-1, -1)

After

CLT ORIGINAL common marginal exact value square1 = 1

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

CLT ORIGINAL common marginal ORIGINAL square times probability1 := mul(1, 1/2)

After

CLT ORIGINAL common marginal ORIGINAL square times probability1 = 1/2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

CLT ORIGINAL common marginal ORIGINAL ascending second moment sum1 := sum(2, 1/2)

After

CLT ORIGINAL common marginal ORIGINAL ascending second moment sum1 = 5/2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

CLT ORIGINAL common marginal ORIGINAL mean square := mul(-3/2, -3/2)

After

CLT ORIGINAL common marginal ORIGINAL mean square = 9/4

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

CLT ORIGINAL common marginal ORIGINAL second minus mean square := sub(5/2, 9/4)

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · implies

Before

source roles = [CLT ORIGINAL common marginal, [[-2, 1/2], [-1, 1/2]]]

After

derived = [-3/2, 5/2, 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 dependencies

Rule: CLT_ALL_finite_marginal_moments_not_a_process_or_limit_proof · Uses: G001, S017

S019 · calculation

Before

CLT joint ORIGINAL word0 sum := sum(-2, -2)

After

CLT joint ORIGINAL word0 sum = -4

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

CLT joint ORIGINAL sum-4 bucket word0 := sum(0, 1/4)

After

CLT joint ORIGINAL sum-4 bucket word0 = 1/4

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · implies

Before

source roles = [2, 0, [-2, -2], [1/2, 1/2]]

After

derived = [-4, 1/4, 0, 1/4]

Conditions: none

The declared IID premise gives each finite word the product of coordinate probabilities.Enumerate EVERY word in source order and add it to its actual sum fibre,retaining zero-mass words/buckets.Do not guess the distribution from samples or moments.

Step identifiers and dependencies

Rule: CLT_EVERY_joint_word_original_coordinate_products_AND_full_sum_fibre · Uses: G001, S022

S024 · calculation

Before

CLT joint ORIGINAL word1 sum := sum(-2, -1)

After

CLT joint ORIGINAL word1 sum = -3

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · calculation

Before

CLT joint ORIGINAL sum-3 bucket word1 := sum(0, 1/4)

After

CLT joint ORIGINAL sum-3 bucket word1 = 1/4

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · implies

Before

source roles = [2, 1, [-2, -1], [1/2, 1/2]]

After

derived = [-3, 1/4, 0, 1/4]

Conditions: none

Step identifiers and dependencies

Rule: CLT_EVERY_joint_word_original_coordinate_products_AND_full_sum_fibre · Uses: G001, S027

S029 · calculation

Before

CLT joint ORIGINAL word2 sum := sum(-1, -2)

After

CLT joint ORIGINAL word2 sum = -3

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · calculation

Before

CLT joint ORIGINAL sum-3 bucket word2 := sum(1/4, 1/4)

After

CLT joint ORIGINAL sum-3 bucket word2 = 1/2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · implies

Before

source roles = [2, 2, [-1, -2], [1/2, 1/2]]

After

derived = [-3, 1/4, 1/4, 1/2]

Conditions: none

Step identifiers and dependencies

Rule: CLT_EVERY_joint_word_original_coordinate_products_AND_full_sum_fibre · Uses: G001, S032

S034 · calculation

Before

CLT joint ORIGINAL word3 sum := sum(-1, -1)

After

CLT joint ORIGINAL word3 sum = -2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · calculation

Before

CLT joint ORIGINAL sum-2 bucket word3 := sum(0, 1/4)

After

CLT joint ORIGINAL sum-2 bucket word3 = 1/4

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · implies

Before

source roles = [2, 3, [-1, -1], [1/2, 1/2]]

After

derived = [-2, 1/4, 0, 1/4]

Conditions: none

Step identifiers and dependencies

Rule: CLT_EVERY_joint_word_original_coordinate_products_AND_full_sum_fibre · Uses: G001, S037

S039 · calculation

Before

CLT SUM ORIGINAL total distribution mass0 := sum(0, 1/4)

After

CLT SUM ORIGINAL total distribution mass0 = 1/4

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · calculation

Before

CLT SUM ORIGINAL total distribution mass1 := sum(1/4, 1/2)

After

CLT SUM ORIGINAL total distribution mass1 = 3/4

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S039

S041 · calculation

Before

CLT SUM ORIGINAL total distribution mass2 := sum(3/4, 1/4)

After

CLT SUM ORIGINAL total distribution mass2 = 1

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S040

S042 · calculation

Before

CLT SUM ORIGINAL value times probability0 := mul(-4, 1/4)

After

CLT SUM 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S041

S043 · calculation

Before

CLT SUM ORIGINAL ascending mean sum0 := sum(0, -1)

After

CLT SUM 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · calculation

Before

CLT SUM exact value square0 := mul(-4, -4)

After

CLT SUM exact value square0 = 16

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S043

S045 · calculation

Before

CLT SUM ORIGINAL square times probability0 := mul(16, 1/4)

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S044

S046 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S045

S047 · calculation

Before

CLT SUM ORIGINAL value times probability1 := mul(-3, 1/2)

After

CLT SUM 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S046

S048 · calculation

Before

CLT SUM ORIGINAL ascending mean sum1 := sum(-1, -3/2)

After

CLT SUM 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S047

S049 · calculation

Before

CLT SUM exact value square1 := mul(-3, -3)

After

CLT SUM 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S048

S050 · calculation

Before

CLT SUM ORIGINAL square times probability1 := mul(9, 1/2)

After

CLT SUM 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S049

S051 · calculation

Before

CLT SUM ORIGINAL ascending second moment sum1 := sum(4, 9/2)

After

CLT SUM ORIGINAL ascending second moment sum1 = 17/2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S050

S052 · calculation

Before

CLT SUM ORIGINAL value times probability2 := mul(-2, 1/4)

After

CLT SUM ORIGINAL value times probability2 = -1/2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S051

S053 · calculation

Before

CLT SUM ORIGINAL ascending mean sum2 := sum(-5/2, -1/2)

After

CLT SUM ORIGINAL ascending mean sum2 = -3

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S052

S054 · calculation

Before

CLT SUM exact value square2 := mul(-2, -2)

After

CLT SUM exact value square2 = 4

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S053

S055 · calculation

Before

CLT SUM ORIGINAL square times probability2 := mul(4, 1/4)

After

CLT SUM ORIGINAL square times probability2 = 1

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S054

S056 · calculation

Before

CLT SUM ORIGINAL ascending second moment sum2 := sum(17/2, 1)

After

CLT SUM ORIGINAL ascending second moment sum2 = 19/2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S055

S057 · calculation

Before

CLT SUM ORIGINAL mean square := mul(-3, -3)

After

CLT SUM ORIGINAL mean square = 9

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S056

S058 · calculation

Before

CLT SUM ORIGINAL second minus mean square := sub(19/2, 9)

After

CLT SUM ORIGINAL second minus mean square = 1/2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S057

S059 · calculation

Before

CLT ORIGINAL n times common mean := mul(2, -3/2)

After

CLT ORIGINAL n times common mean = -3

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S058

S060 · calculation

Before

CLT ORIGINAL n times common variance := mul(2, 1/4)

After

CLT ORIGINAL n times common variance = 1/2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S059

S061 · implies

Before

source roles = [2, -3/2, 1/4, [(-4, 1/4), (-3, 1/2), (-2, 1/4)], -3, 1/2]

After

derived = [1, -3, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n))]

Conditions: none

sigma²>0 and n>0 make the denominator nonzero.E[Z]=(nmu-nmu)/(sigma sqrt(n))=0 and Var(Z)=n sigma²/(sigma² n)=1.These exact moments belong to this complete FIXED-n law;they do not make it normal or prove the general CLT.

Step identifiers and dependencies

Rule: CLT_standardization_EXACT_0_1_moments_NOT_normality_or_limit · Uses: G001, S060

Reverse checks

S061 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
S060 · premise_and_result_audit — Recompute CLT ORIGINAL n times common variance with exact rational arithmetic, not rounded decimals.
S059 · premise_and_result_audit — Recompute CLT ORIGINAL n times common mean with exact rational arithmetic, not rounded decimals.
S058 · premise_and_result_audit — Recompute CLT SUM ORIGINAL second minus mean square with exact rational arithmetic, not rounded decimals.
S057 · premise_and_result_audit — Recompute CLT SUM ORIGINAL mean square with exact rational arithmetic, not rounded decimals.
S056 · premise_and_result_audit — Recompute CLT SUM ORIGINAL ascending second moment sum2 with exact rational arithmetic, not rounded decimals.
S055 · premise_and_result_audit — Recompute CLT SUM ORIGINAL square times probability2 with exact rational arithmetic, not rounded decimals.
S054 · premise_and_result_audit — Recompute CLT SUM exact value square2 with exact rational arithmetic, not rounded decimals.
S053 · premise_and_result_audit — Recompute CLT SUM ORIGINAL ascending mean sum2 with exact rational arithmetic, not rounded decimals.
S052 · premise_and_result_audit — Recompute CLT SUM ORIGINAL value times probability2 with exact rational arithmetic, not rounded decimals.
S051 · premise_and_result_audit — Recompute CLT SUM ORIGINAL ascending second moment sum1 with exact rational arithmetic, not rounded decimals.
S050 · premise_and_result_audit — Recompute CLT SUM ORIGINAL square times probability1 with exact rational arithmetic, not rounded decimals.
S049 · premise_and_result_audit — Recompute CLT SUM exact value square1 with exact rational arithmetic, not rounded decimals.
S048 · premise_and_result_audit — Recompute CLT SUM ORIGINAL ascending mean sum1 with exact rational arithmetic, not rounded decimals.
S047 · premise_and_result_audit — Recompute CLT SUM ORIGINAL value times probability1 with exact rational arithmetic, not rounded decimals.
S046 · premise_and_result_audit — Recompute CLT SUM ORIGINAL ascending second moment sum0 with exact rational arithmetic, not rounded decimals.
S045 · premise_and_result_audit — Recompute CLT SUM ORIGINAL square times probability0 with exact rational arithmetic, not rounded decimals.
S044 · premise_and_result_audit — Recompute CLT SUM exact value square0 with exact rational arithmetic, not rounded decimals.
S043 · premise_and_result_audit — Recompute CLT SUM ORIGINAL ascending mean sum0 with exact rational arithmetic, not rounded decimals.
S042 · premise_and_result_audit — Recompute CLT SUM ORIGINAL value times probability0 with exact rational arithmetic, not rounded decimals.
S041 · premise_and_result_audit — Recompute CLT SUM ORIGINAL total distribution mass2 with exact rational arithmetic, not rounded decimals.
S040 · premise_and_result_audit — Recompute CLT SUM ORIGINAL total distribution mass1 with exact rational arithmetic, not rounded decimals.
S039 · premise_and_result_audit — Recompute CLT SUM ORIGINAL total distribution mass0 with exact rational arithmetic, not rounded decimals.
S038 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
S037 · premise_and_result_audit — Recompute CLT joint ORIGINAL sum-2 bucket word3 with exact rational arithmetic, not rounded decimals.
S036 · premise_and_result_audit — Recompute CLT joint ORIGINAL word3 probability factor1 with exact rational arithmetic, not rounded decimals.
S035 · premise_and_result_audit — Recompute CLT joint ORIGINAL word3 probability factor0 with exact rational arithmetic, not rounded decimals.
S034 · premise_and_result_audit — Recompute CLT joint ORIGINAL word3 sum with exact rational arithmetic, not rounded decimals.
S033 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
S032 · premise_and_result_audit — Recompute CLT joint ORIGINAL sum-3 bucket word2 with exact rational arithmetic, not rounded decimals.
S031 · premise_and_result_audit — Recompute CLT joint ORIGINAL word2 probability factor1 with exact rational arithmetic, not rounded decimals.
S030 · premise_and_result_audit — Recompute CLT joint ORIGINAL word2 probability factor0 with exact rational arithmetic, not rounded decimals.
S029 · premise_and_result_audit — Recompute CLT joint ORIGINAL word2 sum with exact rational arithmetic, not rounded decimals.
S028 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
S027 · premise_and_result_audit — Recompute CLT joint ORIGINAL sum-3 bucket word1 with exact rational arithmetic, not rounded decimals.
S026 · premise_and_result_audit — Recompute CLT joint ORIGINAL word1 probability factor1 with exact rational arithmetic, not rounded decimals.
S025 · premise_and_result_audit — Recompute CLT joint ORIGINAL word1 probability factor0 with exact rational arithmetic, not rounded decimals.
S024 · premise_and_result_audit — Recompute CLT joint ORIGINAL word1 sum with exact rational arithmetic, not rounded decimals.
S023 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
S022 · premise_and_result_audit — Recompute CLT joint ORIGINAL sum-4 bucket word0 with exact rational arithmetic, not rounded decimals.
S021 · premise_and_result_audit — Recompute CLT joint ORIGINAL word0 probability factor1 with exact rational arithmetic, not rounded decimals.
S020 · premise_and_result_audit — Recompute CLT joint ORIGINAL word0 probability factor0 with exact rational arithmetic, not rounded decimals.
S019 · premise_and_result_audit — Recompute CLT joint ORIGINAL word0 sum with exact rational arithmetic, not rounded decimals.
S018 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
S017 · premise_and_result_audit — Recompute CLT ORIGINAL common marginal ORIGINAL second minus mean square with exact rational arithmetic, not rounded decimals.
S016 · premise_and_result_audit — Recompute CLT ORIGINAL common marginal ORIGINAL mean square with exact rational arithmetic, not rounded decimals.
S015 · premise_and_result_audit — Recompute CLT ORIGINAL common marginal ORIGINAL ascending second moment sum1 with exact rational arithmetic, not rounded decimals.
S014 · premise_and_result_audit — Recompute CLT ORIGINAL common marginal ORIGINAL square times probability1 with exact rational arithmetic, not rounded decimals.
S013 · premise_and_result_audit — Recompute CLT ORIGINAL common marginal exact value square1 with exact rational arithmetic, not rounded decimals.
S012 · premise_and_result_audit — Recompute CLT ORIGINAL common marginal ORIGINAL ascending mean sum1 with exact rational arithmetic, not rounded decimals.
S011 · premise_and_result_audit — Recompute CLT ORIGINAL common marginal ORIGINAL value times probability1 with exact rational arithmetic, not rounded decimals.
S010 · premise_and_result_audit — Recompute CLT ORIGINAL common marginal ORIGINAL ascending second moment sum0 with exact rational arithmetic, not rounded decimals.
S009 · premise_and_result_audit — Recompute CLT ORIGINAL common marginal ORIGINAL square times probability0 with exact rational arithmetic, not rounded decimals.
S008 · premise_and_result_audit — Recompute CLT ORIGINAL common marginal exact value square0 with exact rational arithmetic, not rounded decimals.
S007 · premise_and_result_audit — Recompute CLT ORIGINAL common marginal ORIGINAL ascending mean sum0 with exact rational arithmetic, not rounded decimals.
S006 · premise_and_result_audit — Recompute CLT ORIGINAL common marginal ORIGINAL value times probability0 with exact rational arithmetic, not rounded decimals.
S005 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
S004 · premise_and_result_audit — Recompute CLT ORIGINAL common marginal ORIGINAL law mass sum1 with exact rational arithmetic, not rounded decimals.
S003 · premise_and_result_audit — Recompute CLT ORIGINAL common marginal ORIGINAL law mass sum0 with exact rational arithmetic, not rounded decimals.
S002 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
S001 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.

Result

(2, [[-4, 1/4], [-3, 1/2], [-2, 1/4]], 1, -3, 1/2, -3, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts

input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
original question SHA-256: 6335b2041eb095f2ec142e0a6a8df2e755f6f3b9a11e82da5ce5bba0bef07a8f
candidate record SHA-256: b0806813125571721ebb4858bba5037d610e62b8549035261d3bc54e74258352
semantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62
question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1
Lean: NOT_RUN · human review: NOT_PERFORMED

Same-source full JSON