OCMEBilingual mathematics & derivations繁體中文

Explore related knowledge and questions →

Question 27222: Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

Return to catalogue page 137

← Question derivation index · Original question batch

R0053-27222 · q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0097

Original question and givens

Use mode=non_iid and complete model={"values_a":[-1,0],"probabilities_a":[[1,5],[4,5]],"values_b":[-1,0],"probabilities_b":[[1,7],[6,7]]}. Give (mode, first_marginal_law, second_marginal_law, identical_marginals, this_IID_theorem_applicable, scope).

  • G001 — Use mode=non_iid and complete model={"values_a":[-1,0],"probabilities_a":[[1,5],[4,5]],"values_b":[-1,0],"probabilities_b":[[1,7],[6,7]]}. Give (mode, first_marginal_law, second_marginal_law, identical_marginals, this_IID_theorem_applicable, 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.

Step-by-step derivation

S001 · implies

Before

source roles = [clt-application-boundary]

After

derived = [[mode, first_marginal_law, second_marginal_law, identical_marginals, this_IID_theorem_applicable, 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 · calculation

Before

CLT ORIGINAL first marginal ORIGINAL law mass sum0 := sum(0, 1/5)

After

CLT ORIGINAL first marginal ORIGINAL law mass sum0 = 1/5

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, S001

S003 · calculation

Before

CLT ORIGINAL first marginal ORIGINAL law mass sum1 := sum(1/5, 4/5)

After

CLT ORIGINAL first 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, S002

S004 · implies

Before

source roles = [CLT ORIGINAL first marginal, [-1, 0], [[1, 5], [4, 5]]]

After

derived = [[[-1, 1/5], [0, 4/5]], 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, S003

S005 · calculation

Before

CLT ORIGINAL first marginal ORIGINAL value times probability0 := mul(-1, 1/5)

After

CLT ORIGINAL first marginal ORIGINAL value times probability0 = -1/5

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, S004

S006 · calculation

Before

CLT ORIGINAL first marginal ORIGINAL ascending mean sum0 := sum(0, -1/5)

After

CLT ORIGINAL first marginal ORIGINAL ascending mean sum0 = -1/5

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 first marginal exact value square0 := mul(-1, -1)

After

CLT ORIGINAL first marginal exact value square0 = 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 first marginal ORIGINAL square times probability0 := mul(1, 1/5)

After

CLT ORIGINAL first marginal ORIGINAL square times probability0 = 1/5

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 first marginal ORIGINAL ascending second moment sum0 := sum(0, 1/5)

After

CLT ORIGINAL first marginal ORIGINAL ascending second moment sum0 = 1/5

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 first marginal ORIGINAL value times probability1 := mul(0, 4/5)

After

CLT ORIGINAL first marginal ORIGINAL value times probability1 = 0

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 first marginal ORIGINAL ascending mean sum1 := sum(-1/5, 0)

After

CLT ORIGINAL first marginal ORIGINAL ascending mean sum1 = -1/5

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 first marginal exact value square1 := mul(0, 0)

After

CLT ORIGINAL first marginal exact value square1 = 0

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 first marginal ORIGINAL square times probability1 := mul(0, 4/5)

After

CLT ORIGINAL first marginal ORIGINAL square times probability1 = 0

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 first marginal ORIGINAL ascending second moment sum1 := sum(1/5, 0)

After

CLT ORIGINAL first marginal ORIGINAL ascending second moment sum1 = 1/5

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 first marginal ORIGINAL mean square := mul(-1/5, -1/5)

After

CLT ORIGINAL first marginal ORIGINAL mean square = 1/25

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 first marginal ORIGINAL second minus mean square := sub(1/5, 1/25)

After

CLT ORIGINAL first marginal ORIGINAL second minus mean square = 4/25

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 · implies

Before

source roles = [CLT ORIGINAL first marginal, [[-1, 1/5], [0, 4/5]]]

After

derived = [-1/5, 1/5, 4/25]

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, S016

S018 · calculation

Before

CLT ORIGINAL second marginal ORIGINAL law mass sum0 := sum(0, 1/7)

After

CLT ORIGINAL second marginal ORIGINAL law mass sum0 = 1/7

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, S017

S019 · calculation

Before

CLT ORIGINAL second marginal ORIGINAL law mass sum1 := sum(1/7, 6/7)

After

CLT ORIGINAL second 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, S018

S020 · implies

Before

source roles = [CLT ORIGINAL second marginal, [-1, 0], [[1, 7], [6, 7]]]

After

derived = [[[-1, 1/7], [0, 6/7]], 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, S019

S021 · calculation

Before

CLT ORIGINAL second marginal ORIGINAL value times probability0 := mul(-1, 1/7)

After

CLT ORIGINAL second marginal ORIGINAL value times probability0 = -1/7

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 ORIGINAL second marginal ORIGINAL ascending mean sum0 := sum(0, -1/7)

After

CLT ORIGINAL second marginal ORIGINAL ascending mean sum0 = -1/7

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 · calculation

Before

CLT ORIGINAL second marginal exact value square0 := mul(-1, -1)

After

CLT ORIGINAL second marginal exact value square0 = 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, S022

S024 · calculation

Before

CLT ORIGINAL second marginal ORIGINAL square times probability0 := mul(1, 1/7)

After

CLT ORIGINAL second marginal ORIGINAL square times probability0 = 1/7

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 ORIGINAL second marginal ORIGINAL ascending second moment sum0 := sum(0, 1/7)

After

CLT ORIGINAL second marginal ORIGINAL ascending second moment sum0 = 1/7

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 ORIGINAL second marginal ORIGINAL value times probability1 := mul(0, 6/7)

After

CLT ORIGINAL second marginal ORIGINAL value times probability1 = 0

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 ORIGINAL second marginal ORIGINAL ascending mean sum1 := sum(-1/7, 0)

After

CLT ORIGINAL second marginal ORIGINAL ascending mean sum1 = -1/7

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 · calculation

Before

CLT ORIGINAL second marginal exact value square1 := mul(0, 0)

After

CLT ORIGINAL second marginal exact value square1 = 0

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, S027

S029 · calculation

Before

CLT ORIGINAL second marginal ORIGINAL square times probability1 := mul(0, 6/7)

After

CLT ORIGINAL second marginal ORIGINAL square times probability1 = 0

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 ORIGINAL second marginal ORIGINAL ascending second moment sum1 := sum(1/7, 0)

After

CLT ORIGINAL second marginal ORIGINAL ascending second moment sum1 = 1/7

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 ORIGINAL second marginal ORIGINAL mean square := mul(-1/7, -1/7)

After

CLT ORIGINAL second marginal ORIGINAL mean square = 1/49

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 ORIGINAL second marginal ORIGINAL second minus mean square := sub(1/7, 1/49)

After

CLT ORIGINAL second marginal ORIGINAL second minus mean square = 6/49

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 = [CLT ORIGINAL second marginal, [[-1, 1/7], [0, 6/7]]]

After

derived = [-1/7, 1/7, 6/49]

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, S032

S034 · implies

Before

source roles = [[[-1, 1/5], [0, 4/5]], [[-1, 1/7], [0, 6/7]]]

After

derived = [No, No]

Conditions: none

The complete source marginal rows actually differ and fail this identical-distribution premise.No rejects this scoped IID theorem,not every non-IID CLT or actual distributional limit.

Step identifiers and dependencies

Rule: CLT_ACTUAL_different_full_marginal_rows_scope_NOT_all_CLTs · Uses: G001, S033

Reverse checks

  • S034 · 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.
  • 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 ORIGINAL second marginal ORIGINAL second minus mean square with exact rational arithmetic, not rounded decimals.
  • S031 · premise_and_result_audit — Recompute CLT ORIGINAL second marginal ORIGINAL mean square with exact rational arithmetic, not rounded decimals.
  • S030 · premise_and_result_audit — Recompute CLT ORIGINAL second marginal ORIGINAL ascending second moment sum1 with exact rational arithmetic, not rounded decimals.
  • S029 · premise_and_result_audit — Recompute CLT ORIGINAL second marginal ORIGINAL square times probability1 with exact rational arithmetic, not rounded decimals.
  • S028 · premise_and_result_audit — Recompute CLT ORIGINAL second marginal exact value square1 with exact rational arithmetic, not rounded decimals.
  • S027 · premise_and_result_audit — Recompute CLT ORIGINAL second marginal ORIGINAL ascending mean sum1 with exact rational arithmetic, not rounded decimals.
  • S026 · premise_and_result_audit — Recompute CLT ORIGINAL second marginal ORIGINAL value times probability1 with exact rational arithmetic, not rounded decimals.
  • S025 · premise_and_result_audit — Recompute CLT ORIGINAL second marginal ORIGINAL ascending second moment sum0 with exact rational arithmetic, not rounded decimals.
  • S024 · premise_and_result_audit — Recompute CLT ORIGINAL second marginal ORIGINAL square times probability0 with exact rational arithmetic, not rounded decimals.
  • S023 · premise_and_result_audit — Recompute CLT ORIGINAL second marginal exact value square0 with exact rational arithmetic, not rounded decimals.
  • S022 · premise_and_result_audit — Recompute CLT ORIGINAL second marginal ORIGINAL ascending mean sum0 with exact rational arithmetic, not rounded decimals.
  • S021 · premise_and_result_audit — Recompute CLT ORIGINAL second marginal ORIGINAL value times probability0 with exact rational arithmetic, not rounded decimals.
  • S020 · 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.
  • S019 · premise_and_result_audit — Recompute CLT ORIGINAL second marginal ORIGINAL law mass sum1 with exact rational arithmetic, not rounded decimals.
  • S018 · premise_and_result_audit — Recompute CLT ORIGINAL second marginal ORIGINAL law mass sum0 with exact rational arithmetic, not rounded decimals.
  • S017 · 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.
  • S016 · premise_and_result_audit — Recompute CLT ORIGINAL first marginal ORIGINAL second minus mean square with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Recompute CLT ORIGINAL first marginal ORIGINAL mean square with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Recompute CLT ORIGINAL first marginal ORIGINAL ascending second moment sum1 with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Recompute CLT ORIGINAL first marginal ORIGINAL square times probability1 with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute CLT ORIGINAL first marginal exact value square1 with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute CLT ORIGINAL first marginal ORIGINAL ascending mean sum1 with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute CLT ORIGINAL first marginal ORIGINAL value times probability1 with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute CLT ORIGINAL first marginal ORIGINAL ascending second moment sum0 with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute CLT ORIGINAL first marginal ORIGINAL square times probability0 with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Recompute CLT ORIGINAL first marginal exact value square0 with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute CLT ORIGINAL first marginal ORIGINAL ascending mean sum0 with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute CLT ORIGINAL first marginal ORIGINAL value times probability0 with exact rational arithmetic, not rounded decimals.
  • S004 · 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.
  • S003 · premise_and_result_audit — Recompute CLT ORIGINAL first marginal ORIGINAL law mass sum1 with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute CLT ORIGINAL first marginal ORIGINAL law mass sum0 with exact rational arithmetic, not rounded decimals.
  • 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

(non_iid, [[-1, 1/5], [0, 4/5]], [[-1, 1/7], [0, 6/7]], No, No, this_IID_CLT_not_applicable_not_a_general_counterexample)

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: 1fbd99de225e1a33fa031ef3c6f5ff34a87a939a728f46089fac5091008b840f
  • candidate record SHA-256: 6c070444d66027b8761b7dabe9cfe9e19c1716dd4c5d5db64b4df99e188240ac
  • semantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62
  • question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON