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R0053-27211 · q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0086
Use mode=symbolic_application and complete model={"values":[-2,-1],"probabilities":[[1,4],[3,4]],"process_certificate":"infinite_iid","n":2}. Give (mode, declared_process_certificate, mean, variance, n, standardization_description, named_theorem_conclusion, conditional_application_qualified).
G001 — Use mode=symbolic_application and complete model={"values":[-2,-1],"probabilities":[[1,4],[3,4]],"process_certificate":"infinite_iid","n":2}. Give (mode, declared_process_certificate, mean, variance, n, standardization_description, named_theorem_conclusion, conditional_application_qualified).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.
Before
source roles = [clt-application-boundary]
After
derived = [[mode, declared_process_certificate, mean, variance, n, standardization_description, named_theorem_conclusion, conditional_application_qualified]]
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.
Rule: source_ordered_fields_without_JS · Uses: G001
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.
Rule: CLT_infinite_iid_is_EXACT_GIVEN_premise_not_constructed_process · Uses: G001, S001
Before
CLT ORIGINAL conditional marginal ORIGINAL law mass sum0 := sum(0, 1/4)
After
CLT ORIGINAL conditional marginal ORIGINAL law mass sum0 = 1/4
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
CLT ORIGINAL conditional marginal ORIGINAL law mass sum1 := sum(1/4, 3/4)
After
CLT ORIGINAL conditional marginal ORIGINAL law mass sum1 = 1
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
source roles = [CLT ORIGINAL conditional marginal, [-2, -1], [[1, 4], [3, 4]]]
After
derived = [[[-2, 1/4], [-1, 3/4]], 1]
Conditions: none
Pair ORIGINAL values/probabilities before numeric sorting,retaining zero-mass labels and checking total mass1.This qualifies a finite marginal,not an infinite IID process inferred from samples.
Rule: CLT_ORIGINAL_value_probability_pairs_NUMERIC_order_normalized · Uses: G001, S004
Before
CLT ORIGINAL conditional marginal ORIGINAL value times probability0 := mul(-2, 1/4)
After
CLT ORIGINAL conditional marginal ORIGINAL value times probability0 = -1/2
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
CLT ORIGINAL conditional marginal ORIGINAL ascending mean sum0 := sum(0, -1/2)
After
CLT ORIGINAL conditional marginal ORIGINAL ascending mean sum0 = -1/2
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
CLT ORIGINAL conditional marginal exact value square0 := mul(-2, -2)
After
CLT ORIGINAL conditional marginal exact value square0 = 4
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
CLT ORIGINAL conditional marginal ORIGINAL square times probability0 := mul(4, 1/4)
After
CLT ORIGINAL conditional marginal ORIGINAL square times probability0 = 1
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
CLT ORIGINAL conditional marginal ORIGINAL ascending second moment sum0 := sum(0, 1)
After
CLT ORIGINAL conditional marginal ORIGINAL ascending second moment sum0 = 1
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
CLT ORIGINAL conditional marginal ORIGINAL value times probability1 := mul(-1, 3/4)
After
CLT ORIGINAL conditional marginal ORIGINAL value times probability1 = -3/4
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
CLT ORIGINAL conditional marginal ORIGINAL ascending mean sum1 := sum(-1/2, -3/4)
After
CLT ORIGINAL conditional marginal ORIGINAL ascending mean sum1 = -5/4
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
CLT ORIGINAL conditional marginal exact value square1 := mul(-1, -1)
After
CLT ORIGINAL conditional marginal exact value square1 = 1
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
CLT ORIGINAL conditional marginal ORIGINAL square times probability1 := mul(1, 3/4)
After
CLT ORIGINAL conditional marginal ORIGINAL square times probability1 = 3/4
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
CLT ORIGINAL conditional marginal ORIGINAL ascending second moment sum1 := sum(1, 3/4)
After
CLT ORIGINAL conditional marginal ORIGINAL ascending second moment sum1 = 7/4
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
CLT ORIGINAL conditional marginal ORIGINAL mean square := mul(-5/4, -5/4)
After
CLT ORIGINAL conditional marginal ORIGINAL mean square = 25/16
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
CLT ORIGINAL conditional marginal ORIGINAL second minus mean square := sub(7/4, 25/16)
After
CLT ORIGINAL conditional marginal ORIGINAL second minus mean square = 3/16
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
source roles = [CLT ORIGINAL conditional marginal, [[-2, 1/4], [-1, 3/4]]]
After
derived = [-5/4, 7/4, 3/16]
Conditions: none
Use the SAME complete law for E[X],E[X²] and Var(X)=E[X²]-E[X]².Keep value² as an exact teaching intermediate and apply the source profile to its product with probability,not an extra rejection for a large zero-mass value square.Finite moments prove neither independence,an infinite process nor a distributional limit.
Rule: CLT_ALL_finite_marginal_moments_not_a_process_or_limit_proof · Uses: G001, S017
Before
source roles = [infinite_iid, [[-2, 1/4], [-1, 3/4]], -5/4, 3/16, 2]
After
derived = [Z_n=(S_n-n*mu)/(sigma*sqrt(n)), Z_n_converges_in_distribution_to_N(0,1), Yes]
Conditions: none
Given the WHOLE infinite IID premise,common mu and0<sigma²<infinity,the named CLT gives convergence in distribution of Z_n to N(0,1).This is a conditional theorem invocation,not a proof performed here,a constructed infinite process or normality at a fixed n.
Rule: CLT_NAMED_theorem_CONDITIONAL_application_NOT_performed_proof · Uses: G001, S018
S019 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.S018 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.S017 · premise_and_result_audit — Recompute CLT ORIGINAL conditional marginal ORIGINAL second minus mean square with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Recompute CLT ORIGINAL conditional marginal ORIGINAL mean square with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Recompute CLT ORIGINAL conditional marginal ORIGINAL ascending second moment sum1 with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Recompute CLT ORIGINAL conditional marginal ORIGINAL square times probability1 with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute CLT ORIGINAL conditional marginal exact value square1 with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute CLT ORIGINAL conditional marginal ORIGINAL ascending mean sum1 with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute CLT ORIGINAL conditional marginal ORIGINAL value times probability1 with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute CLT ORIGINAL conditional marginal ORIGINAL ascending second moment sum0 with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute CLT ORIGINAL conditional marginal ORIGINAL square times probability0 with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute CLT ORIGINAL conditional marginal exact value square0 with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute CLT ORIGINAL conditional marginal ORIGINAL ascending mean sum0 with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute CLT ORIGINAL conditional marginal ORIGINAL value times probability0 with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.S004 · premise_and_result_audit — Recompute CLT ORIGINAL conditional marginal ORIGINAL law mass sum1 with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute CLT ORIGINAL conditional marginal ORIGINAL law mass sum0 with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.S001 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.(symbolic_application, infinite_iid, -5/4, 3/16, 2, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), Z_n_converges_in_distribution_to_N(0,1), Yes)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: c3556898f8259ea46dbc4fb208e532fad625d818b700fad59b6a46827486519bcandidate record SHA-256: d2d6914efff66feba3c8177b60b9c3450a15941360bc216f6a88349a6a215b6dsemantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1Lean: NOT_RUN · human review: NOT_PERFORMED