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R0054-27611 · q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0061
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 n=3 and epsilon=2/4>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).
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 n=3 and epsilon=2/4>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).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 = [slln-fixed-n-deviation]
After
derived = [[n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary]]
Conditions: none
List the complete operation/mode field order without JavaScript or answer-shape inference;Boolean,string,null and an unclaimed result remain distinct.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
source roles = [infinite_iid]
After
derived = [ASSUME whole infinite sequence is IID with this common marginal]
Conditions: none
infinite_iid is a GIVEN whole-process premise,not independence constructed or proved by finite data/this program.
Rule: SLLN_infinite_IID_EXACT_GIVEN_premise_not_finite_observation_inference · Uses: G001, S001
Before
SLLN ORIGINAL common marginal ORIGINAL mass sum0 := sum(0, 1/2)
After
SLLN ORIGINAL common marginal ORIGINAL mass sum0 = 1/2
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
SLLN ORIGINAL common marginal ORIGINAL mass sum1 := sum(1/2, 1/2)
After
SLLN ORIGINAL common marginal ORIGINAL mass sum1 = 1
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
SLLN ORIGINAL common marginal ORIGINAL weighted value0 := mul(-2, 1/2)
After
SLLN ORIGINAL common marginal ORIGINAL weighted value0 = -1
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
SLLN ORIGINAL common marginal ORIGINAL mean sum0 := sum(0, -1)
After
SLLN ORIGINAL common marginal ORIGINAL mean sum0 = -1
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
SLLN ORIGINAL common marginal exact unbounded value square0 := mul(-2, -2)
After
SLLN ORIGINAL common marginal exact unbounded value square0 = 4
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
SLLN ORIGINAL common marginal ORIGINAL weighted square0 := mul(4, 1/2)
After
SLLN ORIGINAL common marginal ORIGINAL weighted square0 = 2
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
SLLN ORIGINAL common marginal ORIGINAL second moment sum0 := sum(0, 2)
After
SLLN ORIGINAL common marginal ORIGINAL second moment sum0 = 2
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
SLLN ORIGINAL common marginal ORIGINAL weighted value1 := mul(-1, 1/2)
After
SLLN ORIGINAL common marginal ORIGINAL weighted value1 = -1/2
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
SLLN ORIGINAL common marginal ORIGINAL mean sum1 := sum(-1, -1/2)
After
SLLN ORIGINAL common marginal ORIGINAL mean sum1 = -3/2
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
SLLN ORIGINAL common marginal exact unbounded value square1 := mul(-1, -1)
After
SLLN ORIGINAL common marginal exact unbounded value square1 = 1
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
SLLN ORIGINAL common marginal ORIGINAL weighted square1 := mul(1, 1/2)
After
SLLN ORIGINAL common marginal ORIGINAL weighted square1 = 1/2
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
SLLN ORIGINAL common marginal ORIGINAL second moment sum1 := sum(2, 1/2)
After
SLLN ORIGINAL common marginal ORIGINAL second moment sum1 = 5/2
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
SLLN ORIGINAL common marginal ORIGINAL mean squared := mul(-3/2, -3/2)
After
SLLN ORIGINAL common marginal ORIGINAL mean squared = 9/4
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
SLLN ORIGINAL common marginal ORIGINAL second minus mean squared := sub(5/2, 9/4)
After
SLLN ORIGINAL common marginal ORIGINAL second minus mean squared = 1/4
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
source roles = [SLLN ORIGINAL common marginal, [-2, -1], [[1, 2], [1, 2]]]
After
derived = [[[-2, 1/2], [-1, 1/2]], 1, -3/2, 5/2, 1/4]
Conditions: none
Pair ORIGINAL values/probabilities before numeric sorting and retain zero-mass labels.Qualify the complete nonnegative mass1 law and its source second-minus-mean-square variance.SLLN allows variance0;do not import the CLT positive-variance condition.
Rule: SLLN_complete_ORIGINAL_numeric_value_probability_alignment_ALL_moments · Uses: G001, S016
Before
SLLN ORIGINAL joint word0 integer sum := sum(-2, -2, -2)
After
SLLN ORIGINAL joint word0 integer sum = -6
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
SLLN ORIGINAL joint word0 mean := div(-6, 3)
After
SLLN ORIGINAL joint word0 mean = -2
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
SLLN ORIGINAL joint word0 mean minus mu := sub(-2, -3/2)
After
SLLN ORIGINAL joint word0 mean minus mu = -1/2
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
SLLN ORIGINAL deviating word0 probability factor0 := mul(1, 1/2)
After
SLLN ORIGINAL deviating word0 probability factor0 = 1/2
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
SLLN ORIGINAL deviating word0 probability factor1 := mul(1/2, 1/2)
After
SLLN ORIGINAL deviating word0 probability factor1 = 1/4
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
SLLN ORIGINAL deviating word0 probability factor2 := mul(1/4, 1/2)
After
SLLN ORIGINAL deviating word0 probability factor2 = 1/8
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
SLLN ORIGINAL deviation probability word0 := sum(0, 1/8)
After
SLLN ORIGINAL deviation probability word0 = 1/8
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
source roles = [3, 0, [-2, -2, -2], -3/2, 1/2]
After
derived = [-2, 1/2, Yes, 1/8, 1, 1/8]
Conditions: none
For EVERY word test |mean-mu|>=epsilon,equality included.A zero-mass deviating word still contributes to the combinatorial count,while probability is weighted separately.Non-deviating words need no unused probability product under the source.This finite event is not an almost-sure limit certificate.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S024
Before
SLLN ORIGINAL joint word1 integer sum := sum(-2, -2, -1)
After
SLLN ORIGINAL joint word1 integer sum = -5
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
SLLN ORIGINAL joint word1 mean := div(-5, 3)
After
SLLN ORIGINAL joint word1 mean = -5/3
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S026
Before
SLLN ORIGINAL joint word1 mean minus mu := sub(-5/3, -3/2)
After
SLLN ORIGINAL joint word1 mean minus mu = -1/6
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
source roles = [3, 1, [-2, -2, -1], -3/2, 1/2]
After
derived = [-5/3, 1/6, No, null, 1, 1/8]
Conditions: none
For EVERY word test |mean-mu|>=epsilon,equality included.A zero-mass deviating word still contributes to the combinatorial count,while probability is weighted separately.Non-deviating words need no unused probability product under the source.This finite event is not an almost-sure limit certificate.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S028
Before
SLLN ORIGINAL joint word2 integer sum := sum(-2, -1, -2)
After
SLLN ORIGINAL joint word2 integer sum = -5
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
SLLN ORIGINAL joint word2 mean := div(-5, 3)
After
SLLN ORIGINAL joint word2 mean = -5/3
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
SLLN ORIGINAL joint word2 mean minus mu := sub(-5/3, -3/2)
After
SLLN ORIGINAL joint word2 mean minus mu = -1/6
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
source roles = [3, 2, [-2, -1, -2], -3/2, 1/2]
After
derived = [-5/3, 1/6, No, null, 1, 1/8]
Conditions: none
For EVERY word test |mean-mu|>=epsilon,equality included.A zero-mass deviating word still contributes to the combinatorial count,while probability is weighted separately.Non-deviating words need no unused probability product under the source.This finite event is not an almost-sure limit certificate.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S032
Before
SLLN ORIGINAL joint word3 integer sum := sum(-2, -1, -1)
After
SLLN ORIGINAL joint word3 integer sum = -4
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S033
Before
SLLN ORIGINAL joint word3 mean := div(-4, 3)
After
SLLN ORIGINAL joint word3 mean = -4/3
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
SLLN ORIGINAL joint word3 mean minus mu := sub(-4/3, -3/2)
After
SLLN ORIGINAL joint word3 mean minus mu = 1/6
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
source roles = [3, 3, [-2, -1, -1], -3/2, 1/2]
After
derived = [-4/3, 1/6, No, null, 1, 1/8]
Conditions: none
For EVERY word test |mean-mu|>=epsilon,equality included.A zero-mass deviating word still contributes to the combinatorial count,while probability is weighted separately.Non-deviating words need no unused probability product under the source.This finite event is not an almost-sure limit certificate.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S036
Before
SLLN ORIGINAL joint word4 integer sum := sum(-1, -2, -2)
After
SLLN ORIGINAL joint word4 integer sum = -5
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
SLLN ORIGINAL joint word4 mean := div(-5, 3)
After
SLLN ORIGINAL joint word4 mean = -5/3
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S038
Before
SLLN ORIGINAL joint word4 mean minus mu := sub(-5/3, -3/2)
After
SLLN ORIGINAL joint word4 mean minus mu = -1/6
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S039
Before
source roles = [3, 4, [-1, -2, -2], -3/2, 1/2]
After
derived = [-5/3, 1/6, No, null, 1, 1/8]
Conditions: none
For EVERY word test |mean-mu|>=epsilon,equality included.A zero-mass deviating word still contributes to the combinatorial count,while probability is weighted separately.Non-deviating words need no unused probability product under the source.This finite event is not an almost-sure limit certificate.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S040
Before
SLLN ORIGINAL joint word5 integer sum := sum(-1, -2, -1)
After
SLLN ORIGINAL joint word5 integer sum = -4
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S041
Before
SLLN ORIGINAL joint word5 mean := div(-4, 3)
After
SLLN ORIGINAL joint word5 mean = -4/3
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S042
Before
SLLN ORIGINAL joint word5 mean minus mu := sub(-4/3, -3/2)
After
SLLN ORIGINAL joint word5 mean minus mu = 1/6
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S043
Before
source roles = [3, 5, [-1, -2, -1], -3/2, 1/2]
After
derived = [-4/3, 1/6, No, null, 1, 1/8]
Conditions: none
For EVERY word test |mean-mu|>=epsilon,equality included.A zero-mass deviating word still contributes to the combinatorial count,while probability is weighted separately.Non-deviating words need no unused probability product under the source.This finite event is not an almost-sure limit certificate.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S044
Before
SLLN ORIGINAL joint word6 integer sum := sum(-1, -1, -2)
After
SLLN ORIGINAL joint word6 integer sum = -4
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S045
Before
SLLN ORIGINAL joint word6 mean := div(-4, 3)
After
SLLN ORIGINAL joint word6 mean = -4/3
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S046
Before
SLLN ORIGINAL joint word6 mean minus mu := sub(-4/3, -3/2)
After
SLLN ORIGINAL joint word6 mean minus mu = 1/6
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S047
Before
source roles = [3, 6, [-1, -1, -2], -3/2, 1/2]
After
derived = [-4/3, 1/6, No, null, 1, 1/8]
Conditions: none
For EVERY word test |mean-mu|>=epsilon,equality included.A zero-mass deviating word still contributes to the combinatorial count,while probability is weighted separately.Non-deviating words need no unused probability product under the source.This finite event is not an almost-sure limit certificate.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S048
Before
SLLN ORIGINAL joint word7 integer sum := sum(-1, -1, -1)
After
SLLN ORIGINAL joint word7 integer sum = -3
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S049
Before
SLLN ORIGINAL joint word7 mean := div(-3, 3)
After
SLLN ORIGINAL joint word7 mean = -1
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S050
Before
SLLN ORIGINAL joint word7 mean minus mu := sub(-1, -3/2)
After
SLLN ORIGINAL joint word7 mean minus mu = 1/2
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S051
Before
SLLN ORIGINAL deviating word7 probability factor0 := mul(1, 1/2)
After
SLLN ORIGINAL deviating word7 probability factor0 = 1/2
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S052
Before
SLLN ORIGINAL deviating word7 probability factor1 := mul(1/2, 1/2)
After
SLLN ORIGINAL deviating word7 probability factor1 = 1/4
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S053
Before
SLLN ORIGINAL deviating word7 probability factor2 := mul(1/4, 1/2)
After
SLLN ORIGINAL deviating word7 probability factor2 = 1/8
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S054
Before
SLLN ORIGINAL deviation probability word7 := sum(1/8, 1/8)
After
SLLN ORIGINAL deviation probability word7 = 1/4
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S055
Before
source roles = [3, 7, [-1, -1, -1], -3/2, 1/2]
After
derived = [-1, 1/2, Yes, 1/8, 2, 1/4]
Conditions: none
For EVERY word test |mean-mu|>=epsilon,equality included.A zero-mass deviating word still contributes to the combinatorial count,while probability is weighted separately.Non-deviating words need no unused probability product under the source.This finite event is not an almost-sure limit certificate.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S056
S057 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S056 · premise_and_result_audit — Recompute SLLN ORIGINAL deviation probability word7 with exact rational arithmetic, not rounded decimals.S055 · premise_and_result_audit — Recompute SLLN ORIGINAL deviating word7 probability factor2 with exact rational arithmetic, not rounded decimals.S054 · premise_and_result_audit — Recompute SLLN ORIGINAL deviating word7 probability factor1 with exact rational arithmetic, not rounded decimals.S053 · premise_and_result_audit — Recompute SLLN ORIGINAL deviating word7 probability factor0 with exact rational arithmetic, not rounded decimals.S052 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word7 mean minus mu with exact rational arithmetic, not rounded decimals.S051 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word7 mean with exact rational arithmetic, not rounded decimals.S050 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word7 integer sum with exact rational arithmetic, not rounded decimals.S049 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S048 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word6 mean minus mu with exact rational arithmetic, not rounded decimals.S047 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word6 mean with exact rational arithmetic, not rounded decimals.S046 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word6 integer sum with exact rational arithmetic, not rounded decimals.S045 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S044 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word5 mean minus mu with exact rational arithmetic, not rounded decimals.S043 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word5 mean with exact rational arithmetic, not rounded decimals.S042 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word5 integer sum with exact rational arithmetic, not rounded decimals.S041 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S040 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word4 mean minus mu with exact rational arithmetic, not rounded decimals.S039 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word4 mean with exact rational arithmetic, not rounded decimals.S038 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word4 integer sum with exact rational arithmetic, not rounded decimals.S037 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S036 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word3 mean minus mu with exact rational arithmetic, not rounded decimals.S035 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word3 mean with exact rational arithmetic, not rounded decimals.S034 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word3 integer sum with exact rational arithmetic, not rounded decimals.S033 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S032 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word2 mean minus mu with exact rational arithmetic, not rounded decimals.S031 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word2 mean with exact rational arithmetic, not rounded decimals.S030 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word2 integer sum with exact rational arithmetic, not rounded decimals.S029 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S028 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word1 mean minus mu with exact rational arithmetic, not rounded decimals.S027 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word1 mean with exact rational arithmetic, not rounded decimals.S026 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word1 integer sum with exact rational arithmetic, not rounded decimals.S025 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S024 · premise_and_result_audit — Recompute SLLN ORIGINAL deviation probability word0 with exact rational arithmetic, not rounded decimals.S023 · premise_and_result_audit — Recompute SLLN ORIGINAL deviating word0 probability factor2 with exact rational arithmetic, not rounded decimals.S022 · premise_and_result_audit — Recompute SLLN ORIGINAL deviating word0 probability factor1 with exact rational arithmetic, not rounded decimals.S021 · premise_and_result_audit — Recompute SLLN ORIGINAL deviating word0 probability factor0 with exact rational arithmetic, not rounded decimals.S020 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word0 mean minus mu with exact rational arithmetic, not rounded decimals.S019 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word0 mean with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word0 integer sum with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S016 · premise_and_result_audit — Recompute SLLN ORIGINAL common marginal ORIGINAL second minus mean squared with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Recompute SLLN ORIGINAL common marginal ORIGINAL mean squared with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Recompute SLLN ORIGINAL common marginal ORIGINAL second moment sum1 with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute SLLN ORIGINAL common marginal ORIGINAL weighted square1 with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute SLLN ORIGINAL common marginal exact unbounded value square1 with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute SLLN ORIGINAL common marginal ORIGINAL mean sum1 with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute SLLN ORIGINAL common marginal ORIGINAL weighted value1 with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute SLLN ORIGINAL common marginal ORIGINAL second moment sum0 with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute SLLN ORIGINAL common marginal ORIGINAL weighted square0 with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute SLLN ORIGINAL common marginal exact unbounded value square0 with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute SLLN ORIGINAL common marginal ORIGINAL mean sum0 with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute SLLN ORIGINAL common marginal ORIGINAL weighted value0 with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute SLLN ORIGINAL common marginal ORIGINAL mass sum1 with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute SLLN ORIGINAL common marginal ORIGINAL mass sum0 with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S001 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.(3, -3/2, 1/2, 2, 1/4, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)
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: b5def7c9e95851b697f0bd7d8e81d520b5747e300396241d64c3b93db524ad89candidate record SHA-256: ac80b58dba11a3ea0930907c02bc27560ba104d5e2cb18f3f80d63e589001fecsemantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437Lean: NOT_RUN · human review: NOT_PERFORMED