# Question 27619: The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit [Return to catalogue page 139](https://ocme.evemisslab.com/en/questions/page/139/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations/#q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0069) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0054-27619 · q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0069 ## Original question and givens Use the aligned finite law values=[1, 2], 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=4 and epsilon=1/3>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=[1, 2], 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=4 and epsilon=1/3>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). ## 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-strong-law-large-numbers-finite-valued-iid](https://ocme.evemisslab.com/en/mko/mko-strong-law-large-numbers-finite-valued-iid/) ## Step-by-step derivation S001 · impliesBefore 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. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · impliesBefore 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. Step identifiers and dependenciesRule: SLLN_infinite_IID_EXACT_GIVEN_premise_not_finite_observation_inference · Uses: G001, S001 S003 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore SLLN ORIGINAL common marginal ORIGINAL weighted value0 := mul(1, 1/2) After SLLN ORIGINAL common marginal ORIGINAL weighted value0 = 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore SLLN ORIGINAL common marginal ORIGINAL mean sum0 := sum(0, 1/2) After SLLN ORIGINAL common marginal ORIGINAL mean 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore SLLN ORIGINAL common marginal exact unbounded value square0 := mul(1, 1) After SLLN ORIGINAL common marginal exact unbounded value square0 = 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore SLLN ORIGINAL common marginal ORIGINAL weighted square0 := mul(1, 1/2) After SLLN ORIGINAL common marginal ORIGINAL weighted square0 = 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore SLLN ORIGINAL common marginal ORIGINAL second moment sum0 := sum(0, 1/2) After SLLN ORIGINAL common marginal ORIGINAL second moment 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore SLLN ORIGINAL common marginal ORIGINAL weighted value1 := mul(2, 1/2) After SLLN ORIGINAL common marginal ORIGINAL weighted value1 = 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore SLLN ORIGINAL common marginal ORIGINAL mean sum1 := sum(1/2, 1) 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore SLLN ORIGINAL common marginal exact unbounded value square1 := mul(2, 2) After SLLN ORIGINAL common marginal exact unbounded value square1 = 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore SLLN ORIGINAL common marginal ORIGINAL weighted square1 := mul(4, 1/2) After SLLN ORIGINAL common marginal ORIGINAL weighted square1 = 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore SLLN ORIGINAL common marginal ORIGINAL second moment sum1 := sum(1/2, 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · impliesBefore source roles = [SLLN ORIGINAL common marginal, [1, 2], [[1, 2], [1, 2]]] After derived = [[[1, 1/2], [2, 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. Step identifiers and dependenciesRule: SLLN_complete_ORIGINAL_numeric_value_probability_alignment_ALL_moments · Uses: G001, S016 S018 · calculationBefore SLLN ORIGINAL joint word0 integer sum := sum(1, 1, 1, 1) After SLLN ORIGINAL joint word0 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore SLLN ORIGINAL joint word0 mean := div(4, 4) After SLLN ORIGINAL joint word0 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore SLLN ORIGINAL joint word0 mean minus mu := sub(1, 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore SLLN ORIGINAL deviating word0 probability factor3 := mul(1/8, 1/2) After SLLN ORIGINAL deviating word0 probability factor3 = 1/16 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore SLLN ORIGINAL deviation probability word0 := sum(0, 1/16) After SLLN ORIGINAL deviation probability word0 = 1/16 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · impliesBefore source roles = [4, 0, [1, 1, 1, 1], 3/2, 1/3] After derived = [1, 1/2, Yes, 1/16, 1, 1/16] 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. Step identifiers and dependenciesRule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S025 S027 · calculationBefore SLLN ORIGINAL joint word1 integer sum := sum(1, 1, 1, 2) 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore SLLN ORIGINAL joint word1 mean := div(5, 4) After SLLN ORIGINAL joint word1 mean = 5/4 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore SLLN ORIGINAL joint word1 mean minus mu := sub(5/4, 3/2) After SLLN ORIGINAL joint word1 mean minus mu = -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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · impliesBefore source roles = [4, 1, [1, 1, 1, 2], 3/2, 1/3] After derived = [5/4, 1/4, No, null, 1, 1/16] 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. Step identifiers and dependenciesRule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S029 S031 · calculationBefore SLLN ORIGINAL joint word2 integer sum := sum(1, 1, 2, 1) 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore SLLN ORIGINAL joint word2 mean := div(5, 4) After SLLN ORIGINAL joint word2 mean = 5/4 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore SLLN ORIGINAL joint word2 mean minus mu := sub(5/4, 3/2) After SLLN ORIGINAL joint word2 mean minus mu = -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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · impliesBefore source roles = [4, 2, [1, 1, 2, 1], 3/2, 1/3] After derived = [5/4, 1/4, No, null, 1, 1/16] 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. Step identifiers and dependenciesRule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S033 S035 · calculationBefore SLLN ORIGINAL joint word3 integer sum := sum(1, 1, 2, 2) After SLLN ORIGINAL joint word3 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · calculationBefore SLLN ORIGINAL joint word3 mean := div(6, 4) After SLLN ORIGINAL joint word3 mean = 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · calculationBefore SLLN ORIGINAL joint word3 mean minus mu := sub(3/2, 3/2) After SLLN ORIGINAL joint word3 mean minus mu = 0 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · impliesBefore source roles = [4, 3, [1, 1, 2, 2], 3/2, 1/3] After derived = [3/2, 0, No, null, 1, 1/16] 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. Step identifiers and dependenciesRule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S037 S039 · calculationBefore SLLN ORIGINAL joint word4 integer sum := sum(1, 2, 1, 1) 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 S040 · calculationBefore SLLN ORIGINAL joint word4 mean := div(5, 4) After SLLN ORIGINAL joint word4 mean = 5/4 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · calculationBefore SLLN ORIGINAL joint word4 mean minus mu := sub(5/4, 3/2) After SLLN ORIGINAL joint word4 mean minus mu = -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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · impliesBefore source roles = [4, 4, [1, 2, 1, 1], 3/2, 1/3] After derived = [5/4, 1/4, No, null, 1, 1/16] 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. Step identifiers and dependenciesRule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S041 S043 · calculationBefore SLLN ORIGINAL joint word5 integer sum := sum(1, 2, 1, 2) After SLLN ORIGINAL joint word5 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · calculationBefore SLLN ORIGINAL joint word5 mean := div(6, 4) After SLLN ORIGINAL joint word5 mean = 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043 S045 · calculationBefore SLLN ORIGINAL joint word5 mean minus mu := sub(3/2, 3/2) After SLLN ORIGINAL joint word5 mean minus mu = 0 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044 S046 · impliesBefore source roles = [4, 5, [1, 2, 1, 2], 3/2, 1/3] After derived = [3/2, 0, No, null, 1, 1/16] 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. Step identifiers and dependenciesRule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S045 S047 · calculationBefore SLLN ORIGINAL joint word6 integer sum := sum(1, 2, 2, 1) After SLLN ORIGINAL joint word6 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S046 S048 · calculationBefore SLLN ORIGINAL joint word6 mean := div(6, 4) After SLLN ORIGINAL joint word6 mean = 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S047 S049 · calculationBefore SLLN ORIGINAL joint word6 mean minus mu := sub(3/2, 3/2) After SLLN ORIGINAL joint word6 mean minus mu = 0 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S048 S050 · impliesBefore source roles = [4, 6, [1, 2, 2, 1], 3/2, 1/3] After derived = [3/2, 0, No, null, 1, 1/16] 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. Step identifiers and dependenciesRule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S049 S051 · calculationBefore SLLN ORIGINAL joint word7 integer sum := sum(1, 2, 2, 2) After SLLN ORIGINAL joint word7 integer sum = 7 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S050 S052 · calculationBefore SLLN ORIGINAL joint word7 mean := div(7, 4) After SLLN ORIGINAL joint word7 mean = 7/4 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S051 S053 · calculationBefore SLLN ORIGINAL joint word7 mean minus mu := sub(7/4, 3/2) After SLLN ORIGINAL joint word7 mean minus mu = 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S052 S054 · impliesBefore source roles = [4, 7, [1, 2, 2, 2], 3/2, 1/3] After derived = [7/4, 1/4, No, null, 1, 1/16] 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. Step identifiers and dependenciesRule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S053 S055 · calculationBefore SLLN ORIGINAL joint word8 integer sum := sum(2, 1, 1, 1) After SLLN ORIGINAL joint word8 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S054 S056 · calculationBefore SLLN ORIGINAL joint word8 mean := div(5, 4) After SLLN ORIGINAL joint word8 mean = 5/4 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S055 S057 · calculationBefore SLLN ORIGINAL joint word8 mean minus mu := sub(5/4, 3/2) After SLLN ORIGINAL joint word8 mean minus mu = -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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S056 S058 · impliesBefore source roles = [4, 8, [2, 1, 1, 1], 3/2, 1/3] After derived = [5/4, 1/4, No, null, 1, 1/16] 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. Step identifiers and dependenciesRule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S057 S059 · calculationBefore SLLN ORIGINAL joint word9 integer sum := sum(2, 1, 1, 2) After SLLN ORIGINAL joint word9 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S058 S060 · calculationBefore SLLN ORIGINAL joint word9 mean := div(6, 4) After SLLN ORIGINAL joint word9 mean = 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S059 S061 · calculationBefore SLLN ORIGINAL joint word9 mean minus mu := sub(3/2, 3/2) After SLLN ORIGINAL joint word9 mean minus mu = 0 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S060 S062 · impliesBefore source roles = [4, 9, [2, 1, 1, 2], 3/2, 1/3] After derived = [3/2, 0, No, null, 1, 1/16] 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. Step identifiers and dependenciesRule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S061 S063 · calculationBefore SLLN ORIGINAL joint word10 integer sum := sum(2, 1, 2, 1) After SLLN ORIGINAL joint word10 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S062 S064 · calculationBefore SLLN ORIGINAL joint word10 mean := div(6, 4) After SLLN ORIGINAL joint word10 mean = 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S063 S065 · calculationBefore SLLN ORIGINAL joint word10 mean minus mu := sub(3/2, 3/2) After SLLN ORIGINAL joint word10 mean minus mu = 0 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S064 S066 · impliesBefore source roles = [4, 10, [2, 1, 2, 1], 3/2, 1/3] After derived = [3/2, 0, No, null, 1, 1/16] 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. Step identifiers and dependenciesRule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S065 S067 · calculationBefore SLLN ORIGINAL joint word11 integer sum := sum(2, 1, 2, 2) After SLLN ORIGINAL joint word11 integer sum = 7 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S066 S068 · calculationBefore SLLN ORIGINAL joint word11 mean := div(7, 4) After SLLN ORIGINAL joint word11 mean = 7/4 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S067 S069 · calculationBefore SLLN ORIGINAL joint word11 mean minus mu := sub(7/4, 3/2) After SLLN ORIGINAL joint word11 mean minus mu = 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S068 S070 · impliesBefore source roles = [4, 11, [2, 1, 2, 2], 3/2, 1/3] After derived = [7/4, 1/4, No, null, 1, 1/16] 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. Step identifiers and dependenciesRule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S069 S071 · calculationBefore SLLN ORIGINAL joint word12 integer sum := sum(2, 2, 1, 1) After SLLN ORIGINAL joint word12 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S070 S072 · calculationBefore SLLN ORIGINAL joint word12 mean := div(6, 4) After SLLN ORIGINAL joint word12 mean = 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S071 S073 · calculationBefore SLLN ORIGINAL joint word12 mean minus mu := sub(3/2, 3/2) After SLLN ORIGINAL joint word12 mean minus mu = 0 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S072 S074 · impliesBefore source roles = [4, 12, [2, 2, 1, 1], 3/2, 1/3] After derived = [3/2, 0, No, null, 1, 1/16] 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. Step identifiers and dependenciesRule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S073 S075 · calculationBefore SLLN ORIGINAL joint word13 integer sum := sum(2, 2, 1, 2) After SLLN ORIGINAL joint word13 integer sum = 7 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S074 S076 · calculationBefore SLLN ORIGINAL joint word13 mean := div(7, 4) After SLLN ORIGINAL joint word13 mean = 7/4 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S075 S077 · calculationBefore SLLN ORIGINAL joint word13 mean minus mu := sub(7/4, 3/2) After SLLN ORIGINAL joint word13 mean minus mu = 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S076 S078 · impliesBefore source roles = [4, 13, [2, 2, 1, 2], 3/2, 1/3] After derived = [7/4, 1/4, No, null, 1, 1/16] 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. Step identifiers and dependenciesRule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S077 S079 · calculationBefore SLLN ORIGINAL joint word14 integer sum := sum(2, 2, 2, 1) After SLLN ORIGINAL joint word14 integer sum = 7 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S078 S080 · calculationBefore SLLN ORIGINAL joint word14 mean := div(7, 4) After SLLN ORIGINAL joint word14 mean = 7/4 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S079 S081 · calculationBefore SLLN ORIGINAL joint word14 mean minus mu := sub(7/4, 3/2) After SLLN ORIGINAL joint word14 mean minus mu = 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S080 S082 · impliesBefore source roles = [4, 14, [2, 2, 2, 1], 3/2, 1/3] After derived = [7/4, 1/4, No, null, 1, 1/16] 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. Step identifiers and dependenciesRule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S081 S083 · calculationBefore SLLN ORIGINAL joint word15 integer sum := sum(2, 2, 2, 2) After SLLN ORIGINAL joint word15 integer sum = 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S082 S084 · calculationBefore SLLN ORIGINAL joint word15 mean := div(8, 4) After SLLN ORIGINAL joint word15 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S083 S085 · calculationBefore SLLN ORIGINAL joint word15 mean minus mu := sub(2, 3/2) After SLLN ORIGINAL joint word15 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S084 S086 · calculationBefore SLLN ORIGINAL deviating word15 probability factor0 := mul(1, 1/2) After SLLN ORIGINAL deviating word15 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S085 S087 · calculationBefore SLLN ORIGINAL deviating word15 probability factor1 := mul(1/2, 1/2) After SLLN ORIGINAL deviating word15 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S086 S088 · calculationBefore SLLN ORIGINAL deviating word15 probability factor2 := mul(1/4, 1/2) After SLLN ORIGINAL deviating word15 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S087 S089 · calculationBefore SLLN ORIGINAL deviating word15 probability factor3 := mul(1/8, 1/2) After SLLN ORIGINAL deviating word15 probability factor3 = 1/16 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S088 S090 · calculationBefore SLLN ORIGINAL deviation probability word15 := sum(1/16, 1/16) After SLLN ORIGINAL deviation probability word15 = 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S089 S091 · impliesBefore source roles = [4, 15, [2, 2, 2, 2], 3/2, 1/3] After derived = [2, 1/2, Yes, 1/16, 2, 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. Step identifiers and dependenciesRule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S090 ## Reverse checks - S091 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S090 · premise_and_result_audit — Recompute SLLN ORIGINAL deviation probability word15 with exact rational arithmetic, not rounded decimals. - S089 · premise_and_result_audit — Recompute SLLN ORIGINAL deviating word15 probability factor3 with exact rational arithmetic, not rounded decimals. - S088 · premise_and_result_audit — Recompute SLLN ORIGINAL deviating word15 probability factor2 with exact rational arithmetic, not rounded decimals. - S087 · premise_and_result_audit — Recompute SLLN ORIGINAL deviating word15 probability factor1 with exact rational arithmetic, not rounded decimals. - S086 · premise_and_result_audit — Recompute SLLN ORIGINAL deviating word15 probability factor0 with exact rational arithmetic, not rounded decimals. - S085 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word15 mean minus mu with exact rational arithmetic, not rounded decimals. - S084 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word15 mean with exact rational arithmetic, not rounded decimals. - S083 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word15 integer sum with exact rational arithmetic, not rounded decimals. - S082 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S081 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word14 mean minus mu with exact rational arithmetic, not rounded decimals. - S080 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word14 mean with exact rational arithmetic, not rounded decimals. - S079 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word14 integer sum with exact rational arithmetic, not rounded decimals. - S078 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S077 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word13 mean minus mu with exact rational arithmetic, not rounded decimals. - S076 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word13 mean with exact rational arithmetic, not rounded decimals. - S075 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word13 integer sum with exact rational arithmetic, not rounded decimals. - S074 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S073 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word12 mean minus mu with exact rational arithmetic, not rounded decimals. - S072 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word12 mean with exact rational arithmetic, not rounded decimals. - S071 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word12 integer sum with exact rational arithmetic, not rounded decimals. - S070 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S069 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word11 mean minus mu with exact rational arithmetic, not rounded decimals. - S068 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word11 mean with exact rational arithmetic, not rounded decimals. - S067 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word11 integer sum with exact rational arithmetic, not rounded decimals. - S066 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S065 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word10 mean minus mu with exact rational arithmetic, not rounded decimals. - S064 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word10 mean with exact rational arithmetic, not rounded decimals. - S063 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word10 integer sum with exact rational arithmetic, not rounded decimals. - S062 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S061 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word9 mean minus mu with exact rational arithmetic, not rounded decimals. - S060 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word9 mean with exact rational arithmetic, not rounded decimals. - S059 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word9 integer sum with exact rational arithmetic, not rounded decimals. - S058 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S057 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word8 mean minus mu with exact rational arithmetic, not rounded decimals. - S056 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word8 mean with exact rational arithmetic, not rounded decimals. - S055 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word8 integer sum with exact rational arithmetic, not rounded decimals. - S054 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S053 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word7 mean minus mu with exact rational arithmetic, not rounded decimals. - S052 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word7 mean with exact rational arithmetic, not rounded decimals. - S051 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word7 integer sum with exact rational arithmetic, not rounded decimals. - S050 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S049 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word6 mean minus mu with exact rational arithmetic, not rounded decimals. - S048 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word6 mean with exact rational arithmetic, not rounded decimals. - S047 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word6 integer sum with exact rational arithmetic, not rounded decimals. - S046 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S045 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word5 mean minus mu with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word5 mean with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word5 integer sum with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S041 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word4 mean minus mu with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word4 mean with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word4 integer sum with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S037 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word3 mean minus mu with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word3 mean with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word3 integer sum with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S033 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word2 mean minus mu with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word2 mean with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word2 integer sum with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S029 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word1 mean minus mu with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word1 mean with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word1 integer sum with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S025 · premise_and_result_audit — Recompute SLLN ORIGINAL deviation probability word0 with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute SLLN ORIGINAL deviating word0 probability factor3 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. ## Result (4, 3/2, 1/3, 2, 1/8, 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. ## 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: f96a957ede28f8fe5a8635bf245f00c71c590d39ab0e7456abd6cfa9da3dd25b - candidate record SHA-256: 03e5c61a19393fab164aca81ca41d847a676d6d3526318e04aaa8999278f0fd4 - semantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54 - question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0069.json)