# Question 27622: 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-0072) 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. R0054-27622 · q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0072 ## Original question and givens Use the aligned finite law values=[-1, 0], 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=5 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, 0], 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=5 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(0, 1/2) After SLLN ORIGINAL common marginal ORIGINAL weighted value1 = 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, S009 S011 · calculationBefore SLLN ORIGINAL common marginal ORIGINAL mean sum1 := sum(-1/2, 0) After SLLN ORIGINAL common marginal ORIGINAL mean sum1 = -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, S010 S012 · calculationBefore SLLN ORIGINAL common marginal exact unbounded value square1 := mul(0, 0) After SLLN ORIGINAL common marginal exact unbounded value square1 = 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, S011 S013 · calculationBefore SLLN ORIGINAL common marginal ORIGINAL weighted square1 := mul(0, 1/2) After SLLN ORIGINAL common marginal ORIGINAL weighted square1 = 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, S012 S014 · calculationBefore SLLN ORIGINAL common marginal ORIGINAL second moment sum1 := sum(1/2, 0) After SLLN ORIGINAL common marginal ORIGINAL second moment sum1 = 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, S013 S015 · calculationBefore SLLN ORIGINAL common marginal ORIGINAL mean squared := mul(-1/2, -1/2) After SLLN ORIGINAL common marginal ORIGINAL 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, S014 S016 · calculationBefore SLLN ORIGINAL common marginal ORIGINAL second minus mean squared := sub(1/2, 1/4) 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, 0], [[1, 2], [1, 2]]] After derived = [[[-1, 1/2], [0, 1/2]], 1, -1/2, 1/2, 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, -1) After SLLN ORIGINAL joint word0 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, S017 S019 · calculationBefore SLLN ORIGINAL joint word0 mean := div(-5, 5) 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, -1/2) 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 deviating word0 probability factor4 := mul(1/16, 1/2) After SLLN ORIGINAL deviating word0 probability factor4 = 1/32 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 · calculationBefore SLLN ORIGINAL deviation probability word0 := sum(0, 1/32) After SLLN ORIGINAL deviation probability word0 = 1/32 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, S025 S027 · impliesBefore source roles = [5, 0, [-1, -1, -1, -1, -1], -1/2, 1/3] After derived = [-1, 1/2, Yes, 1/32, 1, 1/32] 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, S026 S028 · calculationBefore SLLN ORIGINAL joint word1 integer sum := sum(-1, -1, -1, -1, 0) After SLLN ORIGINAL joint word1 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, S027 S029 · calculationBefore SLLN ORIGINAL joint word1 mean := div(-4, 5) After SLLN ORIGINAL joint word1 mean = -4/5 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, S028 S030 · calculationBefore SLLN ORIGINAL joint word1 mean minus mu := sub(-4/5, -1/2) After SLLN ORIGINAL joint word1 mean minus mu = -3/10 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, S029 S031 · impliesBefore source roles = [5, 1, [-1, -1, -1, -1, 0], -1/2, 1/3] After derived = [-4/5, 3/10, No, null, 1, 1/32] 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, S030 S032 · calculationBefore SLLN ORIGINAL joint word2 integer sum := sum(-1, -1, -1, 0, -1) After SLLN ORIGINAL joint word2 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, S031 S033 · calculationBefore SLLN ORIGINAL joint word2 mean := div(-4, 5) After SLLN ORIGINAL joint word2 mean = -4/5 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, S032 S034 · calculationBefore SLLN ORIGINAL joint word2 mean minus mu := sub(-4/5, -1/2) After SLLN ORIGINAL joint word2 mean minus mu = -3/10 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, S033 S035 · impliesBefore source roles = [5, 2, [-1, -1, -1, 0, -1], -1/2, 1/3] After derived = [-4/5, 3/10, No, null, 1, 1/32] 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, S034 S036 · calculationBefore SLLN ORIGINAL joint word3 integer sum := sum(-1, -1, -1, 0, 0) After SLLN ORIGINAL joint word3 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · calculationBefore SLLN ORIGINAL joint word3 mean := div(-3, 5) After SLLN ORIGINAL joint word3 mean = -3/5 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, S036 S038 · calculationBefore SLLN ORIGINAL joint word3 mean minus mu := sub(-3/5, -1/2) After SLLN ORIGINAL joint word3 mean minus mu = -1/10 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, S037 S039 · impliesBefore source roles = [5, 3, [-1, -1, -1, 0, 0], -1/2, 1/3] After derived = [-3/5, 1/10, No, null, 1, 1/32] 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, S038 S040 · calculationBefore SLLN ORIGINAL joint word4 integer sum := sum(-1, -1, 0, -1, -1) After SLLN ORIGINAL joint word4 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, S039 S041 · calculationBefore SLLN ORIGINAL joint word4 mean := div(-4, 5) After SLLN ORIGINAL joint word4 mean = -4/5 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, S040 S042 · calculationBefore SLLN ORIGINAL joint word4 mean minus mu := sub(-4/5, -1/2) After SLLN ORIGINAL joint word4 mean minus mu = -3/10 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, S041 S043 · impliesBefore source roles = [5, 4, [-1, -1, 0, -1, -1], -1/2, 1/3] After derived = [-4/5, 3/10, No, null, 1, 1/32] 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, S042 S044 · calculationBefore SLLN ORIGINAL joint word5 integer sum := sum(-1, -1, 0, -1, 0) After SLLN ORIGINAL joint word5 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043 S045 · calculationBefore SLLN ORIGINAL joint word5 mean := div(-3, 5) After SLLN ORIGINAL joint word5 mean = -3/5 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, S044 S046 · calculationBefore SLLN ORIGINAL joint word5 mean minus mu := sub(-3/5, -1/2) After SLLN ORIGINAL joint word5 mean minus mu = -1/10 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, S045 S047 · impliesBefore source roles = [5, 5, [-1, -1, 0, -1, 0], -1/2, 1/3] After derived = [-3/5, 1/10, No, null, 1, 1/32] 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, S046 S048 · calculationBefore SLLN ORIGINAL joint word6 integer sum := sum(-1, -1, 0, 0, -1) After SLLN ORIGINAL joint word6 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S047 S049 · calculationBefore SLLN ORIGINAL joint word6 mean := div(-3, 5) After SLLN ORIGINAL joint word6 mean = -3/5 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, S048 S050 · calculationBefore SLLN ORIGINAL joint word6 mean minus mu := sub(-3/5, -1/2) After SLLN ORIGINAL joint word6 mean minus mu = -1/10 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, S049 S051 · impliesBefore source roles = [5, 6, [-1, -1, 0, 0, -1], -1/2, 1/3] After derived = [-3/5, 1/10, No, null, 1, 1/32] 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, S050 S052 · calculationBefore SLLN ORIGINAL joint word7 integer sum := sum(-1, -1, 0, 0, 0) After SLLN ORIGINAL joint word7 integer sum = -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, S051 S053 · calculationBefore SLLN ORIGINAL joint word7 mean := div(-2, 5) After SLLN ORIGINAL joint word7 mean = -2/5 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, S052 S054 · calculationBefore SLLN ORIGINAL joint word7 mean minus mu := sub(-2/5, -1/2) After SLLN ORIGINAL joint word7 mean minus mu = 1/10 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, S053 S055 · impliesBefore source roles = [5, 7, [-1, -1, 0, 0, 0], -1/2, 1/3] After derived = [-2/5, 1/10, No, null, 1, 1/32] 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, S054 S056 · calculationBefore SLLN ORIGINAL joint word8 integer sum := sum(-1, 0, -1, -1, -1) After SLLN ORIGINAL joint word8 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, S055 S057 · calculationBefore SLLN ORIGINAL joint word8 mean := div(-4, 5) After SLLN ORIGINAL joint word8 mean = -4/5 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, S056 S058 · calculationBefore SLLN ORIGINAL joint word8 mean minus mu := sub(-4/5, -1/2) After SLLN ORIGINAL joint word8 mean minus mu = -3/10 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, S057 S059 · impliesBefore source roles = [5, 8, [-1, 0, -1, -1, -1], -1/2, 1/3] After derived = [-4/5, 3/10, No, null, 1, 1/32] 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, S058 S060 · calculationBefore SLLN ORIGINAL joint word9 integer sum := sum(-1, 0, -1, -1, 0) After SLLN ORIGINAL joint word9 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S059 S061 · calculationBefore SLLN ORIGINAL joint word9 mean := div(-3, 5) After SLLN ORIGINAL joint word9 mean = -3/5 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, S060 S062 · calculationBefore SLLN ORIGINAL joint word9 mean minus mu := sub(-3/5, -1/2) After SLLN ORIGINAL joint word9 mean minus mu = -1/10 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, S061 S063 · impliesBefore source roles = [5, 9, [-1, 0, -1, -1, 0], -1/2, 1/3] After derived = [-3/5, 1/10, No, null, 1, 1/32] 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, S062 S064 · calculationBefore SLLN ORIGINAL joint word10 integer sum := sum(-1, 0, -1, 0, -1) After SLLN ORIGINAL joint word10 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S063 S065 · calculationBefore SLLN ORIGINAL joint word10 mean := div(-3, 5) After SLLN ORIGINAL joint word10 mean = -3/5 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, S064 S066 · calculationBefore SLLN ORIGINAL joint word10 mean minus mu := sub(-3/5, -1/2) After SLLN ORIGINAL joint word10 mean minus mu = -1/10 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, S065 S067 · impliesBefore source roles = [5, 10, [-1, 0, -1, 0, -1], -1/2, 1/3] After derived = [-3/5, 1/10, No, null, 1, 1/32] 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, S066 S068 · calculationBefore SLLN ORIGINAL joint word11 integer sum := sum(-1, 0, -1, 0, 0) After SLLN ORIGINAL joint word11 integer sum = -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, S067 S069 · calculationBefore SLLN ORIGINAL joint word11 mean := div(-2, 5) After SLLN ORIGINAL joint word11 mean = -2/5 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, S068 S070 · calculationBefore SLLN ORIGINAL joint word11 mean minus mu := sub(-2/5, -1/2) After SLLN ORIGINAL joint word11 mean minus mu = 1/10 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, S069 S071 · impliesBefore source roles = [5, 11, [-1, 0, -1, 0, 0], -1/2, 1/3] After derived = [-2/5, 1/10, No, null, 1, 1/32] 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, S070 S072 · calculationBefore SLLN ORIGINAL joint word12 integer sum := sum(-1, 0, 0, -1, -1) After SLLN ORIGINAL joint word12 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S071 S073 · calculationBefore SLLN ORIGINAL joint word12 mean := div(-3, 5) After SLLN ORIGINAL joint word12 mean = -3/5 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, S072 S074 · calculationBefore SLLN ORIGINAL joint word12 mean minus mu := sub(-3/5, -1/2) After SLLN ORIGINAL joint word12 mean minus mu = -1/10 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, S073 S075 · impliesBefore source roles = [5, 12, [-1, 0, 0, -1, -1], -1/2, 1/3] After derived = [-3/5, 1/10, No, null, 1, 1/32] 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, S074 S076 · calculationBefore SLLN ORIGINAL joint word13 integer sum := sum(-1, 0, 0, -1, 0) After SLLN ORIGINAL joint word13 integer sum = -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, S075 S077 · calculationBefore SLLN ORIGINAL joint word13 mean := div(-2, 5) After SLLN ORIGINAL joint word13 mean = -2/5 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, S076 S078 · calculationBefore SLLN ORIGINAL joint word13 mean minus mu := sub(-2/5, -1/2) After SLLN ORIGINAL joint word13 mean minus mu = 1/10 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, S077 S079 · impliesBefore source roles = [5, 13, [-1, 0, 0, -1, 0], -1/2, 1/3] After derived = [-2/5, 1/10, No, null, 1, 1/32] 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, S078 S080 · calculationBefore SLLN ORIGINAL joint word14 integer sum := sum(-1, 0, 0, 0, -1) After SLLN ORIGINAL joint word14 integer sum = -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, S079 S081 · calculationBefore SLLN ORIGINAL joint word14 mean := div(-2, 5) After SLLN ORIGINAL joint word14 mean = -2/5 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, S080 S082 · calculationBefore SLLN ORIGINAL joint word14 mean minus mu := sub(-2/5, -1/2) After SLLN ORIGINAL joint word14 mean minus mu = 1/10 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, S081 S083 · impliesBefore source roles = [5, 14, [-1, 0, 0, 0, -1], -1/2, 1/3] After derived = [-2/5, 1/10, No, null, 1, 1/32] 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, S082 S084 · calculationBefore SLLN ORIGINAL joint word15 integer sum := sum(-1, 0, 0, 0, 0) After SLLN ORIGINAL joint word15 integer sum = -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, S083 S085 · calculationBefore SLLN ORIGINAL joint word15 mean := div(-1, 5) After SLLN ORIGINAL joint word15 mean = -1/5 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, S084 S086 · calculationBefore SLLN ORIGINAL joint word15 mean minus mu := sub(-1/5, -1/2) After SLLN ORIGINAL joint word15 mean minus mu = 3/10 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 · impliesBefore source roles = [5, 15, [-1, 0, 0, 0, 0], -1/2, 1/3] After derived = [-1/5, 3/10, No, null, 1, 1/32] 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, S086 S088 · calculationBefore SLLN ORIGINAL joint word16 integer sum := sum(0, -1, -1, -1, -1) After SLLN ORIGINAL joint word16 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, S087 S089 · calculationBefore SLLN ORIGINAL joint word16 mean := div(-4, 5) After SLLN ORIGINAL joint word16 mean = -4/5 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, S088 S090 · calculationBefore SLLN ORIGINAL joint word16 mean minus mu := sub(-4/5, -1/2) After SLLN ORIGINAL joint word16 mean minus mu = -3/10 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 = [5, 16, [0, -1, -1, -1, -1], -1/2, 1/3] After derived = [-4/5, 3/10, No, null, 1, 1/32] 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 S092 · calculationBefore SLLN ORIGINAL joint word17 integer sum := sum(0, -1, -1, -1, 0) After SLLN ORIGINAL joint word17 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S091 S093 · calculationBefore SLLN ORIGINAL joint word17 mean := div(-3, 5) After SLLN ORIGINAL joint word17 mean = -3/5 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, S092 S094 · calculationBefore SLLN ORIGINAL joint word17 mean minus mu := sub(-3/5, -1/2) After SLLN ORIGINAL joint word17 mean minus mu = -1/10 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, S093 S095 · impliesBefore source roles = [5, 17, [0, -1, -1, -1, 0], -1/2, 1/3] After derived = [-3/5, 1/10, No, null, 1, 1/32] 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, S094 S096 · calculationBefore SLLN ORIGINAL joint word18 integer sum := sum(0, -1, -1, 0, -1) After SLLN ORIGINAL joint word18 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S095 S097 · calculationBefore SLLN ORIGINAL joint word18 mean := div(-3, 5) After SLLN ORIGINAL joint word18 mean = -3/5 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, S096 S098 · calculationBefore SLLN ORIGINAL joint word18 mean minus mu := sub(-3/5, -1/2) After SLLN ORIGINAL joint word18 mean minus mu = -1/10 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, S097 S099 · impliesBefore source roles = [5, 18, [0, -1, -1, 0, -1], -1/2, 1/3] After derived = [-3/5, 1/10, No, null, 1, 1/32] 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, S098 S100 · calculationBefore SLLN ORIGINAL joint word19 integer sum := sum(0, -1, -1, 0, 0) After SLLN ORIGINAL joint word19 integer sum = -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, S099 S101 · calculationBefore SLLN ORIGINAL joint word19 mean := div(-2, 5) After SLLN ORIGINAL joint word19 mean = -2/5 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, S100 S102 · calculationBefore SLLN ORIGINAL joint word19 mean minus mu := sub(-2/5, -1/2) After SLLN ORIGINAL joint word19 mean minus mu = 1/10 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, S101 S103 · impliesBefore source roles = [5, 19, [0, -1, -1, 0, 0], -1/2, 1/3] After derived = [-2/5, 1/10, No, null, 1, 1/32] 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, S102 S104 · calculationBefore SLLN ORIGINAL joint word20 integer sum := sum(0, -1, 0, -1, -1) After SLLN ORIGINAL joint word20 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S103 S105 · calculationBefore SLLN ORIGINAL joint word20 mean := div(-3, 5) After SLLN ORIGINAL joint word20 mean = -3/5 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, S104 S106 · calculationBefore SLLN ORIGINAL joint word20 mean minus mu := sub(-3/5, -1/2) After SLLN ORIGINAL joint word20 mean minus mu = -1/10 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, S105 S107 · impliesBefore source roles = [5, 20, [0, -1, 0, -1, -1], -1/2, 1/3] After derived = [-3/5, 1/10, No, null, 1, 1/32] 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, S106 S108 · calculationBefore SLLN ORIGINAL joint word21 integer sum := sum(0, -1, 0, -1, 0) After SLLN ORIGINAL joint word21 integer sum = -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, S107 S109 · calculationBefore SLLN ORIGINAL joint word21 mean := div(-2, 5) After SLLN ORIGINAL joint word21 mean = -2/5 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, S108 S110 · calculationBefore SLLN ORIGINAL joint word21 mean minus mu := sub(-2/5, -1/2) After SLLN ORIGINAL joint word21 mean minus mu = 1/10 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, S109 S111 · impliesBefore source roles = [5, 21, [0, -1, 0, -1, 0], -1/2, 1/3] After derived = [-2/5, 1/10, No, null, 1, 1/32] 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, S110 S112 · calculationBefore SLLN ORIGINAL joint word22 integer sum := sum(0, -1, 0, 0, -1) After SLLN ORIGINAL joint word22 integer sum = -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, S111 S113 · calculationBefore SLLN ORIGINAL joint word22 mean := div(-2, 5) After SLLN ORIGINAL joint word22 mean = -2/5 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, S112 S114 · calculationBefore SLLN ORIGINAL joint word22 mean minus mu := sub(-2/5, -1/2) After SLLN ORIGINAL joint word22 mean minus mu = 1/10 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, S113 S115 · impliesBefore source roles = [5, 22, [0, -1, 0, 0, -1], -1/2, 1/3] After derived = [-2/5, 1/10, No, null, 1, 1/32] 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, S114 S116 · calculationBefore SLLN ORIGINAL joint word23 integer sum := sum(0, -1, 0, 0, 0) After SLLN ORIGINAL joint word23 integer sum = -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, S115 S117 · calculationBefore SLLN ORIGINAL joint word23 mean := div(-1, 5) After SLLN ORIGINAL joint word23 mean = -1/5 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, S116 S118 · calculationBefore SLLN ORIGINAL joint word23 mean minus mu := sub(-1/5, -1/2) After SLLN ORIGINAL joint word23 mean minus mu = 3/10 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, S117 S119 · impliesBefore source roles = [5, 23, [0, -1, 0, 0, 0], -1/2, 1/3] After derived = [-1/5, 3/10, No, null, 1, 1/32] 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, S118 S120 · calculationBefore SLLN ORIGINAL joint word24 integer sum := sum(0, 0, -1, -1, -1) After SLLN ORIGINAL joint word24 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S119 S121 · calculationBefore SLLN ORIGINAL joint word24 mean := div(-3, 5) After SLLN ORIGINAL joint word24 mean = -3/5 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, S120 S122 · calculationBefore SLLN ORIGINAL joint word24 mean minus mu := sub(-3/5, -1/2) After SLLN ORIGINAL joint word24 mean minus mu = -1/10 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, S121 S123 · impliesBefore source roles = [5, 24, [0, 0, -1, -1, -1], -1/2, 1/3] After derived = [-3/5, 1/10, No, null, 1, 1/32] 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, S122 S124 · calculationBefore SLLN ORIGINAL joint word25 integer sum := sum(0, 0, -1, -1, 0) After SLLN ORIGINAL joint word25 integer sum = -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, S123 S125 · calculationBefore SLLN ORIGINAL joint word25 mean := div(-2, 5) After SLLN ORIGINAL joint word25 mean = -2/5 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, S124 S126 · calculationBefore SLLN ORIGINAL joint word25 mean minus mu := sub(-2/5, -1/2) After SLLN ORIGINAL joint word25 mean minus mu = 1/10 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, S125 S127 · impliesBefore source roles = [5, 25, [0, 0, -1, -1, 0], -1/2, 1/3] After derived = [-2/5, 1/10, No, null, 1, 1/32] 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, S126 S128 · calculationBefore SLLN ORIGINAL joint word26 integer sum := sum(0, 0, -1, 0, -1) After SLLN ORIGINAL joint word26 integer sum = -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, S127 S129 · calculationBefore SLLN ORIGINAL joint word26 mean := div(-2, 5) After SLLN ORIGINAL joint word26 mean = -2/5 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, S128 S130 · calculationBefore SLLN ORIGINAL joint word26 mean minus mu := sub(-2/5, -1/2) After SLLN ORIGINAL joint word26 mean minus mu = 1/10 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, S129 S131 · impliesBefore source roles = [5, 26, [0, 0, -1, 0, -1], -1/2, 1/3] After derived = [-2/5, 1/10, No, null, 1, 1/32] 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, S130 S132 · calculationBefore SLLN ORIGINAL joint word27 integer sum := sum(0, 0, -1, 0, 0) After SLLN ORIGINAL joint word27 integer sum = -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, S131 S133 · calculationBefore SLLN ORIGINAL joint word27 mean := div(-1, 5) After SLLN ORIGINAL joint word27 mean = -1/5 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, S132 S134 · calculationBefore SLLN ORIGINAL joint word27 mean minus mu := sub(-1/5, -1/2) After SLLN ORIGINAL joint word27 mean minus mu = 3/10 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, S133 S135 · impliesBefore source roles = [5, 27, [0, 0, -1, 0, 0], -1/2, 1/3] After derived = [-1/5, 3/10, No, null, 1, 1/32] 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, S134 S136 · calculationBefore SLLN ORIGINAL joint word28 integer sum := sum(0, 0, 0, -1, -1) After SLLN ORIGINAL joint word28 integer sum = -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, S135 S137 · calculationBefore SLLN ORIGINAL joint word28 mean := div(-2, 5) After SLLN ORIGINAL joint word28 mean = -2/5 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, S136 S138 · calculationBefore SLLN ORIGINAL joint word28 mean minus mu := sub(-2/5, -1/2) After SLLN ORIGINAL joint word28 mean minus mu = 1/10 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, S137 S139 · impliesBefore source roles = [5, 28, [0, 0, 0, -1, -1], -1/2, 1/3] After derived = [-2/5, 1/10, No, null, 1, 1/32] 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, S138 S140 · calculationBefore SLLN ORIGINAL joint word29 integer sum := sum(0, 0, 0, -1, 0) After SLLN ORIGINAL joint word29 integer sum = -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, S139 S141 · calculationBefore SLLN ORIGINAL joint word29 mean := div(-1, 5) After SLLN ORIGINAL joint word29 mean = -1/5 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, S140 S142 · calculationBefore SLLN ORIGINAL joint word29 mean minus mu := sub(-1/5, -1/2) After SLLN ORIGINAL joint word29 mean minus mu = 3/10 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, S141 S143 · impliesBefore source roles = [5, 29, [0, 0, 0, -1, 0], -1/2, 1/3] After derived = [-1/5, 3/10, No, null, 1, 1/32] 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, S142 S144 · calculationBefore SLLN ORIGINAL joint word30 integer sum := sum(0, 0, 0, 0, -1) After SLLN ORIGINAL joint word30 integer sum = -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, S143 S145 · calculationBefore SLLN ORIGINAL joint word30 mean := div(-1, 5) After SLLN ORIGINAL joint word30 mean = -1/5 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, S144 S146 · calculationBefore SLLN ORIGINAL joint word30 mean minus mu := sub(-1/5, -1/2) After SLLN ORIGINAL joint word30 mean minus mu = 3/10 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, S145 S147 · impliesBefore source roles = [5, 30, [0, 0, 0, 0, -1], -1/2, 1/3] After derived = [-1/5, 3/10, No, null, 1, 1/32] 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, S146 S148 · calculationBefore SLLN ORIGINAL joint word31 integer sum := sum(0, 0, 0, 0, 0) After SLLN ORIGINAL joint word31 integer sum = 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, S147 S149 · calculationBefore SLLN ORIGINAL joint word31 mean := div(0, 5) After SLLN ORIGINAL joint word31 mean = 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, S148 S150 · calculationBefore SLLN ORIGINAL joint word31 mean minus mu := sub(0, -1/2) After SLLN ORIGINAL joint word31 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, S149 S151 · calculationBefore SLLN ORIGINAL deviating word31 probability factor0 := mul(1, 1/2) After SLLN ORIGINAL deviating word31 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, S150 S152 · calculationBefore SLLN ORIGINAL deviating word31 probability factor1 := mul(1/2, 1/2) After SLLN ORIGINAL deviating word31 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, S151 S153 · calculationBefore SLLN ORIGINAL deviating word31 probability factor2 := mul(1/4, 1/2) After SLLN ORIGINAL deviating word31 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, S152 S154 · calculationBefore SLLN ORIGINAL deviating word31 probability factor3 := mul(1/8, 1/2) After SLLN ORIGINAL deviating word31 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, S153 S155 · calculationBefore SLLN ORIGINAL deviating word31 probability factor4 := mul(1/16, 1/2) After SLLN ORIGINAL deviating word31 probability factor4 = 1/32 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, S154 S156 · calculationBefore SLLN ORIGINAL deviation probability word31 := sum(1/32, 1/32) After SLLN ORIGINAL deviation probability word31 = 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, S155 S157 · impliesBefore source roles = [5, 31, [0, 0, 0, 0, 0], -1/2, 1/3] After derived = [0, 1/2, Yes, 1/32, 2, 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, S156 ## Reverse checks - S157 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S156 · premise_and_result_audit — Recompute SLLN ORIGINAL deviation probability word31 with exact rational arithmetic, not rounded decimals. - S155 · premise_and_result_audit — Recompute SLLN ORIGINAL deviating word31 probability factor4 with exact rational arithmetic, not rounded decimals. - S154 · premise_and_result_audit — Recompute SLLN ORIGINAL deviating word31 probability factor3 with exact rational arithmetic, not rounded decimals. - S153 · premise_and_result_audit — Recompute SLLN ORIGINAL deviating word31 probability factor2 with exact rational arithmetic, not rounded decimals. - S152 · premise_and_result_audit — Recompute SLLN ORIGINAL deviating word31 probability factor1 with exact rational arithmetic, not rounded decimals. - S151 · premise_and_result_audit — Recompute SLLN ORIGINAL deviating word31 probability factor0 with exact rational arithmetic, not rounded decimals. - S150 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word31 mean minus mu with exact rational arithmetic, not rounded decimals. - S149 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word31 mean with exact rational arithmetic, not rounded decimals. - S148 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word31 integer sum with exact rational arithmetic, not rounded decimals. - S147 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S146 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word30 mean minus mu with exact rational arithmetic, not rounded decimals. - S145 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word30 mean with exact rational arithmetic, not rounded decimals. - S144 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word30 integer sum with exact rational arithmetic, not rounded decimals. - S143 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S142 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word29 mean minus mu with exact rational arithmetic, not rounded decimals. - S141 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word29 mean with exact rational arithmetic, not rounded decimals. - S140 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word29 integer sum with exact rational arithmetic, not rounded decimals. - S139 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S138 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word28 mean minus mu with exact rational arithmetic, not rounded decimals. - S137 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word28 mean with exact rational arithmetic, not rounded decimals. - S136 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word28 integer sum with exact rational arithmetic, not rounded decimals. - S135 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S134 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word27 mean minus mu with exact rational arithmetic, not rounded decimals. - S133 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word27 mean with exact rational arithmetic, not rounded decimals. - S132 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word27 integer sum with exact rational arithmetic, not rounded decimals. - S131 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S130 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word26 mean minus mu with exact rational arithmetic, not rounded decimals. - S129 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word26 mean with exact rational arithmetic, not rounded decimals. - S128 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word26 integer sum with exact rational arithmetic, not rounded decimals. - S127 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S126 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word25 mean minus mu with exact rational arithmetic, not rounded decimals. - S125 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word25 mean with exact rational arithmetic, not rounded decimals. - S124 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word25 integer sum with exact rational arithmetic, not rounded decimals. - S123 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S122 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word24 mean minus mu with exact rational arithmetic, not rounded decimals. - S121 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word24 mean with exact rational arithmetic, not rounded decimals. - S120 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word24 integer sum with exact rational arithmetic, not rounded decimals. - S119 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S118 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word23 mean minus mu with exact rational arithmetic, not rounded decimals. - S117 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word23 mean with exact rational arithmetic, not rounded decimals. - S116 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word23 integer sum with exact rational arithmetic, not rounded decimals. - S115 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S114 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word22 mean minus mu with exact rational arithmetic, not rounded decimals. - S113 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word22 mean with exact rational arithmetic, not rounded decimals. - S112 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word22 integer sum with exact rational arithmetic, not rounded decimals. - S111 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S110 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word21 mean minus mu with exact rational arithmetic, not rounded decimals. - S109 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word21 mean with exact rational arithmetic, not rounded decimals. - S108 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word21 integer sum with exact rational arithmetic, not rounded decimals. - S107 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S106 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word20 mean minus mu with exact rational arithmetic, not rounded decimals. - S105 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word20 mean with exact rational arithmetic, not rounded decimals. - S104 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word20 integer sum with exact rational arithmetic, not rounded decimals. - S103 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S102 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word19 mean minus mu with exact rational arithmetic, not rounded decimals. - S101 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word19 mean with exact rational arithmetic, not rounded decimals. - S100 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word19 integer sum with exact rational arithmetic, not rounded decimals. - S099 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S098 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word18 mean minus mu with exact rational arithmetic, not rounded decimals. - S097 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word18 mean with exact rational arithmetic, not rounded decimals. - S096 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word18 integer sum with exact rational arithmetic, not rounded decimals. - S095 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S094 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word17 mean minus mu with exact rational arithmetic, not rounded decimals. - S093 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word17 mean with exact rational arithmetic, not rounded decimals. - S092 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word17 integer sum with exact rational arithmetic, not rounded decimals. - 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 joint word16 mean minus mu with exact rational arithmetic, not rounded decimals. - S089 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word16 mean with exact rational arithmetic, not rounded decimals. - S088 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word16 integer sum with exact rational arithmetic, not rounded decimals. - S087 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S086 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word15 mean minus mu with exact rational arithmetic, not rounded decimals. - S085 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word15 mean with exact rational arithmetic, not rounded decimals. - S084 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word15 integer sum with exact rational arithmetic, not rounded decimals. - S083 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S082 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word14 mean minus mu with exact rational arithmetic, not rounded decimals. - S081 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word14 mean with exact rational arithmetic, not rounded decimals. - S080 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word14 integer sum with exact rational arithmetic, not rounded decimals. - S079 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S078 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word13 mean minus mu with exact rational arithmetic, not rounded decimals. - S077 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word13 mean with exact rational arithmetic, not rounded decimals. - S076 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word13 integer sum with exact rational arithmetic, not rounded decimals. - S075 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S074 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word12 mean minus mu with exact rational arithmetic, not rounded decimals. - S073 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word12 mean with exact rational arithmetic, not rounded decimals. - S072 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word12 integer sum with exact rational arithmetic, not rounded decimals. - S071 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S070 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word11 mean minus mu with exact rational arithmetic, not rounded decimals. - S069 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word11 mean with exact rational arithmetic, not rounded decimals. - S068 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word11 integer sum with exact rational arithmetic, not rounded decimals. - S067 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S066 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word10 mean minus mu with exact rational arithmetic, not rounded decimals. - S065 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word10 mean with exact rational arithmetic, not rounded decimals. - S064 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word10 integer sum with exact rational arithmetic, not rounded decimals. - S063 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S062 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word9 mean minus mu with exact rational arithmetic, not rounded decimals. - S061 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word9 mean with exact rational arithmetic, not rounded decimals. - S060 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word9 integer sum with exact rational arithmetic, not rounded decimals. - S059 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S058 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word8 mean minus mu with exact rational arithmetic, not rounded decimals. - S057 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word8 mean with exact rational arithmetic, not rounded decimals. - S056 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word8 integer sum with exact rational arithmetic, not rounded decimals. - S055 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S054 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word7 mean minus mu with exact rational arithmetic, not rounded decimals. - S053 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word7 mean with exact rational arithmetic, not rounded decimals. - S052 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word7 integer sum with exact rational arithmetic, not rounded decimals. - S051 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S050 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word6 mean minus mu with exact rational arithmetic, not rounded decimals. - S049 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word6 mean with exact rational arithmetic, not rounded decimals. - S048 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word6 integer sum with exact rational arithmetic, not rounded decimals. - S047 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S046 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word5 mean minus mu with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word5 mean with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word5 integer sum with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S042 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word4 mean minus mu with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word4 mean with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word4 integer sum with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S038 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word3 mean minus mu with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word3 mean with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word3 integer sum with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S034 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word2 mean minus mu with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word2 mean with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word2 integer sum with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S030 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word1 mean minus mu with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word1 mean with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word1 integer sum with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S026 · premise_and_result_audit — Recompute SLLN ORIGINAL deviation probability word0 with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute SLLN ORIGINAL deviating word0 probability factor4 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 (5, -1/2, 1/3, 2, 1/16, 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: c3851c6726c0b296fdc025b8682eb2ad772716c4cce3576573105b425ad2215c - candidate record SHA-256: 0459b6141458953b6e7b95060838949733736369c7091baeadcd2c9ffba916d2 - 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-0072.json)