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R0054-27622 · q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0072
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).Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
Before
source roles = [slln-fixed-n-deviation]
After
derived = [[n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary]]
Conditions: none
List the complete operation/mode field order without JavaScript or answer-shape inference;Boolean,string,null and an unclaimed result remain distinct.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
source roles = [infinite_iid]
After
derived = [ASSUME whole infinite sequence is IID with this common marginal]
Conditions: none
infinite_iid is a GIVEN whole-process premise,not independence constructed or proved by finite data/this program.
Rule: SLLN_infinite_IID_EXACT_GIVEN_premise_not_finite_observation_inference · Uses: G001, S001
Before
SLLN ORIGINAL common marginal ORIGINAL mass sum0 := sum(0, 1/2)
After
SLLN ORIGINAL common marginal ORIGINAL mass sum0 = 1/2
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
SLLN ORIGINAL common marginal ORIGINAL mass sum1 := sum(1/2, 1/2)
After
SLLN ORIGINAL common marginal ORIGINAL mass sum1 = 1
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
SLLN ORIGINAL common marginal ORIGINAL weighted value0 := mul(-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.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
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.
Rule: SLLN_complete_ORIGINAL_numeric_value_probability_alignment_ALL_moments · Uses: G001, S016
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
SLLN ORIGINAL deviating word0 probability factor0 := mul(1, 1/2)
After
SLLN ORIGINAL deviating word0 probability factor0 = 1/2
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
SLLN ORIGINAL deviating word0 probability factor1 := mul(1/2, 1/2)
After
SLLN ORIGINAL deviating word0 probability factor1 = 1/4
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
SLLN ORIGINAL deviating word0 probability factor2 := mul(1/4, 1/2)
After
SLLN ORIGINAL deviating word0 probability factor2 = 1/8
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
SLLN ORIGINAL 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.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
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.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S026
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
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.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S030
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S032
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S033
Before
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.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S034
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S036
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
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.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S038
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S039
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S040
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S041
Before
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.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S042
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S043
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S044
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S045
Before
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.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S046
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S047
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S048
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S049
Before
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.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S050
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S051
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S052
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S053
Before
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.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S054
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S055
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S056
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S057
Before
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.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S058
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S059
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S060
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S061
Before
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.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S062
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S063
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S064
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S065
Before
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.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S066
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S067
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S068
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S069
Before
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.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S070
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S071
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S072
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S073
Before
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.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S074
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S075
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S076
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S077
Before
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.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S078
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S079
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S080
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S081
Before
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.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S082
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S083
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S084
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S085
Before
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.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S086
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S087
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S088
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S089
Before
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.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S090
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S091
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S092
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S093
Before
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.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S094
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S095
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S096
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S097
Before
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.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S098
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S099
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S100
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S101
Before
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.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S102
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S103
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S104
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S105
Before
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.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S106
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S107
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S108
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S109
Before
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.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S110
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S111
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S112
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S113
Before
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.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S114
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S115
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S116
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S117
Before
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.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S118
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S119
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S120
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S121
Before
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.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S122
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S123
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S124
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S125
Before
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.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S126
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S127
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S128
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S129
Before
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.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S130
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S131
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S132
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S133
Before
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.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S134
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S135
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S136
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S137
Before
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.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S138
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S139
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S140
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S141
Before
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.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S142
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S143
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S144
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S145
Before
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.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S146
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S147
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S148
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S149
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S150
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S151
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S152
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S153
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S154
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S155
Before
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.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S156
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.(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.
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: c3851c6726c0b296fdc025b8682eb2ad772716c4cce3576573105b425ad2215ccandidate record SHA-256: 0459b6141458953b6e7b95060838949733736369c7091baeadcd2c9ffba916d2semantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437Lean: NOT_RUN · human review: NOT_PERFORMED