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R0054-27608 · q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0058
Use the aligned finite law values=[0, 1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=2 and epsilon=2/4>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).
G001 — Use the aligned finite law values=[0, 1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=2 and epsilon=2/4>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
Before
source roles = [slln-fixed-n-deviation]
After
derived = [[n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary]]
Conditions: none
List the complete operation/mode field order without JavaScript or answer-shape inference;Boolean,string,null and an unclaimed result remain distinct.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
source roles = [infinite_iid]
After
derived = [ASSUME whole infinite sequence is IID with this common marginal]
Conditions: none
infinite_iid is a GIVEN whole-process premise,not independence constructed or proved by finite data/this program.
Rule: SLLN_infinite_IID_EXACT_GIVEN_premise_not_finite_observation_inference · Uses: G001, S001
Before
SLLN ORIGINAL common marginal ORIGINAL mass sum0 := sum(0, 1/2)
After
SLLN ORIGINAL common marginal ORIGINAL mass sum0 = 1/2
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
SLLN ORIGINAL common marginal ORIGINAL mass sum1 := sum(1/2, 1/2)
After
SLLN ORIGINAL common marginal ORIGINAL mass sum1 = 1
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
SLLN ORIGINAL common marginal ORIGINAL weighted value0 := mul(0, 1/2)
After
SLLN ORIGINAL common marginal ORIGINAL weighted value0 = 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, S004
Before
SLLN ORIGINAL common marginal ORIGINAL mean sum0 := sum(0, 0)
After
SLLN ORIGINAL common marginal ORIGINAL mean sum0 = 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, S005
Before
SLLN ORIGINAL common marginal exact unbounded value square0 := mul(0, 0)
After
SLLN ORIGINAL common marginal exact unbounded value square0 = 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, S006
Before
SLLN ORIGINAL common marginal ORIGINAL weighted square0 := mul(0, 1/2)
After
SLLN ORIGINAL common marginal ORIGINAL weighted square0 = 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, S007
Before
SLLN ORIGINAL common marginal ORIGINAL second moment sum0 := sum(0, 0)
After
SLLN ORIGINAL common marginal ORIGINAL second moment sum0 = 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, S008
Before
SLLN ORIGINAL common marginal ORIGINAL weighted value1 := mul(1, 1/2)
After
SLLN ORIGINAL common marginal ORIGINAL weighted value1 = 1/2
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
SLLN ORIGINAL common marginal ORIGINAL mean sum1 := sum(0, 1/2)
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(1, 1)
After
SLLN ORIGINAL common marginal exact unbounded value square1 = 1
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
SLLN ORIGINAL common marginal ORIGINAL weighted square1 := mul(1, 1/2)
After
SLLN ORIGINAL common marginal ORIGINAL weighted square1 = 1/2
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
SLLN ORIGINAL common marginal ORIGINAL second moment sum1 := sum(0, 1/2)
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, [0, 1], [[1, 2], [1, 2]]]
After
derived = [[[0, 1/2], [1, 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(0, 0)
After
SLLN ORIGINAL joint word0 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, S017
Before
SLLN ORIGINAL joint word0 mean := div(0, 2)
After
SLLN ORIGINAL joint word0 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, S018
Before
SLLN ORIGINAL joint word0 mean minus mu := sub(0, 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 deviation probability word0 := sum(0, 1/4)
After
SLLN ORIGINAL deviation probability word0 = 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, S022
Before
source roles = [2, 0, [0, 0], 1/2, 1/2]
After
derived = [0, 1/2, Yes, 1/4, 1, 1/4]
Conditions: none
For EVERY word test |mean-mu|>=epsilon,equality included.A zero-mass deviating word still contributes to the combinatorial count,while probability is weighted separately.Non-deviating words need no unused probability product under the source.This finite event is not an almost-sure limit certificate.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S023
Before
SLLN ORIGINAL joint word1 integer sum := sum(0, 1)
After
SLLN ORIGINAL joint word1 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, S024
Before
SLLN ORIGINAL joint word1 mean := div(1, 2)
After
SLLN ORIGINAL joint word1 mean = 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, S025
Before
SLLN ORIGINAL joint word1 mean minus mu := sub(1/2, 1/2)
After
SLLN ORIGINAL joint word1 mean minus mu = 0
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S026
Before
source roles = [2, 1, [0, 1], 1/2, 1/2]
After
derived = [1/2, 0, No, null, 1, 1/4]
Conditions: none
For EVERY word test |mean-mu|>=epsilon,equality included.A zero-mass deviating word still contributes to the combinatorial count,while probability is weighted separately.Non-deviating words need no unused probability product under the source.This finite event is not an almost-sure limit certificate.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S027
Before
SLLN ORIGINAL joint word2 integer sum := sum(1, 0)
After
SLLN ORIGINAL joint word2 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, S028
Before
SLLN ORIGINAL joint word2 mean := div(1, 2)
After
SLLN ORIGINAL joint word2 mean = 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, S029
Before
SLLN ORIGINAL joint word2 mean minus mu := sub(1/2, 1/2)
After
SLLN ORIGINAL joint word2 mean minus mu = 0
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
source roles = [2, 2, [1, 0], 1/2, 1/2]
After
derived = [1/2, 0, No, null, 1, 1/4]
Conditions: none
For EVERY word test |mean-mu|>=epsilon,equality included.A zero-mass deviating word still contributes to the combinatorial count,while probability is weighted separately.Non-deviating words need no unused probability product under the source.This finite event is not an almost-sure limit certificate.
Rule: SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability · Uses: G001, S031
Before
SLLN ORIGINAL joint word3 integer sum := sum(1, 1)
After
SLLN ORIGINAL joint word3 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, S032
Before
SLLN ORIGINAL joint word3 mean := div(2, 2)
After
SLLN ORIGINAL joint word3 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, S033
Before
SLLN ORIGINAL joint word3 mean minus mu := sub(1, 1/2)
After
SLLN ORIGINAL joint word3 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, S034
Before
SLLN ORIGINAL deviating word3 probability factor0 := mul(1, 1/2)
After
SLLN ORIGINAL deviating word3 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, S035
Before
SLLN ORIGINAL deviating word3 probability factor1 := mul(1/2, 1/2)
After
SLLN ORIGINAL deviating word3 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, S036
Before
SLLN ORIGINAL deviation probability word3 := sum(1/4, 1/4)
After
SLLN ORIGINAL deviation probability word3 = 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, S037
Before
source roles = [2, 3, [1, 1], 1/2, 1/2]
After
derived = [1, 1/2, Yes, 1/4, 2, 1/2]
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
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 deviation probability word3 with exact rational arithmetic, not rounded decimals.S037 · premise_and_result_audit — Recompute SLLN ORIGINAL deviating word3 probability factor1 with exact rational arithmetic, not rounded decimals.S036 · premise_and_result_audit — Recompute SLLN ORIGINAL deviating word3 probability factor0 with exact rational arithmetic, not rounded decimals.S035 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word3 mean minus mu with exact rational arithmetic, not rounded decimals.S034 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word3 mean with exact rational arithmetic, not rounded decimals.S033 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word3 integer sum with exact rational arithmetic, not rounded decimals.S032 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S031 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word2 mean minus mu with exact rational arithmetic, not rounded decimals.S030 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word2 mean with exact rational arithmetic, not rounded decimals.S029 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word2 integer sum with exact rational arithmetic, not rounded decimals.S028 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S027 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word1 mean minus mu with exact rational arithmetic, not rounded decimals.S026 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word1 mean with exact rational arithmetic, not rounded decimals.S025 · premise_and_result_audit — Recompute SLLN ORIGINAL joint word1 integer sum with exact rational arithmetic, not rounded decimals.S024 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S023 · premise_and_result_audit — Recompute SLLN ORIGINAL deviation probability word0 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.(2, 1/2, 1/2, 2, 1/2, 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: a2c73adc778d1c0a98bdac8d9d90bed9040a77bf60bd7edc77ebfb7b476deb79candidate record SHA-256: c2e2675f14bf66c0a17af05eaad3fc51fdaa600cb5045f34a2f4396bf43f1d64semantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437Lean: NOT_RUN · human review: NOT_PERFORMED