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R0054-27649 · q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0049
Use the aligned finite law values=[1, 2], probabilities=[1/5, 4/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. The observed finite prefix is=[2, 1, 2, 1, 2, 1]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).
G001 — Use the aligned finite law values=[1, 2], probabilities=[1/5, 4/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. The observed finite prefix is=[2, 1, 2, 1, 2, 1]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_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-prefix-sample-mean]
After
derived = [[observations, common_mean, prefix_sample_mean, absolute_error, evidence_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, 1/5)
After
SLLN ORIGINAL common marginal ORIGINAL mass sum0 = 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, S002
Before
SLLN ORIGINAL common marginal ORIGINAL mass sum1 := sum(1/5, 4/5)
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, 1/5)
After
SLLN ORIGINAL common marginal ORIGINAL weighted value0 = 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, S004
Before
SLLN ORIGINAL common marginal ORIGINAL mean sum0 := sum(0, 1/5)
After
SLLN ORIGINAL common marginal ORIGINAL mean sum0 = 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, 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, 1/5)
After
SLLN ORIGINAL common marginal ORIGINAL weighted square0 = 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, S007
Before
SLLN ORIGINAL common marginal ORIGINAL second moment sum0 := sum(0, 1/5)
After
SLLN ORIGINAL common marginal ORIGINAL second moment sum0 = 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, S008
Before
SLLN ORIGINAL common marginal ORIGINAL weighted value1 := mul(2, 4/5)
After
SLLN ORIGINAL common marginal ORIGINAL weighted value1 = 8/5
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, S009
Before
SLLN ORIGINAL common marginal ORIGINAL mean sum1 := sum(1/5, 8/5)
After
SLLN ORIGINAL common marginal ORIGINAL mean sum1 = 9/5
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, S010
Before
SLLN ORIGINAL common marginal exact unbounded value square1 := mul(2, 2)
After
SLLN ORIGINAL common marginal exact unbounded value square1 = 4
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
SLLN ORIGINAL common marginal ORIGINAL weighted square1 := mul(4, 4/5)
After
SLLN ORIGINAL common marginal ORIGINAL weighted square1 = 16/5
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, S012
Before
SLLN ORIGINAL common marginal ORIGINAL second moment sum1 := sum(1/5, 16/5)
After
SLLN ORIGINAL common marginal ORIGINAL second moment sum1 = 17/5
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, S013
Before
SLLN ORIGINAL common marginal ORIGINAL mean squared := mul(9/5, 9/5)
After
SLLN ORIGINAL common marginal ORIGINAL mean squared = 81/25
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(17/5, 81/25)
After
SLLN ORIGINAL common marginal ORIGINAL second minus mean squared = 4/25
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, 2], [[1, 5], [4, 5]]]
After
derived = [[[1, 1/5], [2, 4/5]], 1, 9/5, 17/5, 4/25]
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 prefix integer sum := sum(2, 1, 2, 1, 2, 1)
After
SLLN ORIGINAL prefix integer sum = 9
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 prefix sample mean := div(9, 6)
After
SLLN ORIGINAL prefix sample mean = 3/2
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
SLLN ORIGINAL sample mean minus common mean := sub(3/2, 9/5)
After
SLLN ORIGINAL sample mean minus common mean = -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, S019
Before
source roles = [[2, 1, 2, 1, 2, 1], 9/5, 3/2, -3/10]
After
derived = [3/10, No]
Conditions: none
Every observation is correctly typed and in the carrier;the prefix must be nonempty.The source uses the UNREDUCED[sum,length]1e6 profile,not an invented8-term cap.One close/equal prefix does not prove an almost-sure limit.
Rule: SLLN_fixed_nonempty_prefix_mean_error_NOT_as_limit_certificate · Uses: G001, S020
S021 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S020 · premise_and_result_audit — Recompute SLLN ORIGINAL sample mean minus common mean with exact rational arithmetic, not rounded decimals.S019 · premise_and_result_audit — Recompute SLLN ORIGINAL prefix sample mean with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Recompute SLLN ORIGINAL prefix 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, 1], 9/5, 3/2, 3/10, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)
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: 5c37edf2a2d33f1d6604ff0596246312e5939c32d2fa31bcda6e9b028811212fcandidate record SHA-256: 588910c91b081ce04fa09035ed92348dec7b45f2ca5e5939f891bc8a76315f26semantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437Lean: NOT_RUN · human review: NOT_PERFORMED