# Question 27680: Outside-Scope Inputs Are Not General Strong-Law Counterexamples [Return to catalogue page 139](https://ocme.evemisslab.com/en/questions/page/139/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations/#q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0080) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0054-27680 · q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0080 ## Original question and givens Use mode=non_iid, complete model={"values_a":[2,3],"probabilities_a":[[1,2],[1,2]],"values_b":[2,3],"probabilities_b":[[1,7],[6,7]]}. Give (mode, identical_marginals, this_finite_valued_IID_theorem_applicable, scope). - G001 — Use mode=non_iid, complete model={"values_a":[2,3],"probabilities_a":[[1,2],[1,2]],"values_b":[2,3],"probabilities_b":[[1,7],[6,7]]}. Give (mode, identical_marginals, this_finite_valued_IID_theorem_applicable, scope). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-strong-law-large-numbers-finite-valued-iid](https://ocme.evemisslab.com/en/mko/mko-strong-law-large-numbers-finite-valued-iid/) ## Step-by-step derivation S001 · impliesBefore source roles = [slln-scope-boundary] After derived = [[mode, identical_marginals, this_finite_valued_IID_theorem_applicable, scope]] Conditions: none List the complete operation/mode field order without JavaScript or answer-shape inference;Boolean,string,null and an unclaimed result remain distinct. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · calculationBefore SLLN boundary first marginal ORIGINAL mass sum0 := sum(0, 1/2) After SLLN boundary first marginal ORIGINAL mass sum0 = 1/2 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore SLLN boundary first marginal ORIGINAL mass sum1 := sum(1/2, 1/2) After SLLN boundary first marginal ORIGINAL mass sum1 = 1 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore SLLN boundary first marginal ORIGINAL weighted value0 := mul(2, 1/2) After SLLN boundary first marginal ORIGINAL weighted value0 = 1 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore SLLN boundary first marginal ORIGINAL mean sum0 := sum(0, 1) After SLLN boundary first marginal ORIGINAL mean sum0 = 1 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore SLLN boundary first marginal exact unbounded value square0 := mul(2, 2) After SLLN boundary first marginal exact unbounded value square0 = 4 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore SLLN boundary first marginal ORIGINAL weighted square0 := mul(4, 1/2) After SLLN boundary first marginal ORIGINAL weighted square0 = 2 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore SLLN boundary first marginal ORIGINAL second moment sum0 := sum(0, 2) After SLLN boundary first marginal ORIGINAL second moment sum0 = 2 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore SLLN boundary first marginal ORIGINAL weighted value1 := mul(3, 1/2) After SLLN boundary first marginal ORIGINAL weighted value1 = 3/2 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore SLLN boundary first marginal ORIGINAL mean sum1 := sum(1, 3/2) After SLLN boundary first marginal ORIGINAL mean sum1 = 5/2 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore SLLN boundary first marginal exact unbounded value square1 := mul(3, 3) After SLLN boundary first marginal exact unbounded value square1 = 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore SLLN boundary first marginal ORIGINAL weighted square1 := mul(9, 1/2) After SLLN boundary first marginal ORIGINAL weighted square1 = 9/2 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore SLLN boundary first marginal ORIGINAL second moment sum1 := sum(2, 9/2) After SLLN boundary first marginal ORIGINAL second moment sum1 = 13/2 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore SLLN boundary first marginal ORIGINAL mean squared := mul(5/2, 5/2) After SLLN boundary first marginal ORIGINAL mean squared = 25/4 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore SLLN boundary first marginal ORIGINAL second minus mean squared := sub(13/2, 25/4) After SLLN boundary first marginal ORIGINAL second minus mean squared = 1/4 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · impliesBefore source roles = [SLLN boundary first marginal, [2, 3], [[1, 2], [1, 2]]] After derived = [[[2, 1/2], [3, 1/2]], 1, 5/2, 13/2, 1/4] Conditions: none Pair ORIGINAL values/probabilities before numeric sorting and retain zero-mass labels.Qualify the complete nonnegative mass1 law and its source second-minus-mean-square variance.SLLN allows variance0;do not import the CLT positive-variance condition. Step identifiers and dependenciesRule: SLLN_complete_ORIGINAL_numeric_value_probability_alignment_ALL_moments · Uses: G001, S015 S017 · calculationBefore SLLN boundary second marginal ORIGINAL mass sum0 := sum(0, 1/7) After SLLN boundary second marginal ORIGINAL mass sum0 = 1/7 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore SLLN boundary second marginal ORIGINAL mass sum1 := sum(1/7, 6/7) After SLLN boundary second marginal ORIGINAL mass sum1 = 1 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore SLLN boundary second marginal ORIGINAL weighted value0 := mul(2, 1/7) After SLLN boundary second marginal ORIGINAL weighted value0 = 2/7 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore SLLN boundary second marginal ORIGINAL mean sum0 := sum(0, 2/7) After SLLN boundary second marginal ORIGINAL mean sum0 = 2/7 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore SLLN boundary second marginal exact unbounded value square0 := mul(2, 2) After SLLN boundary second marginal exact unbounded value square0 = 4 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore SLLN boundary second marginal ORIGINAL weighted square0 := mul(4, 1/7) After SLLN boundary second marginal ORIGINAL weighted square0 = 4/7 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore SLLN boundary second marginal ORIGINAL second moment sum0 := sum(0, 4/7) After SLLN boundary second marginal ORIGINAL second moment sum0 = 4/7 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore SLLN boundary second marginal ORIGINAL weighted value1 := mul(3, 6/7) After SLLN boundary second marginal ORIGINAL weighted value1 = 18/7 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore SLLN boundary second marginal ORIGINAL mean sum1 := sum(2/7, 18/7) After SLLN boundary second marginal ORIGINAL mean sum1 = 20/7 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore SLLN boundary second marginal exact unbounded value square1 := mul(3, 3) After SLLN boundary second marginal exact unbounded value square1 = 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore SLLN boundary second marginal ORIGINAL weighted square1 := mul(9, 6/7) After SLLN boundary second marginal ORIGINAL weighted square1 = 54/7 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore SLLN boundary second marginal ORIGINAL second moment sum1 := sum(4/7, 54/7) After SLLN boundary second marginal ORIGINAL second moment sum1 = 58/7 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore SLLN boundary second marginal ORIGINAL mean squared := mul(20/7, 20/7) After SLLN boundary second marginal ORIGINAL mean squared = 400/49 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · calculationBefore SLLN boundary second marginal ORIGINAL second minus mean squared := sub(58/7, 400/49) After SLLN boundary second marginal ORIGINAL second minus mean squared = 6/49 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · impliesBefore source roles = [SLLN boundary second marginal, [2, 3], [[1, 7], [6, 7]]] After derived = [[[2, 1/7], [3, 6/7]], 1, 20/7, 58/7, 6/49] Conditions: none Pair ORIGINAL values/probabilities before numeric sorting and retain zero-mass labels.Qualify the complete nonnegative mass1 law and its source second-minus-mean-square variance.SLLN allows variance0;do not import the CLT positive-variance condition. Step identifiers and dependenciesRule: SLLN_complete_ORIGINAL_numeric_value_probability_alignment_ALL_moments · Uses: G001, S030 S032 · impliesBefore source roles = [[[2, 1/2], [3, 1/2]], [[2, 1/7], [3, 6/7]]] After derived = [No, No] Conditions: none The complete source marginal rows actually differ,withholding this IID version,not every non-IID strong law. Step identifiers and dependenciesRule: SLLN_ACTUAL_different_full_marginal_rows_NOT_all_strong_laws · Uses: G001, S031 ## Reverse checks - 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 — 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 boundary second marginal ORIGINAL second minus mean squared with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute SLLN boundary second marginal ORIGINAL mean squared with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute SLLN boundary second marginal ORIGINAL second moment sum1 with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute SLLN boundary second marginal ORIGINAL weighted square1 with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute SLLN boundary second marginal exact unbounded value square1 with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute SLLN boundary second marginal ORIGINAL mean sum1 with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute SLLN boundary second marginal ORIGINAL weighted value1 with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute SLLN boundary second marginal ORIGINAL second moment sum0 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute SLLN boundary second marginal ORIGINAL weighted square0 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute SLLN boundary second marginal exact unbounded value square0 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute SLLN boundary second marginal ORIGINAL mean sum0 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute SLLN boundary second marginal ORIGINAL weighted value0 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute SLLN boundary second marginal ORIGINAL mass sum1 with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute SLLN boundary second marginal ORIGINAL mass sum0 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S015 · premise_and_result_audit — Recompute SLLN boundary first marginal ORIGINAL second minus mean squared with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute SLLN boundary first marginal ORIGINAL mean squared with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute SLLN boundary first marginal ORIGINAL second moment sum1 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute SLLN boundary first marginal ORIGINAL weighted square1 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute SLLN boundary first marginal exact unbounded value square1 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute SLLN boundary first marginal ORIGINAL mean sum1 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute SLLN boundary first marginal ORIGINAL weighted value1 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute SLLN boundary first marginal ORIGINAL second moment sum0 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute SLLN boundary first marginal ORIGINAL weighted square0 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute SLLN boundary first marginal exact unbounded value square0 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute SLLN boundary first marginal ORIGINAL mean sum0 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute SLLN boundary first marginal ORIGINAL weighted value0 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute SLLN boundary first marginal ORIGINAL mass sum1 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute SLLN boundary first marginal ORIGINAL mass sum0 with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. ## Result (non_iid, No, No, finite_valued_IID_SLLN_not_applicable_not_general_failure) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 2df278b606e28d10e53d29886d505b5a843f4e952af11997f61114533d4afe7d - candidate record SHA-256: cd1d3e584a9607b741d309b530a345171c559803810b8d1f0deb1cc3886a6841 - semantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54 - question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0080.json)