# Question 27691: 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-0091) 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. R0054-27691 · q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0091 ## Original question and givens Use mode=wrong_target, complete model={"values":[-2,-1],"probabilities":[[2,5],[3,5]],"target":17}. Give (mode, common_mean, proposed_target, target_equals_common_mean, proposed_target_claim, boundary). - G001 — Use mode=wrong_target, complete model={"values":[-2,-1],"probabilities":[[2,5],[3,5]],"target":17}. Give (mode, common_mean, proposed_target, target_equals_common_mean, proposed_target_claim, boundary). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-strong-law-large-numbers-finite-valued-iid](https://ocme.evemisslab.com/en/mko/mko-strong-law-large-numbers-finite-valued-iid/) ## Step-by-step derivation S001 · impliesBefore source roles = [slln-scope-boundary] After derived = [[mode, common_mean, proposed_target, target_equals_common_mean, proposed_target_claim, boundary]] Conditions: none List the complete operation/mode field order without JavaScript or answer-shape inference;Boolean,string,null and an unclaimed result remain distinct. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · calculationBefore SLLN boundary target law ORIGINAL mass sum0 := sum(0, 2/5) After SLLN boundary target law ORIGINAL mass sum0 = 2/5 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore SLLN boundary target law ORIGINAL mass sum1 := sum(2/5, 3/5) After SLLN boundary target law 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 target law ORIGINAL weighted value0 := mul(-2, 2/5) After SLLN boundary target law ORIGINAL weighted value0 = -4/5 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore SLLN boundary target law ORIGINAL mean sum0 := sum(0, -4/5) After SLLN boundary target law ORIGINAL mean sum0 = -4/5 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore SLLN boundary target law exact unbounded value square0 := mul(-2, -2) After SLLN boundary target law 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 target law ORIGINAL weighted square0 := mul(4, 2/5) After SLLN boundary target law ORIGINAL weighted square0 = 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore SLLN boundary target law ORIGINAL second moment sum0 := sum(0, 8/5) After SLLN boundary target law ORIGINAL second moment sum0 = 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore SLLN boundary target law ORIGINAL weighted value1 := mul(-1, 3/5) After SLLN boundary target law ORIGINAL weighted value1 = -3/5 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore SLLN boundary target law ORIGINAL mean sum1 := sum(-4/5, -3/5) After SLLN boundary target law ORIGINAL mean sum1 = -7/5 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore SLLN boundary target law exact unbounded value square1 := mul(-1, -1) After SLLN boundary target law 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore SLLN boundary target law ORIGINAL weighted square1 := mul(1, 3/5) After SLLN boundary target law ORIGINAL weighted square1 = 3/5 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore SLLN boundary target law ORIGINAL second moment sum1 := sum(8/5, 3/5) After SLLN boundary target law ORIGINAL second moment sum1 = 11/5 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore SLLN boundary target law ORIGINAL mean squared := mul(-7/5, -7/5) After SLLN boundary target law ORIGINAL mean squared = 49/25 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore SLLN boundary target law ORIGINAL second minus mean squared := sub(11/5, 49/25) After SLLN boundary target law ORIGINAL second minus mean squared = 6/25 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · impliesBefore source roles = [SLLN boundary target law, [-2, -1], [[2, 5], [3, 5]]] After derived = [[[-2, 2/5], [-1, 3/5]], 1, -7/5, 11/5, 6/25] 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. Step identifiers and dependenciesRule: SLLN_complete_ORIGINAL_numeric_value_probability_alignment_ALL_moments · Uses: G001, S015 S017 · impliesBefore source roles = [[[-2, 2/5], [-1, 3/5]], -7/5, 17] After derived = [No, No] Conditions: none First verify the proposed target actually differs from the common mean.The SLLN target is that mean,not a substituted value. Step identifiers and dependenciesRule: SLLN_ACTUAL_wrong_target_not_common_expectation · Uses: G001, S016 ## Reverse checks - 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 — 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 target law ORIGINAL second minus mean squared with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute SLLN boundary target law ORIGINAL mean squared with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute SLLN boundary target law ORIGINAL second moment sum1 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute SLLN boundary target law ORIGINAL weighted square1 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute SLLN boundary target law exact unbounded value square1 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute SLLN boundary target law ORIGINAL mean sum1 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute SLLN boundary target law ORIGINAL weighted value1 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute SLLN boundary target law ORIGINAL second moment sum0 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute SLLN boundary target law ORIGINAL weighted square0 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute SLLN boundary target law exact unbounded value square0 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute SLLN boundary target law ORIGINAL mean sum0 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute SLLN boundary target law ORIGINAL weighted value0 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute SLLN boundary target law ORIGINAL mass sum1 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute SLLN boundary target law 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 (wrong_target, -7/5, 17, No, No, SLLN_target_is_common_expectation) 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: dfcc259e2d3f8d361e67af54b3e329b35a1a117963313901ecdb4ea999cc0fc7 - candidate record SHA-256: 5df9096a8679e26510cec91f07af60e3fca87e113fbaf5324ab03f1eb4d85202 - 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-0091.json)