{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0042", "pilot_number": "R0054-27642", "number": 27642, "family_id": "qf-mod49-slln-prefix-sample-mean", "title": { "zh-Hant": "第 27642 題：有限值IID強大數律", "en": "Question 27642: A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0042", "batch_id": "batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-strong-law-large-numbers-finite-valued-iid" ], "title_zh": "有限值IID強大數律", "stem_zh": "無限IID process certificate、finite marginal law與almost-sure limit責任。操作=slln-prefix-sample-mean；資料={\"values\":[-1,0],\"probabilities\":[[2,5],[3,5]],\"process_certificate\":\"infinite_iid\",\"observations\":[-1,0,-1,0,-1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "([-1, 0, -1, 0, -1], -2/5, -3/5, 1/5, fixed_prefix_only, 否, finite_prefix_does_not_prove_almost_sure_convergence)", "explanation_zh": "Wave6第十二層核對=([-1, 0, -1, 0, -1], -2/5, -3/5, 1/5, fixed_prefix_only, 否, finite_prefix_does_not_prove_almost_sure_convergence)；依完整明示資料獨立重算；固定prefix mean與deviation event只屬fixed-n audit；non-IID/nonfinite-support是scoped theorem不適用，finite prefix不證a.s. convergence。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "有限值IID強大數律" ] }, "variant": { "family_id": "qf-mod49-slln-prefix-sample-mean", "seed": 202611182042, "parameters": { "values": [ -1, 0 ], "probabilities": [ [ 2, 5 ], [ 3, 5 ] ], "process_certificate": "infinite_iid", "observations": [ -1, 0, -1, 0, -1 ], "domain": "Exact_finite_valued_SLLN_boundary", "operation": "slln-prefix-sample-mean", "semantic_template": "slln-prefix-sample-mean|{law:[[-1,[2,5]],[0,[3,5]]],observations:[-1,0,-1,0,-1],process_certificate:\"infinite_iid\"}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-15", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0042", "batch_id": "batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "3870a60b42b76ebf4e70d56099fdde869b2f9e405afb0fffb5e1d0cbb154262e", "family_id": "qf-mod49-slln-prefix-sample-mean", "seed": 202611182042, "semantic_template": "slln-prefix-sample-mean|{law:[[-1,[2,5]],[0,[3,5]]],observations:[-1,0,-1,0,-1],process_certificate:\"infinite_iid\"}" }, "fields": { "title": "A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate", "stem": "Use the aligned finite law values=[-1, 0], probabilities=[2/5, 3/5], 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=[-1, 0, -1, 0, -1]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).", "answer": { "kind": "expression", "value": "([-1, 0, -1, 0, -1], -2/5, -3/5, 1/5, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)", "explanation": "Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([-1, 0, -1, 0, -1], -2/5, -3/5, 1/5, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)." }, "skills": [ "A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "無限IID process certificate、finite marginal law與almost-sure limit責任。操作=slln-prefix-sample-mean；資料={\"values\":[-1,0],\"probabilities\":[[2,5],[3,5]],\"process_certificate\":\"infinite_iid\",\"observations\":[-1,0,-1,0,-1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use the aligned finite law values=[-1, 0], probabilities=[2/5, 3/5], 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=[-1, 0, -1, 0, -1]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "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]]", "relation": "implies", "rule_id": "source_ordered_fields_without_JS", "conditions": [], "explanation": { "zh-Hant": "直接明列此操作與mode完整欄序，不依JavaScript、不由答案形狀猜角色；布林、字串、null與未作聲稱保持不同。", "en": "List the complete operation/mode field order without JavaScript or answer-shape inference;Boolean,string,null and an unclaimed result remain distinct." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。", "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [infinite_iid]", "after": "derived = [ASSUME whole infinite sequence is IID with this common marginal]", "relation": "implies", "rule_id": "SLLN_infinite_IID_EXACT_GIVEN_premise_not_finite_observation_inference", "conditions": [], "explanation": { "zh-Hant": "infinite_iid是題設無限過程前提，非有限資料或本程式已構造／證明的independence。", "en": "infinite_iid is a GIVEN whole-process premise,not independence constructed or proved by finite data/this program." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。", "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "SLLN ORIGINAL common marginal ORIGINAL mass sum0 := sum(0, 2/5)", "after": "SLLN ORIGINAL common marginal ORIGINAL mass sum0 = 2/5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。", "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL mass sum0，不以小數近似替代。", "en": "Recompute SLLN ORIGINAL common marginal ORIGINAL mass sum0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "2/5" ], "output": "2/5", "label": "SLLN ORIGINAL common marginal ORIGINAL mass sum0" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "SLLN ORIGINAL common marginal ORIGINAL mass sum1 := sum(2/5, 3/5)", "after": "SLLN ORIGINAL common marginal ORIGINAL mass sum1 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。", "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL mass sum1，不以小數近似替代。", "en": "Recompute SLLN ORIGINAL common marginal ORIGINAL mass sum1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "2/5", "3/5" ], "output": "1", "label": "SLLN ORIGINAL common marginal ORIGINAL mass sum1" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "SLLN ORIGINAL common marginal ORIGINAL weighted value0 := mul(-1, 2/5)", "after": "SLLN ORIGINAL common marginal ORIGINAL weighted value0 = -2/5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。", "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL weighted value0，不以小數近似替代。", "en": "Recompute SLLN ORIGINAL common marginal ORIGINAL weighted value0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-1", "2/5" ], "output": "-2/5", "label": "SLLN ORIGINAL common marginal ORIGINAL weighted value0" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "SLLN ORIGINAL common marginal ORIGINAL mean sum0 := sum(0, -2/5)", "after": "SLLN ORIGINAL common marginal ORIGINAL mean sum0 = -2/5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。", "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL mean sum0，不以小數近似替代。", "en": "Recompute SLLN ORIGINAL common marginal ORIGINAL mean sum0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "-2/5" ], "output": "-2/5", "label": "SLLN ORIGINAL common marginal ORIGINAL mean sum0" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "SLLN ORIGINAL common marginal exact unbounded value square0 := mul(-1, -1)", "after": "SLLN ORIGINAL common marginal exact unbounded value square0 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。", "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 SLLN ORIGINAL common marginal exact unbounded value square0，不以小數近似替代。", "en": "Recompute SLLN ORIGINAL common marginal exact unbounded value square0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-1", "-1" ], "output": "1", "label": "SLLN ORIGINAL common marginal exact unbounded value square0" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "SLLN ORIGINAL common marginal ORIGINAL weighted square0 := mul(1, 2/5)", "after": "SLLN ORIGINAL common marginal ORIGINAL weighted square0 = 2/5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。", "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL weighted square0，不以小數近似替代。", "en": "Recompute SLLN ORIGINAL common marginal ORIGINAL weighted square0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2/5" ], "output": "2/5", "label": "SLLN ORIGINAL common marginal ORIGINAL weighted square0" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "SLLN ORIGINAL common marginal ORIGINAL second moment sum0 := sum(0, 2/5)", "after": "SLLN ORIGINAL common marginal ORIGINAL second moment sum0 = 2/5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。", "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL second moment sum0，不以小數近似替代。", "en": "Recompute SLLN ORIGINAL common marginal ORIGINAL second moment sum0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "2/5" ], "output": "2/5", "label": "SLLN ORIGINAL common marginal ORIGINAL second moment sum0" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "SLLN ORIGINAL common marginal ORIGINAL weighted value1 := mul(0, 3/5)", "after": "SLLN ORIGINAL common marginal ORIGINAL weighted value1 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。", "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL weighted value1，不以小數近似替代。", "en": "Recompute SLLN ORIGINAL common marginal ORIGINAL weighted value1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "3/5" ], "output": "0", "label": "SLLN ORIGINAL common marginal ORIGINAL weighted value1" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "SLLN ORIGINAL common marginal ORIGINAL mean sum1 := sum(-2/5, 0)", "after": "SLLN ORIGINAL common marginal ORIGINAL mean sum1 = -2/5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。", "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL mean sum1，不以小數近似替代。", "en": "Recompute SLLN ORIGINAL common marginal ORIGINAL mean sum1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-2/5", "0" ], "output": "-2/5", "label": "SLLN ORIGINAL common marginal ORIGINAL mean sum1" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "SLLN ORIGINAL common marginal exact unbounded value square1 := mul(0, 0)", "after": "SLLN ORIGINAL common marginal exact unbounded value square1 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。", "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 SLLN ORIGINAL common marginal exact unbounded value square1，不以小數近似替代。", "en": "Recompute SLLN ORIGINAL common marginal exact unbounded value square1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "SLLN ORIGINAL common marginal exact unbounded value square1" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "SLLN ORIGINAL common marginal ORIGINAL weighted square1 := mul(0, 3/5)", "after": "SLLN ORIGINAL common marginal ORIGINAL weighted square1 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。", "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL weighted square1，不以小數近似替代。", "en": "Recompute SLLN ORIGINAL common marginal ORIGINAL weighted square1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "3/5" ], "output": "0", "label": "SLLN ORIGINAL common marginal ORIGINAL weighted square1" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "SLLN ORIGINAL common marginal ORIGINAL second moment sum1 := sum(2/5, 0)", "after": "SLLN ORIGINAL common marginal ORIGINAL second moment sum1 = 2/5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。", "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL second moment sum1，不以小數近似替代。", "en": "Recompute SLLN ORIGINAL common marginal ORIGINAL second moment sum1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "2/5", "0" ], "output": "2/5", "label": "SLLN ORIGINAL common marginal ORIGINAL second moment sum1" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "SLLN ORIGINAL common marginal ORIGINAL mean squared := mul(-2/5, -2/5)", "after": "SLLN ORIGINAL common marginal ORIGINAL mean squared = 4/25", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。", "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL mean squared，不以小數近似替代。", "en": "Recompute SLLN ORIGINAL common marginal ORIGINAL mean squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2/5", "-2/5" ], "output": "4/25", "label": "SLLN ORIGINAL common marginal ORIGINAL mean squared" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "SLLN ORIGINAL common marginal ORIGINAL second minus mean squared := sub(2/5, 4/25)", "after": "SLLN ORIGINAL common marginal ORIGINAL second minus mean squared = 6/25", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。", "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL second minus mean squared，不以小數近似替代。", "en": "Recompute SLLN ORIGINAL common marginal ORIGINAL second minus mean squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "2/5", "4/25" ], "output": "6/25", "label": "SLLN ORIGINAL common marginal ORIGINAL second minus mean squared" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "source roles = [SLLN ORIGINAL common marginal, [-1, 0], [[2, 5], [3, 5]]]", "after": "derived = [[[-1, 2/5], [0, 3/5]], 1, -2/5, 2/5, 6/25]", "relation": "implies", "rule_id": "SLLN_complete_ORIGINAL_numeric_value_probability_alignment_ALL_moments", "conditions": [], "explanation": { "zh-Hant": "先綁原始value與probability再numeric排序，保零質量labels。完整law核非負與mass1，且來源仍計E[X²]-E[X]²；SLLN允許variance0，不照搬CLT的strict-positive條件。", "en": "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." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。", "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct." } }, "arithmetic": null }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "SLLN ORIGINAL prefix integer sum := sum(-1, 0, -1, 0, -1)", "after": "SLLN ORIGINAL prefix integer sum = -3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。", "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 SLLN ORIGINAL prefix integer sum，不以小數近似替代。", "en": "Recompute SLLN ORIGINAL prefix integer sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-1", "0", "-1", "0", "-1" ], "output": "-3", "label": "SLLN ORIGINAL prefix integer sum" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "SLLN ORIGINAL prefix sample mean := div(-3, 5)", "after": "SLLN ORIGINAL prefix sample mean = -3/5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。", "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 SLLN ORIGINAL prefix sample mean，不以小數近似替代。", "en": "Recompute SLLN ORIGINAL prefix sample mean with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "-3", "5" ], "output": "-3/5", "label": "SLLN ORIGINAL prefix sample mean" } }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "SLLN ORIGINAL sample mean minus common mean := sub(-3/5, -2/5)", "after": "SLLN ORIGINAL sample mean minus common mean = -1/5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。", "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 SLLN ORIGINAL sample mean minus common mean，不以小數近似替代。", "en": "Recompute SLLN ORIGINAL sample mean minus common mean with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-3/5", "-2/5" ], "output": "-1/5", "label": "SLLN ORIGINAL sample mean minus common mean" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "source roles = [[-1, 0, -1, 0, -1], -2/5, -3/5, -1/5]", "after": "derived = [1/5, 否]", "relation": "implies", "rule_id": "SLLN_fixed_nonempty_prefix_mean_error_NOT_as_limit_certificate", "conditions": [], "explanation": { "zh-Hant": "觀測值逐個合型且來自完整carrier，prefix須非空；來源用未約分[sum,length]的1e6 profile，不私加fixed-n的8項限制。一次接近或等於mean都不證a.s. limit。", "en": "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." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。", "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct." } }, "arithmetic": null } ], "math_model": { "result": [ [ -1, 0, -1, 0, -1 ], "-2/5", "-3/5", "1/5", "fixed_prefix_only", false, "finite_prefix_does_not_prove_almost_sure_convergence" ] }, "conclusion": { "canonical_value": "([-1, 0, -1, 0, -1], -2/5, -3/5, 1/5, fixed_prefix_only, 否, finite_prefix_does_not_prove_almost_sure_convergence)", "explanation": { "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。", "en": "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." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations", "batch_path": "public/data/questions/batches/batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations/questions.jsonl", "batch_sha256": "ddb49de712a058f17353bffb58a7d023ec4a62dc04e72fd2aa4d83570a11cec2", "original_line": 42, "original_line_sha256": "6046df193082e0b5a913de1b9419018300ded06460e438038b0a27d252b3363d", "question_compact_sha256": "3870a60b42b76ebf4e70d56099fdde869b2f9e405afb0fffb5e1d0cbb154262e", "english_row_compact_sha256": "e9b7c9eb5903c20079f4baf842e64ef13dacec6753c5a6e1c095c19a74b794b6", "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations.en.jsonl" }, "mko_bindings": { "mko-strong-law-large-numbers-finite-valued-iid": { "sha256": "14ad2e3637b7d7efdbf46bd6d640bf2637a5ab65b1f9ceafa6ff1a08565e9037", "english_sha256": "d7006cf0173ce35dcb02c4b0f80110b0124038bb778c5edc2903ca25a2569553", "original_path": "public/data/mko/strong-law-large-numbers-finite-valued-iid.json", "english_path": "public/data/i18n/editions/mko-strong-law-large-numbers-finite-valued-iid.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "ad40cc18f9292c43924a7679f3af46574a5284aa5447f5f7f78a0de98840acb8", "original_candidate_sha256": "2c3cc5f58380e1fd8d75780b591fb82e2b0cca26a18dc7a5b12c4084f53a6365", "original_candidate_line_sha256": "2c3cc5f58380e1fd8d75780b591fb82e2b0cca26a18dc7a5b12c4084f53a6365" }, "historical_source_state": { "review_state": { "derivation": "candidate_authored_and_locally_checked", "source_question_state": "candidate", "independent_AI_or_human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN" }, "scope_note": { "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。", "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged." }, "label": "Frozen producer metadata; current per-question review is separate below." }, "revision": { "id": "review-20261026-010-p55", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.", "review_receipts": { "semantic_review_sha256": "0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54", "question_first_witness_sha256": "09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 21, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "c03a5a5f0109dc1ab07a3dac271d991cb1e2eb7d093e84e175f4449c75b2c5a8", "arithmetic_steps": 17 }, "uses_answer_as_premise": false, "human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN", "Canonical_truth_promotion": false, "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged." } }