{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261114-r04-standard-normal-distribution-0084", "pilot_number": "R0049-25259", "number": 25259, "family_id": "qf-mod45-normaldist-normalization-moment-boundary", "title": { "zh-Hant": "第 25259 題：標準常態分布", "en": "Question 25259: Finite Windows and Panel Counts Do Not Prove Gaussian Normalization or Moments" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261114-r04-standard-normal-distribution-0084", "batch_id": "batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-standard-normal-distribution" ], "title_zh": "標準常態分布", "stem_zh": "standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={\"window\":9,\"panels\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(9, 18, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)", "explanation_zh": "Wave6第八層核對=(9, 18, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "標準常態分布" ] }, "variant": { "family_id": "qf-mod45-normaldist-normalization-moment-boundary", "seed": 202611144084, "parameters": { "window": 9, "panels": 18, "domain": "Symbolic_standard_normal_distribution", "operation": "normaldist-normalization-moment-boundary", "semantic_template": "qf-mod45-normaldist-normalization-moment-boundary/{\"panels\":18,\"window\":[9,1]}" } }, "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；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、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-20261114-r04-standard-normal-distribution-0084", "batch_id": "batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "08e288ed888153dcdaa7dde001fdf5900aa25d596d1eed88424d5703357c6690", "family_id": "qf-mod45-normaldist-normalization-moment-boundary", "seed": 202611144084, "semantic_template": "qf-mod45-normaldist-normalization-moment-boundary/{\"panels\":18,\"window\":[9,1]}" }, "fields": { "title": "Finite Windows and Panel Counts Do Not Prove Gaussian Normalization or Moments", "stem": "Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Declared finite window=9>0, panel count=18. Give (window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary).", "answer": { "kind": "expression", "value": "(9, 18, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)", "explanation": "This boundary records finite-method scope only; it does not even supply a quadrature result. Retain the No proof claim separately from named normalization/mean/variance facts. Those facts need their own theorem-level derivations, not a larger finite window or panel count. The exact result, retaining the requested order, is (9, 18, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)." }, "skills": [ "Finite Windows and Panel Counts Do Not Prove Gaussian Normalization or Moments" ], "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": "standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={\"window\":9,\"panels\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Declared finite window=9>0, panel count=18. Give (window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [normaldist-normalization-moment-boundary]", "after": "derived = [[window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary]]", "relation": "implies", "rule_id": "source_ordered_fields_without_JS", "conditions": [], "explanation": { "zh-Hant": "本頁列完整來源欄序，不需JavaScript；LP三種mode雖同為6欄，角色仍由原mode選，不由答案形狀猜。", "en": "List the full source field order without JavaScript.LP boundary modes all have6 fields but select different roles from the original mode,not answer shape." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [9, 18]", "after": "derived = [finite_quadrature_only, 否, named normalization1,mean0,variance1]", "relation": "implies", "rule_id": "Gaussian_finite_window_panels_NO_quadrature_NO_theorem_proof", "conditions": [], "explanation": { "zh-Hant": "這裡只有window/panels，沒有執行quadrature或給tail/error控制。有限窗遺漏正尾，有限panel數不證exact全域積分或moments；named facts既未由此證成，也不被No否定。", "en": "Only window/panels are supplied;no quadrature or tail/error control is performed.A finite window omits positive tails and a panel count does not prove an exact global integral or moments.Named facts are neither proved here nor negated by No." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null } ], "math_model": { "result": [ "9", 18, "finite_quadrature_only", false, "named_fact:integral_R_phi=1", "named_fact:E[Z]=0", "named_fact:Var(Z)=1", "finite_window_or_panel_count_does_not_prove_normalization_or_moments" ] }, "conclusion": { "canonical_value": "(9, 18, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)", "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-20261114-r04-standard-normal-distribution-foundations", "batch_path": "public/data/questions/batches/batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations/questions.jsonl", "batch_sha256": "fae4e594ca5df267f7b3bb28d68f91a80ba9411fb7e9e2fa68b7dda56db4ee46", "original_line": 84, "original_line_sha256": "fb6bb394ad01b922284d7156f23c23fa61c48ccbc9225c43b9cbbe80a3f39736", "question_compact_sha256": "08e288ed888153dcdaa7dde001fdf5900aa25d596d1eed88424d5703357c6690", "english_row_compact_sha256": "ed2b22e746d3a59c9445637025900dddfbb38a07da60eedeaa60c295c8969729", "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations.en.jsonl" }, "mko_bindings": { "mko-standard-normal-distribution": { "sha256": "5dac4a8d8eb1b38c2dbe86853a836e63d55563a544cfe67d5e6b4095fdaa9ada", "english_sha256": "1aebcb3737fd2d8647b6558c8a8272337f768dad39c6fdb34b76210d9375fdb4", "original_path": "public/data/mko/source13-20261018-standard-normal-distribution.json", "english_path": "public/data/i18n/editions/mko-standard-normal-distribution.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "991c73df066352bfcfb1142a2492e104cd1d20dafe7e9ecea8cff26a4f9d8e50", "original_candidate_sha256": "3aefa9476fe1167468c3acde910bf8c56e08ca1273b69fb103631acbd332418f", "original_candidate_line_sha256": "3aefa9476fe1167468c3acde910bf8c56e08ca1273b69fb103631acbd332418f" }, "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-p50", "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": "6212b8b0b3b273f3a514e26ebd17d7e81219c56a8f47e40047ee2b6bf01d1ba3", "question_first_witness_sha256": "f32a23d5e1b66c7794c60def58741bcdfacff24f3cdfc9bc48109c8f71543cf3" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 2, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "d50153f9088ffd9a6e3abcdc89c26d7d332b3dec36cc066f80d2af4c175c5873", "arithmetic_steps": 0 }, "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." } }