{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261010-advance-20261014-r07-circle-0069", "pilot_number": "R0060-30519", "number": 30519, "family_id": "qf-density14-circle-radius-scale-witness", "title": { "zh-Hant": "第 30519 題：半徑參數的尺度見證", "en": "Question 30519: Radius-Parameter Scaling and Positivity" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261010-advance-20261014-r07-circle-0069", "batch_id": "batch-20261010-advance-20261014-r07-circle-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-circle" ], "title_zh": "半徑參數的尺度見證", "stem_zh": "圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=18、k=1。依序求 (kr, r², (kr)², 2kr, kr>0資格)，只問參數乘算。", "answer": { "kind": "expression", "value": "(18, 324, 324, 36, 是)", "explanation_zh": "半徑尺度平方與直徑=(18, 324, 324, 36, 是)。先乘非負半徑參數，再平方及倍增；參數乘算非相似定理證明。r=0只有中心，有限座標核對非一般圓API形式橋。" }, "difficulty": { "band": "lower_secondary", "level": 3, "skills": [ "半徑參數的尺度見證" ] }, "variant": { "family_id": "qf-density14-circle-radius-scale-witness", "seed": 202610147069, "parameters": { "r": 18, "k": 1, "frame": "orthogonal_equal_unit", "domain": "finite_integer_circle_boundary_protocol", "operation": "density14_circle_radius_scale_witness", "semantic_template": "qf-density14-circle-radius-scale-witness/[18,1]" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_integer_rational_arithmetic", "producer_finite_enumeration", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-10", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261010-advance-20261014-r07-circle-0069", "batch_id": "batch-20261010-advance-20261014-r07-circle-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "1b5d6d170d8ae0bd5d15c6faad833de969983f1815bef25f1993decb11b1cc7d", "family_id": "qf-density14-circle-radius-scale-witness", "seed": 202610147069, "semantic_template": "qf-density14-circle-radius-scale-witness/[18,1]" }, "fields": { "title": "Radius-Parameter Scaling and Positivity", "stem": "Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=18, scale k=1. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.", "answer": { "kind": "expression", "value": "(18, 324, 324, 36, Yes)", "explanation": "Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (18, 324, 324, 36, Yes)." }, "skills": [ "Radius-Parameter Scaling and Positivity" ], "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": "圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=18、k=1。依序求 (kr, r², (kr)², 2kr, kr>0資格)，只問參數乘算。", "en": "Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=18, scale k=1. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-density14-circle-radius-scale-witness, density14_circle_radius_scale_witness, finite_integer_circle_boundary_protocol]", "after": "derived = [[scaled_radius, original_radius_squared, scaled_radius_squared, diameter, scaled_positive]]", "relation": "implies", "rule_id": "R60_EXACT_perbank_operation_domain_FRAME_and_output_roles", "conditions": [], "explanation": { "zh-Hant": "density14/15用full-prefix underscores，density17用local hyphens。coordinate frame與circle frame不同，輸出欄序／型別依本family，不改原中英／JSON／status。", "en": "density14/15 use FULL-prefix underscores;density17 uses LOCAL hyphens.The coordinate and circle frames differ.Retain THIS family ordered answer roles/types and all original bilingual JSON/status." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "用原始資料、型別、索引與本步前提反向代回；一致性審查不等於可由結論唯一還原輸入。有限候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "Circle scale ORIGINAL k times r := mul(1, 18)", "after": "Circle scale ORIGINAL k times r = 18", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。", "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Circle scale ORIGINAL k times r，不以小數近似替代。", "en": "Recompute Circle scale ORIGINAL k times r with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "18" ], "output": "18", "label": "Circle scale ORIGINAL k times r" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Circle scale ORIGINAL r squared := mul(18, 18)", "after": "Circle scale ORIGINAL r squared = 324", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。", "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Circle scale ORIGINAL r squared，不以小數近似替代。", "en": "Recompute Circle scale ORIGINAL r squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "18", "18" ], "output": "324", "label": "Circle scale ORIGINAL r squared" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Circle scale ORIGINAL scaled radius squared := mul(18, 18)", "after": "Circle scale ORIGINAL scaled radius squared = 324", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。", "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Circle scale ORIGINAL scaled radius squared，不以小數近似替代。", "en": "Recompute Circle scale ORIGINAL scaled radius squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "18", "18" ], "output": "324", "label": "Circle scale ORIGINAL scaled radius squared" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Circle scale ORIGINAL twice scaled radius := mul(2, 18)", "after": "Circle scale ORIGINAL twice scaled radius = 36", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。", "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Circle scale ORIGINAL twice scaled radius，不以小數近似替代。", "en": "Recompute Circle scale ORIGINAL twice scaled radius with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "18" ], "output": "36", "label": "Circle scale ORIGINAL twice scaled radius" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "source roles = [18, 1, orthogonal_equal_unit]", "after": "derived = [18, 324, 324, 36, 是]", "relation": "implies", "rule_id": "Circle_radius_parameter_arithmetic_ZEROlegal_NOT_similarity_theorem", "conditions": [], "explanation": { "zh-Hant": "r,k都非負且0合法；逐算kr、r²、(kr)²、2kr，最後只問kr>0。derived可到SAFE^4，不截raw界；這是參數算術，不是相似定理或Circle API形式橋。", "en": "Both r,k are nonnegative and0 is legal.Compute kr,r²,(kr)²,2kr separately,then test ONLY kr>0.Derived values may reach SAFE^4 without raw recapping.This parameter arithmetic is not a similarity theorem or formal Circle API bridge." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "用原始資料、型別、索引與本步前提反向代回；一致性審查不等於可由結論唯一還原輸入。有限候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate." } }, "arithmetic": null } ], "math_model": { "result": [ "18", "324", "324", "36", true ] }, "conclusion": { "canonical_value": "(18, 324, 324, 36, 是)", "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-20261010-advance-20261014-r07-circle-foundations", "batch_path": "public/data/questions/batches/batch-20261010-advance-20261014-r07-circle-foundations/questions.jsonl", "batch_sha256": "364c6501dc54730733b5b6b94cd2a95815a1b4419f33cafb01c9de044038395f", "original_line": 69, "original_line_sha256": "091dcd1b1dfbe2ff4e30614e69516b3ce31e0c5c4e99a82355cafae5d101a872", "question_compact_sha256": "1b5d6d170d8ae0bd5d15c6faad833de969983f1815bef25f1993decb11b1cc7d", "english_row_compact_sha256": "c62612e3e5f2ca38430b1b024d8177e194b284f70233644ef7d3e3c33cbd9cb7", "english_manifest_path": "public/data/i18n/editions/batch-20261010-advance-20261014-r07-circle-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261010-advance-20261014-r07-circle-foundations.en.jsonl" }, "mko_bindings": { "mko-circle": { "sha256": "f5666d01288a4c7140be59abad5dbf093293c9136d7d60dc752a79a497d99d2e", "english_sha256": "e3c801529b17bbcfc6b73fef34a12a234a6dedbbaf8f1e283d3371fd2c7290d0", "original_path": "public/data/mko/circle.json", "english_path": "public/data/i18n/editions/mko-circle.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "322533190c1eb10144c4c59b6c70cc2f8105108b5f4f3a39dd577712e7185c02", "original_candidate_sha256": "62eff16762f5614c7fab2597641a155b549350fd7ee4613a6a6fad041c17cfc6", "original_candidate_line_sha256": "62eff16762f5614c7fab2597641a155b549350fd7ee4613a6a6fad041c17cfc6" }, "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-20261026-011-p61", "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": "d6b5a409bf933281a46617cd715d8016fd5ca3d6fb76f77d179ac2ecd9d608bf", "question_first_witness_sha256": "f80ecf598c6893a580af018e1a82849b33ee6887efb48a6a1d69569aa17f12e4" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 6, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "c5face1697ade41043a358367de5d1743e39b75214843549e7a8652fe89f8181", "arithmetic_steps": 4 }, "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." } }