{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261005-advance-20261006-r04-distance-formula-0032", "pilot_number": "R0061-31382", "number": 31382, "family_id": "qf-distance-perfect-square", "title": { "zh-Hant": "第 31382 題：完全平方切片的非負距離", "en": "Question 31382: Nonnegative Distance in a Perfect-Square Slice" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261005-advance-20261006-r04-distance-formula-0032", "batch_id": "batch-20261005-advance-20261006-r04-distance-formula-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-distance-formula" ], "title_zh": "完全平方切片的非負距離", "stem_zh": "在等單位正交座標系中，兩軸使用同一長度單位，P=(-6, -1)、Q=(-6, 6)。已明示座標差平方和為完全平方，求非負整數距離 d(P,Q)。", "answer": { "kind": "short_text", "value": "7", "explanation_zh": "距離=7；平方距離=49。產生端以整數二分搜尋確認根的平方等於平方和，採非負主根；重合點距離零合法。" }, "difficulty": { "band": "upper_secondary", "level": 1, "skills": [ "完全平方切片的非負距離" ] }, "variant": { "family_id": "qf-distance-perfect-square", "seed": 202610064032, "parameters": { "P": [ -6, -1 ], "Q": [ -6, 6 ], "frame": "euclidean_orthonormal_equal_units", "operation": "perfect_square_distance", "semantic_template": "qf-distance-perfect-square/{\"inputs\":[[\"0\",\"7\"]],\"operation\":\"perfect_square_distance\"}" } }, "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": "2026-10-05", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261005-advance-20261006-r04-distance-formula-0032", "batch_id": "batch-20261005-advance-20261006-r04-distance-formula-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "8a841cad721cbc5cebf1579d104eb4daf5215a98860e99c2ce57b5c94700f849", "family_id": "qf-distance-perfect-square", "seed": 202610064032, "semantic_template": "qf-distance-perfect-square/{\"inputs\":[[\"0\",\"7\"]],\"operation\":\"perfect_square_distance\"}" }, "fields": { "title": "Nonnegative Distance in a Perfect-Square Slice", "stem": "Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-6, -1) and Q=(-6, 6). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.", "answer": { "kind": "short_text", "value": "7", "explanation": "The squared distance is 49, and its nonnegative principal root is 7. A negative root is not a distance, and no floating approximation is used." }, "skills": [ "Nonnegative Distance in a Perfect-Square Slice" ], "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": "在等單位正交座標系中，兩軸使用同一長度單位，P=(-6, -1)、Q=(-6, 6)。已明示座標差平方和為完全平方，求非負整數距離 d(P,Q)。", "en": "Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-6, -1) and Q=(-6, 6). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-distance-perfect-square, perfect_square_distance, frame, euclidean_orthonormal_equal_units]", "after": "derived = [[integer_principal_root]]", "relation": "implies", "rule_id": "R61_SOURCE_exact_keys_operation_domain_OR_frame_and_output_roles", "conditions": [], "explanation": { "zh-Hant": "先核本題欄位與來源協議：distance只用完整座標frame，其他用domain；離散運算local-hyphen，舊bank用underscore。輸出欄序及型別依題族，保留原始中英資料與狀態。", "en": "Check THIS source interface:distance uses ONLY its complete coordinate frame;other families use a domain.Discrete operations use local hyphens,legacy banks underscores.Retain family-specific ordered output types and original bilingual data/status." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。", "en": "Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "Distance PQ ORIGINAL Q minus P axis0 := sub(-6, -6)", "after": "Distance PQ ORIGINAL Q minus P axis0 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。", "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Distance PQ ORIGINAL Q minus P axis0，不以小數近似替代。", "en": "Recompute Distance PQ ORIGINAL Q minus P axis0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-6", "-6" ], "output": "0", "label": "Distance PQ ORIGINAL Q minus P axis0" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Distance PQ ORIGINAL Q minus P axis1 := sub(6, -1)", "after": "Distance PQ ORIGINAL Q minus P axis1 = 7", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。", "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Distance PQ ORIGINAL Q minus P axis1，不以小數近似替代。", "en": "Recompute Distance PQ ORIGINAL Q minus P axis1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "6", "-1" ], "output": "7", "label": "Distance PQ ORIGINAL Q minus P axis1" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Distance PQ ORIGINAL signed delta squared axis0 := mul(0, 0)", "after": "Distance PQ ORIGINAL signed delta squared axis0 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。", "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Distance PQ ORIGINAL signed delta squared axis0，不以小數近似替代。", "en": "Recompute Distance PQ ORIGINAL signed delta squared axis0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "Distance PQ ORIGINAL signed delta squared axis0" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Distance PQ ORIGINAL signed delta squared axis1 := mul(7, 7)", "after": "Distance PQ ORIGINAL signed delta squared axis1 = 49", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。", "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Distance PQ ORIGINAL signed delta squared axis1，不以小數近似替代。", "en": "Recompute Distance PQ ORIGINAL signed delta squared axis1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "7", "7" ], "output": "49", "label": "Distance PQ ORIGINAL signed delta squared axis1" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Distance PQ ORIGINAL sum of two squares := sum(0, 49)", "after": "Distance PQ ORIGINAL sum of two squares = 49", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。", "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Distance PQ ORIGINAL sum of two squares，不以小數近似替代。", "en": "Recompute Distance PQ ORIGINAL sum of two squares with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "49" ], "output": "49", "label": "Distance PQ ORIGINAL sum of two squares" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "source roles = [Distance PQ, [-6, -1], [-6, 6]]", "after": "derived = [[0, 7], [0, 49], 49]", "relation": "implies", "rule_id": "Distance_ORTHONORMAL_equal_units_two_squares_NONNEGATIVE", "conditions": [], "explanation": { "zh-Hant": "在明示正交單位且等尺度的座標frame，距離平方為(Δx)²+(Δy)²≥0；重合點可為0。分量差保sign，平方和不把signed分量當距離。", "en": "In the DECLARED orthonormal equal-unit frame,distance squared is(Δx)²+(Δy)²≥0,including coincident points.Keep signed differences;do not mistake a signed component for distance." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。", "en": "Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof." } }, "arithmetic": null }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "Distance reverse principal root square := mul(7, 7)", "after": "Distance reverse principal root square = 49", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。", "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Distance reverse principal root square，不以小數近似替代。", "en": "Recompute Distance reverse principal root square with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "7", "7" ], "output": "49", "label": "Distance reverse principal root square" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "source roles = [49]", "after": "derived = [7, 49, 是]", "relation": "implies", "rule_id": "Distance_PERFECT_square_guard_principal_NONNEGATIVE_root", "conditions": [], "explanation": { "zh-Hant": "此題族先驗N真為完全平方，再取唯一非負平方根；0合法，不取±兩值或浮點近似。root是衍生值，可超過raw SAFE。", "en": "THIS family requires an actual perfect square,then takes its UNIQUE nonnegative root,including0.Do not return± roots or floating approximations.A DERIVED root may exceed RAW SAFE." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。", "en": "Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof." } }, "arithmetic": null } ], "math_model": { "result": "7" }, "conclusion": { "canonical_value": "7", "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-20261005-advance-20261006-r04-distance-formula-foundations", "batch_path": "public/data/questions/batches/batch-20261005-advance-20261006-r04-distance-formula-foundations/questions.jsonl", "batch_sha256": "db00f4c65329007cf957a8e824cabec9929a493ea4774276e8d837604ebdd905", "original_line": 32, "original_line_sha256": "080b4f3e732753e2c6369e61b8f22396c1569f4c1811d7505705ff231df6efb1", "question_compact_sha256": "8a841cad721cbc5cebf1579d104eb4daf5215a98860e99c2ce57b5c94700f849", "english_row_compact_sha256": "524704eb2b029228c5e74a1ba4634ab57853e65f2f90f68d60bbb84e8f619cf4", "english_manifest_path": "public/data/i18n/editions/batch-20261005-advance-20261006-r04-distance-formula-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261005-advance-20261006-r04-distance-formula-foundations.en.jsonl" }, "mko_bindings": { "mko-distance-formula": { "sha256": "153d0d0feabadaf33e08081677be0f81709df28ffa5bffaf0176a0f7875e5904", "english_sha256": "d297d8ae3afb0c4aa0a2de4cbd9e17c6355f2ae3f0fbf1510b9dc7cea3b27827", "original_path": "public/data/mko/distance-formula.json", "english_path": "public/data/i18n/editions/mko-distance-formula.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "912db26eb6c90aa6e256f4356cb5ee94cd0a352f04f667876ae6739f8181a17c", "original_candidate_sha256": "ffb02bdbcc660dc5e640f24a29cfebcfcc7c672206565aada92951f156d3225f", "original_candidate_line_sha256": "ffb02bdbcc660dc5e640f24a29cfebcfcc7c672206565aada92951f156d3225f" }, "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-p62", "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": "5403317467ff7c6b74c18ae7211530d0156ebc67d63fc00a3e07b81a201d3d69", "question_first_witness_sha256": "c2b5022aa77f803cd3e1ba9ce162a707da0281aa5bb5289111a97de51fe8c495" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 9, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "a38fd7df8f3f260943d021b0e68aa8c06646af7195cf9283cfc886271986dcd2", "arithmetic_steps": 6 }, "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." } }